Int. J. Logistics Systems and Management, Vol. 13, No. 1, 2012 51
An integrated single-vendor single-buyer targeting problem with time-dependent process mean
M.A. Darwish* and Fawaz AbdulmalekDepartment of Industrial and Management Systems Engineering,College of Engineering and Petroleum, Kuwait University,P.O. Box 5969, 13060, Safat, KuwaitE-mail: [email protected]: [email protected]*Corresponding author
Abstract: The inventory/production decisions in single-vendor single-buyer supply chain are integrated with the targeting problem in this paper. Traditionally, the process mean is assumed to be constant over time. However, it is common that production processes deteriorate with time and constant process mean assumption is not appropriate in this situation. In the proposed model, this restrictive assumption is relaxed and the process mean is considered to drift with time. Also, solution method is discussed and sensitivity analysis on the model’s key parameters is conducted. In particular, the effect of the shift in the process mean and the variation in the fi lling process are examined.
Reference to this paper should be made as follows: Darwish, M.A. and Abdulmalek, F. (2012) ‘An integrated single-vendor single-buyer targeting problem with time-dependent process mean’, Int. J. Logistics Systems and Management, Vol. 13, No. 1, pp.51–64.
Biographical notes: Mohammed A. Darwish is an Associate Professor in Industrial and Management Systems Engineering Department at Kuwait University. His research interests include supply chain management, production planning and control, and quality control. He published in several journals including international journal of production economics, computers and operations research, and European journal of operational research, international journal of logistics systems and management, international journal of quality and reliability management, international journal of services and operations management, international journal of mathematics in operational research and journal of quality in maintenance engineering. He received the CCSE outstanding research performance award (2007/2008). He also received the KFUPM distinguished teaching and advising award of (2004/2005). He is a member in the engineering honour society (Tau Beta Pi) and industrial engineering honour society (Alpha Pi Mu).
Fawaz Abdulmalek is an Assistant Professor in the Industrial and Management Systems Engineering Department at the Kuwait University. He received his BS in Mechanical Engineering from the University of South Carolina, and his MS and PhD in Industrial Engineering from the University of Pittsburgh. His areas of interest are production planning, supply chain management and simulation.
52 M.A. Darwish and F. Abdulmalek
1 Introduction
The problem of selecting the most economical process mean is important and challenging for quality and production managers (Hariga and Al-Fawzan, 2005). This problem is referred to in quality control literature as the targeting problem. Typically in this problem, containers are fi lled with material and a lower-specifi cation limit is set on the amount of the material in a can (Gong et al., 1998). An item is classifi ed as conforming if its amount of material is larger than or equal to a lower-specifi cation limit. Otherwise, the item is classifi ed as nonconforming, which would be sold at a reduced price, reprocessed or scrapped. Process targeting is especially important to industries, which are governed by laws and regulations on the net content labelling, such as food, diary, drug and cosmetic industries (Kloos and Clark, 1981).
In practice, it is common that producers set a high-process mean (fi lling amount) to conform to specifi cations, this strategy leads to high material cost (Roan et al., 2000). However, very tight process setting is also harmful, for then, the cost of producing nonconforming items may be high (Al-Sultan and Pulak, 2000). Hence, process targeting is a trade-off between material cost and the cost associated with producing nonconforming items. Specifi cally, it deals with the determination of the optimum process mean to achieve some economic objectives, such as maximising profi t and minimising process cost.
In the past, process targeting and production/inventory decisions were treated separately. Roan et al. (2000) jointly determined the process mean, production lot size and material order quantity for a can-fi lling process. They assumed that the process mean is static (constant over time) during the production cycle. However, it is well-known that production processes deteriorate with time. This deterioration is usually refl ected on the fi lling amount in a can. Therefore, relaxing the constant process mean assumption would lead to more realistic results. In this direction, Al-Fawzan and Hariga (2002) extended the model in Roan et al. (2000) by considering a time varying process mean. The objective of their model is to fi nd the optimum process mean and production lot size, which minimise the expected total cost. They showed by numerical examples that the type of the process drift affects lot-sizing decisions signifi cantly. It is important to indicate that this model is suitable for producers who order raw material to manufacture a product and directly satisfy the demand of end customers. However, rapidly changing markets forced companies to provide better products at reduced cost for customers with high expectations. As a result, companies are pushed towards not only integrating different decision processes within their borders but also closely cooperating with their retailers and suppliers to reduce the total cost of the supply chain. In particular, managing inventories across the entire supply chain can be more effi cient through cooperation and better coordination (Ben-Daya et al., 2008).
Integrating targeting-inventory decisions in the context of supply chain management was investigated by Darwish (2004). He developed an integrated targeting-inventory model for a two-layer supply chain. In this model, the vendor is restricted to deliver equal-sized shipments to a retailer. Recently, Darwish (2009) proposed a model that integrates single-vendor single-buyer supply chain and process targeting. He determines the optimal process mean, production lot size and number of deliveries received by a retailer. In this model, the performance variable of the product has a lower specifi cation limit, and items that do not conform to specifi cation limit are reprocessed. In addition, he considers the case where shipments to retailer are not necessarily of equal size. Most importantly, he assumes that the process mean is fi xed during the production cycle. This specifi c assumption may lead to
An integrated single-vendor single-buyer targeting problem 53
signifi cant deviation from the optimal decisions. This is particularly true when the process under consideration deteriorates with time causing the process mean to drift.
In this paper, we develop a mathematical model for simultaneously determining the optimal setting of process mean, production lot size and shipments schedule, which minimise the average total cost of vendor-buyer supply chain. Unlike Darwish (2009), the container-fi lling process under investigation is assumed to deteriorate during the production run, that is, the mean of the process is considered to be time dependent. The proposed model allows the vendor to ship the produced lot to a buyer in number of unequal-sized shipments. Moreover, a produced container is considered conforming if its performance variable (weight, volume, concentration, …) is larger than or equal to a lower specifi cation limit. Conforming containers are used to satisfy demand, while non-conforming items are scraped with no salvage value. Sensitivity analysis is conducted to investigate the effect of the model key parameters on the optimal solution. In particular, the effect of the shift in the process mean and the variation in the fi lling process are examined.
The organisation of the paper is as follows. Literature review is presented in the next section. Then, we state the assumptions and formulate the model in Section 3. Solution method and sensitivity analysis are given in Section 4. Finally, we conclude the paper in Section 5.
2 Literature review
In supply chain management literature, researchers have proposed many shipment policies for the single-vendor single-buyer problem. Goyal (1977) suggested a lot-for-lot policy with the assumption of infi nite production rate. Then, Banerjee (1986) proposed a lot-for-lot policy with the assumption of fi nite production rate. Goyal (1988) relaxes lot-for-lot assumption and assumed that the vendor ships the lot in a number of equal size shipments. Goyal (1995) developed a policy where the shipment sizes increase by a factor geometrically. Hill (1997) generalised the model developed by Goyal (1995) by considering the geometric-growth factor as a decision variable. Hill (1999) found the optimal solution of the problem without any assumptions about the shipment policy. Goyal and Nebebe (2000) considered a policy where the fi rst shipment is small and the following shipments are larger and of equal size. Extensions in many directions can be found for single-vendor single-buyer problem. For example, Hoque and Goyal (2006) developed a heuristic solution procedure to minimise the total cost of setup, inventory holding and lead-time crashing for an integrated inventory system under controllable lead-time between a vendor and a buyer. Further, Hill and Omar (2006) relaxed an assumption regarding holding costs. They allowed for decreasing holding costs down the supply chain. This model has been modifi ed by Zhou and Wang (2006). They indicated that when the holding cost of the vendor is greater than that of the buyer, the optimal shipment policy consists of unequal-sized shipments with successive shipment sizes increasing by a fi xed factor equal to the ratio of the production rate to the demand rate. Darwish and Goyal (2010) addressed permissible delay in payments conditions for single-vendor single-buyer supply chain. Furthermore, Darwish and Goyal (2011) developed a model for single-vendor single-buyer supply chain under vendor managed inventory programme. We refer the reader to Ben-Daya et al. (2008) for a comprehensive review on the single-vendor single-buyer problem. Recently, Melacini et al. (2011) analysed the centralisation of supply chain for multinational companies. Diopenes
54 M.A. Darwish and F. Abdulmalek
and Laptaned (2011) considered a case for a supply chain from Thailand and analysed the related costs. Pettersson and Segerstedt (2011) developed performance measures in supply chains in Swedish industry. Moreover, Ho et. al. (2011) studied the effect of defective products in integrated vendor–buyer inventory model.
It is assumed in the models mentioned above that the production process has one setting for the mean and the variation among products is neglected. Usually, in can-fi lling processes, the mean fi lling amount can be controlled by the producer and the variation among containers is not negligible in this type of processes. In these situations, process targeting is used as an important tool in controlling the variation of the process output. Targeting problem has been investigated and discussed for more than 60 years. Springer (1951) was the fi rst to consider the problem of process targeting, in which the process mean that minimises the total cost is obtained. Bettes (1962) presented a similar model by determining the optimal process mean and upper specifi cation limit simultaneously. In this model, the nonconforming items are scrapped with no salvage value. This model is extended by Hunter and Kartha (1977) by considering the problem of selecting the optimum target value when nonconforming items are sold at a secondary market. Then, Bisgaard et al. (1984) modifi ed Hunter and Kartha’s model to a situation where nonconforming products are sold at a price proportional to the amount of material used. Duffuaa and Siddiqui (2003) discussed a model for a multi class screening when measurement error existed and Darwish and Duffuaa (2010) developed a model that determines the optimal process mean and sampling plan’s parameters. The model maximises producer expected profi t while protecting the consumer through a constraint on the probability of accepting lots with low-incoming quality. Duffuaa and Darwish (2011) extended this model by considering inspection errors. A reverse programming routine that identifi es the relationship between the process mean and the settings within an experimental factor space was established by Goethals and Cho (2011). Chen and Lai (2007) developed a model that fi nds the economic process mean based on quadratic quality loss function and rectifying inspection plan. Also, on the basis of the quality loss function, Chen and Kao (2009) determined the optimal process mean and screening limits. Hong and Cho (2007) found the optimal process mean and tolerance limits with measurement errors under multi-decision alternatives.
3 Model assumptions and development
A fi nished product having a constant demand rate D per unit of time is produced by a vendor at a constant production rate r per unit time. The vendor manufactures the product in lots and incurs a set-up cost. Each produced lot is delivered to a buyer in number of shipments. The buyer incurs a fi xed order/delivery cost associated with each shipment. The vendor and the buyer incur holding costs at different rates. We have to indicate that the produced items are not all conforming to specifi cations. Let the quality characteristic of the product is represented by a random variable X , which follows normal distribution with a time-dependent mean, µ(t), and a constant variance, σ2. It is assumed that the process mean takes the form µ(t) = µ0 + θtm, where µ0 and θ are the initial mean and the drift rate, respectively, and m is the shape parameter (Al-Fawzan and Hariga, 2002). A unit of the product is conforming if its X value is larger than or equal to the lower-specifi cation limit L. Conforming items are used to satisfy the demand, while non-conforming items are scraped with no salvage value. Therefore, the yield production rate at a given time t is λ(t) = rp(t, µ0)
An integrated single-vendor single-buyer targeting problem 55
where p(t, µ0) is the conforming rate of the process at time t and initial target value µ0, and is given by
0( )
( , ) ( ) 1 ( ( ))z t
p t x dx z tµ ϕ∞
= = −Φ∫ (1)
where ( )ϕ ⋅ is the standard normal probability density function, ( )⋅ is the standard normal cumulative distribution, and z(t) = (L – µ(t))/σ. We also consider the direct production cost as a linear function of the item’s material cost and is given by
( ) α= +g X b cX where b > 0 and a > 1.
Many shipment policies can be found for the single-vendor single-buyer problem in the literature of supply chain management (Ben-Daya et al. 2008). However, the policy proposed by Goyal (1995) is selected in this paper because it reduces the diffi culty of the proposed model and renders the mathematics tractable. Also, this policy gives satisfactory results in many cases (Zhou and Wang 2006). According to this policy, the time needed to consume a shipment of size qi-1 by the buyer is equal to the time needed to produce the next shipment of size qi by the vendor (as shown by Figure 1), that is
11 , 1,...,
( )λ−
−− = = =j jj j
q qt t j n
D t (2)
where n is number of shipments delivered from vendor to buyer and t0 = 0. Since some produced items are non-conforming and only conforming items are used to satisfy demand, it follows, therefore, that
1
1
1( )
( ) , 2,...,ϕ−
−∞
= =
∫ ∫j
j
jt
jt z t
rq q z dzdt j nD
(3)
The costs of the supply chain in this model are related to raw material, vendor and buyer. They are itemised as follows:
3.1 Raw material related costs
The costs associated with raw material are holding, ordering and acquisition costs. From Figure 1, the inventory of the raw material at time t is given by
( ) ( )µ= ,∫pt
r tI t r y dy
where tp is the production run length. Therefore, the holding cost of the raw material per cycle can be found as follows
0( )µ= ,∫ ∫
p pt t
r r tHC h r y dydt (4)
hence,
56 M.A. Darwish and F. Abdulmalek
2 2
0( )2 2
µ θ+
= ++
mp p
r r
t tHC h r
m (5)
Moreover, the raw material acquisition cost per cycle is (0)rAC cI= which yields 1
0( )1
µ θ+
= ++
mp
p
tAC cr t
m (6)
3.2 Vendor related costs
The vendor incurs holding cost, production cost and set-up cost. The inventory of the vendor at a given time t ∈ 1[ ]− ,j jt t can be set out using Figure 1 as follows
10 1( ) ( )µ
−−= , , ≤ ≤∫
j
t
j j jtIV t r p t dt t t t (7)
Therefore, the cost of holding the fi nished products inventory during the inventory cycle [0, T] is given by
1 10
1
( ( ) )µ− −=
= ,∑ ∫ ∫j
j j
n t x
v v t tj
HC rh p t dt dx (8)
Equation (8) can be simplifi ed as follows:
00 1
2 21
11
( ) ( ) ( )2 2
µ θµ θ
σ σ
φ φ− −− − −
∞ ∞−−
=
= − −∑ ∫ ∫ mm L tL t j j
nj j
v v j jj
t tHC rh x dx t t x dx
1
0( )µ θθ
φσ σ−
− −− ∫
j
j
mt mj t
L tm t t dt
1
1 0( )2
µ θθφ
σ σ−
+ − −+ ∫
j
j
mt m
t
L tm t dt (9)
In addition, the production cost can be found to be
0( ( ))µ= ,∫
ptPC r g t dt
which yields 1
0( )1
α µ α θ+
= + + ,+
mp
p
tPC r b c t r c
m (10)
3.3 Buyer related costs
The costs associated with the buyer are the holding cost and ordering cost. The inventory of the buyer at time t ∈ [ , ]j j 1t t + as shown by Figure 1 is given by
An integrated single-vendor single-buyer targeting problem 57
1 1( ) ( ) 2 3+ += − ≤ ≤ = , ,...,j j j jIB t D t t t t t j n (11)
Therefore, the cost of holding the fi nished products inventory during the inventory cycle [0, T] is as follows:
1
11
( )+
+=
= −∑ ∫j
j
n t
b b jtj
HC Dh t t dt
which gives
21
1
( )2 +
=
= −∑n
bb j j
j
DhHC t t (12)
where 1n 1t T t+ = +
3.4 Average total cost
The average total cost per unit time is the costs associated with the raw material, the vendor and the buyer, therefore,
( )= + + + + + + +r r v v b bDTC A H AC A H PC A HQ
(13)
where. 1=
= ∑n
jj
Q q The objective is to determine the production schedule, shipment schedule and the process mean, which minimise the average total cost per unit time, TC.
Figure 1 Inventory of vendor, buyer and raw material against time for n = 3
58 M.A. Darwish and F. Abdulmalek
4 Solution method and sensitivity analysis
In this section, we discuss how the optimal solution can be found. The decision variables in this model are the time needed to produce the fi rst shipment, t1, number of shipments, n and process initial mean, μ0. Given the mathematical complexity of the objective function in equation (13), it is diffi cult to prove its convexity. However, we used Hooke and Jeeves (1961) optimisation algorithm to solve the model. To avoid local minima, we used different starting points.
The effects of the following key parameters on the optimal solution are studied: (1) the shape parameter m, (2) the drift rate θ, (3) the process standard deviation σ, (4) demand rate D and (5) the lower specifi cation limit L. The data in Table 1 is used as the basis for all examples unless specifi ed otherwise.
Table 1 Basic data
Lower specifi cation limit L 1.00Standard deviation of the amount of raw material used in an item σ 0.40Expected value of the demand per unit time D 2500Standard deviation of the demand per unit time r 3000Production rate Av 120Ordering cost of buyer Ab 70Ordering cost of raw material Ar 30Setup cost for the production process hv 0.25Holding cost for the vendor per item per unit time hb 0.50Holding cost for the raw material per unit per unit time hr 0.20Fixed penalty cost θ –0.30Fixed production cost b 0.10Value-added factor α 1.10Unit material cost c 0.10Shape factor m 1.00
4.1 Effect of shape factor
To investigate the effect of the shape factor, m, on the total cost of the supply chain, we obtained optimal solutions for a selection of values of m ranging from 0 to 1.4 with an increment of 0.2. Table 2 shows that as the shape factor increases the optimal target value, µ0, also increases. This is true because as the shape factor increases, the process mean, µ(t), will decrease (note that θ = –0.3). Therefore, to make up for this loss, the optimal value of µ0 must increase. It can also be noticed in Table 2 that as m increases the total cost increases. This is owing to the fact that the increase in the value of µ0 would lead to higher material cost. Therefore, higher shape factor leads to higher total cost. To assess the effect of m, we report the percent increase in the total cost (δ1) with respect to a base case where m = 0, that is
01 100%δ
−= ×
mTC TCTC
where is the total cost when m = 0. One may observe from Table 2 that the percentage increase in total cost is higher for large values of m.
An integrated single-vendor single-buyer targeting problem 59
Table 3 shows how the change in drift rate, θ, affects the optimal solution. The drift rate was incremented by 0.1 in a range of values from –0.4 to 0.4. The case with θ = 0 represents the no drift case and we refer to its total cost as 0
θTC . Therefore, the percent increase in total cost owing to change in drift rate is defi ned as follows:
02 100%
θ
δ−
= ×TC TC
TC
Table 3 illustrates that any deviation from the no drift case (θ = 0), increases the total cost and thus the least cost is associated with θ = 0. Hence, the value of δ2 increases with the deviation of drift rate from the base case where θ = 0. Also, in Table 3, one can observe that as the drift rate increases in the negative direction the total cost starts to increase as well. This is factual because as the drift rate becomes more negative, the value of µ0 increases, which in turn causes the cost of raw material to go up. Furthermore, the total cost increases when drift rate increases in the positive direction. This is also true because more deviation in the drift rate in the positive direction means more material in a can and, therefore more material cost.
Another important observation, as the drift rate increases in the negative direction (–0.1 to –0.4) the optimal value of µ0 increases. This is intuitive and is owing to the fact that as the fi lling process starts to under fi ll containers, one needs to increase µ0 so that the proportion of conforming items is higher. On the other hand, as the drift rate increases in the positive direction (0.1 to 0.4) the value µ0 decreases. Yet, again this happens when the fi lling process starts to overfi ll cans and consequently the optimal µ0 is reduced to avoid overfi lling.
Table 4 shows the effect of the variation of the fi lling process on the optimal target value and total system cost. In Table 4 it is clear that as the variance of fi lling process increases it becomes crucial to increase the target value. This is true because for high values of σ the likelihood of producing a nonconforming container increases, this leads to high-optimal target value. As the process mean increases, the material cost will be higher. Therefore, high variation in the fi lling process leads to higher total cost.
It is important here to examine the effect of variation of fi lling process on total cost when variation exists with that of perfect fi lling process where variation is neglected. We, therefore, defi ne the percentage increase in the expected total cost owing to variation as follows:
02 100%
θ
δ−
= ×TC TC
TC
where TCPP is the total cost of the perfect process (σ = 0). It can be observed in Table 4 that Ω increases with σ. For example, ignoring the variation in the fi lling process when σ is actually 0.7, can cause the total cost to be underestimated by approximately 40%. This indicates that, when the effect of variation in fi lling process is ignored, the vendor notably underestimates the actual cost of the produced items. Consequently, it is important to consider the fi lling process variation in determining the total system cost.
The results of changing the demand rate are presented in Table 5. To examine the effect of changing the demand rate on the model, we obtained optimal solutions for some chosen values of demand ranging from 2000 to 2900 with an increment of 100. As the demand rate increases, the process mean increases because as the demand becomes closer to production capacity, it becomes necessary to increase process yield rate to satisfy the demand. In addition, it is expected that, for high-demand rate, the total cost is also high. This is true because of the need for a higher a yield rate to cover the increasing demand, which will directly infl uence the inventory, order and material costs.
An integrated single-vendor single-buyer targeting problem 61
Table 6 gives the optimal solution for selected values of L ranging from 0.4 to 1.8. The results indicate that when the lower-specifi cation limit increases, the process mean increases. This is true because high-target value leads to large yield rate, which is needed to satisfy the demand. As expected the total cost is high for high values of lower-specifi cation limit. This is owing to the increase in material cost.
Process mean selection and inventory/production decisions in a two-layer supply chain have been investigated in this paper. Unlike the hitherto the existing targeting-inventory models for the vendor–buyer supply chain, the present model does not require the process mean to be constant. In fact, the constant process mean assumption may be valid if the process does not deteriorate with time. However, even when the process is in state of statistical control, it possesses some type of deterioration leading to increased nonconforming rate with time. The model in this paper relaxes the constant process mean assumption and considers it changing with time. This model simultaneously determines the optimal process mean setting, production lot size and number of deliveries received by the buyer such that the expected total cost of the supply chain is minimised.
62 M.A. Darwish and F. Abdulmalek
Sensitivity analysis is conducted to investigate the effect of the model’s key parameters on the optimal total cost. In particular, numerical results show that the shape factor and drift rate have signifi cant impact on the total cost. Moreover, the sensitivity analysis shows that the minimum cost increases considerably as the process variation increases, therefore, the performance of a process can be improved by reducing its inherent variation. Also, it is worthwhile to point out that the total cost is high when demand rate approaches the production rate. Hence, the design of production capacity is important in controlling the supply chain total cost. The work in this paper is expected to have a main implication on the management of supply chain. It involves a framework for simultaneous determination of optimal short-term decision (process target) and intermediate-term decisions (production and inventory control).
The line of research in this paper can be extended in many directions. For example, the assumption regarding deterministic demand is valid if variation in demand is low, which does not hold in many practical situations. Hence, studying the effect of random demand on integrated inventory-targeting decisions would be interesting. Also, the production process considered in this paper is perfect. In general, production processes are not perfect, therefore, developing a model for imperfect production processes would lead to more realistic results.
Acknowledgements
The authors are grateful to the anonymous referees for their constructive comments and helpful suggestions. They would like also to acknowledge Kuwait University.
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Notations
We use the following notation in developing the model:D Demand rater Production rateX A random variable represents the amount of raw material an item receivesL Lower specifi cation limitµ0 Initial process meanp(t, µ0 ) Conforming rate of the process at time t and initial target value µ0λ Yield rate of process (λ = rp)ρ Ratio of yield rate to demand rate (ρ = λ/D)Av Vendor setup costAb Buyer ordering costAr Ordering cost of raw materialhv Holding cost for the vendor per item per unit timehb Holding cost for the buyer per item per unit timehr Holding cost for the raw material per unit per unit timeb Fixed production cost (b > 0)α Value-added factor (α ≥ 1)c Unit material costqi Size of ith shipmenttj Time needed to produce the jth shipmentQ Production lot sizeT Inventory cycle lengthn Number of shipments per produced lottP Production cycle lengthm Shape factorθ Drift rateTC Average total cost per unit time
