Research ArticleAn Inventory Model under Trapezoidal Type DemandWeibull-Distributed Deterioration and Partial Backlogging
Lianxia Zhao
School of Management Shanghai University Shanghai 200444 China
Correspondence should be addressed to Lianxia Zhao zhaolianxiastaffshueducn
Received 7 November 2013 Revised 19 January 2014 Accepted 21 January 2014 Published 6 March 2014
Academic Editor Nachamada Blamah
Copyright copy 2014 Lianxia ZhaoThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper studies an inventory model for Weibull-distributed deterioration items with trapezoidal type demand rate in whichshortages are allowed and partially backlogging depends on the waiting time for the next replenishment The inventory modelsstarting with no shortage is are to be discussed and an optimal inventory replenishment policy of the model is proposed Finallynumerical examples are provided to illustrate the theoretical results and a sensitivity analysis of the major parameters with respectto the optimal solution is also carried out
1 Introduction
The effect of deteriorating for items cannot be disregarded inmany inventory systems and it is a general phenomenon inreal life Deterioration is defined as any process that decreasesthe usefulness or the value of the original item such asdecay or physical depletion For example fruits vegetablesor foodstuffs are subject to spoilage directly while beingkept in store and electronic products radioactive substancesand photographic film deteriorate through a gradual loss ofpotential or utility with the passage of time
Due to the variability in economic circumstances thebasic assumptions of the EOQ model should be constantlymodified according to the studied inventory model In recentyears many researchers have studied kinds of EOQ modelsfor deteriorating items Ghare and Schrader [1] establishedthe classical no-shortage inventorymodelwith a constant rateof decay Wu et al [2] studied an inventory model with aWeibull-distributed deteriorating rate for items and assumedthe demand rate with a continuous function of time Wee[3] developed an inventory model with quantity discountpricing and partial backordering when the product in stockdeteriorates with time Related literature also includes Skouriand Papachristos [4] Wee [5] and Dye et al [6]
practically the demand rate of deterioration items isimpossible to increase continuously all the time Hill [7]
proposed an inventory model with ramp type demand rateMandal and Pal [8] extended the inventory model withramp type demand for deterioration items and allowedshortageWu [9] considered an inventorymodelwithWeibulldistribution deterioration and ramp type demand rate inwhich shortages are allowed and the backlogging rate isdependent on waiting time Giri et al [10] extended theramp type demand inventory model with a more generalizedWeibull deterioration distribution Manna and Chaudhuri[11] developed an inventory model for time-dependent dete-riorating items with ramp type demand rate Skouri et al[12] considered an inventory model with general ramp typedemand rate partial backlogging and Weibull deteriorationrate Hung [13] extended their inventory model from ramptype demand rate and Weibull deterioration rate to arbitrarydemand rate and arbitrary deterioration rate Kumar et al[14] studied fuzzy EOQ models with ramp type demandrate partial backlogging and time-dependent deteriorationrat Cheng et al [15] considered an inventory model fortime-dependent deteriorating items with trapezoidal typedemand rate and partial backlogging Uthayakumar andRameswari [16] studied an inventory model for defectiveitems with trapezoidal type demand rate to determine theoptimal product reliability Tan and Weng [17] considered adiscrete-in-time inventorymodel for deteriorating itemswithpartially backloggedAhmed et al [18] proposed amethod for
Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 747419 10 pageshttpdxdoiorg1011552014747419
2 Journal of Applied Mathematics
finding the economic order quantity for an inventory modelwith ramp type demand rate partial backlogging and generaldeterioration rate Lin [19] explored the inventorymodel witha general demand rate in which both the Weibull-distributeddeterioration and partial backlogging are considered
In the above mentioned research one of assumptionswas considered the ramp type demand rate partial back-logging andWeibull-distributed deterioration rateHoweverfor fashionable commodities high-tech products and othershort life cycle products the demand rate should increasewith the time up to certain point at first stage then reacha stabilized period and finally the demand rate decreaseto zero and the products retreat from market in theirproduct life cycle that is the demand rate with continuoustrapezoidal function of time On the other hand inmany realsituations customers encountering shortages will responddifferently Some customers are willing to wait until thenext replenishment while others may be impatient and goelsewhere as waiting time increases that is the willingnessfor a customer to wait for backlogging is diminishing withthe length of the waiting time In this paper we consideran inventory model with Weibull-distributed deteriorationitems trapezoidal type demand rate and time-dependentpartial backlogging By analyzing the inventory model auseful inventory replenishment policy is proposed Finallynumerical examples are provided to illustrate the theoreticalresults and a sensitivity analysis of the optimal solution withrespect to major parameters is also carried out
The rest of the paper is organized as follows Section 2describes the notation and assumptions used throughout thispaper Section 3 analyzes the inventory model and somenumerical examples to illustrate the solution procedure areprovided Sensitivity analysis of the major parameters is alsocarried out in Section 4 and the final Section concludes thispaper
2 Notations and Assumptions
The fundamental notations and assumptions used in inven-tory model and considered in this paper are given as below
(i) 119868(119905) the level of inventory at time 119905 0 le 119905 le 119879(ii) 119879 the fixed length of each ordering cycle(iii) 1199051the time when the inventory level reaches zero for
the inventory model(iv) 119905lowast1the optimal point
(v) 119878 the maximum inventory level for each orderingcycle
(vi) 119876lowast the optimal ordering quantity(vii) 119860
0the fixed cost per order
(viii) 1198881the cost of each deteriorated item
(ix) 1198882the inventory holding cost per unit per unit of time
(x) 1198883the shortage cost per unit per unit of time
(xi) 1198884the lost sales cost per unit
(xii) 119862119894(1199051) 119894 = 1 2 3 the average total cost per unit time
under different conditions respectively
(xiii) 119879119862(1199051) the average total cost per unit time
(xiv) The demand rate 119863(119905) which is positive and consec-utive is assumed to be a trapezoidal type function oftime that is
119863 (119905) =
119891 (119905) 119905 le 1205831
1198630 120583
1lt 119905 lt 120583
2
119892 (119905) 1205832le 119905 lt 119879
(1)
where 1205831is time point changing from the increasing
demand function119891(119905) to constant demand1198630 and 120583
2
is time point changing from the constant demand1198630
to the decreasing demand function 119892(119905)(xv) The replenishment rate is infinite that is replenish-
ment is instantaneous(xvi) The deterioration rate of the item is defined as
Weibull (120572 120573) that is the deterioration rate is 120579(119905) =120572120573119905120573minus1
(120572 gt 0 120573 gt 0 119905 gt 0)(xvii) Shortages are allowed and they adopt the notation
used in Abad [20] where the unsatisfied demandis backlogged and the fraction of shortages backo-rdered is 119890minus120575119905 where 119905 is the waiting time up to thenext replenishment We also assume that 119905119890minus120575119905 is anincreasing function which had appeared in Skouri etal [12]
(xviii) The time horizon of the inventory model is finite
3 Model Formulation
In this section we consider an inventory model startingwith no shortage The behavior of the model during a givencycle is depicted in Figure 1 Replenishment occurs at time119905 = 0 and the inventory level attains its maximum From119905 = 0 to 119905
1 the inventory level reduces due to demand
and deterioration At 1199051 the inventory level achieves zero
then shortage is allowed to occur during the time interval(1199051 119879) and all of the demand during the shortage period
(1199051 119879) is partially backlogged According to the notations and
assumptions mentioned above the behavior of the modelat any time can be described by the following differentialequations
(119879 minus 119905) 119890120575(119905minus119879)
119892 (119905) 119889119905]
+ 1198884[int
119879
1199051
(1 minus 119890120575(119905minus119879)
) 119892 (119905) 119889119905]
(26)
From the above analysis we obtain that the total average costof the model over the time interval [0 119879] is
119879119862 (1199051) =
1198621(1199051) 0 lt 119905
1le 1205831
1198622(1199051) 120583
1lt 1199051le 1205832
1198623(1199051) 120583
2lt 1199051lt 119879
(27)
where 1198621(1199051) 1198622(1199051) and 119862
3(1199051) are obtained from (10) (18)
and (26) respectivelyIn the following we will provide the results which ensure
the existence of a unique 1199051 say 119905lowast1 so as to minimize the total
average cost for the model system starting with no shortagesIf 0 lt 119905
1le 1205831 taking the first-order derivative of 119862
1(1199051)
with respect to 1199051 we obtain
1198891198621(1199051)
1198891199051
=119891 (1199051)
119879ℎ (1199051) (28)
where
ℎ (1199051) = 1198881(119890120572119905120573
1 minus 1)
+ 1198882int
1199051
0
119890120572(119905120573
1minus119905120573)119889119905 + 119888
3(1199051minus 119879) 119890
120575(1199051minus119879)
+ 1198884(119890120575(1199051minus119879)
minus 1)
(29)
then we can obtain ℎ(0) lt 0 and ℎ(119879) gt 0 By using theassumption (119905119890minus120575119905 is an increasing function where 119905 is thewaiting time up to the next replenishment) we have
The function ℎ(1199051) is given by (31) and (40) implies that
1198623(1199051) can obtain its minimum value at 119905lowast
1
The optimal value of the order level 119878 = 119868(0) is
119878lowast= int
1205831
0
119890120572119909120573
119891 (119909) 119889119909 + 1198630int
1205832
1205831
119890120572119909120573
119889119909 + int
119905lowast
1
1205832
119890120572119909120573
119892 (119909) 119889119909
(41)
and the optimal order quantity 119876lowast is
119876lowast= 119878lowast+ int
119879
119905lowast
1
119890120575(119905minus119879)
119892 (119905) 119889119905 (42)
If 119905lowast1
le 1205832 then the optimal value of 119862
3(1199051) is obtained at
119905lowast
1= 1205832
The above analysis shows that the three average cost func-tions 119862
1(1199051) 1198622(1199052) and 119862
3(1199051) can obtain their minimum
value at 119905lowast1isin (0 119879) which is determined by (31) Therefore
based on the results analyzed above this paper derives aprocedure to locate the optimal replenishment policy startingwith no shortage for the three cases The procedure is asfollowsStep 1 Solve 119905lowast
1from (31)
Step 2 Compare 119905lowast1to 1205831and 120583
2 respectively
Step 21 If 119905lowast1isin (0 120583
1] then the optimal total average cost and
the optimal order quantity can be obtained by (10) and (34)respectivelyStep 22 If 119905lowast
1isin (1205831 1205832] then the optimal total average cost
and the optimal order quantity can be obtained by (18) and(38) respectively
Step 23 If 119905lowast1isin (1205832 119879] then the optimal total average cost
and the optimal order quantity can be obtained by (26) and(42) respectively
Remark 1 In such considered inventory model starting withno shortage if 119905
1satisfies 120583
1lt 1199051le 119879 lt 120583
2 the considered
inventory model reduces to that of Skouri et al [12]
4 Numerical Example
In order to demonstrate the above procedure which can beapplied to obtain the optimal solution of the model this
paper presents several examples for the model respectivelyExamples are based on piecewise demand rate such as119891(119905) =1198861+ 1198871119905 and 119892(119905) = 119886
2119890minus1198872119905
Example 1 The parameter values are given as follows 119879 = 12
weeks1205831= 4weeks120583
2= 8weeks120572 = 0005120573 = 2120575 = 004
1198861= 30 unit 119887
1= 5 unit 119886
2= 100 unit 119860
0= $500 119888
1= $2
1198882= $3 119888
3= $12 and 119888
4= $8
The model starting with no shortage by solving theequation ℎ(119905
1) = 0 we have 119905
lowast
1= 87622 From (42) and
(26) we obtain 119876lowast
= 5825217 and 119879119862(119905lowast
1) = 7939986
respectively
Example 2 Theparameter values are given as follows119879 = 12
weeks1205831= 4weeks120583
2= 8weeks120572 = 0005120573 = 2120575 = 002
1198861= 30 unit 119887
1= 5 unit 119886
2= 100 unit 119860
0= $500 119888
1= $5
1198882= $10 119888
3= $12 and 119888
4= $8
The model starting with no shortage by solving theequation ℎ(119905
1) = 0 we have 119905
lowast
1= 58330 From (38) and
(18) we obtain 119876lowast
= 5284725 and 119879119862(119905lowast
1) = 16014013
respectively
Example 3 Theparameter values are given as follows119879 = 12
weeks 1205831= 4 weeks 120583
2= 6 weeks 120572 = 0005 120573 = 16
120575 = 02 1198861= 30 unit 119887
1= 5 unit 119886
2= 100 unit 119860
0= $500
1198881= $5 119888
2= $10 119888
3= $12 and 119888
4= $8
Themodel starting with no shortage solving the equationℎ(1199051) = 0 the optimal value of 119905
1is 119905lowast1= 23235 The optimal
ordering quantity is 119876lowast = 2726678 and the minimum cost119879119862(119905lowast
1) = 125882
In order to clearly indicate the effects of parameters suchas 120575 120572 120573 119888
1 1198882 1198883 and 119888
4on the optimal on-hand inventory
119878lowast the optimal ordering quantity 119876
lowast and the optimal totalcost 119879119862(119905lowast
1) respectively the paper will study the sensitivity
of the optimal solution to changes in the value of differentparameter associated with the studied inventory model Thesensitivity analysis is performed on the base of Example 1 andthe results are shown in Table 1ndash7
By studying the results of Table 1 it is found that theshortage time 119905
lowast
1 inventory level 119878
lowast order quantity 119876lowast
and the total average cost 119879119862(119905lowast1) gradually decrease as the
shortage parameter 120575 increases for the model respectivelyWe also find that the percentage increase of 120575 from 143to 100 causes 119879119862(119905
lowast
1) to decrease from 045 to 034
119876lowast decrease from 075 to 052 119905lowast
1decrease from 078
to 053 and 119878lowast decrease from 102 to 069 It is also
observed that the value of 119905lowast1 119878lowast119876lowast and 119879119862(119905lowast
1) all are lowly
sensitive to the changes of 120575 for the considered inventorymodel
By studying the results of Table 2 it is found that 119878lowast 119876lowastand 119879119862(119905lowast
1) coordinates to the deterioration parameter 120572 the
shortage time 119905lowast1decreases as 120572 increases for the model It is
also found that the percentage increase of 120572 from 167 to100 causes 119879119862(119905lowast
1) to decrease by 2066ndash2595 119876lowast to
increase by 1597ndash244 the shortage time 119905lowast1to decrease
by 1513ndash1455 and 119878lowast to increase by 0917ndash1651 It
also observes that the value of 119905lowast1 119878lowast 119876lowast and 119879119862(119905
lowast
1) all are
moderately sensitive to the changes of 120572 for the consideredinventory model
8 Journal of Applied Mathematics
Table 1 The sensitivity of 120575 for the models in Example 1
By studying the results of Table 3 it is found that 119878lowast119876lowast and 119879119862(119905
lowast
1) coordinate to the deterioration parameter 120573
while the shortage time 119905lowast
1decreases as 120573 increases for the
model It is also found that the increase of 120573 from 14 to 22causes 119878
lowast to increase while the increase of 120573 from 24 to28 causes 119878lowast to decrease 119876lowast to increase by 244ndash1597119879119862(119905lowast
1) to increase by 2595ndash2066 and the shortage time
119905lowast
1to decrease by 1513ndash1455 It is also observed that the
value of 119905lowast1 119878lowast119876lowast and 119879119862(119905lowast
1) all are moderately sensitive to
the changes of 120573 for the considered inventory modelBy studying the results of Table 4 it is found that 119879119862(119905lowast
1)
coordinate to 1198881 while the shortage time 119905
lowast
1 119878lowast and 119876
lowast
decrease as 1198881increases for the model It is also found that 119888
1
increases from 83 to 150 119879119862(119905lowast1) decreases by 0269ndash
0383 119876lowast decreases by 0083ndash0058 119905lowast1decreases by
0264ndash0181 and 119878lowast decreases by 0203ndash0138 respec-tively It is also observed that the values of 119905lowast
1 119878lowast 119876lowast and
119879119862(119905lowast
1) all are lowly sensitive to the changes of 119888
1for the
considered inventory modelBy studying the results of Table 5 it is found that
119879119862(119905lowast
1) coordinates to 119888
2 while 119878
lowast 119876lowastand 119905lowast
1decrease as
1198881increases for the model It is also found that 119888
2increases
by 100 119879119862(119905lowast1) decreases by 0269ndash0383 119876lowast decreases
by 0083ndash0058 119905lowast1decreases by 0264ndash0181 and 119878
lowast
decreases by 0203ndash0138By studying the results of Table 6 it is found that 119905lowast
1 119878lowast
119876lowast and 119879119862(119905
lowast
1) coordinate to 119888
3 It is also observed that the
value of 119905lowast1 119878lowast 119876lowast and 119879119862(119905
lowast
1) all are lowly sensitive to the
changes of 1198883for the inventory models that is 119888
3increases
from 19 to 32 the change of all the parameters is nomorethan 1
By studying the results of Table 7 it is found that 119905lowast1 119878lowast
119876lowast and 119879119862(119905
lowast
1) coordinate to 119888
4 It is also observed that the
value of 119905lowast1 119878lowast 119876lowast and 119879119862(119905
lowast
1) all are lowly sensitive to the
changes of 1198884for the inventory models that is 119888
4increases
from 143 to 100 the change of all the parameters is nomore than 1
5 Conclusion
An inventory model starting without shortage for Weibull-distributed deterioration with trapezoidal type demand rateand partial backlogging is considered in this paper Theoptimal replenishment policy for the inventory model isproposed and numerical examples are provided to illustratethe theoretical results A sensitivity analysis of the optimalsolution with respect to major parameters is also carried outFrom Table 1ndash7 it can be found that the shortage time point119905lowast
1 order quantity 119876lowast and the total average cost 119879119862(119905lowast
1) are
moderately sensitive to the changes of 120572 and 120573 and lowlysensitive to the changes of 120575 119888
119894(119894 = 1 2 3 4) respectively
The paper provides an interesting topic for further studysuch that the joint influence from some of these parametersmay be investigated to show the effects the model startingwith shortage will be studied and other types of models fordeteriorating items in supply chain situation are also to bestudied in the future
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author is grateful to the anonymous referees who pro-vided valuable comments and suggestions to significantlyimprove the quality of the paper This work was supportedpartly byHumanities and Social Science Fund of theMinistryof Education of China (no 11YJCZH019)
References
[1] P M Ghare and G F Schrader ldquoA model for exponentiallydecaying inventoriesrdquo Journal of Industrial Engineering vol 14pp 238ndash243 1963
[2] J-W Wu C Lin B Tan and W-C Lee ldquoAn EOQ inventorymodel with time-varying demand and Weibull deteriorationwith shortagesrdquo International Journal of Systems Science vol 31no 6 pp 677ndash683 2000
[3] H-M Wee ldquoDeteriorating inventory model with quantity dis-count pricing and partial backorderingrdquo International Journalof Production Economics vol 59 no 1 pp 511ndash518 1999
[4] K Skouri and S Papachristos ldquoA continuous review inven-tory model with deteriorating items time-varying demandlinear replenishment cost partially time-varying backloggingrdquoApplied Mathematical Modelling vol 26 no 5 pp 603ndash6172002
[5] H-MWee J C P Yu and S T Law ldquoTwo-warehouse inventorymodel with partial backordering and Weibull distributiondeterioration under inflationrdquo Journal of the Chinese Institute ofIndustrial Engineers vol 22 no 6 pp 451ndash462 2005
[6] C-Y Dye T-P Hsieh and L-Y Ouyang ldquoDetermining optimalselling price and lot size with a varying rate of deteriorationand exponential partial backloggingrdquo European Journal ofOperational Research vol 181 no 2 pp 668ndash678 2007
[7] R M Hill ldquoInventory models for increasing demand followedby level demandrdquo Journal of the Operational Research Societyvol 46 no 10 pp 1250ndash1259 1995
[8] B Mandal and A K Pal ldquoOrder level inventory system withramp type demand rate for deteriorating itemsrdquo Journal ofInterdisciplinary Mathematics vol 1 no 1 pp 49ndash66 1998
[9] K-S Wu ldquoAn EOQ inventory model for items with Weibulldistribution deterioration ramp type demand rate and partialbackloggingrdquo Production Planning amp Control vol 12 no 8 pp787ndash793 2001
[10] B C Giri A K Jalan and K S Chaudhuri ldquoEconomic orderquantity model with Weibull deterioration distribution short-age and ramp-type demandrdquo International Journal of SystemsScience vol 34 no 4 pp 237ndash243 2003
[11] S K Manna and K S Chaudhuri ldquoAn EOQ model withramp type demand rate time dependent deterioration rate unitproduction cost and shortagesrdquoEuropean Journal ofOperationalResearch vol 171 no 2 pp 557ndash566 2006
[12] K Skouri I Konstantaras S Papachristos and I Ganas ldquoInven-tory models with ramp type demand rate partial backloggingandWeibull deterioration raterdquoEuropean Journal ofOperationalResearch vol 192 no 1 pp 79ndash92 2009
10 Journal of Applied Mathematics
[13] K-C Hung ldquoAn inventory model with generalized typedemand deterioration and backorder ratesrdquo European Journalof Operational Research vol 208 no 3 pp 239ndash242 2011
[14] R S Kumar S K De and A Goswami ldquoFuzzy EOQ modelswith ramp type demand rate partial backlogging and timedependent deterioration raterdquo International Journal of Mathe-matics in Operational Research vol 4 no 5 pp 473ndash502 2012
[15] M B Cheng B X Zhang and G QWang ldquoOptimal policy fordeteriorating items with trapezoidal type demand and partialbackloggingrdquoAppliedMathematical Modelling vol 35 no 7 pp3552ndash3560 2011
[16] R Uthayakumar and M Rameswari ldquoAn economic produc-tion quantity model for defective items with trapezoidal typedemand raterdquo Journal of Optimization Theory and Applicationsvol 154 no 3 pp 1055ndash1079 2012
[17] Y Tan and M X Weng ldquoA discrete-in-time deterioratinginventory model with time-varying demand variable deterio-ration rate and waiting-time-dependent partial backloggingrdquoInternational Journal of Systems Science vol 44 no 8 pp 1483ndash1493 2013
[18] M A Ahmed T A Al-Khamis and L Benkherouf ldquoInventorymodels with ramp type demand rate partial backlogging andgeneral deterioration raterdquo Applied Mathematics and Computa-tion vol 219 no 9 pp 4288ndash4307 2013
[19] K-P Lin ldquoAn extended inventory models with trapezoidal typedemandsrdquo Applied Mathematics and Computation vol 219 no24 pp 11414ndash11419 2013
[20] P L Abad ldquoOptimal pricing and lot-sizing under conditionsof perishability and partial backorderingrdquoManagement Sciencevol 42 no 8 pp 1093ndash1104 1996
finding the economic order quantity for an inventory modelwith ramp type demand rate partial backlogging and generaldeterioration rate Lin [19] explored the inventorymodel witha general demand rate in which both the Weibull-distributeddeterioration and partial backlogging are considered
In the above mentioned research one of assumptionswas considered the ramp type demand rate partial back-logging andWeibull-distributed deterioration rateHoweverfor fashionable commodities high-tech products and othershort life cycle products the demand rate should increasewith the time up to certain point at first stage then reacha stabilized period and finally the demand rate decreaseto zero and the products retreat from market in theirproduct life cycle that is the demand rate with continuoustrapezoidal function of time On the other hand inmany realsituations customers encountering shortages will responddifferently Some customers are willing to wait until thenext replenishment while others may be impatient and goelsewhere as waiting time increases that is the willingnessfor a customer to wait for backlogging is diminishing withthe length of the waiting time In this paper we consideran inventory model with Weibull-distributed deteriorationitems trapezoidal type demand rate and time-dependentpartial backlogging By analyzing the inventory model auseful inventory replenishment policy is proposed Finallynumerical examples are provided to illustrate the theoreticalresults and a sensitivity analysis of the optimal solution withrespect to major parameters is also carried out
The rest of the paper is organized as follows Section 2describes the notation and assumptions used throughout thispaper Section 3 analyzes the inventory model and somenumerical examples to illustrate the solution procedure areprovided Sensitivity analysis of the major parameters is alsocarried out in Section 4 and the final Section concludes thispaper
2 Notations and Assumptions
The fundamental notations and assumptions used in inven-tory model and considered in this paper are given as below
(i) 119868(119905) the level of inventory at time 119905 0 le 119905 le 119879(ii) 119879 the fixed length of each ordering cycle(iii) 1199051the time when the inventory level reaches zero for
the inventory model(iv) 119905lowast1the optimal point
(v) 119878 the maximum inventory level for each orderingcycle
(vi) 119876lowast the optimal ordering quantity(vii) 119860
0the fixed cost per order
(viii) 1198881the cost of each deteriorated item
(ix) 1198882the inventory holding cost per unit per unit of time
(x) 1198883the shortage cost per unit per unit of time
(xi) 1198884the lost sales cost per unit
(xii) 119862119894(1199051) 119894 = 1 2 3 the average total cost per unit time
under different conditions respectively
(xiii) 119879119862(1199051) the average total cost per unit time
(xiv) The demand rate 119863(119905) which is positive and consec-utive is assumed to be a trapezoidal type function oftime that is
119863 (119905) =
119891 (119905) 119905 le 1205831
1198630 120583
1lt 119905 lt 120583
2
119892 (119905) 1205832le 119905 lt 119879
(1)
where 1205831is time point changing from the increasing
demand function119891(119905) to constant demand1198630 and 120583
2
is time point changing from the constant demand1198630
to the decreasing demand function 119892(119905)(xv) The replenishment rate is infinite that is replenish-
ment is instantaneous(xvi) The deterioration rate of the item is defined as
Weibull (120572 120573) that is the deterioration rate is 120579(119905) =120572120573119905120573minus1
(120572 gt 0 120573 gt 0 119905 gt 0)(xvii) Shortages are allowed and they adopt the notation
used in Abad [20] where the unsatisfied demandis backlogged and the fraction of shortages backo-rdered is 119890minus120575119905 where 119905 is the waiting time up to thenext replenishment We also assume that 119905119890minus120575119905 is anincreasing function which had appeared in Skouri etal [12]
(xviii) The time horizon of the inventory model is finite
3 Model Formulation
In this section we consider an inventory model startingwith no shortage The behavior of the model during a givencycle is depicted in Figure 1 Replenishment occurs at time119905 = 0 and the inventory level attains its maximum From119905 = 0 to 119905
1 the inventory level reduces due to demand
and deterioration At 1199051 the inventory level achieves zero
then shortage is allowed to occur during the time interval(1199051 119879) and all of the demand during the shortage period
(1199051 119879) is partially backlogged According to the notations and
assumptions mentioned above the behavior of the modelat any time can be described by the following differentialequations
(119879 minus 119905) 119890120575(119905minus119879)
119892 (119905) 119889119905]
+ 1198884[int
119879
1199051
(1 minus 119890120575(119905minus119879)
) 119892 (119905) 119889119905]
(26)
From the above analysis we obtain that the total average costof the model over the time interval [0 119879] is
119879119862 (1199051) =
1198621(1199051) 0 lt 119905
1le 1205831
1198622(1199051) 120583
1lt 1199051le 1205832
1198623(1199051) 120583
2lt 1199051lt 119879
(27)
where 1198621(1199051) 1198622(1199051) and 119862
3(1199051) are obtained from (10) (18)
and (26) respectivelyIn the following we will provide the results which ensure
the existence of a unique 1199051 say 119905lowast1 so as to minimize the total
average cost for the model system starting with no shortagesIf 0 lt 119905
1le 1205831 taking the first-order derivative of 119862
1(1199051)
with respect to 1199051 we obtain
1198891198621(1199051)
1198891199051
=119891 (1199051)
119879ℎ (1199051) (28)
where
ℎ (1199051) = 1198881(119890120572119905120573
1 minus 1)
+ 1198882int
1199051
0
119890120572(119905120573
1minus119905120573)119889119905 + 119888
3(1199051minus 119879) 119890
120575(1199051minus119879)
+ 1198884(119890120575(1199051minus119879)
minus 1)
(29)
then we can obtain ℎ(0) lt 0 and ℎ(119879) gt 0 By using theassumption (119905119890minus120575119905 is an increasing function where 119905 is thewaiting time up to the next replenishment) we have
The function ℎ(1199051) is given by (31) and (40) implies that
1198623(1199051) can obtain its minimum value at 119905lowast
1
The optimal value of the order level 119878 = 119868(0) is
119878lowast= int
1205831
0
119890120572119909120573
119891 (119909) 119889119909 + 1198630int
1205832
1205831
119890120572119909120573
119889119909 + int
119905lowast
1
1205832
119890120572119909120573
119892 (119909) 119889119909
(41)
and the optimal order quantity 119876lowast is
119876lowast= 119878lowast+ int
119879
119905lowast
1
119890120575(119905minus119879)
119892 (119905) 119889119905 (42)
If 119905lowast1
le 1205832 then the optimal value of 119862
3(1199051) is obtained at
119905lowast
1= 1205832
The above analysis shows that the three average cost func-tions 119862
1(1199051) 1198622(1199052) and 119862
3(1199051) can obtain their minimum
value at 119905lowast1isin (0 119879) which is determined by (31) Therefore
based on the results analyzed above this paper derives aprocedure to locate the optimal replenishment policy startingwith no shortage for the three cases The procedure is asfollowsStep 1 Solve 119905lowast
1from (31)
Step 2 Compare 119905lowast1to 1205831and 120583
2 respectively
Step 21 If 119905lowast1isin (0 120583
1] then the optimal total average cost and
the optimal order quantity can be obtained by (10) and (34)respectivelyStep 22 If 119905lowast
1isin (1205831 1205832] then the optimal total average cost
and the optimal order quantity can be obtained by (18) and(38) respectively
Step 23 If 119905lowast1isin (1205832 119879] then the optimal total average cost
and the optimal order quantity can be obtained by (26) and(42) respectively
Remark 1 In such considered inventory model starting withno shortage if 119905
1satisfies 120583
1lt 1199051le 119879 lt 120583
2 the considered
inventory model reduces to that of Skouri et al [12]
4 Numerical Example
In order to demonstrate the above procedure which can beapplied to obtain the optimal solution of the model this
paper presents several examples for the model respectivelyExamples are based on piecewise demand rate such as119891(119905) =1198861+ 1198871119905 and 119892(119905) = 119886
2119890minus1198872119905
Example 1 The parameter values are given as follows 119879 = 12
weeks1205831= 4weeks120583
2= 8weeks120572 = 0005120573 = 2120575 = 004
1198861= 30 unit 119887
1= 5 unit 119886
2= 100 unit 119860
0= $500 119888
1= $2
1198882= $3 119888
3= $12 and 119888
4= $8
The model starting with no shortage by solving theequation ℎ(119905
1) = 0 we have 119905
lowast
1= 87622 From (42) and
(26) we obtain 119876lowast
= 5825217 and 119879119862(119905lowast
1) = 7939986
respectively
Example 2 Theparameter values are given as follows119879 = 12
weeks1205831= 4weeks120583
2= 8weeks120572 = 0005120573 = 2120575 = 002
1198861= 30 unit 119887
1= 5 unit 119886
2= 100 unit 119860
0= $500 119888
1= $5
1198882= $10 119888
3= $12 and 119888
4= $8
The model starting with no shortage by solving theequation ℎ(119905
1) = 0 we have 119905
lowast
1= 58330 From (38) and
(18) we obtain 119876lowast
= 5284725 and 119879119862(119905lowast
1) = 16014013
respectively
Example 3 Theparameter values are given as follows119879 = 12
weeks 1205831= 4 weeks 120583
2= 6 weeks 120572 = 0005 120573 = 16
120575 = 02 1198861= 30 unit 119887
1= 5 unit 119886
2= 100 unit 119860
0= $500
1198881= $5 119888
2= $10 119888
3= $12 and 119888
4= $8
Themodel starting with no shortage solving the equationℎ(1199051) = 0 the optimal value of 119905
1is 119905lowast1= 23235 The optimal
ordering quantity is 119876lowast = 2726678 and the minimum cost119879119862(119905lowast
1) = 125882
In order to clearly indicate the effects of parameters suchas 120575 120572 120573 119888
1 1198882 1198883 and 119888
4on the optimal on-hand inventory
119878lowast the optimal ordering quantity 119876
lowast and the optimal totalcost 119879119862(119905lowast
1) respectively the paper will study the sensitivity
of the optimal solution to changes in the value of differentparameter associated with the studied inventory model Thesensitivity analysis is performed on the base of Example 1 andthe results are shown in Table 1ndash7
By studying the results of Table 1 it is found that theshortage time 119905
lowast
1 inventory level 119878
lowast order quantity 119876lowast
and the total average cost 119879119862(119905lowast1) gradually decrease as the
shortage parameter 120575 increases for the model respectivelyWe also find that the percentage increase of 120575 from 143to 100 causes 119879119862(119905
lowast
1) to decrease from 045 to 034
119876lowast decrease from 075 to 052 119905lowast
1decrease from 078
to 053 and 119878lowast decrease from 102 to 069 It is also
observed that the value of 119905lowast1 119878lowast119876lowast and 119879119862(119905lowast
1) all are lowly
sensitive to the changes of 120575 for the considered inventorymodel
By studying the results of Table 2 it is found that 119878lowast 119876lowastand 119879119862(119905lowast
1) coordinates to the deterioration parameter 120572 the
shortage time 119905lowast1decreases as 120572 increases for the model It is
also found that the percentage increase of 120572 from 167 to100 causes 119879119862(119905lowast
1) to decrease by 2066ndash2595 119876lowast to
increase by 1597ndash244 the shortage time 119905lowast1to decrease
by 1513ndash1455 and 119878lowast to increase by 0917ndash1651 It
also observes that the value of 119905lowast1 119878lowast 119876lowast and 119879119862(119905
lowast
1) all are
moderately sensitive to the changes of 120572 for the consideredinventory model
8 Journal of Applied Mathematics
Table 1 The sensitivity of 120575 for the models in Example 1
By studying the results of Table 3 it is found that 119878lowast119876lowast and 119879119862(119905
lowast
1) coordinate to the deterioration parameter 120573
while the shortage time 119905lowast
1decreases as 120573 increases for the
model It is also found that the increase of 120573 from 14 to 22causes 119878
lowast to increase while the increase of 120573 from 24 to28 causes 119878lowast to decrease 119876lowast to increase by 244ndash1597119879119862(119905lowast
1) to increase by 2595ndash2066 and the shortage time
119905lowast
1to decrease by 1513ndash1455 It is also observed that the
value of 119905lowast1 119878lowast119876lowast and 119879119862(119905lowast
1) all are moderately sensitive to
the changes of 120573 for the considered inventory modelBy studying the results of Table 4 it is found that 119879119862(119905lowast
1)
coordinate to 1198881 while the shortage time 119905
lowast
1 119878lowast and 119876
lowast
decrease as 1198881increases for the model It is also found that 119888
1
increases from 83 to 150 119879119862(119905lowast1) decreases by 0269ndash
0383 119876lowast decreases by 0083ndash0058 119905lowast1decreases by
0264ndash0181 and 119878lowast decreases by 0203ndash0138 respec-tively It is also observed that the values of 119905lowast
1 119878lowast 119876lowast and
119879119862(119905lowast
1) all are lowly sensitive to the changes of 119888
1for the
considered inventory modelBy studying the results of Table 5 it is found that
119879119862(119905lowast
1) coordinates to 119888
2 while 119878
lowast 119876lowastand 119905lowast
1decrease as
1198881increases for the model It is also found that 119888
2increases
by 100 119879119862(119905lowast1) decreases by 0269ndash0383 119876lowast decreases
by 0083ndash0058 119905lowast1decreases by 0264ndash0181 and 119878
lowast
decreases by 0203ndash0138By studying the results of Table 6 it is found that 119905lowast
1 119878lowast
119876lowast and 119879119862(119905
lowast
1) coordinate to 119888
3 It is also observed that the
value of 119905lowast1 119878lowast 119876lowast and 119879119862(119905
lowast
1) all are lowly sensitive to the
changes of 1198883for the inventory models that is 119888
3increases
from 19 to 32 the change of all the parameters is nomorethan 1
By studying the results of Table 7 it is found that 119905lowast1 119878lowast
119876lowast and 119879119862(119905
lowast
1) coordinate to 119888
4 It is also observed that the
value of 119905lowast1 119878lowast 119876lowast and 119879119862(119905
lowast
1) all are lowly sensitive to the
changes of 1198884for the inventory models that is 119888
4increases
from 143 to 100 the change of all the parameters is nomore than 1
5 Conclusion
An inventory model starting without shortage for Weibull-distributed deterioration with trapezoidal type demand rateand partial backlogging is considered in this paper Theoptimal replenishment policy for the inventory model isproposed and numerical examples are provided to illustratethe theoretical results A sensitivity analysis of the optimalsolution with respect to major parameters is also carried outFrom Table 1ndash7 it can be found that the shortage time point119905lowast
1 order quantity 119876lowast and the total average cost 119879119862(119905lowast
1) are
moderately sensitive to the changes of 120572 and 120573 and lowlysensitive to the changes of 120575 119888
119894(119894 = 1 2 3 4) respectively
The paper provides an interesting topic for further studysuch that the joint influence from some of these parametersmay be investigated to show the effects the model startingwith shortage will be studied and other types of models fordeteriorating items in supply chain situation are also to bestudied in the future
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author is grateful to the anonymous referees who pro-vided valuable comments and suggestions to significantlyimprove the quality of the paper This work was supportedpartly byHumanities and Social Science Fund of theMinistryof Education of China (no 11YJCZH019)
References
[1] P M Ghare and G F Schrader ldquoA model for exponentiallydecaying inventoriesrdquo Journal of Industrial Engineering vol 14pp 238ndash243 1963
[2] J-W Wu C Lin B Tan and W-C Lee ldquoAn EOQ inventorymodel with time-varying demand and Weibull deteriorationwith shortagesrdquo International Journal of Systems Science vol 31no 6 pp 677ndash683 2000
[3] H-M Wee ldquoDeteriorating inventory model with quantity dis-count pricing and partial backorderingrdquo International Journalof Production Economics vol 59 no 1 pp 511ndash518 1999
[4] K Skouri and S Papachristos ldquoA continuous review inven-tory model with deteriorating items time-varying demandlinear replenishment cost partially time-varying backloggingrdquoApplied Mathematical Modelling vol 26 no 5 pp 603ndash6172002
[5] H-MWee J C P Yu and S T Law ldquoTwo-warehouse inventorymodel with partial backordering and Weibull distributiondeterioration under inflationrdquo Journal of the Chinese Institute ofIndustrial Engineers vol 22 no 6 pp 451ndash462 2005
[6] C-Y Dye T-P Hsieh and L-Y Ouyang ldquoDetermining optimalselling price and lot size with a varying rate of deteriorationand exponential partial backloggingrdquo European Journal ofOperational Research vol 181 no 2 pp 668ndash678 2007
[7] R M Hill ldquoInventory models for increasing demand followedby level demandrdquo Journal of the Operational Research Societyvol 46 no 10 pp 1250ndash1259 1995
[8] B Mandal and A K Pal ldquoOrder level inventory system withramp type demand rate for deteriorating itemsrdquo Journal ofInterdisciplinary Mathematics vol 1 no 1 pp 49ndash66 1998
[9] K-S Wu ldquoAn EOQ inventory model for items with Weibulldistribution deterioration ramp type demand rate and partialbackloggingrdquo Production Planning amp Control vol 12 no 8 pp787ndash793 2001
[10] B C Giri A K Jalan and K S Chaudhuri ldquoEconomic orderquantity model with Weibull deterioration distribution short-age and ramp-type demandrdquo International Journal of SystemsScience vol 34 no 4 pp 237ndash243 2003
[11] S K Manna and K S Chaudhuri ldquoAn EOQ model withramp type demand rate time dependent deterioration rate unitproduction cost and shortagesrdquoEuropean Journal ofOperationalResearch vol 171 no 2 pp 557ndash566 2006
[12] K Skouri I Konstantaras S Papachristos and I Ganas ldquoInven-tory models with ramp type demand rate partial backloggingandWeibull deterioration raterdquoEuropean Journal ofOperationalResearch vol 192 no 1 pp 79ndash92 2009
10 Journal of Applied Mathematics
[13] K-C Hung ldquoAn inventory model with generalized typedemand deterioration and backorder ratesrdquo European Journalof Operational Research vol 208 no 3 pp 239ndash242 2011
[14] R S Kumar S K De and A Goswami ldquoFuzzy EOQ modelswith ramp type demand rate partial backlogging and timedependent deterioration raterdquo International Journal of Mathe-matics in Operational Research vol 4 no 5 pp 473ndash502 2012
[15] M B Cheng B X Zhang and G QWang ldquoOptimal policy fordeteriorating items with trapezoidal type demand and partialbackloggingrdquoAppliedMathematical Modelling vol 35 no 7 pp3552ndash3560 2011
[16] R Uthayakumar and M Rameswari ldquoAn economic produc-tion quantity model for defective items with trapezoidal typedemand raterdquo Journal of Optimization Theory and Applicationsvol 154 no 3 pp 1055ndash1079 2012
[17] Y Tan and M X Weng ldquoA discrete-in-time deterioratinginventory model with time-varying demand variable deterio-ration rate and waiting-time-dependent partial backloggingrdquoInternational Journal of Systems Science vol 44 no 8 pp 1483ndash1493 2013
[18] M A Ahmed T A Al-Khamis and L Benkherouf ldquoInventorymodels with ramp type demand rate partial backlogging andgeneral deterioration raterdquo Applied Mathematics and Computa-tion vol 219 no 9 pp 4288ndash4307 2013
[19] K-P Lin ldquoAn extended inventory models with trapezoidal typedemandsrdquo Applied Mathematics and Computation vol 219 no24 pp 11414ndash11419 2013
[20] P L Abad ldquoOptimal pricing and lot-sizing under conditionsof perishability and partial backorderingrdquoManagement Sciencevol 42 no 8 pp 1093ndash1104 1996
(119879 minus 119905) 119890120575(119905minus119879)
119892 (119905) 119889119905]
+ 1198884[int
119879
1199051
(1 minus 119890120575(119905minus119879)
) 119892 (119905) 119889119905]
(26)
From the above analysis we obtain that the total average costof the model over the time interval [0 119879] is
119879119862 (1199051) =
1198621(1199051) 0 lt 119905
1le 1205831
1198622(1199051) 120583
1lt 1199051le 1205832
1198623(1199051) 120583
2lt 1199051lt 119879
(27)
where 1198621(1199051) 1198622(1199051) and 119862
3(1199051) are obtained from (10) (18)
and (26) respectivelyIn the following we will provide the results which ensure
the existence of a unique 1199051 say 119905lowast1 so as to minimize the total
average cost for the model system starting with no shortagesIf 0 lt 119905
1le 1205831 taking the first-order derivative of 119862
1(1199051)
with respect to 1199051 we obtain
1198891198621(1199051)
1198891199051
=119891 (1199051)
119879ℎ (1199051) (28)
where
ℎ (1199051) = 1198881(119890120572119905120573
1 minus 1)
+ 1198882int
1199051
0
119890120572(119905120573
1minus119905120573)119889119905 + 119888
3(1199051minus 119879) 119890
120575(1199051minus119879)
+ 1198884(119890120575(1199051minus119879)
minus 1)
(29)
then we can obtain ℎ(0) lt 0 and ℎ(119879) gt 0 By using theassumption (119905119890minus120575119905 is an increasing function where 119905 is thewaiting time up to the next replenishment) we have
The function ℎ(1199051) is given by (31) and (40) implies that
1198623(1199051) can obtain its minimum value at 119905lowast
1
The optimal value of the order level 119878 = 119868(0) is
119878lowast= int
1205831
0
119890120572119909120573
119891 (119909) 119889119909 + 1198630int
1205832
1205831
119890120572119909120573
119889119909 + int
119905lowast
1
1205832
119890120572119909120573
119892 (119909) 119889119909
(41)
and the optimal order quantity 119876lowast is
119876lowast= 119878lowast+ int
119879
119905lowast
1
119890120575(119905minus119879)
119892 (119905) 119889119905 (42)
If 119905lowast1
le 1205832 then the optimal value of 119862
3(1199051) is obtained at
119905lowast
1= 1205832
The above analysis shows that the three average cost func-tions 119862
1(1199051) 1198622(1199052) and 119862
3(1199051) can obtain their minimum
value at 119905lowast1isin (0 119879) which is determined by (31) Therefore
based on the results analyzed above this paper derives aprocedure to locate the optimal replenishment policy startingwith no shortage for the three cases The procedure is asfollowsStep 1 Solve 119905lowast
1from (31)
Step 2 Compare 119905lowast1to 1205831and 120583
2 respectively
Step 21 If 119905lowast1isin (0 120583
1] then the optimal total average cost and
the optimal order quantity can be obtained by (10) and (34)respectivelyStep 22 If 119905lowast
1isin (1205831 1205832] then the optimal total average cost
and the optimal order quantity can be obtained by (18) and(38) respectively
Step 23 If 119905lowast1isin (1205832 119879] then the optimal total average cost
and the optimal order quantity can be obtained by (26) and(42) respectively
Remark 1 In such considered inventory model starting withno shortage if 119905
1satisfies 120583
1lt 1199051le 119879 lt 120583
2 the considered
inventory model reduces to that of Skouri et al [12]
4 Numerical Example
In order to demonstrate the above procedure which can beapplied to obtain the optimal solution of the model this
paper presents several examples for the model respectivelyExamples are based on piecewise demand rate such as119891(119905) =1198861+ 1198871119905 and 119892(119905) = 119886
2119890minus1198872119905
Example 1 The parameter values are given as follows 119879 = 12
weeks1205831= 4weeks120583
2= 8weeks120572 = 0005120573 = 2120575 = 004
1198861= 30 unit 119887
1= 5 unit 119886
2= 100 unit 119860
0= $500 119888
1= $2
1198882= $3 119888
3= $12 and 119888
4= $8
The model starting with no shortage by solving theequation ℎ(119905
1) = 0 we have 119905
lowast
1= 87622 From (42) and
(26) we obtain 119876lowast
= 5825217 and 119879119862(119905lowast
1) = 7939986
respectively
Example 2 Theparameter values are given as follows119879 = 12
weeks1205831= 4weeks120583
2= 8weeks120572 = 0005120573 = 2120575 = 002
1198861= 30 unit 119887
1= 5 unit 119886
2= 100 unit 119860
0= $500 119888
1= $5
1198882= $10 119888
3= $12 and 119888
4= $8
The model starting with no shortage by solving theequation ℎ(119905
1) = 0 we have 119905
lowast
1= 58330 From (38) and
(18) we obtain 119876lowast
= 5284725 and 119879119862(119905lowast
1) = 16014013
respectively
Example 3 Theparameter values are given as follows119879 = 12
weeks 1205831= 4 weeks 120583
2= 6 weeks 120572 = 0005 120573 = 16
120575 = 02 1198861= 30 unit 119887
1= 5 unit 119886
2= 100 unit 119860
0= $500
1198881= $5 119888
2= $10 119888
3= $12 and 119888
4= $8
Themodel starting with no shortage solving the equationℎ(1199051) = 0 the optimal value of 119905
1is 119905lowast1= 23235 The optimal
ordering quantity is 119876lowast = 2726678 and the minimum cost119879119862(119905lowast
1) = 125882
In order to clearly indicate the effects of parameters suchas 120575 120572 120573 119888
1 1198882 1198883 and 119888
4on the optimal on-hand inventory
119878lowast the optimal ordering quantity 119876
lowast and the optimal totalcost 119879119862(119905lowast
1) respectively the paper will study the sensitivity
of the optimal solution to changes in the value of differentparameter associated with the studied inventory model Thesensitivity analysis is performed on the base of Example 1 andthe results are shown in Table 1ndash7
By studying the results of Table 1 it is found that theshortage time 119905
lowast
1 inventory level 119878
lowast order quantity 119876lowast
and the total average cost 119879119862(119905lowast1) gradually decrease as the
shortage parameter 120575 increases for the model respectivelyWe also find that the percentage increase of 120575 from 143to 100 causes 119879119862(119905
lowast
1) to decrease from 045 to 034
119876lowast decrease from 075 to 052 119905lowast
1decrease from 078
to 053 and 119878lowast decrease from 102 to 069 It is also
observed that the value of 119905lowast1 119878lowast119876lowast and 119879119862(119905lowast
1) all are lowly
sensitive to the changes of 120575 for the considered inventorymodel
By studying the results of Table 2 it is found that 119878lowast 119876lowastand 119879119862(119905lowast
1) coordinates to the deterioration parameter 120572 the
shortage time 119905lowast1decreases as 120572 increases for the model It is
also found that the percentage increase of 120572 from 167 to100 causes 119879119862(119905lowast
1) to decrease by 2066ndash2595 119876lowast to
increase by 1597ndash244 the shortage time 119905lowast1to decrease
by 1513ndash1455 and 119878lowast to increase by 0917ndash1651 It
also observes that the value of 119905lowast1 119878lowast 119876lowast and 119879119862(119905
lowast
1) all are
moderately sensitive to the changes of 120572 for the consideredinventory model
8 Journal of Applied Mathematics
Table 1 The sensitivity of 120575 for the models in Example 1
By studying the results of Table 3 it is found that 119878lowast119876lowast and 119879119862(119905
lowast
1) coordinate to the deterioration parameter 120573
while the shortage time 119905lowast
1decreases as 120573 increases for the
model It is also found that the increase of 120573 from 14 to 22causes 119878
lowast to increase while the increase of 120573 from 24 to28 causes 119878lowast to decrease 119876lowast to increase by 244ndash1597119879119862(119905lowast
1) to increase by 2595ndash2066 and the shortage time
119905lowast
1to decrease by 1513ndash1455 It is also observed that the
value of 119905lowast1 119878lowast119876lowast and 119879119862(119905lowast
1) all are moderately sensitive to
the changes of 120573 for the considered inventory modelBy studying the results of Table 4 it is found that 119879119862(119905lowast
1)
coordinate to 1198881 while the shortage time 119905
lowast
1 119878lowast and 119876
lowast
decrease as 1198881increases for the model It is also found that 119888
1
increases from 83 to 150 119879119862(119905lowast1) decreases by 0269ndash
0383 119876lowast decreases by 0083ndash0058 119905lowast1decreases by
0264ndash0181 and 119878lowast decreases by 0203ndash0138 respec-tively It is also observed that the values of 119905lowast
1 119878lowast 119876lowast and
119879119862(119905lowast
1) all are lowly sensitive to the changes of 119888
1for the
considered inventory modelBy studying the results of Table 5 it is found that
119879119862(119905lowast
1) coordinates to 119888
2 while 119878
lowast 119876lowastand 119905lowast
1decrease as
1198881increases for the model It is also found that 119888
2increases
by 100 119879119862(119905lowast1) decreases by 0269ndash0383 119876lowast decreases
by 0083ndash0058 119905lowast1decreases by 0264ndash0181 and 119878
lowast
decreases by 0203ndash0138By studying the results of Table 6 it is found that 119905lowast
1 119878lowast
119876lowast and 119879119862(119905
lowast
1) coordinate to 119888
3 It is also observed that the
value of 119905lowast1 119878lowast 119876lowast and 119879119862(119905
lowast
1) all are lowly sensitive to the
changes of 1198883for the inventory models that is 119888
3increases
from 19 to 32 the change of all the parameters is nomorethan 1
By studying the results of Table 7 it is found that 119905lowast1 119878lowast
119876lowast and 119879119862(119905
lowast
1) coordinate to 119888
4 It is also observed that the
value of 119905lowast1 119878lowast 119876lowast and 119879119862(119905
lowast
1) all are lowly sensitive to the
changes of 1198884for the inventory models that is 119888
4increases
from 143 to 100 the change of all the parameters is nomore than 1
5 Conclusion
An inventory model starting without shortage for Weibull-distributed deterioration with trapezoidal type demand rateand partial backlogging is considered in this paper Theoptimal replenishment policy for the inventory model isproposed and numerical examples are provided to illustratethe theoretical results A sensitivity analysis of the optimalsolution with respect to major parameters is also carried outFrom Table 1ndash7 it can be found that the shortage time point119905lowast
1 order quantity 119876lowast and the total average cost 119879119862(119905lowast
1) are
moderately sensitive to the changes of 120572 and 120573 and lowlysensitive to the changes of 120575 119888
119894(119894 = 1 2 3 4) respectively
The paper provides an interesting topic for further studysuch that the joint influence from some of these parametersmay be investigated to show the effects the model startingwith shortage will be studied and other types of models fordeteriorating items in supply chain situation are also to bestudied in the future
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author is grateful to the anonymous referees who pro-vided valuable comments and suggestions to significantlyimprove the quality of the paper This work was supportedpartly byHumanities and Social Science Fund of theMinistryof Education of China (no 11YJCZH019)
References
[1] P M Ghare and G F Schrader ldquoA model for exponentiallydecaying inventoriesrdquo Journal of Industrial Engineering vol 14pp 238ndash243 1963
[2] J-W Wu C Lin B Tan and W-C Lee ldquoAn EOQ inventorymodel with time-varying demand and Weibull deteriorationwith shortagesrdquo International Journal of Systems Science vol 31no 6 pp 677ndash683 2000
[3] H-M Wee ldquoDeteriorating inventory model with quantity dis-count pricing and partial backorderingrdquo International Journalof Production Economics vol 59 no 1 pp 511ndash518 1999
[4] K Skouri and S Papachristos ldquoA continuous review inven-tory model with deteriorating items time-varying demandlinear replenishment cost partially time-varying backloggingrdquoApplied Mathematical Modelling vol 26 no 5 pp 603ndash6172002
[5] H-MWee J C P Yu and S T Law ldquoTwo-warehouse inventorymodel with partial backordering and Weibull distributiondeterioration under inflationrdquo Journal of the Chinese Institute ofIndustrial Engineers vol 22 no 6 pp 451ndash462 2005
[6] C-Y Dye T-P Hsieh and L-Y Ouyang ldquoDetermining optimalselling price and lot size with a varying rate of deteriorationand exponential partial backloggingrdquo European Journal ofOperational Research vol 181 no 2 pp 668ndash678 2007
[7] R M Hill ldquoInventory models for increasing demand followedby level demandrdquo Journal of the Operational Research Societyvol 46 no 10 pp 1250ndash1259 1995
[8] B Mandal and A K Pal ldquoOrder level inventory system withramp type demand rate for deteriorating itemsrdquo Journal ofInterdisciplinary Mathematics vol 1 no 1 pp 49ndash66 1998
[9] K-S Wu ldquoAn EOQ inventory model for items with Weibulldistribution deterioration ramp type demand rate and partialbackloggingrdquo Production Planning amp Control vol 12 no 8 pp787ndash793 2001
[10] B C Giri A K Jalan and K S Chaudhuri ldquoEconomic orderquantity model with Weibull deterioration distribution short-age and ramp-type demandrdquo International Journal of SystemsScience vol 34 no 4 pp 237ndash243 2003
[11] S K Manna and K S Chaudhuri ldquoAn EOQ model withramp type demand rate time dependent deterioration rate unitproduction cost and shortagesrdquoEuropean Journal ofOperationalResearch vol 171 no 2 pp 557ndash566 2006
[12] K Skouri I Konstantaras S Papachristos and I Ganas ldquoInven-tory models with ramp type demand rate partial backloggingandWeibull deterioration raterdquoEuropean Journal ofOperationalResearch vol 192 no 1 pp 79ndash92 2009
10 Journal of Applied Mathematics
[13] K-C Hung ldquoAn inventory model with generalized typedemand deterioration and backorder ratesrdquo European Journalof Operational Research vol 208 no 3 pp 239ndash242 2011
[14] R S Kumar S K De and A Goswami ldquoFuzzy EOQ modelswith ramp type demand rate partial backlogging and timedependent deterioration raterdquo International Journal of Mathe-matics in Operational Research vol 4 no 5 pp 473ndash502 2012
[15] M B Cheng B X Zhang and G QWang ldquoOptimal policy fordeteriorating items with trapezoidal type demand and partialbackloggingrdquoAppliedMathematical Modelling vol 35 no 7 pp3552ndash3560 2011
[16] R Uthayakumar and M Rameswari ldquoAn economic produc-tion quantity model for defective items with trapezoidal typedemand raterdquo Journal of Optimization Theory and Applicationsvol 154 no 3 pp 1055ndash1079 2012
[17] Y Tan and M X Weng ldquoA discrete-in-time deterioratinginventory model with time-varying demand variable deterio-ration rate and waiting-time-dependent partial backloggingrdquoInternational Journal of Systems Science vol 44 no 8 pp 1483ndash1493 2013
[18] M A Ahmed T A Al-Khamis and L Benkherouf ldquoInventorymodels with ramp type demand rate partial backlogging andgeneral deterioration raterdquo Applied Mathematics and Computa-tion vol 219 no 9 pp 4288ndash4307 2013
[19] K-P Lin ldquoAn extended inventory models with trapezoidal typedemandsrdquo Applied Mathematics and Computation vol 219 no24 pp 11414ndash11419 2013
[20] P L Abad ldquoOptimal pricing and lot-sizing under conditionsof perishability and partial backorderingrdquoManagement Sciencevol 42 no 8 pp 1093ndash1104 1996
(119879 minus 119905) 119890120575(119905minus119879)
119892 (119905) 119889119905]
+ 1198884[int
119879
1199051
(1 minus 119890120575(119905minus119879)
) 119892 (119905) 119889119905]
(26)
From the above analysis we obtain that the total average costof the model over the time interval [0 119879] is
119879119862 (1199051) =
1198621(1199051) 0 lt 119905
1le 1205831
1198622(1199051) 120583
1lt 1199051le 1205832
1198623(1199051) 120583
2lt 1199051lt 119879
(27)
where 1198621(1199051) 1198622(1199051) and 119862
3(1199051) are obtained from (10) (18)
and (26) respectivelyIn the following we will provide the results which ensure
the existence of a unique 1199051 say 119905lowast1 so as to minimize the total
average cost for the model system starting with no shortagesIf 0 lt 119905
1le 1205831 taking the first-order derivative of 119862
1(1199051)
with respect to 1199051 we obtain
1198891198621(1199051)
1198891199051
=119891 (1199051)
119879ℎ (1199051) (28)
where
ℎ (1199051) = 1198881(119890120572119905120573
1 minus 1)
+ 1198882int
1199051
0
119890120572(119905120573
1minus119905120573)119889119905 + 119888
3(1199051minus 119879) 119890
120575(1199051minus119879)
+ 1198884(119890120575(1199051minus119879)
minus 1)
(29)
then we can obtain ℎ(0) lt 0 and ℎ(119879) gt 0 By using theassumption (119905119890minus120575119905 is an increasing function where 119905 is thewaiting time up to the next replenishment) we have
The function ℎ(1199051) is given by (31) and (40) implies that
1198623(1199051) can obtain its minimum value at 119905lowast
1
The optimal value of the order level 119878 = 119868(0) is
119878lowast= int
1205831
0
119890120572119909120573
119891 (119909) 119889119909 + 1198630int
1205832
1205831
119890120572119909120573
119889119909 + int
119905lowast
1
1205832
119890120572119909120573
119892 (119909) 119889119909
(41)
and the optimal order quantity 119876lowast is
119876lowast= 119878lowast+ int
119879
119905lowast
1
119890120575(119905minus119879)
119892 (119905) 119889119905 (42)
If 119905lowast1
le 1205832 then the optimal value of 119862
3(1199051) is obtained at
119905lowast
1= 1205832
The above analysis shows that the three average cost func-tions 119862
1(1199051) 1198622(1199052) and 119862
3(1199051) can obtain their minimum
value at 119905lowast1isin (0 119879) which is determined by (31) Therefore
based on the results analyzed above this paper derives aprocedure to locate the optimal replenishment policy startingwith no shortage for the three cases The procedure is asfollowsStep 1 Solve 119905lowast
1from (31)
Step 2 Compare 119905lowast1to 1205831and 120583
2 respectively
Step 21 If 119905lowast1isin (0 120583
1] then the optimal total average cost and
the optimal order quantity can be obtained by (10) and (34)respectivelyStep 22 If 119905lowast
1isin (1205831 1205832] then the optimal total average cost
and the optimal order quantity can be obtained by (18) and(38) respectively
Step 23 If 119905lowast1isin (1205832 119879] then the optimal total average cost
and the optimal order quantity can be obtained by (26) and(42) respectively
Remark 1 In such considered inventory model starting withno shortage if 119905
1satisfies 120583
1lt 1199051le 119879 lt 120583
2 the considered
inventory model reduces to that of Skouri et al [12]
4 Numerical Example
In order to demonstrate the above procedure which can beapplied to obtain the optimal solution of the model this
paper presents several examples for the model respectivelyExamples are based on piecewise demand rate such as119891(119905) =1198861+ 1198871119905 and 119892(119905) = 119886
2119890minus1198872119905
Example 1 The parameter values are given as follows 119879 = 12
weeks1205831= 4weeks120583
2= 8weeks120572 = 0005120573 = 2120575 = 004
1198861= 30 unit 119887
1= 5 unit 119886
2= 100 unit 119860
0= $500 119888
1= $2
1198882= $3 119888
3= $12 and 119888
4= $8
The model starting with no shortage by solving theequation ℎ(119905
1) = 0 we have 119905
lowast
1= 87622 From (42) and
(26) we obtain 119876lowast
= 5825217 and 119879119862(119905lowast
1) = 7939986
respectively
Example 2 Theparameter values are given as follows119879 = 12
weeks1205831= 4weeks120583
2= 8weeks120572 = 0005120573 = 2120575 = 002
1198861= 30 unit 119887
1= 5 unit 119886
2= 100 unit 119860
0= $500 119888
1= $5
1198882= $10 119888
3= $12 and 119888
4= $8
The model starting with no shortage by solving theequation ℎ(119905
1) = 0 we have 119905
lowast
1= 58330 From (38) and
(18) we obtain 119876lowast
= 5284725 and 119879119862(119905lowast
1) = 16014013
respectively
Example 3 Theparameter values are given as follows119879 = 12
weeks 1205831= 4 weeks 120583
2= 6 weeks 120572 = 0005 120573 = 16
120575 = 02 1198861= 30 unit 119887
1= 5 unit 119886
2= 100 unit 119860
0= $500
1198881= $5 119888
2= $10 119888
3= $12 and 119888
4= $8
Themodel starting with no shortage solving the equationℎ(1199051) = 0 the optimal value of 119905
1is 119905lowast1= 23235 The optimal
ordering quantity is 119876lowast = 2726678 and the minimum cost119879119862(119905lowast
1) = 125882
In order to clearly indicate the effects of parameters suchas 120575 120572 120573 119888
1 1198882 1198883 and 119888
4on the optimal on-hand inventory
119878lowast the optimal ordering quantity 119876
lowast and the optimal totalcost 119879119862(119905lowast
1) respectively the paper will study the sensitivity
of the optimal solution to changes in the value of differentparameter associated with the studied inventory model Thesensitivity analysis is performed on the base of Example 1 andthe results are shown in Table 1ndash7
By studying the results of Table 1 it is found that theshortage time 119905
lowast
1 inventory level 119878
lowast order quantity 119876lowast
and the total average cost 119879119862(119905lowast1) gradually decrease as the
shortage parameter 120575 increases for the model respectivelyWe also find that the percentage increase of 120575 from 143to 100 causes 119879119862(119905
lowast
1) to decrease from 045 to 034
119876lowast decrease from 075 to 052 119905lowast
1decrease from 078
to 053 and 119878lowast decrease from 102 to 069 It is also
observed that the value of 119905lowast1 119878lowast119876lowast and 119879119862(119905lowast
1) all are lowly
sensitive to the changes of 120575 for the considered inventorymodel
By studying the results of Table 2 it is found that 119878lowast 119876lowastand 119879119862(119905lowast
1) coordinates to the deterioration parameter 120572 the
shortage time 119905lowast1decreases as 120572 increases for the model It is
also found that the percentage increase of 120572 from 167 to100 causes 119879119862(119905lowast
1) to decrease by 2066ndash2595 119876lowast to
increase by 1597ndash244 the shortage time 119905lowast1to decrease
by 1513ndash1455 and 119878lowast to increase by 0917ndash1651 It
also observes that the value of 119905lowast1 119878lowast 119876lowast and 119879119862(119905
lowast
1) all are
moderately sensitive to the changes of 120572 for the consideredinventory model
8 Journal of Applied Mathematics
Table 1 The sensitivity of 120575 for the models in Example 1
By studying the results of Table 3 it is found that 119878lowast119876lowast and 119879119862(119905
lowast
1) coordinate to the deterioration parameter 120573
while the shortage time 119905lowast
1decreases as 120573 increases for the
model It is also found that the increase of 120573 from 14 to 22causes 119878
lowast to increase while the increase of 120573 from 24 to28 causes 119878lowast to decrease 119876lowast to increase by 244ndash1597119879119862(119905lowast
1) to increase by 2595ndash2066 and the shortage time
119905lowast
1to decrease by 1513ndash1455 It is also observed that the
value of 119905lowast1 119878lowast119876lowast and 119879119862(119905lowast
1) all are moderately sensitive to
the changes of 120573 for the considered inventory modelBy studying the results of Table 4 it is found that 119879119862(119905lowast
1)
coordinate to 1198881 while the shortage time 119905
lowast
1 119878lowast and 119876
lowast
decrease as 1198881increases for the model It is also found that 119888
1
increases from 83 to 150 119879119862(119905lowast1) decreases by 0269ndash
0383 119876lowast decreases by 0083ndash0058 119905lowast1decreases by
0264ndash0181 and 119878lowast decreases by 0203ndash0138 respec-tively It is also observed that the values of 119905lowast
1 119878lowast 119876lowast and
119879119862(119905lowast
1) all are lowly sensitive to the changes of 119888
1for the
considered inventory modelBy studying the results of Table 5 it is found that
119879119862(119905lowast
1) coordinates to 119888
2 while 119878
lowast 119876lowastand 119905lowast
1decrease as
1198881increases for the model It is also found that 119888
2increases
by 100 119879119862(119905lowast1) decreases by 0269ndash0383 119876lowast decreases
by 0083ndash0058 119905lowast1decreases by 0264ndash0181 and 119878
lowast
decreases by 0203ndash0138By studying the results of Table 6 it is found that 119905lowast
1 119878lowast
119876lowast and 119879119862(119905
lowast
1) coordinate to 119888
3 It is also observed that the
value of 119905lowast1 119878lowast 119876lowast and 119879119862(119905
lowast
1) all are lowly sensitive to the
changes of 1198883for the inventory models that is 119888
3increases
from 19 to 32 the change of all the parameters is nomorethan 1
By studying the results of Table 7 it is found that 119905lowast1 119878lowast
119876lowast and 119879119862(119905
lowast
1) coordinate to 119888
4 It is also observed that the
value of 119905lowast1 119878lowast 119876lowast and 119879119862(119905
lowast
1) all are lowly sensitive to the
changes of 1198884for the inventory models that is 119888
4increases
from 143 to 100 the change of all the parameters is nomore than 1
5 Conclusion
An inventory model starting without shortage for Weibull-distributed deterioration with trapezoidal type demand rateand partial backlogging is considered in this paper Theoptimal replenishment policy for the inventory model isproposed and numerical examples are provided to illustratethe theoretical results A sensitivity analysis of the optimalsolution with respect to major parameters is also carried outFrom Table 1ndash7 it can be found that the shortage time point119905lowast
1 order quantity 119876lowast and the total average cost 119879119862(119905lowast
1) are
moderately sensitive to the changes of 120572 and 120573 and lowlysensitive to the changes of 120575 119888
119894(119894 = 1 2 3 4) respectively
The paper provides an interesting topic for further studysuch that the joint influence from some of these parametersmay be investigated to show the effects the model startingwith shortage will be studied and other types of models fordeteriorating items in supply chain situation are also to bestudied in the future
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author is grateful to the anonymous referees who pro-vided valuable comments and suggestions to significantlyimprove the quality of the paper This work was supportedpartly byHumanities and Social Science Fund of theMinistryof Education of China (no 11YJCZH019)
References
[1] P M Ghare and G F Schrader ldquoA model for exponentiallydecaying inventoriesrdquo Journal of Industrial Engineering vol 14pp 238ndash243 1963
[2] J-W Wu C Lin B Tan and W-C Lee ldquoAn EOQ inventorymodel with time-varying demand and Weibull deteriorationwith shortagesrdquo International Journal of Systems Science vol 31no 6 pp 677ndash683 2000
[3] H-M Wee ldquoDeteriorating inventory model with quantity dis-count pricing and partial backorderingrdquo International Journalof Production Economics vol 59 no 1 pp 511ndash518 1999
[4] K Skouri and S Papachristos ldquoA continuous review inven-tory model with deteriorating items time-varying demandlinear replenishment cost partially time-varying backloggingrdquoApplied Mathematical Modelling vol 26 no 5 pp 603ndash6172002
[5] H-MWee J C P Yu and S T Law ldquoTwo-warehouse inventorymodel with partial backordering and Weibull distributiondeterioration under inflationrdquo Journal of the Chinese Institute ofIndustrial Engineers vol 22 no 6 pp 451ndash462 2005
[6] C-Y Dye T-P Hsieh and L-Y Ouyang ldquoDetermining optimalselling price and lot size with a varying rate of deteriorationand exponential partial backloggingrdquo European Journal ofOperational Research vol 181 no 2 pp 668ndash678 2007
[7] R M Hill ldquoInventory models for increasing demand followedby level demandrdquo Journal of the Operational Research Societyvol 46 no 10 pp 1250ndash1259 1995
[8] B Mandal and A K Pal ldquoOrder level inventory system withramp type demand rate for deteriorating itemsrdquo Journal ofInterdisciplinary Mathematics vol 1 no 1 pp 49ndash66 1998
[9] K-S Wu ldquoAn EOQ inventory model for items with Weibulldistribution deterioration ramp type demand rate and partialbackloggingrdquo Production Planning amp Control vol 12 no 8 pp787ndash793 2001
[10] B C Giri A K Jalan and K S Chaudhuri ldquoEconomic orderquantity model with Weibull deterioration distribution short-age and ramp-type demandrdquo International Journal of SystemsScience vol 34 no 4 pp 237ndash243 2003
[11] S K Manna and K S Chaudhuri ldquoAn EOQ model withramp type demand rate time dependent deterioration rate unitproduction cost and shortagesrdquoEuropean Journal ofOperationalResearch vol 171 no 2 pp 557ndash566 2006
[12] K Skouri I Konstantaras S Papachristos and I Ganas ldquoInven-tory models with ramp type demand rate partial backloggingandWeibull deterioration raterdquoEuropean Journal ofOperationalResearch vol 192 no 1 pp 79ndash92 2009
10 Journal of Applied Mathematics
[13] K-C Hung ldquoAn inventory model with generalized typedemand deterioration and backorder ratesrdquo European Journalof Operational Research vol 208 no 3 pp 239ndash242 2011
[14] R S Kumar S K De and A Goswami ldquoFuzzy EOQ modelswith ramp type demand rate partial backlogging and timedependent deterioration raterdquo International Journal of Mathe-matics in Operational Research vol 4 no 5 pp 473ndash502 2012
[15] M B Cheng B X Zhang and G QWang ldquoOptimal policy fordeteriorating items with trapezoidal type demand and partialbackloggingrdquoAppliedMathematical Modelling vol 35 no 7 pp3552ndash3560 2011
[16] R Uthayakumar and M Rameswari ldquoAn economic produc-tion quantity model for defective items with trapezoidal typedemand raterdquo Journal of Optimization Theory and Applicationsvol 154 no 3 pp 1055ndash1079 2012
[17] Y Tan and M X Weng ldquoA discrete-in-time deterioratinginventory model with time-varying demand variable deterio-ration rate and waiting-time-dependent partial backloggingrdquoInternational Journal of Systems Science vol 44 no 8 pp 1483ndash1493 2013
[18] M A Ahmed T A Al-Khamis and L Benkherouf ldquoInventorymodels with ramp type demand rate partial backlogging andgeneral deterioration raterdquo Applied Mathematics and Computa-tion vol 219 no 9 pp 4288ndash4307 2013
[19] K-P Lin ldquoAn extended inventory models with trapezoidal typedemandsrdquo Applied Mathematics and Computation vol 219 no24 pp 11414ndash11419 2013
[20] P L Abad ldquoOptimal pricing and lot-sizing under conditionsof perishability and partial backorderingrdquoManagement Sciencevol 42 no 8 pp 1093ndash1104 1996
(119879 minus 119905) 119890120575(119905minus119879)
119892 (119905) 119889119905]
+ 1198884[int
119879
1199051
(1 minus 119890120575(119905minus119879)
) 119892 (119905) 119889119905]
(26)
From the above analysis we obtain that the total average costof the model over the time interval [0 119879] is
119879119862 (1199051) =
1198621(1199051) 0 lt 119905
1le 1205831
1198622(1199051) 120583
1lt 1199051le 1205832
1198623(1199051) 120583
2lt 1199051lt 119879
(27)
where 1198621(1199051) 1198622(1199051) and 119862
3(1199051) are obtained from (10) (18)
and (26) respectivelyIn the following we will provide the results which ensure
the existence of a unique 1199051 say 119905lowast1 so as to minimize the total
average cost for the model system starting with no shortagesIf 0 lt 119905
1le 1205831 taking the first-order derivative of 119862
1(1199051)
with respect to 1199051 we obtain
1198891198621(1199051)
1198891199051
=119891 (1199051)
119879ℎ (1199051) (28)
where
ℎ (1199051) = 1198881(119890120572119905120573
1 minus 1)
+ 1198882int
1199051
0
119890120572(119905120573
1minus119905120573)119889119905 + 119888
3(1199051minus 119879) 119890
120575(1199051minus119879)
+ 1198884(119890120575(1199051minus119879)
minus 1)
(29)
then we can obtain ℎ(0) lt 0 and ℎ(119879) gt 0 By using theassumption (119905119890minus120575119905 is an increasing function where 119905 is thewaiting time up to the next replenishment) we have
The function ℎ(1199051) is given by (31) and (40) implies that
1198623(1199051) can obtain its minimum value at 119905lowast
1
The optimal value of the order level 119878 = 119868(0) is
119878lowast= int
1205831
0
119890120572119909120573
119891 (119909) 119889119909 + 1198630int
1205832
1205831
119890120572119909120573
119889119909 + int
119905lowast
1
1205832
119890120572119909120573
119892 (119909) 119889119909
(41)
and the optimal order quantity 119876lowast is
119876lowast= 119878lowast+ int
119879
119905lowast
1
119890120575(119905minus119879)
119892 (119905) 119889119905 (42)
If 119905lowast1
le 1205832 then the optimal value of 119862
3(1199051) is obtained at
119905lowast
1= 1205832
The above analysis shows that the three average cost func-tions 119862
1(1199051) 1198622(1199052) and 119862
3(1199051) can obtain their minimum
value at 119905lowast1isin (0 119879) which is determined by (31) Therefore
based on the results analyzed above this paper derives aprocedure to locate the optimal replenishment policy startingwith no shortage for the three cases The procedure is asfollowsStep 1 Solve 119905lowast
1from (31)
Step 2 Compare 119905lowast1to 1205831and 120583
2 respectively
Step 21 If 119905lowast1isin (0 120583
1] then the optimal total average cost and
the optimal order quantity can be obtained by (10) and (34)respectivelyStep 22 If 119905lowast
1isin (1205831 1205832] then the optimal total average cost
and the optimal order quantity can be obtained by (18) and(38) respectively
Step 23 If 119905lowast1isin (1205832 119879] then the optimal total average cost
and the optimal order quantity can be obtained by (26) and(42) respectively
Remark 1 In such considered inventory model starting withno shortage if 119905
1satisfies 120583
1lt 1199051le 119879 lt 120583
2 the considered
inventory model reduces to that of Skouri et al [12]
4 Numerical Example
In order to demonstrate the above procedure which can beapplied to obtain the optimal solution of the model this
paper presents several examples for the model respectivelyExamples are based on piecewise demand rate such as119891(119905) =1198861+ 1198871119905 and 119892(119905) = 119886
2119890minus1198872119905
Example 1 The parameter values are given as follows 119879 = 12
weeks1205831= 4weeks120583
2= 8weeks120572 = 0005120573 = 2120575 = 004
1198861= 30 unit 119887
1= 5 unit 119886
2= 100 unit 119860
0= $500 119888
1= $2
1198882= $3 119888
3= $12 and 119888
4= $8
The model starting with no shortage by solving theequation ℎ(119905
1) = 0 we have 119905
lowast
1= 87622 From (42) and
(26) we obtain 119876lowast
= 5825217 and 119879119862(119905lowast
1) = 7939986
respectively
Example 2 Theparameter values are given as follows119879 = 12
weeks1205831= 4weeks120583
2= 8weeks120572 = 0005120573 = 2120575 = 002
1198861= 30 unit 119887
1= 5 unit 119886
2= 100 unit 119860
0= $500 119888
1= $5
1198882= $10 119888
3= $12 and 119888
4= $8
The model starting with no shortage by solving theequation ℎ(119905
1) = 0 we have 119905
lowast
1= 58330 From (38) and
(18) we obtain 119876lowast
= 5284725 and 119879119862(119905lowast
1) = 16014013
respectively
Example 3 Theparameter values are given as follows119879 = 12
weeks 1205831= 4 weeks 120583
2= 6 weeks 120572 = 0005 120573 = 16
120575 = 02 1198861= 30 unit 119887
1= 5 unit 119886
2= 100 unit 119860
0= $500
1198881= $5 119888
2= $10 119888
3= $12 and 119888
4= $8
Themodel starting with no shortage solving the equationℎ(1199051) = 0 the optimal value of 119905
1is 119905lowast1= 23235 The optimal
ordering quantity is 119876lowast = 2726678 and the minimum cost119879119862(119905lowast
1) = 125882
In order to clearly indicate the effects of parameters suchas 120575 120572 120573 119888
1 1198882 1198883 and 119888
4on the optimal on-hand inventory
119878lowast the optimal ordering quantity 119876
lowast and the optimal totalcost 119879119862(119905lowast
1) respectively the paper will study the sensitivity
of the optimal solution to changes in the value of differentparameter associated with the studied inventory model Thesensitivity analysis is performed on the base of Example 1 andthe results are shown in Table 1ndash7
By studying the results of Table 1 it is found that theshortage time 119905
lowast
1 inventory level 119878
lowast order quantity 119876lowast
and the total average cost 119879119862(119905lowast1) gradually decrease as the
shortage parameter 120575 increases for the model respectivelyWe also find that the percentage increase of 120575 from 143to 100 causes 119879119862(119905
lowast
1) to decrease from 045 to 034
119876lowast decrease from 075 to 052 119905lowast
1decrease from 078
to 053 and 119878lowast decrease from 102 to 069 It is also
observed that the value of 119905lowast1 119878lowast119876lowast and 119879119862(119905lowast
1) all are lowly
sensitive to the changes of 120575 for the considered inventorymodel
By studying the results of Table 2 it is found that 119878lowast 119876lowastand 119879119862(119905lowast
1) coordinates to the deterioration parameter 120572 the
shortage time 119905lowast1decreases as 120572 increases for the model It is
also found that the percentage increase of 120572 from 167 to100 causes 119879119862(119905lowast
1) to decrease by 2066ndash2595 119876lowast to
increase by 1597ndash244 the shortage time 119905lowast1to decrease
by 1513ndash1455 and 119878lowast to increase by 0917ndash1651 It
also observes that the value of 119905lowast1 119878lowast 119876lowast and 119879119862(119905
lowast
1) all are
moderately sensitive to the changes of 120572 for the consideredinventory model
8 Journal of Applied Mathematics
Table 1 The sensitivity of 120575 for the models in Example 1
By studying the results of Table 3 it is found that 119878lowast119876lowast and 119879119862(119905
lowast
1) coordinate to the deterioration parameter 120573
while the shortage time 119905lowast
1decreases as 120573 increases for the
model It is also found that the increase of 120573 from 14 to 22causes 119878
lowast to increase while the increase of 120573 from 24 to28 causes 119878lowast to decrease 119876lowast to increase by 244ndash1597119879119862(119905lowast
1) to increase by 2595ndash2066 and the shortage time
119905lowast
1to decrease by 1513ndash1455 It is also observed that the
value of 119905lowast1 119878lowast119876lowast and 119879119862(119905lowast
1) all are moderately sensitive to
the changes of 120573 for the considered inventory modelBy studying the results of Table 4 it is found that 119879119862(119905lowast
1)
coordinate to 1198881 while the shortage time 119905
lowast
1 119878lowast and 119876
lowast
decrease as 1198881increases for the model It is also found that 119888
1
increases from 83 to 150 119879119862(119905lowast1) decreases by 0269ndash
0383 119876lowast decreases by 0083ndash0058 119905lowast1decreases by
0264ndash0181 and 119878lowast decreases by 0203ndash0138 respec-tively It is also observed that the values of 119905lowast
1 119878lowast 119876lowast and
119879119862(119905lowast
1) all are lowly sensitive to the changes of 119888
1for the
considered inventory modelBy studying the results of Table 5 it is found that
119879119862(119905lowast
1) coordinates to 119888
2 while 119878
lowast 119876lowastand 119905lowast
1decrease as
1198881increases for the model It is also found that 119888
2increases
by 100 119879119862(119905lowast1) decreases by 0269ndash0383 119876lowast decreases
by 0083ndash0058 119905lowast1decreases by 0264ndash0181 and 119878
lowast
decreases by 0203ndash0138By studying the results of Table 6 it is found that 119905lowast
1 119878lowast
119876lowast and 119879119862(119905
lowast
1) coordinate to 119888
3 It is also observed that the
value of 119905lowast1 119878lowast 119876lowast and 119879119862(119905
lowast
1) all are lowly sensitive to the
changes of 1198883for the inventory models that is 119888
3increases
from 19 to 32 the change of all the parameters is nomorethan 1
By studying the results of Table 7 it is found that 119905lowast1 119878lowast
119876lowast and 119879119862(119905
lowast
1) coordinate to 119888
4 It is also observed that the
value of 119905lowast1 119878lowast 119876lowast and 119879119862(119905
lowast
1) all are lowly sensitive to the
changes of 1198884for the inventory models that is 119888
4increases
from 143 to 100 the change of all the parameters is nomore than 1
5 Conclusion
An inventory model starting without shortage for Weibull-distributed deterioration with trapezoidal type demand rateand partial backlogging is considered in this paper Theoptimal replenishment policy for the inventory model isproposed and numerical examples are provided to illustratethe theoretical results A sensitivity analysis of the optimalsolution with respect to major parameters is also carried outFrom Table 1ndash7 it can be found that the shortage time point119905lowast
1 order quantity 119876lowast and the total average cost 119879119862(119905lowast
1) are
moderately sensitive to the changes of 120572 and 120573 and lowlysensitive to the changes of 120575 119888
119894(119894 = 1 2 3 4) respectively
The paper provides an interesting topic for further studysuch that the joint influence from some of these parametersmay be investigated to show the effects the model startingwith shortage will be studied and other types of models fordeteriorating items in supply chain situation are also to bestudied in the future
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author is grateful to the anonymous referees who pro-vided valuable comments and suggestions to significantlyimprove the quality of the paper This work was supportedpartly byHumanities and Social Science Fund of theMinistryof Education of China (no 11YJCZH019)
References
[1] P M Ghare and G F Schrader ldquoA model for exponentiallydecaying inventoriesrdquo Journal of Industrial Engineering vol 14pp 238ndash243 1963
[2] J-W Wu C Lin B Tan and W-C Lee ldquoAn EOQ inventorymodel with time-varying demand and Weibull deteriorationwith shortagesrdquo International Journal of Systems Science vol 31no 6 pp 677ndash683 2000
[3] H-M Wee ldquoDeteriorating inventory model with quantity dis-count pricing and partial backorderingrdquo International Journalof Production Economics vol 59 no 1 pp 511ndash518 1999
[4] K Skouri and S Papachristos ldquoA continuous review inven-tory model with deteriorating items time-varying demandlinear replenishment cost partially time-varying backloggingrdquoApplied Mathematical Modelling vol 26 no 5 pp 603ndash6172002
[5] H-MWee J C P Yu and S T Law ldquoTwo-warehouse inventorymodel with partial backordering and Weibull distributiondeterioration under inflationrdquo Journal of the Chinese Institute ofIndustrial Engineers vol 22 no 6 pp 451ndash462 2005
[6] C-Y Dye T-P Hsieh and L-Y Ouyang ldquoDetermining optimalselling price and lot size with a varying rate of deteriorationand exponential partial backloggingrdquo European Journal ofOperational Research vol 181 no 2 pp 668ndash678 2007
[7] R M Hill ldquoInventory models for increasing demand followedby level demandrdquo Journal of the Operational Research Societyvol 46 no 10 pp 1250ndash1259 1995
[8] B Mandal and A K Pal ldquoOrder level inventory system withramp type demand rate for deteriorating itemsrdquo Journal ofInterdisciplinary Mathematics vol 1 no 1 pp 49ndash66 1998
[9] K-S Wu ldquoAn EOQ inventory model for items with Weibulldistribution deterioration ramp type demand rate and partialbackloggingrdquo Production Planning amp Control vol 12 no 8 pp787ndash793 2001
[10] B C Giri A K Jalan and K S Chaudhuri ldquoEconomic orderquantity model with Weibull deterioration distribution short-age and ramp-type demandrdquo International Journal of SystemsScience vol 34 no 4 pp 237ndash243 2003
[11] S K Manna and K S Chaudhuri ldquoAn EOQ model withramp type demand rate time dependent deterioration rate unitproduction cost and shortagesrdquoEuropean Journal ofOperationalResearch vol 171 no 2 pp 557ndash566 2006
[12] K Skouri I Konstantaras S Papachristos and I Ganas ldquoInven-tory models with ramp type demand rate partial backloggingandWeibull deterioration raterdquoEuropean Journal ofOperationalResearch vol 192 no 1 pp 79ndash92 2009
10 Journal of Applied Mathematics
[13] K-C Hung ldquoAn inventory model with generalized typedemand deterioration and backorder ratesrdquo European Journalof Operational Research vol 208 no 3 pp 239ndash242 2011
[14] R S Kumar S K De and A Goswami ldquoFuzzy EOQ modelswith ramp type demand rate partial backlogging and timedependent deterioration raterdquo International Journal of Mathe-matics in Operational Research vol 4 no 5 pp 473ndash502 2012
[15] M B Cheng B X Zhang and G QWang ldquoOptimal policy fordeteriorating items with trapezoidal type demand and partialbackloggingrdquoAppliedMathematical Modelling vol 35 no 7 pp3552ndash3560 2011
[16] R Uthayakumar and M Rameswari ldquoAn economic produc-tion quantity model for defective items with trapezoidal typedemand raterdquo Journal of Optimization Theory and Applicationsvol 154 no 3 pp 1055ndash1079 2012
[17] Y Tan and M X Weng ldquoA discrete-in-time deterioratinginventory model with time-varying demand variable deterio-ration rate and waiting-time-dependent partial backloggingrdquoInternational Journal of Systems Science vol 44 no 8 pp 1483ndash1493 2013
[18] M A Ahmed T A Al-Khamis and L Benkherouf ldquoInventorymodels with ramp type demand rate partial backlogging andgeneral deterioration raterdquo Applied Mathematics and Computa-tion vol 219 no 9 pp 4288ndash4307 2013
[19] K-P Lin ldquoAn extended inventory models with trapezoidal typedemandsrdquo Applied Mathematics and Computation vol 219 no24 pp 11414ndash11419 2013
[20] P L Abad ldquoOptimal pricing and lot-sizing under conditionsof perishability and partial backorderingrdquoManagement Sciencevol 42 no 8 pp 1093ndash1104 1996
(119879 minus 119905) 119890120575(119905minus119879)
119892 (119905) 119889119905]
+ 1198884[int
119879
1199051
(1 minus 119890120575(119905minus119879)
) 119892 (119905) 119889119905]
(26)
From the above analysis we obtain that the total average costof the model over the time interval [0 119879] is
119879119862 (1199051) =
1198621(1199051) 0 lt 119905
1le 1205831
1198622(1199051) 120583
1lt 1199051le 1205832
1198623(1199051) 120583
2lt 1199051lt 119879
(27)
where 1198621(1199051) 1198622(1199051) and 119862
3(1199051) are obtained from (10) (18)
and (26) respectivelyIn the following we will provide the results which ensure
the existence of a unique 1199051 say 119905lowast1 so as to minimize the total
average cost for the model system starting with no shortagesIf 0 lt 119905
1le 1205831 taking the first-order derivative of 119862
1(1199051)
with respect to 1199051 we obtain
1198891198621(1199051)
1198891199051
=119891 (1199051)
119879ℎ (1199051) (28)
where
ℎ (1199051) = 1198881(119890120572119905120573
1 minus 1)
+ 1198882int
1199051
0
119890120572(119905120573
1minus119905120573)119889119905 + 119888
3(1199051minus 119879) 119890
120575(1199051minus119879)
+ 1198884(119890120575(1199051minus119879)
minus 1)
(29)
then we can obtain ℎ(0) lt 0 and ℎ(119879) gt 0 By using theassumption (119905119890minus120575119905 is an increasing function where 119905 is thewaiting time up to the next replenishment) we have
The function ℎ(1199051) is given by (31) and (40) implies that
1198623(1199051) can obtain its minimum value at 119905lowast
1
The optimal value of the order level 119878 = 119868(0) is
119878lowast= int
1205831
0
119890120572119909120573
119891 (119909) 119889119909 + 1198630int
1205832
1205831
119890120572119909120573
119889119909 + int
119905lowast
1
1205832
119890120572119909120573
119892 (119909) 119889119909
(41)
and the optimal order quantity 119876lowast is
119876lowast= 119878lowast+ int
119879
119905lowast
1
119890120575(119905minus119879)
119892 (119905) 119889119905 (42)
If 119905lowast1
le 1205832 then the optimal value of 119862
3(1199051) is obtained at
119905lowast
1= 1205832
The above analysis shows that the three average cost func-tions 119862
1(1199051) 1198622(1199052) and 119862
3(1199051) can obtain their minimum
value at 119905lowast1isin (0 119879) which is determined by (31) Therefore
based on the results analyzed above this paper derives aprocedure to locate the optimal replenishment policy startingwith no shortage for the three cases The procedure is asfollowsStep 1 Solve 119905lowast
1from (31)
Step 2 Compare 119905lowast1to 1205831and 120583
2 respectively
Step 21 If 119905lowast1isin (0 120583
1] then the optimal total average cost and
the optimal order quantity can be obtained by (10) and (34)respectivelyStep 22 If 119905lowast
1isin (1205831 1205832] then the optimal total average cost
and the optimal order quantity can be obtained by (18) and(38) respectively
Step 23 If 119905lowast1isin (1205832 119879] then the optimal total average cost
and the optimal order quantity can be obtained by (26) and(42) respectively
Remark 1 In such considered inventory model starting withno shortage if 119905
1satisfies 120583
1lt 1199051le 119879 lt 120583
2 the considered
inventory model reduces to that of Skouri et al [12]
4 Numerical Example
In order to demonstrate the above procedure which can beapplied to obtain the optimal solution of the model this
paper presents several examples for the model respectivelyExamples are based on piecewise demand rate such as119891(119905) =1198861+ 1198871119905 and 119892(119905) = 119886
2119890minus1198872119905
Example 1 The parameter values are given as follows 119879 = 12
weeks1205831= 4weeks120583
2= 8weeks120572 = 0005120573 = 2120575 = 004
1198861= 30 unit 119887
1= 5 unit 119886
2= 100 unit 119860
0= $500 119888
1= $2
1198882= $3 119888
3= $12 and 119888
4= $8
The model starting with no shortage by solving theequation ℎ(119905
1) = 0 we have 119905
lowast
1= 87622 From (42) and
(26) we obtain 119876lowast
= 5825217 and 119879119862(119905lowast
1) = 7939986
respectively
Example 2 Theparameter values are given as follows119879 = 12
weeks1205831= 4weeks120583
2= 8weeks120572 = 0005120573 = 2120575 = 002
1198861= 30 unit 119887
1= 5 unit 119886
2= 100 unit 119860
0= $500 119888
1= $5
1198882= $10 119888
3= $12 and 119888
4= $8
The model starting with no shortage by solving theequation ℎ(119905
1) = 0 we have 119905
lowast
1= 58330 From (38) and
(18) we obtain 119876lowast
= 5284725 and 119879119862(119905lowast
1) = 16014013
respectively
Example 3 Theparameter values are given as follows119879 = 12
weeks 1205831= 4 weeks 120583
2= 6 weeks 120572 = 0005 120573 = 16
120575 = 02 1198861= 30 unit 119887
1= 5 unit 119886
2= 100 unit 119860
0= $500
1198881= $5 119888
2= $10 119888
3= $12 and 119888
4= $8
Themodel starting with no shortage solving the equationℎ(1199051) = 0 the optimal value of 119905
1is 119905lowast1= 23235 The optimal
ordering quantity is 119876lowast = 2726678 and the minimum cost119879119862(119905lowast
1) = 125882
In order to clearly indicate the effects of parameters suchas 120575 120572 120573 119888
1 1198882 1198883 and 119888
4on the optimal on-hand inventory
119878lowast the optimal ordering quantity 119876
lowast and the optimal totalcost 119879119862(119905lowast
1) respectively the paper will study the sensitivity
of the optimal solution to changes in the value of differentparameter associated with the studied inventory model Thesensitivity analysis is performed on the base of Example 1 andthe results are shown in Table 1ndash7
By studying the results of Table 1 it is found that theshortage time 119905
lowast
1 inventory level 119878
lowast order quantity 119876lowast
and the total average cost 119879119862(119905lowast1) gradually decrease as the
shortage parameter 120575 increases for the model respectivelyWe also find that the percentage increase of 120575 from 143to 100 causes 119879119862(119905
lowast
1) to decrease from 045 to 034
119876lowast decrease from 075 to 052 119905lowast
1decrease from 078
to 053 and 119878lowast decrease from 102 to 069 It is also
observed that the value of 119905lowast1 119878lowast119876lowast and 119879119862(119905lowast
1) all are lowly
sensitive to the changes of 120575 for the considered inventorymodel
By studying the results of Table 2 it is found that 119878lowast 119876lowastand 119879119862(119905lowast
1) coordinates to the deterioration parameter 120572 the
shortage time 119905lowast1decreases as 120572 increases for the model It is
also found that the percentage increase of 120572 from 167 to100 causes 119879119862(119905lowast
1) to decrease by 2066ndash2595 119876lowast to
increase by 1597ndash244 the shortage time 119905lowast1to decrease
by 1513ndash1455 and 119878lowast to increase by 0917ndash1651 It
also observes that the value of 119905lowast1 119878lowast 119876lowast and 119879119862(119905
lowast
1) all are
moderately sensitive to the changes of 120572 for the consideredinventory model
8 Journal of Applied Mathematics
Table 1 The sensitivity of 120575 for the models in Example 1
By studying the results of Table 3 it is found that 119878lowast119876lowast and 119879119862(119905
lowast
1) coordinate to the deterioration parameter 120573
while the shortage time 119905lowast
1decreases as 120573 increases for the
model It is also found that the increase of 120573 from 14 to 22causes 119878
lowast to increase while the increase of 120573 from 24 to28 causes 119878lowast to decrease 119876lowast to increase by 244ndash1597119879119862(119905lowast
1) to increase by 2595ndash2066 and the shortage time
119905lowast
1to decrease by 1513ndash1455 It is also observed that the
value of 119905lowast1 119878lowast119876lowast and 119879119862(119905lowast
1) all are moderately sensitive to
the changes of 120573 for the considered inventory modelBy studying the results of Table 4 it is found that 119879119862(119905lowast
1)
coordinate to 1198881 while the shortage time 119905
lowast
1 119878lowast and 119876
lowast
decrease as 1198881increases for the model It is also found that 119888
1
increases from 83 to 150 119879119862(119905lowast1) decreases by 0269ndash
0383 119876lowast decreases by 0083ndash0058 119905lowast1decreases by
0264ndash0181 and 119878lowast decreases by 0203ndash0138 respec-tively It is also observed that the values of 119905lowast
1 119878lowast 119876lowast and
119879119862(119905lowast
1) all are lowly sensitive to the changes of 119888
1for the
considered inventory modelBy studying the results of Table 5 it is found that
119879119862(119905lowast
1) coordinates to 119888
2 while 119878
lowast 119876lowastand 119905lowast
1decrease as
1198881increases for the model It is also found that 119888
2increases
by 100 119879119862(119905lowast1) decreases by 0269ndash0383 119876lowast decreases
by 0083ndash0058 119905lowast1decreases by 0264ndash0181 and 119878
lowast
decreases by 0203ndash0138By studying the results of Table 6 it is found that 119905lowast
1 119878lowast
119876lowast and 119879119862(119905
lowast
1) coordinate to 119888
3 It is also observed that the
value of 119905lowast1 119878lowast 119876lowast and 119879119862(119905
lowast
1) all are lowly sensitive to the
changes of 1198883for the inventory models that is 119888
3increases
from 19 to 32 the change of all the parameters is nomorethan 1
By studying the results of Table 7 it is found that 119905lowast1 119878lowast
119876lowast and 119879119862(119905
lowast
1) coordinate to 119888
4 It is also observed that the
value of 119905lowast1 119878lowast 119876lowast and 119879119862(119905
lowast
1) all are lowly sensitive to the
changes of 1198884for the inventory models that is 119888
4increases
from 143 to 100 the change of all the parameters is nomore than 1
5 Conclusion
An inventory model starting without shortage for Weibull-distributed deterioration with trapezoidal type demand rateand partial backlogging is considered in this paper Theoptimal replenishment policy for the inventory model isproposed and numerical examples are provided to illustratethe theoretical results A sensitivity analysis of the optimalsolution with respect to major parameters is also carried outFrom Table 1ndash7 it can be found that the shortage time point119905lowast
1 order quantity 119876lowast and the total average cost 119879119862(119905lowast
1) are
moderately sensitive to the changes of 120572 and 120573 and lowlysensitive to the changes of 120575 119888
119894(119894 = 1 2 3 4) respectively
The paper provides an interesting topic for further studysuch that the joint influence from some of these parametersmay be investigated to show the effects the model startingwith shortage will be studied and other types of models fordeteriorating items in supply chain situation are also to bestudied in the future
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author is grateful to the anonymous referees who pro-vided valuable comments and suggestions to significantlyimprove the quality of the paper This work was supportedpartly byHumanities and Social Science Fund of theMinistryof Education of China (no 11YJCZH019)
References
[1] P M Ghare and G F Schrader ldquoA model for exponentiallydecaying inventoriesrdquo Journal of Industrial Engineering vol 14pp 238ndash243 1963
[2] J-W Wu C Lin B Tan and W-C Lee ldquoAn EOQ inventorymodel with time-varying demand and Weibull deteriorationwith shortagesrdquo International Journal of Systems Science vol 31no 6 pp 677ndash683 2000
[3] H-M Wee ldquoDeteriorating inventory model with quantity dis-count pricing and partial backorderingrdquo International Journalof Production Economics vol 59 no 1 pp 511ndash518 1999
[4] K Skouri and S Papachristos ldquoA continuous review inven-tory model with deteriorating items time-varying demandlinear replenishment cost partially time-varying backloggingrdquoApplied Mathematical Modelling vol 26 no 5 pp 603ndash6172002
[5] H-MWee J C P Yu and S T Law ldquoTwo-warehouse inventorymodel with partial backordering and Weibull distributiondeterioration under inflationrdquo Journal of the Chinese Institute ofIndustrial Engineers vol 22 no 6 pp 451ndash462 2005
[6] C-Y Dye T-P Hsieh and L-Y Ouyang ldquoDetermining optimalselling price and lot size with a varying rate of deteriorationand exponential partial backloggingrdquo European Journal ofOperational Research vol 181 no 2 pp 668ndash678 2007
[7] R M Hill ldquoInventory models for increasing demand followedby level demandrdquo Journal of the Operational Research Societyvol 46 no 10 pp 1250ndash1259 1995
[8] B Mandal and A K Pal ldquoOrder level inventory system withramp type demand rate for deteriorating itemsrdquo Journal ofInterdisciplinary Mathematics vol 1 no 1 pp 49ndash66 1998
[9] K-S Wu ldquoAn EOQ inventory model for items with Weibulldistribution deterioration ramp type demand rate and partialbackloggingrdquo Production Planning amp Control vol 12 no 8 pp787ndash793 2001
[10] B C Giri A K Jalan and K S Chaudhuri ldquoEconomic orderquantity model with Weibull deterioration distribution short-age and ramp-type demandrdquo International Journal of SystemsScience vol 34 no 4 pp 237ndash243 2003
[11] S K Manna and K S Chaudhuri ldquoAn EOQ model withramp type demand rate time dependent deterioration rate unitproduction cost and shortagesrdquoEuropean Journal ofOperationalResearch vol 171 no 2 pp 557ndash566 2006
[12] K Skouri I Konstantaras S Papachristos and I Ganas ldquoInven-tory models with ramp type demand rate partial backloggingandWeibull deterioration raterdquoEuropean Journal ofOperationalResearch vol 192 no 1 pp 79ndash92 2009
10 Journal of Applied Mathematics
[13] K-C Hung ldquoAn inventory model with generalized typedemand deterioration and backorder ratesrdquo European Journalof Operational Research vol 208 no 3 pp 239ndash242 2011
[14] R S Kumar S K De and A Goswami ldquoFuzzy EOQ modelswith ramp type demand rate partial backlogging and timedependent deterioration raterdquo International Journal of Mathe-matics in Operational Research vol 4 no 5 pp 473ndash502 2012
[15] M B Cheng B X Zhang and G QWang ldquoOptimal policy fordeteriorating items with trapezoidal type demand and partialbackloggingrdquoAppliedMathematical Modelling vol 35 no 7 pp3552ndash3560 2011
[16] R Uthayakumar and M Rameswari ldquoAn economic produc-tion quantity model for defective items with trapezoidal typedemand raterdquo Journal of Optimization Theory and Applicationsvol 154 no 3 pp 1055ndash1079 2012
[17] Y Tan and M X Weng ldquoA discrete-in-time deterioratinginventory model with time-varying demand variable deterio-ration rate and waiting-time-dependent partial backloggingrdquoInternational Journal of Systems Science vol 44 no 8 pp 1483ndash1493 2013
[18] M A Ahmed T A Al-Khamis and L Benkherouf ldquoInventorymodels with ramp type demand rate partial backlogging andgeneral deterioration raterdquo Applied Mathematics and Computa-tion vol 219 no 9 pp 4288ndash4307 2013
[19] K-P Lin ldquoAn extended inventory models with trapezoidal typedemandsrdquo Applied Mathematics and Computation vol 219 no24 pp 11414ndash11419 2013
[20] P L Abad ldquoOptimal pricing and lot-sizing under conditionsof perishability and partial backorderingrdquoManagement Sciencevol 42 no 8 pp 1093ndash1104 1996
The function ℎ(1199051) is given by (31) and (40) implies that
1198623(1199051) can obtain its minimum value at 119905lowast
1
The optimal value of the order level 119878 = 119868(0) is
119878lowast= int
1205831
0
119890120572119909120573
119891 (119909) 119889119909 + 1198630int
1205832
1205831
119890120572119909120573
119889119909 + int
119905lowast
1
1205832
119890120572119909120573
119892 (119909) 119889119909
(41)
and the optimal order quantity 119876lowast is
119876lowast= 119878lowast+ int
119879
119905lowast
1
119890120575(119905minus119879)
119892 (119905) 119889119905 (42)
If 119905lowast1
le 1205832 then the optimal value of 119862
3(1199051) is obtained at
119905lowast
1= 1205832
The above analysis shows that the three average cost func-tions 119862
1(1199051) 1198622(1199052) and 119862
3(1199051) can obtain their minimum
value at 119905lowast1isin (0 119879) which is determined by (31) Therefore
based on the results analyzed above this paper derives aprocedure to locate the optimal replenishment policy startingwith no shortage for the three cases The procedure is asfollowsStep 1 Solve 119905lowast
1from (31)
Step 2 Compare 119905lowast1to 1205831and 120583
2 respectively
Step 21 If 119905lowast1isin (0 120583
1] then the optimal total average cost and
the optimal order quantity can be obtained by (10) and (34)respectivelyStep 22 If 119905lowast
1isin (1205831 1205832] then the optimal total average cost
and the optimal order quantity can be obtained by (18) and(38) respectively
Step 23 If 119905lowast1isin (1205832 119879] then the optimal total average cost
and the optimal order quantity can be obtained by (26) and(42) respectively
Remark 1 In such considered inventory model starting withno shortage if 119905
1satisfies 120583
1lt 1199051le 119879 lt 120583
2 the considered
inventory model reduces to that of Skouri et al [12]
4 Numerical Example
In order to demonstrate the above procedure which can beapplied to obtain the optimal solution of the model this
paper presents several examples for the model respectivelyExamples are based on piecewise demand rate such as119891(119905) =1198861+ 1198871119905 and 119892(119905) = 119886
2119890minus1198872119905
Example 1 The parameter values are given as follows 119879 = 12
weeks1205831= 4weeks120583
2= 8weeks120572 = 0005120573 = 2120575 = 004
1198861= 30 unit 119887
1= 5 unit 119886
2= 100 unit 119860
0= $500 119888
1= $2
1198882= $3 119888
3= $12 and 119888
4= $8
The model starting with no shortage by solving theequation ℎ(119905
1) = 0 we have 119905
lowast
1= 87622 From (42) and
(26) we obtain 119876lowast
= 5825217 and 119879119862(119905lowast
1) = 7939986
respectively
Example 2 Theparameter values are given as follows119879 = 12
weeks1205831= 4weeks120583
2= 8weeks120572 = 0005120573 = 2120575 = 002
1198861= 30 unit 119887
1= 5 unit 119886
2= 100 unit 119860
0= $500 119888
1= $5
1198882= $10 119888
3= $12 and 119888
4= $8
The model starting with no shortage by solving theequation ℎ(119905
1) = 0 we have 119905
lowast
1= 58330 From (38) and
(18) we obtain 119876lowast
= 5284725 and 119879119862(119905lowast
1) = 16014013
respectively
Example 3 Theparameter values are given as follows119879 = 12
weeks 1205831= 4 weeks 120583
2= 6 weeks 120572 = 0005 120573 = 16
120575 = 02 1198861= 30 unit 119887
1= 5 unit 119886
2= 100 unit 119860
0= $500
1198881= $5 119888
2= $10 119888
3= $12 and 119888
4= $8
Themodel starting with no shortage solving the equationℎ(1199051) = 0 the optimal value of 119905
1is 119905lowast1= 23235 The optimal
ordering quantity is 119876lowast = 2726678 and the minimum cost119879119862(119905lowast
1) = 125882
In order to clearly indicate the effects of parameters suchas 120575 120572 120573 119888
1 1198882 1198883 and 119888
4on the optimal on-hand inventory
119878lowast the optimal ordering quantity 119876
lowast and the optimal totalcost 119879119862(119905lowast
1) respectively the paper will study the sensitivity
of the optimal solution to changes in the value of differentparameter associated with the studied inventory model Thesensitivity analysis is performed on the base of Example 1 andthe results are shown in Table 1ndash7
By studying the results of Table 1 it is found that theshortage time 119905
lowast
1 inventory level 119878
lowast order quantity 119876lowast
and the total average cost 119879119862(119905lowast1) gradually decrease as the
shortage parameter 120575 increases for the model respectivelyWe also find that the percentage increase of 120575 from 143to 100 causes 119879119862(119905
lowast
1) to decrease from 045 to 034
119876lowast decrease from 075 to 052 119905lowast
1decrease from 078
to 053 and 119878lowast decrease from 102 to 069 It is also
observed that the value of 119905lowast1 119878lowast119876lowast and 119879119862(119905lowast
1) all are lowly
sensitive to the changes of 120575 for the considered inventorymodel
By studying the results of Table 2 it is found that 119878lowast 119876lowastand 119879119862(119905lowast
1) coordinates to the deterioration parameter 120572 the
shortage time 119905lowast1decreases as 120572 increases for the model It is
also found that the percentage increase of 120572 from 167 to100 causes 119879119862(119905lowast
1) to decrease by 2066ndash2595 119876lowast to
increase by 1597ndash244 the shortage time 119905lowast1to decrease
by 1513ndash1455 and 119878lowast to increase by 0917ndash1651 It
also observes that the value of 119905lowast1 119878lowast 119876lowast and 119879119862(119905
lowast
1) all are
moderately sensitive to the changes of 120572 for the consideredinventory model
8 Journal of Applied Mathematics
Table 1 The sensitivity of 120575 for the models in Example 1
By studying the results of Table 3 it is found that 119878lowast119876lowast and 119879119862(119905
lowast
1) coordinate to the deterioration parameter 120573
while the shortage time 119905lowast
1decreases as 120573 increases for the
model It is also found that the increase of 120573 from 14 to 22causes 119878
lowast to increase while the increase of 120573 from 24 to28 causes 119878lowast to decrease 119876lowast to increase by 244ndash1597119879119862(119905lowast
1) to increase by 2595ndash2066 and the shortage time
119905lowast
1to decrease by 1513ndash1455 It is also observed that the
value of 119905lowast1 119878lowast119876lowast and 119879119862(119905lowast
1) all are moderately sensitive to
the changes of 120573 for the considered inventory modelBy studying the results of Table 4 it is found that 119879119862(119905lowast
1)
coordinate to 1198881 while the shortage time 119905
lowast
1 119878lowast and 119876
lowast
decrease as 1198881increases for the model It is also found that 119888
1
increases from 83 to 150 119879119862(119905lowast1) decreases by 0269ndash
0383 119876lowast decreases by 0083ndash0058 119905lowast1decreases by
0264ndash0181 and 119878lowast decreases by 0203ndash0138 respec-tively It is also observed that the values of 119905lowast
1 119878lowast 119876lowast and
119879119862(119905lowast
1) all are lowly sensitive to the changes of 119888
1for the
considered inventory modelBy studying the results of Table 5 it is found that
119879119862(119905lowast
1) coordinates to 119888
2 while 119878
lowast 119876lowastand 119905lowast
1decrease as
1198881increases for the model It is also found that 119888
2increases
by 100 119879119862(119905lowast1) decreases by 0269ndash0383 119876lowast decreases
by 0083ndash0058 119905lowast1decreases by 0264ndash0181 and 119878
lowast
decreases by 0203ndash0138By studying the results of Table 6 it is found that 119905lowast
1 119878lowast
119876lowast and 119879119862(119905
lowast
1) coordinate to 119888
3 It is also observed that the
value of 119905lowast1 119878lowast 119876lowast and 119879119862(119905
lowast
1) all are lowly sensitive to the
changes of 1198883for the inventory models that is 119888
3increases
from 19 to 32 the change of all the parameters is nomorethan 1
By studying the results of Table 7 it is found that 119905lowast1 119878lowast
119876lowast and 119879119862(119905
lowast
1) coordinate to 119888
4 It is also observed that the
value of 119905lowast1 119878lowast 119876lowast and 119879119862(119905
lowast
1) all are lowly sensitive to the
changes of 1198884for the inventory models that is 119888
4increases
from 143 to 100 the change of all the parameters is nomore than 1
5 Conclusion
An inventory model starting without shortage for Weibull-distributed deterioration with trapezoidal type demand rateand partial backlogging is considered in this paper Theoptimal replenishment policy for the inventory model isproposed and numerical examples are provided to illustratethe theoretical results A sensitivity analysis of the optimalsolution with respect to major parameters is also carried outFrom Table 1ndash7 it can be found that the shortage time point119905lowast
1 order quantity 119876lowast and the total average cost 119879119862(119905lowast
1) are
moderately sensitive to the changes of 120572 and 120573 and lowlysensitive to the changes of 120575 119888
119894(119894 = 1 2 3 4) respectively
The paper provides an interesting topic for further studysuch that the joint influence from some of these parametersmay be investigated to show the effects the model startingwith shortage will be studied and other types of models fordeteriorating items in supply chain situation are also to bestudied in the future
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author is grateful to the anonymous referees who pro-vided valuable comments and suggestions to significantlyimprove the quality of the paper This work was supportedpartly byHumanities and Social Science Fund of theMinistryof Education of China (no 11YJCZH019)
References
[1] P M Ghare and G F Schrader ldquoA model for exponentiallydecaying inventoriesrdquo Journal of Industrial Engineering vol 14pp 238ndash243 1963
[2] J-W Wu C Lin B Tan and W-C Lee ldquoAn EOQ inventorymodel with time-varying demand and Weibull deteriorationwith shortagesrdquo International Journal of Systems Science vol 31no 6 pp 677ndash683 2000
[3] H-M Wee ldquoDeteriorating inventory model with quantity dis-count pricing and partial backorderingrdquo International Journalof Production Economics vol 59 no 1 pp 511ndash518 1999
[4] K Skouri and S Papachristos ldquoA continuous review inven-tory model with deteriorating items time-varying demandlinear replenishment cost partially time-varying backloggingrdquoApplied Mathematical Modelling vol 26 no 5 pp 603ndash6172002
[5] H-MWee J C P Yu and S T Law ldquoTwo-warehouse inventorymodel with partial backordering and Weibull distributiondeterioration under inflationrdquo Journal of the Chinese Institute ofIndustrial Engineers vol 22 no 6 pp 451ndash462 2005
[6] C-Y Dye T-P Hsieh and L-Y Ouyang ldquoDetermining optimalselling price and lot size with a varying rate of deteriorationand exponential partial backloggingrdquo European Journal ofOperational Research vol 181 no 2 pp 668ndash678 2007
[7] R M Hill ldquoInventory models for increasing demand followedby level demandrdquo Journal of the Operational Research Societyvol 46 no 10 pp 1250ndash1259 1995
[8] B Mandal and A K Pal ldquoOrder level inventory system withramp type demand rate for deteriorating itemsrdquo Journal ofInterdisciplinary Mathematics vol 1 no 1 pp 49ndash66 1998
[9] K-S Wu ldquoAn EOQ inventory model for items with Weibulldistribution deterioration ramp type demand rate and partialbackloggingrdquo Production Planning amp Control vol 12 no 8 pp787ndash793 2001
[10] B C Giri A K Jalan and K S Chaudhuri ldquoEconomic orderquantity model with Weibull deterioration distribution short-age and ramp-type demandrdquo International Journal of SystemsScience vol 34 no 4 pp 237ndash243 2003
[11] S K Manna and K S Chaudhuri ldquoAn EOQ model withramp type demand rate time dependent deterioration rate unitproduction cost and shortagesrdquoEuropean Journal ofOperationalResearch vol 171 no 2 pp 557ndash566 2006
[12] K Skouri I Konstantaras S Papachristos and I Ganas ldquoInven-tory models with ramp type demand rate partial backloggingandWeibull deterioration raterdquoEuropean Journal ofOperationalResearch vol 192 no 1 pp 79ndash92 2009
10 Journal of Applied Mathematics
[13] K-C Hung ldquoAn inventory model with generalized typedemand deterioration and backorder ratesrdquo European Journalof Operational Research vol 208 no 3 pp 239ndash242 2011
[14] R S Kumar S K De and A Goswami ldquoFuzzy EOQ modelswith ramp type demand rate partial backlogging and timedependent deterioration raterdquo International Journal of Mathe-matics in Operational Research vol 4 no 5 pp 473ndash502 2012
[15] M B Cheng B X Zhang and G QWang ldquoOptimal policy fordeteriorating items with trapezoidal type demand and partialbackloggingrdquoAppliedMathematical Modelling vol 35 no 7 pp3552ndash3560 2011
[16] R Uthayakumar and M Rameswari ldquoAn economic produc-tion quantity model for defective items with trapezoidal typedemand raterdquo Journal of Optimization Theory and Applicationsvol 154 no 3 pp 1055ndash1079 2012
[17] Y Tan and M X Weng ldquoA discrete-in-time deterioratinginventory model with time-varying demand variable deterio-ration rate and waiting-time-dependent partial backloggingrdquoInternational Journal of Systems Science vol 44 no 8 pp 1483ndash1493 2013
[18] M A Ahmed T A Al-Khamis and L Benkherouf ldquoInventorymodels with ramp type demand rate partial backlogging andgeneral deterioration raterdquo Applied Mathematics and Computa-tion vol 219 no 9 pp 4288ndash4307 2013
[19] K-P Lin ldquoAn extended inventory models with trapezoidal typedemandsrdquo Applied Mathematics and Computation vol 219 no24 pp 11414ndash11419 2013
[20] P L Abad ldquoOptimal pricing and lot-sizing under conditionsof perishability and partial backorderingrdquoManagement Sciencevol 42 no 8 pp 1093ndash1104 1996
By studying the results of Table 3 it is found that 119878lowast119876lowast and 119879119862(119905
lowast
1) coordinate to the deterioration parameter 120573
while the shortage time 119905lowast
1decreases as 120573 increases for the
model It is also found that the increase of 120573 from 14 to 22causes 119878
lowast to increase while the increase of 120573 from 24 to28 causes 119878lowast to decrease 119876lowast to increase by 244ndash1597119879119862(119905lowast
1) to increase by 2595ndash2066 and the shortage time
119905lowast
1to decrease by 1513ndash1455 It is also observed that the
value of 119905lowast1 119878lowast119876lowast and 119879119862(119905lowast
1) all are moderately sensitive to
the changes of 120573 for the considered inventory modelBy studying the results of Table 4 it is found that 119879119862(119905lowast
1)
coordinate to 1198881 while the shortage time 119905
lowast
1 119878lowast and 119876
lowast
decrease as 1198881increases for the model It is also found that 119888
1
increases from 83 to 150 119879119862(119905lowast1) decreases by 0269ndash
0383 119876lowast decreases by 0083ndash0058 119905lowast1decreases by
0264ndash0181 and 119878lowast decreases by 0203ndash0138 respec-tively It is also observed that the values of 119905lowast
1 119878lowast 119876lowast and
119879119862(119905lowast
1) all are lowly sensitive to the changes of 119888
1for the
considered inventory modelBy studying the results of Table 5 it is found that
119879119862(119905lowast
1) coordinates to 119888
2 while 119878
lowast 119876lowastand 119905lowast
1decrease as
1198881increases for the model It is also found that 119888
2increases
by 100 119879119862(119905lowast1) decreases by 0269ndash0383 119876lowast decreases
by 0083ndash0058 119905lowast1decreases by 0264ndash0181 and 119878
lowast
decreases by 0203ndash0138By studying the results of Table 6 it is found that 119905lowast
1 119878lowast
119876lowast and 119879119862(119905
lowast
1) coordinate to 119888
3 It is also observed that the
value of 119905lowast1 119878lowast 119876lowast and 119879119862(119905
lowast
1) all are lowly sensitive to the
changes of 1198883for the inventory models that is 119888
3increases
from 19 to 32 the change of all the parameters is nomorethan 1
By studying the results of Table 7 it is found that 119905lowast1 119878lowast
119876lowast and 119879119862(119905
lowast
1) coordinate to 119888
4 It is also observed that the
value of 119905lowast1 119878lowast 119876lowast and 119879119862(119905
lowast
1) all are lowly sensitive to the
changes of 1198884for the inventory models that is 119888
4increases
from 143 to 100 the change of all the parameters is nomore than 1
5 Conclusion
An inventory model starting without shortage for Weibull-distributed deterioration with trapezoidal type demand rateand partial backlogging is considered in this paper Theoptimal replenishment policy for the inventory model isproposed and numerical examples are provided to illustratethe theoretical results A sensitivity analysis of the optimalsolution with respect to major parameters is also carried outFrom Table 1ndash7 it can be found that the shortage time point119905lowast
1 order quantity 119876lowast and the total average cost 119879119862(119905lowast
1) are
moderately sensitive to the changes of 120572 and 120573 and lowlysensitive to the changes of 120575 119888
119894(119894 = 1 2 3 4) respectively
The paper provides an interesting topic for further studysuch that the joint influence from some of these parametersmay be investigated to show the effects the model startingwith shortage will be studied and other types of models fordeteriorating items in supply chain situation are also to bestudied in the future
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author is grateful to the anonymous referees who pro-vided valuable comments and suggestions to significantlyimprove the quality of the paper This work was supportedpartly byHumanities and Social Science Fund of theMinistryof Education of China (no 11YJCZH019)
References
[1] P M Ghare and G F Schrader ldquoA model for exponentiallydecaying inventoriesrdquo Journal of Industrial Engineering vol 14pp 238ndash243 1963
[2] J-W Wu C Lin B Tan and W-C Lee ldquoAn EOQ inventorymodel with time-varying demand and Weibull deteriorationwith shortagesrdquo International Journal of Systems Science vol 31no 6 pp 677ndash683 2000
[3] H-M Wee ldquoDeteriorating inventory model with quantity dis-count pricing and partial backorderingrdquo International Journalof Production Economics vol 59 no 1 pp 511ndash518 1999
[4] K Skouri and S Papachristos ldquoA continuous review inven-tory model with deteriorating items time-varying demandlinear replenishment cost partially time-varying backloggingrdquoApplied Mathematical Modelling vol 26 no 5 pp 603ndash6172002
[5] H-MWee J C P Yu and S T Law ldquoTwo-warehouse inventorymodel with partial backordering and Weibull distributiondeterioration under inflationrdquo Journal of the Chinese Institute ofIndustrial Engineers vol 22 no 6 pp 451ndash462 2005
[6] C-Y Dye T-P Hsieh and L-Y Ouyang ldquoDetermining optimalselling price and lot size with a varying rate of deteriorationand exponential partial backloggingrdquo European Journal ofOperational Research vol 181 no 2 pp 668ndash678 2007
[7] R M Hill ldquoInventory models for increasing demand followedby level demandrdquo Journal of the Operational Research Societyvol 46 no 10 pp 1250ndash1259 1995
[8] B Mandal and A K Pal ldquoOrder level inventory system withramp type demand rate for deteriorating itemsrdquo Journal ofInterdisciplinary Mathematics vol 1 no 1 pp 49ndash66 1998
[9] K-S Wu ldquoAn EOQ inventory model for items with Weibulldistribution deterioration ramp type demand rate and partialbackloggingrdquo Production Planning amp Control vol 12 no 8 pp787ndash793 2001
[10] B C Giri A K Jalan and K S Chaudhuri ldquoEconomic orderquantity model with Weibull deterioration distribution short-age and ramp-type demandrdquo International Journal of SystemsScience vol 34 no 4 pp 237ndash243 2003
[11] S K Manna and K S Chaudhuri ldquoAn EOQ model withramp type demand rate time dependent deterioration rate unitproduction cost and shortagesrdquoEuropean Journal ofOperationalResearch vol 171 no 2 pp 557ndash566 2006
[12] K Skouri I Konstantaras S Papachristos and I Ganas ldquoInven-tory models with ramp type demand rate partial backloggingandWeibull deterioration raterdquoEuropean Journal ofOperationalResearch vol 192 no 1 pp 79ndash92 2009
10 Journal of Applied Mathematics
[13] K-C Hung ldquoAn inventory model with generalized typedemand deterioration and backorder ratesrdquo European Journalof Operational Research vol 208 no 3 pp 239ndash242 2011
[14] R S Kumar S K De and A Goswami ldquoFuzzy EOQ modelswith ramp type demand rate partial backlogging and timedependent deterioration raterdquo International Journal of Mathe-matics in Operational Research vol 4 no 5 pp 473ndash502 2012
[15] M B Cheng B X Zhang and G QWang ldquoOptimal policy fordeteriorating items with trapezoidal type demand and partialbackloggingrdquoAppliedMathematical Modelling vol 35 no 7 pp3552ndash3560 2011
[16] R Uthayakumar and M Rameswari ldquoAn economic produc-tion quantity model for defective items with trapezoidal typedemand raterdquo Journal of Optimization Theory and Applicationsvol 154 no 3 pp 1055ndash1079 2012
[17] Y Tan and M X Weng ldquoA discrete-in-time deterioratinginventory model with time-varying demand variable deterio-ration rate and waiting-time-dependent partial backloggingrdquoInternational Journal of Systems Science vol 44 no 8 pp 1483ndash1493 2013
[18] M A Ahmed T A Al-Khamis and L Benkherouf ldquoInventorymodels with ramp type demand rate partial backlogging andgeneral deterioration raterdquo Applied Mathematics and Computa-tion vol 219 no 9 pp 4288ndash4307 2013
[19] K-P Lin ldquoAn extended inventory models with trapezoidal typedemandsrdquo Applied Mathematics and Computation vol 219 no24 pp 11414ndash11419 2013
[20] P L Abad ldquoOptimal pricing and lot-sizing under conditionsof perishability and partial backorderingrdquoManagement Sciencevol 42 no 8 pp 1093ndash1104 1996
By studying the results of Table 3 it is found that 119878lowast119876lowast and 119879119862(119905
lowast
1) coordinate to the deterioration parameter 120573
while the shortage time 119905lowast
1decreases as 120573 increases for the
model It is also found that the increase of 120573 from 14 to 22causes 119878
lowast to increase while the increase of 120573 from 24 to28 causes 119878lowast to decrease 119876lowast to increase by 244ndash1597119879119862(119905lowast
1) to increase by 2595ndash2066 and the shortage time
119905lowast
1to decrease by 1513ndash1455 It is also observed that the
value of 119905lowast1 119878lowast119876lowast and 119879119862(119905lowast
1) all are moderately sensitive to
the changes of 120573 for the considered inventory modelBy studying the results of Table 4 it is found that 119879119862(119905lowast
1)
coordinate to 1198881 while the shortage time 119905
lowast
1 119878lowast and 119876
lowast
decrease as 1198881increases for the model It is also found that 119888
1
increases from 83 to 150 119879119862(119905lowast1) decreases by 0269ndash
0383 119876lowast decreases by 0083ndash0058 119905lowast1decreases by
0264ndash0181 and 119878lowast decreases by 0203ndash0138 respec-tively It is also observed that the values of 119905lowast
1 119878lowast 119876lowast and
119879119862(119905lowast
1) all are lowly sensitive to the changes of 119888
1for the
considered inventory modelBy studying the results of Table 5 it is found that
119879119862(119905lowast
1) coordinates to 119888
2 while 119878
lowast 119876lowastand 119905lowast
1decrease as
1198881increases for the model It is also found that 119888
2increases
by 100 119879119862(119905lowast1) decreases by 0269ndash0383 119876lowast decreases
by 0083ndash0058 119905lowast1decreases by 0264ndash0181 and 119878
lowast
decreases by 0203ndash0138By studying the results of Table 6 it is found that 119905lowast
1 119878lowast
119876lowast and 119879119862(119905
lowast
1) coordinate to 119888
3 It is also observed that the
value of 119905lowast1 119878lowast 119876lowast and 119879119862(119905
lowast
1) all are lowly sensitive to the
changes of 1198883for the inventory models that is 119888
3increases
from 19 to 32 the change of all the parameters is nomorethan 1
By studying the results of Table 7 it is found that 119905lowast1 119878lowast
119876lowast and 119879119862(119905
lowast
1) coordinate to 119888
4 It is also observed that the
value of 119905lowast1 119878lowast 119876lowast and 119879119862(119905
lowast
1) all are lowly sensitive to the
changes of 1198884for the inventory models that is 119888
4increases
from 143 to 100 the change of all the parameters is nomore than 1
5 Conclusion
An inventory model starting without shortage for Weibull-distributed deterioration with trapezoidal type demand rateand partial backlogging is considered in this paper Theoptimal replenishment policy for the inventory model isproposed and numerical examples are provided to illustratethe theoretical results A sensitivity analysis of the optimalsolution with respect to major parameters is also carried outFrom Table 1ndash7 it can be found that the shortage time point119905lowast
1 order quantity 119876lowast and the total average cost 119879119862(119905lowast
1) are
moderately sensitive to the changes of 120572 and 120573 and lowlysensitive to the changes of 120575 119888
119894(119894 = 1 2 3 4) respectively
The paper provides an interesting topic for further studysuch that the joint influence from some of these parametersmay be investigated to show the effects the model startingwith shortage will be studied and other types of models fordeteriorating items in supply chain situation are also to bestudied in the future
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author is grateful to the anonymous referees who pro-vided valuable comments and suggestions to significantlyimprove the quality of the paper This work was supportedpartly byHumanities and Social Science Fund of theMinistryof Education of China (no 11YJCZH019)
References
[1] P M Ghare and G F Schrader ldquoA model for exponentiallydecaying inventoriesrdquo Journal of Industrial Engineering vol 14pp 238ndash243 1963
[2] J-W Wu C Lin B Tan and W-C Lee ldquoAn EOQ inventorymodel with time-varying demand and Weibull deteriorationwith shortagesrdquo International Journal of Systems Science vol 31no 6 pp 677ndash683 2000
[3] H-M Wee ldquoDeteriorating inventory model with quantity dis-count pricing and partial backorderingrdquo International Journalof Production Economics vol 59 no 1 pp 511ndash518 1999
[4] K Skouri and S Papachristos ldquoA continuous review inven-tory model with deteriorating items time-varying demandlinear replenishment cost partially time-varying backloggingrdquoApplied Mathematical Modelling vol 26 no 5 pp 603ndash6172002
[5] H-MWee J C P Yu and S T Law ldquoTwo-warehouse inventorymodel with partial backordering and Weibull distributiondeterioration under inflationrdquo Journal of the Chinese Institute ofIndustrial Engineers vol 22 no 6 pp 451ndash462 2005
[6] C-Y Dye T-P Hsieh and L-Y Ouyang ldquoDetermining optimalselling price and lot size with a varying rate of deteriorationand exponential partial backloggingrdquo European Journal ofOperational Research vol 181 no 2 pp 668ndash678 2007
[7] R M Hill ldquoInventory models for increasing demand followedby level demandrdquo Journal of the Operational Research Societyvol 46 no 10 pp 1250ndash1259 1995
[8] B Mandal and A K Pal ldquoOrder level inventory system withramp type demand rate for deteriorating itemsrdquo Journal ofInterdisciplinary Mathematics vol 1 no 1 pp 49ndash66 1998
[9] K-S Wu ldquoAn EOQ inventory model for items with Weibulldistribution deterioration ramp type demand rate and partialbackloggingrdquo Production Planning amp Control vol 12 no 8 pp787ndash793 2001
[10] B C Giri A K Jalan and K S Chaudhuri ldquoEconomic orderquantity model with Weibull deterioration distribution short-age and ramp-type demandrdquo International Journal of SystemsScience vol 34 no 4 pp 237ndash243 2003
[11] S K Manna and K S Chaudhuri ldquoAn EOQ model withramp type demand rate time dependent deterioration rate unitproduction cost and shortagesrdquoEuropean Journal ofOperationalResearch vol 171 no 2 pp 557ndash566 2006
[12] K Skouri I Konstantaras S Papachristos and I Ganas ldquoInven-tory models with ramp type demand rate partial backloggingandWeibull deterioration raterdquoEuropean Journal ofOperationalResearch vol 192 no 1 pp 79ndash92 2009
10 Journal of Applied Mathematics
[13] K-C Hung ldquoAn inventory model with generalized typedemand deterioration and backorder ratesrdquo European Journalof Operational Research vol 208 no 3 pp 239ndash242 2011
[14] R S Kumar S K De and A Goswami ldquoFuzzy EOQ modelswith ramp type demand rate partial backlogging and timedependent deterioration raterdquo International Journal of Mathe-matics in Operational Research vol 4 no 5 pp 473ndash502 2012
[15] M B Cheng B X Zhang and G QWang ldquoOptimal policy fordeteriorating items with trapezoidal type demand and partialbackloggingrdquoAppliedMathematical Modelling vol 35 no 7 pp3552ndash3560 2011
[16] R Uthayakumar and M Rameswari ldquoAn economic produc-tion quantity model for defective items with trapezoidal typedemand raterdquo Journal of Optimization Theory and Applicationsvol 154 no 3 pp 1055ndash1079 2012
[17] Y Tan and M X Weng ldquoA discrete-in-time deterioratinginventory model with time-varying demand variable deterio-ration rate and waiting-time-dependent partial backloggingrdquoInternational Journal of Systems Science vol 44 no 8 pp 1483ndash1493 2013
[18] M A Ahmed T A Al-Khamis and L Benkherouf ldquoInventorymodels with ramp type demand rate partial backlogging andgeneral deterioration raterdquo Applied Mathematics and Computa-tion vol 219 no 9 pp 4288ndash4307 2013
[19] K-P Lin ldquoAn extended inventory models with trapezoidal typedemandsrdquo Applied Mathematics and Computation vol 219 no24 pp 11414ndash11419 2013
[20] P L Abad ldquoOptimal pricing and lot-sizing under conditionsof perishability and partial backorderingrdquoManagement Sciencevol 42 no 8 pp 1093ndash1104 1996
[13] K-C Hung ldquoAn inventory model with generalized typedemand deterioration and backorder ratesrdquo European Journalof Operational Research vol 208 no 3 pp 239ndash242 2011
[14] R S Kumar S K De and A Goswami ldquoFuzzy EOQ modelswith ramp type demand rate partial backlogging and timedependent deterioration raterdquo International Journal of Mathe-matics in Operational Research vol 4 no 5 pp 473ndash502 2012
[15] M B Cheng B X Zhang and G QWang ldquoOptimal policy fordeteriorating items with trapezoidal type demand and partialbackloggingrdquoAppliedMathematical Modelling vol 35 no 7 pp3552ndash3560 2011
[16] R Uthayakumar and M Rameswari ldquoAn economic produc-tion quantity model for defective items with trapezoidal typedemand raterdquo Journal of Optimization Theory and Applicationsvol 154 no 3 pp 1055ndash1079 2012
[17] Y Tan and M X Weng ldquoA discrete-in-time deterioratinginventory model with time-varying demand variable deterio-ration rate and waiting-time-dependent partial backloggingrdquoInternational Journal of Systems Science vol 44 no 8 pp 1483ndash1493 2013
[18] M A Ahmed T A Al-Khamis and L Benkherouf ldquoInventorymodels with ramp type demand rate partial backlogging andgeneral deterioration raterdquo Applied Mathematics and Computa-tion vol 219 no 9 pp 4288ndash4307 2013
[19] K-P Lin ldquoAn extended inventory models with trapezoidal typedemandsrdquo Applied Mathematics and Computation vol 219 no24 pp 11414ndash11419 2013
[20] P L Abad ldquoOptimal pricing and lot-sizing under conditionsof perishability and partial backorderingrdquoManagement Sciencevol 42 no 8 pp 1093ndash1104 1996
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