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MION40 Simulation of industrial processes and logistical systems

HT 2019

Final Project Introduction This final project consists of two distinct parts. Part I is an applied simulation project in ExtendSim based on a real case at Vårdapoteket. Part II deals with simulation and analytical analysis of a two level spare parts distribution system. The project will be graded on a five grade scale 3 (pass), 4 and 5. In case a passing grade is not achieved initially you will be given the chance to correct or complement it to reach a passing grade. In order to receive a course grade all assignments and the final project must have been completed and received a passing grade.

General Instructions The final project is to be carried out in groups of no more than three students. The analysis and results should be well documented in a short technical report to be turned in for grading. The report should be provided in hard copy to the instructor! The report should also be provided in electronic form as a pdf file e-mailed to both johan.marklund@iml.lth.se and Johan.Marklund.lu@analys.urkund.se. This file should be titled

“MION40 Final Project 2019 - X_Y_Z”,

where X_Y_Z are the surnames of the students in the group. The report itself should clearly state the names of all group members on the front page, and all group members must sign the report. The report should explain the problem, your approach to solve it, your findings, results and conclusions. It is important to explain and motivate each step in your analysis. Just an answer or a number that appears to be taken out of the blue is not acceptable! Imagine that you are writing the report to your boss which has little insight into the methods you are using. Thus, they need to be thoroughly explained in order for your results to be trusted. Relevant simulation models and computer files (Excel, Matlab etc) should be e-mailed to the instructor together with the report. Use the program WinZip to compress the files and put them in a common folder. The e-mail should clearly state the names of all group members in the subject field. Furthermore, the models should be labeled and referenced in the report. To receive a passing grade a student must have been actively involved in the analysis and reporting of the assignment. Random checks may be carried out to assure that all group members have a thorough understanding of the work that is turned in. Each group must hand in one peer evaluation report together with the project material. This peer evaluation report enables a differentiation of the grade achieved for the report between the different group members depending on their contributions.

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1. Part I: Live Case at the pharmacy ”Vårdapoteket” This first part of the final project concerns the analysis of a pharmacy store in Lund. This is a real world case, and you can visit the pharmacy located at the university hospital in Lund (close to the main bus stop). In short, Vårdapoteket wants to analyze different queueing configurations in order to provide customers with sufficiently good service using as little resources (i.e. personnel) as possible. The objective is to keep costs low and stay competitive. The assignment given by the logistics team at Vårdapoteket is to use simulation to analyze and compare three alternative process configuration or cases; Case 1, Case 2 and Case 3 at Vårdapoteket in Lund. Case descriptions with further details needed to solve the assignment is provided in Appendix A. The task involves development and analysis of appropriate simulation models in ExtendSim. The problem is relatively loosely defined, as most real world problems are. Based on the provided case description, necessary model assumptions are up to you to decide. However, your assumptions should “make sense” and be well explained and motivated. It is not acceptable to make extreme assumptions that render the problems trivial or completely unrealistic. Your work should be well documented in a short and concise technical report. You can imagine that it is written for the logistics-team at Vårdapoteket. They intend to use this as the basis for deciding on the future process configuration of the pharmacy store. Therefore, you must explain your model, how you solved it, and suggest a clear plan for how the systems should be designed and optimized. It is important to explain and motivate each step in your analysis!

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2. Part II: Simulation of a supply chain system In many industries spare parts and aftersales are big business. For example, the US automotive aftermarket was estimated to be worth $188.6 billion in 2007 (US Automotive Parts Industry Annual Assessment, 2009). Another example is the aviation industry, which stocks spare parts for several billion US-dollars. However, although it may be very costly to invest in expensive spare parts, it is crucial to have spare parts available when needed. Obviously, delays and downtimes of bottleneck production equipment may be very costly. 2.1 A two stage distribution inventory system with complete

backordering Consider a two-level continuous time spare parts inventory system with one central warehouse and two local sites, see Figure 1.

Figure 1: The distribution inventory system consisting of one warehouse and two local sites (retailers).

Customer demands occur only at the local sites and are assumed to follow independent Poisson processes with rate λi (in this setting, an arriving customer only demands one unit). SAs the focus is on spare parts, we assume that replenishments are made according to base-stock ordering policies (i.e., every demand triggers a corresponding replenishment order). The transportation time from the warehouse to local site i is Li days (deterministic), and the transportation time from the external supplier to the warehouse is L0 days (deterministic). The base-stock ordering rule (often called an (S − 1, S) policy) is simple and works as follows. Whenever a customer arrives at a local site, one unit is removed (if there is stock on hand) and a new unit is directly ordered from the central warehouse. If there is no stock on hand upon a customer arrival, the customer demand will be backordered, and a new unit is ordered. A backordered customer demand means that the customer will wait until a unit becomes available. Backordered customers are served on a first-come-first-served basis. The warehouse ordering rule follows exactly the same pattern, i.e., each time a local site demands a unit from the warehouse, the warehouse orders a new unit from an external supplier with ample stock.

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Note that, for each inventory location i ∈ {0, 1, 2} (the central warehouse corresponds to i=0), the inventory position (number of units in stock + number of outstanding units ordered but not yet delivered − number of backordered customer demands) is equal to Si. For example, this means that there can be a maximum of Si units in stock.

The specific input data necessary to analyze the system is: S0 = base-stock level for the warehouse = 11, S1 = base-stock level for location 1 = 5, S2 = base-stock level for location 2 = 5, λ1 = 0.8 units per day, λ2 = 0.8 units per day, L0 = 6 days, L1 = 2 days, L2 = 2 days.

a) Develop a simulation model in ExtendSim of the described inventory system. b) Use the developed model to estimate (with confidence intervals):

Expected stock on hand at locations 1 and 2, Expected number of backorders at locations 1 and 2, Expected stock on hand at the warehouse, The service level P(IL0 > 0), where IL0 is the inventory level at the warehouse. The expected delay at the warehouse (note that, an order placed by location i

arrives after Li + ∆ time units, where ∆ is the random delay encountered at the warehouse level in case the warehouse is out of stock),

The expected time in backorder for customers at locations 1 and 2.

c) This part of the assignment is only for those who aspire on the highest grade. You do not need to complete this part to pass! The aim of this part is to encourage the ability to profit from reading scientific literature. There exist a large number of interesting models in scientific journals, more or less sophisticated, that may be of interest in your future careers. Thus, the assignment is to read the paper in Appendix B: S. Axsäter. (1990) Simple Solution Procedures for a Class of Two- Echelon Inventory Systems, Operations Research, Vol. 38, pp. 64–69. Then use the results from this scientific paper in order to calculate exact values of (use your own choice of software, but you should show your calculations in detail):

Expected stock on hand at locations 1 and 2, Expected number of backorders at locations 1 and 2, Expected stock on hand at the warehouse, The service level P(IL0 > 0), where IL0 is the inventory level at the warehouse. The expected delay at the warehouse (note that, an order placed by location i

arrives after Li + ∆ time units, where ∆ is the random delay encountered at the warehouse level in case the warehouse is out of stock).

Compare these results to those obtained from simulation in b. Are they within the confidence intervals?

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2.2 A two stage distribution inventory system with lost sales

We now assume that arriving customers (at location 1 and 2) are not willing to wait. This means that if there is no stock on hand when a customer arrives, then the customer demand is lost (no inventory pooling is allowed). Notice that, a lost customer does not trigger a new order from the central warehouse (location i orders only from the warehouse if a unit is removed from inventory at location i). However, the warehouse still allows for backorders. That is, if the warehouse cannot satisfy a demand from location i directly from stock, location i will wait until the warehouse has received an incoming order. Except for this change in customer behavior, the supply chain system is exactly the same as above in Section 2.1.

a) Develop a simulation model of the described inventory system in ExtendSim. b) Use the developed model to estimate (with confidence intervals): Expected stock on hand at locations 1 and 2, Expected stock on hand at the warehouse, The expected rate of arriving orders at the central warehouse The expected delay at the warehouse (note that, an order placed by location i

arrives after Li + ∆ time units, where ∆ is the random delay encountered at the warehouse level in case the warehouse is out of stock).

c) Consider a simple (approximate) model of the system. One major complication of the original system is that the lead-time from the central warehouse to location i is stochastic, due to the random delay ∆. To overcome this difficulty, let us assume (as an approximation) that the lead-time from the central warehouse to location i is exponentially distributed with mean Li + E(∆) (where E(∆) is the expected delay at the warehouse). It is allowed to use ExtendSim in order to get an estimate of E(∆). Given this information, it is possible to use Markov theory (continuous time) in order to calculate performance measures for location i.

Your task is to construct a state graph of the considered Markov process, construct the corresponding Q-matrix, and then solve the corresponding linear system of equations to determine the stationary state probabilities. You should then use this distribution to calculate:

the expected stock on hand at location i, the expected number of lost sales at location i, the expected rate of arriving orders at the central warehouse.

Compare these results to the ones obtained from simulation in b). How well does the approximation perform?

Note: For full credit, you must construct a Q-matrix, and then solve the corresponding linear system of equations. It is not allowed, in this specific problem, to use standard queueing theory to calculate steady state probabilities, unless it is for validation purposes of course.

APPENDIX A

Case description: The pharmacy “Vårdapoteket”

Vårdapoteket (Pharmacy) - Case

Pharmacy in Lund – the current configuration of checkouts

5 Prescription checkouts with pay functionality

2 Pay checkouts

0 Prescription checkouts without pay functionality

Type of customers

• Prescription-customer – Only buys prescribed products

• Self care-customer – Only buys self care products (like shampoo etc)

• Mix-customer – Buy both prescribed and self care products

Types of checkout counters

• Prescription counter with pay-functionality – In this checkout it is possible to get prescriptions

serviced and also make payment (here it is possible to pay for all types of products, including self care products)

• Prescription counter without pay-functionality – In this checkout it is possible to get prescriptions

serviced, but all payments must be done in the Pay checkout

• Pay checkout – In this checkout customers pay for their products (all

types of products are allowed)

Case Lund • Case 1: How many checkouts should be staffed at different time slots during a day, in order to expidite 95% of

the customers within 5 minutes? (if a customer waits in several queues, the service definition should be for each waiting line seperately)

– All checkouts for prescriptions have pay-functionality, and all payments of prescriptions are done in these checkouts – Self care customers always pay in the Pay-checkout, and there must always be at least one staff-person who can

service customers (help customers to find the correct product and to administrate payment) in the Pay-checkouts

• Case 2: How many checkouts should be staffed at different time slots during a day, in order to expidite 95% of the customers within 5 minutes? (if a customer waits in several queues, the service definition should be for each waiting line seperately)

– All checkouts for prescriptions do not have pay-functionality, and all payments of prescriptions are done in the Pay checkouts

– Self care customers always pay in the Pay-checkout, and there must always be at least one staff-person who can service customers (help customers to find the correct product and to administrate payment) in the Pay-checkouts

• Case 3: ”Extended service” - A variant of handling queues

– Extended service means that customers with many prescriptions are directed to a separate prescription counter (only for high volume prescriptions). It is assumed that this particular prescription counter has a pay-functionality

– There is only one prescription counter for extended service in Lund – Does this queueing configuration lead to shorther waiting times for customers? – How many prescriptions should a customer have in order to be directed to extended service?

Some data – Nbr of customers arriving per hour at Vårdapoteket in Lund

Clock-time during the day

Nbr of Self care-customers

Nbr of Prescription- customers

Nbr of Mix-customers

8 11,5 5,5 0,8

9 19,0 13,3 5,3

10 20,5 17,0 5,5

11 24,8 15,3 6,0

12 42,3 15,3 7,5

13 28,3 15,0 6,5

14 25,0 18,0 7,0

15 28,0 19,3 7,5

16 42,8 18,8 8,3

17 27,8 17,3 6,0

Average values for 4 Mondays in March

Opening hours: Weekday 08:00 – 18:00

Service time per customer

Nbr of prescriptions

Average service time (min) Fraction of customers

1 03:00 30%

2 04:01 36%

3 05:31 19%

4 05:52 5%

5 07:30 5%

6 08:59 5%

7 or more Assumed to be negligible

Prescription customers (including payment)

Nbr of products

Average service time (min)

Has no impact

2

Self care customers (including payment)

The payment (only) is approximated to 40 seconds in average (this is also the case for prescription-customers if the payment is done at some Pay checkout)

APPENDIX B

S. Axsäter. (1990), Simple Solution Procedures for a Class of Two- Echelon Inventory Systems, Operations Research, Vol. 38, pp. 64–69.

SIMPLE SOLUTION PROCEDURES FOR A CLASS OF TWO-ECHELON INVENTORY PROBLEMS

SVEN AXSATER Lulea University of Technology, Lulea, Sweden

(Received September 1986; revisions received June 1987, May, September 1988; accepted December 1988)

We consider an inventory system with one warehouse and N retailers. Lead times are constant and the retailers face independent Poisson demand. Replenishments are one-for-one. We provide simple recursive procedures for determining the holding and shortage costs of different control policies.

This paper deals with an inventory system that consists of one warehouse and N retailers, as

shown in Figure 1. Transportation times are deter- ministic and we assume that the retailers face station- ary and independent Poisson demand. Unfilled demand is backordered and the shortage cost is a linear function of the time until delivery, or equiva- lently, a time average of the net inventory when it is negative. Furthermore, we limit ourselves to one-for- one replenishment policies. This means essentially that we assume that ordering costs are low and can be disregarded. Orders that cannot be delivered instan- taneously from the warehouse are ultimately delivered on a first come, first served basis. There are linear holding costs at all locations. Given these assump- tions, we are able to derive the costs for different policies recursively. These cost expressions then can be used for the efficient determination of an optimal one-for-one replenishment policy.

The model and minor variations of it have been analyzed previously by several authors. Sherbrooke's well known METRIC model (1968) approximates outstanding orders at the retailers by Poisson random variables. This assumption also means that the ware- house backorder level is Poisson and permits a simple evaluation of different policies. Graves (1985) deter- mines exactly both the average and the variance of outstanding orders at a retailer. The distribution is then approximated by a negative binomial distribu- tion. This two-parameter approximation is, in general, more accurate than the METRIC approximation. Simon (1971) derives the steady-state distribution for the inventory levels at each site. Svoronos and Zipkin (1986) consider stochastic transportation times that are generated exogenously.

Evaluation methods for more general batch order-

ing policies are necessarily more complex because the demand process seen by the warehouse is then a superposition of Erlang renewal processes, which are difficult to characterize. Different approximate meth- ods for this more general problem have been suggested by Deuermeyer and Schwarz (1981), Moinzadeh and Lee (1986), Lee and Moinzadeh (1987a, b) and Svoronos and Zipkin (1988).

Our approach uses an inventory cost function that reflects costs incurred on an average unit. See also Axsater and Lundell ( 1984). The relative advantage of the approach in this paper is that it focuses directly on evaluating the average costs associated with a stock- age policy. Earlier approaches (METRIC, Simon) focus on characterizing the steady-state behavior of the inventory levels for a stockage policy, and then use the steady-state distribution (or an approxi- mation thereof) to determine the average costs associ- ated with the stockage policy. Our approach is more efficient and direct at finding the optimal stockage policy for the assumed (traditional) cost function; it appears to be the only available approach when the cost is given by a nonlinear function of either the delays experienced by the customers, or the unit's storage time at each of the facilities. A gen- eralization to more than two levels is relatively straightforward.

On the other hand, our approach is inappropriate for cost functions that are expressed as nonlinear functions of the inventory levels and/or backlog sizes; it also does not provide any information on the steady- state distribution.

In Section 1 we give a detailed formulation of the problem, and in Section 2 we show how to derive the relevant cost expressions. Section 3 presents the opti- mization procedure.

Subject classifications: Inventory, multi-echelon: one-for-one replenishments. Inventory, stochastic: Poisson demand, continuous review.

Operations Research0030-364X/90/3801-0064 $01.25 Vol. 38, No. 1, January-February 1990 64 / 1990 Operations Research Society of America

Two-Echelon Inventory Solution Procedures / 65

retailers

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warehouse

Fgr Vv Figure 1. Inventory system.

1. PROBLEM FORMULATION

We introduce the following notation.

N = the number of retailers, Li = the transportation time for an item to arrive at

retailer i from the warehouse, Lo = the transportation time for an item to arrive at

the warehouse, Xi = the demand intensity at retailer i, So = ENIf , Xi = the demand intensity at the warehouse, hi= the holding cost per unit and time unit at

retailer i, ho = the holding cost per unit and time unit at the

warehouse, oi= the shortage cost per unit and unit time at

retailer i.

A one-for-one replenishment policy is completely characterized by the vector (SO, S,, . . ., SN) of order- up-to inventory positions.

Si = the inventory position at retailer i, SO = the inventory position at the central warehouse.

Note that each facility i = 0, 1, .. ., N faces a Poisson distribution with a rate Xi, and let g,(-) denote the density function of the Erlang (Xi, Si) distribution of the time elapsed between the placement of an order and the occurrence of its assigned demand unit

__s\itsi1 e`it 91N(t = (S -1) (1) (S - 0 'ei

The corresponding cumulative distribution function GW'(t) is

GW'(t) = (X.t) e-Ai (2) k=si k!

When a demand occurs at a retailer, a new unit is immediately ordered from the warehouse and the

latter orders a new unit at the same time. If demands occur while the warehouse is empty, shipments to retailers are delayed. When units are again available at the warehouse, the retailers are served according to a first come, first served policy. In such situations, the individual unit is, in fact, already virtually assigned to a retailer when the demand occurs, that is, before it arrives at the warehouse.

2. POLICY EVALUATION

Fix a one-for-one replenishment rule with (SO, SI,..., SN) as the vector of order-up-to-levels.

Lemma. (Main Observation). Any unit ordered by facility i (O S i S N) is used to fill the Sith demand following this order, hereafter, referred to as its demand.

The lemma is an immediate consequence of the order- ing policy and of our assumption that delayed demands and orders are filled on a first come, first served basis.

It is easy to conclude that we can disregard policies with negative inventory positions when looking for the optimal solution. We therefore confine ourselves to the case where all Si - 0 (i = 0, 1, . . ., N). Policies with negative inventory positions may, however, be evaluated in similar ways.

An order placed by retailer i arrives after Li + A time units, where A is the random delay encountered at the warehouse level in case the warehouse is out of stock. (Note that 0 - A < Lo because we assume that SO 2 0.) If the ordered unit arrives prior to its (assigned) demand, it is kept in stock and incurs carrying costs; if it arrives after its assigned demand, this customer demand is backlogged and shortage costs are incurred until the order arrives. Let Hi(So) denote the expected retailer inventory carrying and shortage costs incurred to fill a unit of demand at retailer i, 1 S i S N. We evaluate this quantity by conditioning on A = t. Note that the conditional expected cost 7r,i(t) is independent of S0 and is given by

rLi+t

7r,(t) = J gs(s)(Li + t - s) ds

+ hi gt(s)(s - Li - tL) ds Si > + (3) Li+t

7ri?(t) Oi(Li + t). (4)

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The distribution of A is well known and may be verified directly from the lemma; see, for example, Sherbrooke (1975). For SO > 0

So- I X~L k 5 P O 1 6Lko

p(A ~ = ?)= e-,o Lo = 1- Goso(Lo) (5) k=O k!

and the density function f(t) for 0 < A i Lo is given by

xoso(L0 t)So-1 `_LOt f(t) gso (Lo - t ) = e Ao(Lo-t) (6) (SO I 1)

Note that the warehouse faces a Poisson demand process with rate X0. We thus obtain the following expression for II'(So)

rOLO

HIS,(SO) = J gos(Lo -t)2r(t) dt

+ ( - Goso(Lo))7r i(O). (7)

The long-run average shortage and retailer carrying costs are clearly given by Ei XjHi(So). We are thus left with the derivation of an expression for the long- run warehouse carrying costs. The average warehouse holding costs per unit ly(SO), which depend on the stockage level SO only, may be derived in complete analogy to the 7ri-quantities

00 'y(So) hO goso(s)(s - Lo) ds. (8) 0 For SO 0, A = Lo implying that

Hi,(O)= 7rfi(Lo) (9)

and -y(O) = 0. We conclude that the long-run systemwide costs are

given by

N

C(SO, SI, ..., SN) = > Xirli(So) + Xo'y(So). (10)

Let us simplify expression (3) for 7ir~(t). Since

fgs(u)u du= S+ (u)S Sdu = W+ (t)S (I 1) o i xG (i

we obtain

wrsl(t) = (1 - Gsi+'(L + t))(hi + f) Si

- (1 - Gs(Li + t))(hi + fi)(L, + t)

+ fi(Li + t - Si /Xi) (12)

or

7r'(t) = e-A(Li+t) hi + fi

vSi (S -

E k)(Li + t)kxk k=O k.

+ ji(Li + t - Si/Xi)

(0! = 1 by definition). (1 3)

We proceed with the derivation of a recursive formula for the HW'(So) numbers. To facilitate this derivation, we first derive a differential equation for rSi(t) from a simple probabilistic argument. In an

order lead time of length Li + t, consider the initial time interval of infinitesimal length e and condition on the number of demands that occur in this interval.

Clearly

ir,'(t) = (1 - X1e + o(e))irS(t -)

+ (Xe + o(e))iri-(t_ e) + 0(e) (14)

that is, if exactly one customer arrives in the first e time units of the order lead time, then the expected costs are identical to the situation where the order is placed e time units later with a lead time of Li + t - e and a retailer stockage level of Si - 1. From (14), it is easy to obtain the differential equation

dt (t)= -Xi(irs'(t) - irsl(t)). (15) dt

It remains to simplify expression (7) for HI (So). Let SO 3 1 and consider

llsL(So)- HV'(S0)

CLO 1dr~t

j goso(Lo _ 7 0 (t - -)) dt

+ (1 - Goso(Lo))(w,i(O) - wsi'(0)). (16)

Using (I 15) we obtain

rz ,$ (SO) - Hsli I(SO)

=-( goso(Lo -t)-ld2t dt

+(I - Goso(Lo))(7rii(O) -7r $ '(?)). (17)

Two-Echelon Inventory Solution Procedures / 67

After partial integration we have

II si' (SO ) -- II ,SZ 1 (SO )

I (goso(Lo)Si (0) - goso(O)Sr'(Lo))

xi

I Lo? dgoso(Lo - t)S()d + 7r ~~~~~lsi (t ) dt

+ (1 - Goo(Lo))(7rwi(O) - 7r (O))

_ I (goo(Lo)ir,i(O) - gso(0)7rs)(LO))

+ (goso(Lo - t)r,K(t) dt

_ go rl(LO - t)ir$S(t) dt)

+ (1 - Gos(Lo))(irji(O) - 7.S1(O)). (18)

The term gos-I(Lo - t)ir$(t) vanishes for So = 1. Using (7), we can express the integral in (18) in terms of WII(So) and lI;.(S0 - 1). Then, we obtain the following expression (valid both for So = 1 and So> 1)

Il0i (so) =-i II st (so) + Wqso I (Q1) So-xi

I So-xi

--Xi (1 -Goso(L))(7rii(O) v -r (0)). (19) o0-Ni

For S, = 0, we can easily derive the explicit solution from (4) and (7)

llH(So) = Gos(Lo)fiLo-LGoso+'(LO) + fiL1. (20) i X~~~~~~~0

Using (19), (20) and (9) we can determine II s(So) for any So, S, : 0; start with S0 = 1 and evaluate the I l (1) numbers recursively. Repeat this calculation

for So = 2, 3, . . . etc. Since N0/(No -i) > 1, a direct application of (19) may cause some numerical prob- lems. This is because minor errors for low values of SO may have a large impact on the results for larger values of S0. In order to avoid these difficulties we suggest a slight modification of our computational procedure. We reformulate (19) as

qSi(So_ 1)=Xi0IJs>(S0)+

Il(S0) HS'-(So S~o

+ '( - Goso(Lo))(rJi(0) - 1i(0)). (2 1) N0I I

Furthermore, for large values of So we have

lls'(So) - 7rw(0). (22)

This approximation is asymptotically exact. Our procedure starts by determining SO such that

Goso(Lo) < e (23)

where e is a small positive number. For So - go we apply (22). For S0 < S0 we apply (20) and (21) recur- sively for go - 1, go - 2, . , 0. Since the coefficients in (21), Xi/X0 and (X0 - Xi)/X0, are both positive numbers between zero and one this recursion will not cause any numerical difficulties. In the recursion, it may also be of interest to utilize that

irl(0) - 7r?(0) = e-xiLihi + Oi _ fi (24) Xi Xi

7r (0) -, (0)

eLhi + fi, __i_S.> ? (25) Xi Si!

which follows directly from (13). For Si = 0, (20) gives the solution in closed form. It

is also possible to formulate a corresponding, but more complicated expression, that is valid for Si > 0.

llsI(So) = (1 Goso(Lo))7ir>(0)

00(X0Lo) CXOLO + E(??)e-O

i=so soJ ,

* L (I -So)(X}

(A. Xi )j-So-r __________ max S-r,0'iIj

+ E (XL0) e-XOLO j=S0 J -

i-s so - i jO 1 ) -SO-r

* /i3L0(l )( I- r+li (26)

In (26), the first two terms represent the costs incurred after the item has left the warehouse. The third term corresponds to the case when a backorder occurs at a retailer before the demanded item reaches the

68 / AXSATER

warehouse. In this case, we have to add the backorder cost for the remaining time until the item reaches the warehouse.

We can also simplify (8) and express Xy(So) as

y(So) = (1 - Goso (Lo)) Xo

-hoLo( - Goso(Lo)). (27)

3. OPTIMIZATION

In order to optimize our system we need to determine the one-for-one replenishment rule (SO*, S*, . . ., S *) that minimizes the long-run systemwide costs accord- ing to (1O). Note that for a given SO we can optimize each H S(SO) separately with respect to Si. The optimal Si (for given SO) is denoted S* (SO). Furthermore, from (7) and (25) it is easy to conclude that li(So) is convex in Si.

Let us first derive upper and lower bounds for the optimal inventory positions. It is possible to show from (13) that for Si > 0

dt dt (irS'(t) - s+()

- ei(L+t)hi ,~(Li + t)s1XSi0 -e-Xi(i+t) (hi + Oi) (L, + t)? > O (28)

that is, 7r J(t -7r (t) is an increasing function of t. Furthermore, if we consider the distribution of the random delay encountered at the warehouse, (5) and (6), it is evident that the c.d.f. for SO is stochastically larger than the corresponding c.d.f. for SO + 1. It is then easy to see from (7) that

IS, i(SO) -IS, S+1I (So)

H Ils(S0 + 1) - lS+1(So + 1). (29)

We can therefore conclude that

S* (o) I S* < S*(O) (30)

where Si*(oo) and S*(O) are the values of Si that minimize rsi(0) and r'si(Lo), respectively.

Next we reformulate (29) as

rISi (SO ) r i S, (SO + 1 )

lIi+1(So) - ll'+1 (So + 1). (31)

Since y(So) is independent of Si it follows that an upper bound for SO denoted Su is obtained by mini- mizing C(So, S*(oo), ..., S*(oo)). To see this, note

that due to (31) for k > 0 and Si > Si*(oo)

0 Q C(SU, S* (?),.. , SN(OO))

-C(SO' + k, S* (??), . . S(OO))

C(Su,S1, . . ,SN)-C(Su + k,S1, . . ,SN). (32)

In the same way, a lower bound SO is obtained by minimizing C(S0, S*(0), . . ., S*(0)).

Using (30) and the convexity in Si it is easy to determine the optimal Si*(So). In order to optimize the total costs (10) with respect to So, it remains to evaluate C(So, S*(SO), ..., S*(SO)) for SI < SO < Su

0o.

When using the suggested iterative procedure based on (20) and (21), it is necessary to determine H (SO) for 0 < Si < S* (0) and S' s SO Sgo where go (obtained from 23) is normally considerably larger than S". The systemwide costs are evaluated along the way for SO < SO.

It is relevant to compare our method to the METRIC approximation and to Graves' more accu- rate two-parameter approximation. The computa- tional requirements when using these two approxima- tions are of the same order. If we just want to evaluate a single policy, the approximate methods are faster than our exact method, but if we wish to optimize the system, the computational efforts should be similar because we need to consider all interesting values of SO and S1.

ACKNOWLEDGMENT

I am grateful for the valuable comments and sugges- tions by the Associate Editor and the referees.

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