cohen, lee - 1987 - strategic analysis of integrated production-distribution systems

Strategic Analysis of Integrated Production-Distribution Systems: Models and MethodsAuthor(s): Morris A. Cohen and Hau L. LeeReviewed work(s):Source: Operations Research, Vol. 36, No. 2, Operations Research in Manufacturing (Mar. -Apr., 1988), pp. 216-228Published by: INFORMSStable URL: http://www.jstor.org/stable/171277 .Accessed: 28/02/2012 03:09

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STRATEGIC ANALYSIS OF INTEGRATED PRODUCTION- DISTRIBUTION SYSTEMS: MODELS AND METHODS

MORRIS A. COHEN University of Pennsylvania, Philadelphia, Pennsylvania

HAU L. LEE Stanford University, Stanford, California

(Received October 1986; revision received July 1987; accepted October 1987)

This paper presents a comprehensive model framework for linking decisions and performance throughout the material- production-distribution supply chain. The purpose of the model is to support analysis of alternative manufacturing material/service strategies. A series of linked, approximate submodels and an heuristic optimization procedure are introduced. A prototype software implementation is also discussed.

T he threat of foreign competition has caused many firms in the United States to reevaluate their

basic manufacturing strategies in order to regain a position of competitive advantage. Research to develop methodologies to assist managers in the formulation of competitive operations strategies is important for the revival of U.S. manufacturing in world mar- kets. This paper reports on the development of a model structure that can be used to predict the performance of a firm with respect to: (1) the cost of its products, (2) the level of service provided to its customers, and (3) the responsiveness and flexibility of the production/distribution system. Our analysis takes into account the nature of the products produced, the process technologies used to manufacture the products, the structure of the facility network used to manage the flow of materials, and the competitive environment in which the firm operates.

The problem of interest is concerned with measur- ing cost/service/flexibility tradeoffs in production/ distribution systems for various materials management strategies under alternative environmental and structural conditions. Our methodology specifically considers relationships between production and distribution control policies that affect inventory control, plant product mix and production scheduling. Other decisions associated with manufacturing strategy, such as facility location, capacity planning and technology choice are assumed to be fixed.

The production of goods in a factory is accom- plished by the transformation and/or assembly of inputs through various processing stages. The factory

may be viewed as a network of processing centers and stocking points that are linked by material handling and information systems. The distribution system, which channels material to and between plants and delivers products to customers, consists of a network of stocking locations.

In this paper, we restrict our attention to discrete batch manufacturing operations that can be organized into multistage processing lines and to arborescent distribution networks. These systems make extensive use of intermediate buffer storage facilities. Figure 1 illustrates the structure of such "supply chain" systems. It depicts a number of factories that are linked to suppliers and to a multiechelon finished product distribution network.

Material flow in production/distribution systems is managed by a variety of mechanisms. The inputs to each factory consist of materials and intermediate products which can be sourced from different vendors or other manufacturing facilities belonging to the firm. Such input flows are managed by the firm's material requirement inventory control system. The flow of material within the factory is influenced by plant layout, product routings, production lot sizes and manufacturing schedules. The outputs of finished goods can be stockpiled or shipped directly to appropriate locations within the distribution network. Finally, the flow of material through the distribution network is controlled by the stocking policies used to manage finished goods distribution.

Our objective in this paper is to introduce a model framework and an analytic procedure for

Subject classification: 331 inventory/production, 344 inventory/production operating characteristics, 570 stochastic model applications.

Operations Research 0030-364X/88/3602-0216 $01.25 Vol. 36, No. 2, March-April 1988 216 ? 1988 Operations Research Society of America

Integrated Production-Distribution Systems / 217

evaluating the performance attributes of the class of production/distribution systems illustrated in Figure 1. The purpose of the framework is to predict the impact, on performance, of alternative manufacturing and material strategies.

We are particularly interested in developing an analytically based methodology to answer the following questions:

1. How can production and distribution control policies be coordinated to achieve synergies in performance?

2. How do service level requirements for material input, work-in-process and finished goods availability affect costs, lead-times and flexibility?

The model structure introduced in Section 2 cap- tures many of the stochastic, dynamic interactions of multistage production/distribution systems. At the same time, it retains a level of aggregation that is appropriate for its use as a strategy evaluation tool. In this way, the computational complexities and data requirements of simulating or controlling an opera- tional system are avoided.

The model structure described in this paper was designed to test the validity of the overall approach. It represents, therefore, a first attempt to formulate, link and optimize the complex system of submodels required to analyze integrated manufacturing/distribution systems. Research is on-going to extend and validate the model structure. Field tests of a software system implementation of the model structure described here are currently underway with the active participation of the Operations Management Systems group of Booz, Allen and Hamilton, Inc.

The remainder of the paper is organized as follows. In the next section we review the relevant literature. Model formulations and an overview of the solution algorithm are described in Section 2. Section 3 presents results based on an extensive analysis of an example problem.

1. Literature Review

Many models in the literature are concerned with material procurement, production or distribution activities. Most past efforts, however, treat each stage of the supply chain as a separate system. As a result, many of the complex supply chain interactions are ignored. In this section, we will not review the literature associated with each area noted above in detail. Rather, we will focus our attention on those studies that have attempted to link material management activities across different stages of the production/ distribution supply chain.

Hanssmann (1959) described what was probably the earliest attempt to build an analytical model con- taining material procurement, production and distribution elements. The optimal inventory levels at different stages are identified. A number of simplifying assumptions are made, however. Production times are considered constant and independent of the lot-sizing decisions. The existence of multiple products compet- ing for similar resources, and commonality of use for incoming materials for the production of different products, are not considered. This approach, however, is infeasible for the complex production and distribution systems described earlier.

Gavish and Graves (1980, 198 1) extended a conven- tional production model by allowing for finished goods stockpiles. They consider a one product batch production facility with stochastic demands for finished goods. This model constitutes a first attempt at explicitly linking the management of production and finished goods stockpiles.

Williams (1981) is an extensive study of different heuristic algorithms for solving scheduling problems for multiechelon arborescent production and distribution structures. The model's assumption of deterministic and constant production and demand rates limits its applicability. The existence of random material supply is not considered.

Recently, there have been some significant advances in studying the relationship between lot-sizing and manufacturing lead-times. Karmarkar (1987), Kar- markar, Kekre and Kekre (1983), and Zipkin (1986) independently developed results that use an M/G/I1

Raw Material Intermediate Final Distribution Customer Vendors Product Product Centers Zones

Plants Plants Warehouses

V = Inventory Stocking Point

[I = Production Location

Figure 1. Supply chain network.

218 / COHEN AND LEE

queueing system to characterize congestion phenom- ena in production facilities. The lot-sizing decision affects the production lead-times through the number of required setups. Karmarkar, Kekre and Kekre (1985a) showed via simulation that their models are reasonably accurate for determining optimal lot sizes and estimating associated production lead-times in a manufacturing cell.

In Zipkin; and Karmarkar, Kekre and Kekre (1985b), the approach is extended to multimachine job shops. Further refinements to the basic model include coordination of the production of similar parts on a machine (Kekre 1984), and consideration of multiperiod production planning problems (Kekre and Kekre 1985). Although these models do not consider the relationships between production, material procurement and distribution, they are important because they explicitly develop results to relate lot-sizing to lead-times.

Recent papers by Williams (1984) and Bertrand (1985) extend the results of Gavish and Graves (1980) to deal with multiple product production systems. Williams (1984), in particular, assumes that the finished goods stockpile operates as a (Q, R) continuous review inventory system. Furthermore, Williams considers cases where special products are made to order, and the production facility gives priority to the production of these special products. Bertrand considers multiple work center production systems. In general, closed shop queueing model results (see Solberg 1981) are used to approximate the mean waiting times at various work centers.

Two related papers (Burns et al. 1985, and Blumen- feld et al. 1985) focus on the coordination of distribution efforts to minimize inventory holding, production, and transportation costs associated with production and distribution. Their studies, however, emphasize shipment schedules for finished goods. Ac- cordingly, they use a somewhat simplified picture of the manufacturing systems (especially with regard to time elements and demands for capacity). Moreover, the deterministic nature of the model restricts its applicability.

Note finally that there is a rapidly growing literature on stochastic, multiechelon distribution systems. A comprehensive review of that subject is found in Schwarz (1981), and Cohen, Kleindorfer and Lee (1986).

2. Model Structure and Formulations

A unified, hierarchical, stochastic, network model structure is developed in this section. The structure

consists of the following submodels, where each represents a part of the overall supply chain network: (1) material control, (2) production control, (3) finished goods stockpile, and (4) distribution network control. For each submodel there is a set of control parameters, such as lot sizes, reorder points, safety stock, etc., which will be described in greater detail in later sub- sections. These control parameters are set so that the performance of the submodel meets some specific policy targets set for the submodel. Moreover, the control parameters set at one submodel affect the performance of another submodel.

2.1. Overview of Model Structure

Given a finished goods demand requirement and the bill of material for each product, it is possible to generate the material requirements for production. Since there are many sources of uncertainty involved in both production and distribution, material requirements are not deterministic. Furthermore, resupply lead-times of materials from vendors to plants may also be subject to random fluctuations. As a result, safety stocks for input materials are necessary to minimize production delays due to material shortages. The material control submodel models the random- ness of both the demand process for materials and the resupply times from vendors. The model takes into account the cost of material inventory (setup and holding) and the cost and delay impacts of material shortages on production- processing. It is used to determine ordering policies for all materials required at each plant, which result in service/availability levels (fill rates, stockout frequencies) for each of the raw materials used in the production process. This availability will, in turn, affect the production lead-times of the product since material shortage can lead to delays in production.

The production submodel determines production lot sizes for the various finished products and for each processing line in each plant. Our formulation for this submodel includes queueing relationships for each work center which relate lot size to the job queue time. These relationships are based on the work of Karmarkar, Kekre and Kekre (1983, 1985a, b). In the initial formulation of this paper, we restricted our attention to batch manufacturing processes with parallel lines and multiple stages (Figure 2). Lot size determination is based on the tradeoff between the cost of holding work-in-process inventory and the cost of production processing (both fixed and variable). Production lot size decisions affect material control decisions through their effect on the material demand pattern. Material control decisions affect production


lot size decisions through their effect on total production lead-time.

As finished goods are produced, they may be stored in a finished goods stockpile at each plant. These stockpiles act as central distribution points and are depleted as orders are received from the distribution network. Replenishment of the stockpile constitutes an order for production. Orders to replenish the stockpile are assumed to be a fixed batch size and are filled after the production lead-time (as computed in the production submodel) has elapsed. The lead-time for delivery of stock to the distribution system thus depends on the transportation time from the plants to the distribution centers, the availability level of stockpile inventory, and the production lead-time. This lead-time of resupply from the stockpile to the distribution network acts as the key link between the production and distribution segments of the supply chain network.

The distribution system generates demand for the finished products. It is important to note that stocking decisions and service performance at all stocking points in the distribution network affect the specifi- cation of the demand process from a central distribution node to the finished goods stockpile at the plant. In our modeling of the distribution network submodel, we draw heavily from our related work described in Cohen, Kleindorfer and Lee (1985a, b, 1986). The random lead-time distribution offered to the central distribution node from the finished goods stockpiles will, in turn, affect the desired distribution system stocking policy. This interaction completes the production/distribution linkage for the supply chain.

The ultimate service performance of the whole supply chain is measured by the service rate provided by the lowest echelon of the distribution network to the customers. Hence, the ideal approach is to minimize the costs of operating the complete supply chain to achieve some service target, such as the fill rates, to customers. However, this is often computationally

infeasible. Instead of optimizing the whole system, we propose a decomposition approach so that each submodel is optimized, subject to some service target defined for that submodel. These "local" service targets will also serve as linkages between the various submodels. The fill rate is used as the service performance measure at the individual submodels.

In the formulation of the distribution submodel, a standard fixed cost is used to represent the cost of ordering and shipping items from one stocking location to another. Hence, we avoid the problem of dealing with shipping costs as a nonlinear function of the shipment quantity (which would be the case when there are capacitated truckloads and multiple modes of transportation). We note that for bulky products, where transportation costs are significant, a more detailed model of shipment costs is necessary.

Before we describe the submodels, we define the following subscripts:

i = finished product,

j = plant location,

k = distribution center (DC) location,

r = raw material.

To save notational definitions, we assume that there are distinct sets of indices that correspond to finished products, plant locations, and raw materials, so that definitions of variables with subscripts and summa- tions over subscripts are nonambiguous.

Two key probability distributions that are determined by supply chain interactions are:

prj(n) = Pridemand of material r at plant j over material lead-time = n };

pij(n) = Pridemand requests of product i at plant j from distribution centers during the average production lead-time (in distribution review periods) = n}.

Service requirements are specified by the following policy parameters:

/Trj = minimum fill rate for raw material r at plant j;

#ij = minimum fill rate for product i at plant j;

3ik = minimum fill rate for product i at distribution location k.

2.2. The Material Control Submodel We model material control operations by assuming that material procurement follows a continuous review (nQ, R) inventory control policy, i.e., when the

Line I

tr Station Station ... tto

Line 2

Mterial Station *1 * * 0 FStation .** . iI. e Inpot I God

Figure 2. A parallel line multistage production process.

220 / COHEN AND LEE

inventory position drops to the reorder point R, an order of nQ, where n is a positive integer, brings the inventory position up to the interval [R + 1, R + Q]. For a particular material r, demands for this material arise from the production of end-products that utilize r as inputs. The amount of r requested in each demand depends on the material requirement for the production batch size of the end-product from which the demand is initiated. Material shortages in production can either be backordered or expedited from an exter- nal source, with specified (shorter) lead-times. This paper focuses on the backorder case.

In both the material control and production submodels, we let the time unit be defined as a production planning period (which can be a shift, a day, a week, etc.). Production of product i at plant j is assumed to be in batches of size Qj1, and the requirement for raw material r in the production of this batch is UriQij

(where uri, the unit usage rate of r in i, is based on the bill of material). Define Ir = {i I uri > O} as the set of products which consumes raw material r in their production. Let X1j be the mean production requirement of product i at plant j per period. Hence, the mean rate of demand requests for raw material r at plant j for the production of all the products in Ir is Xrj = ZiEIr (X1j/Qij). The total quantity of raw material r required at plant j is E UriXij.

Therefore, the mean quantity demanded in each request for raw material r at plant j is

Mrj = (ilxrj) Z UriXjj. ieIr

We approximate the initiation of production of batches of end-products as a Poisson process in the spirit of Karmarkar, Kekre and Kekre (1983) and Zipkin. The demand requests of materials for the production of end-products can thus be approximated by a Poisson process with mean rate Xrj per period. The quantity of material r in each demand request, of course, depends on the identity of end-product i, whose production initiated the material request. The probability that a demand request supports production of product i is given by (Xij/Qiy)/Xrj. If the demand request comes from product i, then the quantity demanded will be UriQij. The probability distribution of the quantity in each demand request of material r at plant j can thus be specified as

fj(x) = f(Xij/Qij)/Xrj, if X = UriQij (2) rj0x)- 1O otherwise.

The average number of orders placed at plant j for material r per production planning period is XieIr UriXij/Qij. Note that material demands are

treated as special cases of the Compound Poisson distribution. Equations 1, 2 and the definition of X,j can be used to specify prj(*), the demand distribution for material over material resupply lead-time.

Recall that material inventory is controlled by a (nQ, R) system. Hence, when the demand process is Compound Poisson, the steady-state distribution of the inventory position is uniform (see Simon 1968). Using standard techniques, as in Hadley and Whitin (1963), the costs involved in material control for material r at plant j per period may be specified as

TCmJ = U U riXI/Q111 CK + CH E(Irj) iEIr

+ CB E(Brj) (3)

where Irj and Brj are the inventory and backorder levels, respectively, and C`f, CH and C`; are the fixed cost of replenishing material, unit holding cost per period, and unit backorder penalty cost for material shortage for material r at plant j, respectively.

Note that

I R j+ Qr ??

E(Brk) =1 Rd+Qr, (n -m)prj(n), Qri m=Rr,+1 n=m

and

E(Irj)= r Z (m -n)pr(n). Qrj m=Rrj+ I n=O

The variable cost of purchasing material may be ignored because all demands must eventually be satisfied in the backorder case. The unit backorder penalty cost is normally set at zero, since the penalty for material shortage would be accounted for in terms of production lead-time delays.

Using Little's formula, the expected time spent waiting for material r at plant j is

Expected backorder level TR. = for material r at plant j

Expected usage rate for material r at plant j

_ (Brj)

iEi-I UriXij

The probability that demand arriving at any point in time will find the system empty and have to be backordered, denoted as 1 - 3rj, is given by

R .+ Qr ?? ? rQ Z Prj(n).

Qrj m=Rr,+ 1 n=m


The service level constraint for material r at plant j is thus

d rj 3 Trj. (5)

Our interest is in the amount of time that a finished goods production lot size must wait due to the una- vailability of at least one of its required material inputs. Let Ri be the set of required materials for product i. The computation of finished product material delay time is carried out by considering the probability weighted, material delay time for all materials specified in end-product i's bill of material.

At any random point in time, one or more materials in set Ri may be out of stock. For each material r, this occurs with probability 1 - frj. In general, the probability that two or more materials are out of stock is very small. Hence, an approximation for the product- specific material delay time is

T5j = ( 1 ",rj) rERi

E{Delay time for r j shortage for r) (6)

= ( 1 - ~ry)[r>/( 1 -Iri)] 6 =~~~~#jT Trj .j

rER,

2.3. The Production Submodel Consider a serial multistage, multiline production process. For each line, let I be the subscript for workstations (or stages). We allow a product to be processed on more than one line. The allocation of the proportion of the demand requirements of a product among the multiple lines is assumed to be exogenously determined.

Define

IjIm = the set of all products that require processing at station I of line m at plant j;

Pilm,, = work rate for the processing of product i at station I of line m of plant j (in units/period);

aijm = the proportion of product i that is processed at line m of plant j;

Timjlm = the setup time for product i, line m, station / at plant j;

TjQf/ = the waiting time at line m, station I at plant j;

TLj = the production lead-time for product i at plant.

For each batch of product i processed at plant j, the total production lead-time is given by the weighted sum of setup times, processing times, material delay times, and the waiting times at the workstations. That is,

TL = aijm `

[iJIm + TJQm + Qi-/Pilim] + IJ (7)

TfJ is estimated from (6) in the material control submodel. To estimate TJQ1.7 we use the recent work by Karmarkar, Kekre and Kekre (1983) and Zipkin, which is briefly described below.

For product i at plant j, the arrival pattern to workstation 1 in line m will have a mean batch arrival rate of aijmXij/Quj) per period. When many products are processed at station 1, the arrival pattern to the workstation approximates a Poisson process with mean rate Xj/m = aijm ,isI (Xij/Qij). To be precise, the input process to station I is the departure process of station I - 1, which, in general, is not Poisson. Similar to Karmarkar, Kekre and Kekre (1985b), the approximation is used here to achieve tractability. More accurate models would consider the second or higher moments of interarrival and interdeparture times (see Kuehn 1979, Shanthikumar and Buzacott 1981, and Whitt 1983).

For any batch at workstation I of line m, the probability that this batch is for product i is [Xij/Qij]/Xjlm. Hence, the workstation may be viewed as an MIGII queueing system where the service time distribution is based on sampling from the deterministic, product specific batch processing times with the selection probabilities noted above.

The expected waiting time for a randomly selected batch at workstation I of line m at plant j is (see Karmarkar, Kekre and Kekre 1983)

T = x E TiKjm(l/Qij +aijmXii/PiPjm)2

where s = 1EBI TiKjIm (2/Qij)[1 - ZiEIjl, (1/Qij +

aijmXij/Pijjm)]. The total costs for the production of product i at

plant j per period are

T C KC(X11/Q1j) + CiX + C(XH T- (8 TCPbij i>vj~bjj j X'j 7ij (8)

where CKf, CP, and Cffy are, respectively, the setup, processing and holding costs (per period) for product i at plant j.

2.4. The Stockpile Inventory Submodel

We assume that a (Q, R) inventory control system is used for the operation of the finished goods stockpile.

222 / COHEN AND LEE

The order size is equal to the production lot size, Qij. Let Rij denote the trigger (reorder) point for product i at plant j, at which point a production request is initiated. Define an order cycle as the time between two consecutive initiations of production requests.

We define the time unit for analysis in this submodel to be a "distribution review period" (e.g., a day). Suppose there are B distribution review periods in the previously defined (production planning) period.

The demand process for finished goods held at the stockpile is generated in the distribution submodel. Letfj(n) be the p.d.f. of demand order sizes from the distribution centers for product i at plant j over a single distribution review period. Note, for now, that fj depends on the network stocking policies and the plant/DC sourcing procedures. Let mij be the mean order size per distribution period.

There are different ways to model the unmet demands from the distribution centers for finished goods held at the stockpile. In this paper, we assume that these demands are met by the expedited shipment after production, of the standard production lot sizes at the plant.

If we assume that consecutive demands from the distribution centers are independent, then the steady- state operating characteristics of the (Q, R) stockpile inventory control system can be found as follows. Let Uij be the undershoot of the inventory level at the time when a production order is initiated for product i at plant j. Let gij(u) be the p.d.f. of Uij. Then, an approximation formula for g1j(u) is (see Silver and Peterson 1985):

gjj(U) = Pr[ Uij = U]

I Z f3(n), u<Rij; {U 1iJn=u+1

1 00 00

l- X fjn), u = R. mi1 u=Rij n=u+1

The expected amount of demand requests from the distribution centers to be expedited for product i at plant j in an order cycle is

Eij = 2 A, [n - (Rij - u)p(n)g1u(u)]. u=O n=Rij-u

The expected inventory level just before an order arrives is

Ri Rij-u-i

Iu = 2 >2 (Rij - u - n)p1J(n)g1J(n). u=O n=O

The achieved fill rate at the stockpile for product i at plant j is denoted by fij and ftj = Qij/(Qij + E1j).

Note that pij(n) should be based on the demand over the lead-time to replenish the finished goods stockpile. We use the demand over the average production lead-time to approximate pij(n).

We assume that each DC k is supplied by a plant j. Let

T'jk = normal replenishment transportation lead-time for product i from plantj to central distribution center k (in distribution review periods);

TEJk = expedited transportation time for product i from plant j to central distribution center k (in distribution review periods).

Observe that the lead-time offered by plants to each distribution is a Bernoulli random variable, e.g., it takes on one of two values with a fixed probability, depending on whether the stockpile is out of stock or not. Hence, the expected replenishment lead-time for product i from plant j to central distribution center k is

TLjk = Tfikfij + (Tv + TJk)(1 I _l'j), (9)

where production lead-time TLJ was defined in (7). The variance of the replenishment lead-time for i

from j to k is

2f(TiLk) = ij(1 -_ #ij)[TjkL -Tf +E

By Little's formula, the average inventory cost of product i tied up in the transportation of product i from plant j to distribution center k (pipeline inventory) is thus

X C(ijktik)B[Tijk + Tijk(1 _

#ij)], k

where

Aik= the mean order quantity (replenishment plus expedite) per distribution review period for product i from distribution center k computed in the distribution module;

and

Ck-k= the unit holding cost per production planning period for finished goods i at plant j enroute to distribution center k.

The above derivation of the pipeline inventory holding cost excludes the production time TiLj to avoid double counting because holding costs for work-in-process inventories were accounted for in the production module.


The total costs related to the finished goods stockpile module per production planning period for product i at plant j are thus

TCFJ = Cf1 (Ij + Q,/2)

+ E Cika1jkMikB[Tljkflj + jk(l _

fij)] (10) k

+ CsXijEij(Qj + E1J).

where Cs is the cost of initiating an expedited production order at plant j for product i. The service constraint for the stockpile is

oij~ : (11)

2.5. The Distribution Submodel

In this section we suppress subscript i for the products, since the methodology applies to all products in a similar way. Where appropriate, we also suppress the location and echelon level subscripts (k and 1). The result of this section draws heavily on the work by Cohen, Kleindorfer and Lee (1985, 1988) and Cohen et al. (1986), and the reader is referred to those references for details of the model development. As noted above, the time unit used in this model is termed the "distribution review period."

The distribution network structure was illustrated in Figure 1. Each stocking facility in echelons below the central distribution centers has a unique node to which it is linked for product resupply. Note, however, that each product may have a different structure to describe its resupply channels.

Consider a particular stocking facility at some echelon level and one particular product type. Demands for the product at this stocking facility may be of three kinds: demands for the product from customers served directly (type D); the normal replenishment demand requests for the product from stocking points that use this facility as their resupply point (type R); and the expedited demand requests from the stocking points that use the facility as their resupply point when they are out of stock (type E). On the other hand, the resupply requests sent out of the facility to its resupply point may also be of two types: normal replenishment requests and emergency shipment requests that arise out of stock situations.

We assume that every location operates under an (s, S) inventory control policy. Once the means and variances of the replenishment and total customer related (type C = D + E) demand requests for facility k are defined, the optimal (s, S) policy for this location can be obtained. This policy minimizes expected ordering plus holding plus shortage costs and satisfies a

minimum fill rate requirement. In the formulation of this paper, we utilize a decision function to specify values for s and S in terms of key system parameters. In particular, we define s and S as functions of K, CE, CR, g(R) c2(R), 8(C), c2(C), CH, (g)L, r2(L), and f, where

K = setup cost for ordering at the stocking facility;

CE = unit incremental cost of not meeting the emergency demand at the facility (i.e., the incremental cost of having to refer demand to a higher echelon);

CR = unit incremental cost of not meeting replenishment demand at k;

C" = unit holding cost of the product held at the facility per distribution review period;

L = lead-time of replenishing stock to the facility from its stocking point at the immediate echelon (in distribution review periods);

= desired service level, expressed as the fill rate, at stocking facility k.

The required values of s and S are computed by a power approximation method which is based on a backorder, single demand class model developed by Ehrhardt (1979, 1981). The selected value for reorder point s is increased until the required fill rate I is met. At the central DC, note that if S - s is larger than Q, the production quantity at the finished goods stockpile, then severe shortages may occur. Hence, it is necessary to constrain S - s at the central DC to no greater than the corresponding Q at the plant.

Once parameters for (s, S) are specified, the distribution-related costs and service levels for each stocking facility can be computed. We utilize a series of approximation functions for these computations based on our earlier multiechelon inventory research (see Cohen, Kleindorfer and Lee 1985a, b).

Define the following random variables.

D= total demand per distribution review period (replenishment + customer-related demands),

DC = customer-related demand (emergency plus direct demand) per distribution review period,

DLT L-fold convolution of D",

DLC = L-fold convolution of DC,

YT U+D LT,

yC= U+DLC,

where U represents the undershoot variable at the

224 / COHEN AND LEE

time an order is triggered at the facility. Let Vt=

E(D). The distribution of U is approximated by

PrIU= u)

|F1- E PrD T=n), u<s

| 1 X E PrIDT= nb, u=s. M u=s n=u+l

Average distribution-related cost per distribution review period is thus approximated by

Y yT _.,+ ~~+ (CH/2) S-s + E(U) + E(YT-s)+ (12)

[S-s+E(U)+2E(s- YT)+].

where 0 = [K + (CE - CR)E(YC - s)+ + CRE( yT- S) +]

The achieved level of service, expressed as the fill rate of the product at facility k of level 1, is given by

1 -E(YT-s)++ U-E(U) = S-s + U+ E(YT s)

where

- 1 p T Var(D) ' ri 2 U

+ -T 1 is an adjustment to account for lost sales. We require

Ok:I ik- (14)

Note that (12) and (13) are derived in Cohen, Kleindorfer and Lee (1988) and can be used to define TC/k and fik for all products i and distribution locations k.

Our approach to solve the problem for the entire distribution network proceeds through the computations just described on an echelon-by-echelon basis. We start at the lowest echelons of the network, and then move up to the higher echelons sequentially. The linkage from echelon I to echelon / + 1 is through the probability distribution of excess demands originating from locations in level / and received as customer- related demands of the appropriate locations in level I + 1. The excess demand depends on the ordering policies in force at locations in level 1. We call these excess demands "pass-up" demands. Similarly, level l locations generate replenishment demand requests which are received by locations in level / + 1. The pass-up emergency shipment requests and the replenishment requests from level / are then combined with the direct local customer demands in level 1.

To find the distributions of the pass-up demands, we utilize the results of Cohen, Kleindorfer, Lee and Tekerian. In this paper explicit formulas were developed to relate pass-up means and variances to the (s, S) parameters, costs and lead-times of the lower level stocking points.

The total cost of the distribution network is obtained by summing the relevant TC T over all locations k. Of course, since the total costs in the other components of the supply chain are defined in terms of the production planning period, the total cost of distribution per production planning period, for product i, is scaled up, and

TC[=BETCT, (15) k

where B is the number of distribution review periods per production planning period. Figure 3 illustrates the data, the decisions, and the outputs of the submodels described in this section.

3. Determination of Operating Policies

For given operating policies (such as lot sizes, reorder points, etc.), the submodels in the previous section can be used to predict cost/service performance of the supply chain. The problem of determining optimal ordering policies is defined in terms of a mathematical program to minimize the sum of costs defined by (3), (8), (10) and (15) subject to the constraints defined by

Material Order Quantities

and Reorder Points

Sutpplier Lead Time I Material F~~~~~radstctiats Fill Rate Targets 0 Material Matepial Productioat Recltiretssent

Bill of Material Titsses C Cost Data Control Canrarl CatDa

Cost Data - > < Prodartion Titlie Data

Pratlstrcints Sstbnsodel Las Size Sstbnadl Fraress Reqstiretsent

LtSbroe Strsctrsre

Prodtrction Prodtrctiont Lot Size Lead Titsses

Tr-altsportation Finished Goods Finisled Titlie Data

Distribution Lead Times Goods 4 Cost Data Detsand Data

Cst Data SStockpile 4 Fill Rate Submodel Demand for Targets

Network Data Fittishled Goods Sabntodel

Fill Rate Tareets

Distribution Stockpile Iltsentorv Orderine Reorder

Policies Points

Figure 3. Relationships among the submodels.


( 14). This section describes a simple, hierarchical heuristic to determine reasonably good operating policies for the various components of the supply chain.

The hierarchical heuristic decomposes the overall problem into subproblems that correspond to the submodels described above. Each subproblem is optimized separately in a given sequence. The outputs of a submodel solution are used as the input data to all other subproblems. The service levels for the individual submodels, as given by (5), (11) and (14), are used to guide the optimal solution procedure in its search for the overall answer to the problem. In other words, the original problem becomes that of finding appropriate service levels for the individual submodels so that the overall objective is optimized.

To determine operating policies at the production submodel, note that production lot sizes affect both material control operations and production lead- times. Such lead-times, in turn, affect the stockpile and distribution submodels. Finally, production lot sizes and reorder points affect the demand process of the material submodel. Truly optimal production lot sizes can only be obtained by optimizing over all of these submodels simultaneously. Unfortunately, the resulting constrained, nonlinear optimization problem is not tractable. The approach used here calls for the determination of approximately optimal lot sizes for both production and material resupply. We obtain approximately optimal lot sizes by searching over a range of lot sizes centered at the Economic Ordering Quality (EOQ) value to be determined below. The lot sizes that give the best performance with respect to a combined (material plus production plus finished goods) cost function are then selected. The system carries out a search for lot size multipliers which are applied uniformly to all products. Reorder points for each item are then computed to meet the service constraint.

Consider first the material stockpile subproblem (minimize (3) subject to (5)). The solution algorithm used for this problem is

(i) Compute V2XrmrjKrj/C', i.e., the EOQ for each r and j.

(ii) Setup an interval of search for multipliers [NI, N2] (based on input specifications).

(iii) Using the Fibonnaci method, select a value of n E [N1, N2] and set Qrj ln EOQ.

(iv) Find the minimum Rrj that satisfies the service level constraint for the selected value of Qrj. If the-total cost TCrj at this value of Rrj is increasing, go to (vi).

(v) Increase Rj from (iv) to find the value that minimizes TCm4.

(vi) Check for cost convergence with respect to pre- specified tolerance. If convergence is achieved, STOP; otherwise, go to step (iii).

Note that this optimization problem is dependent on the values of production lot sizes Q'j and the exoge- nous production period volumes Xij through their impact on the material demand process.

The overall optimization logic solves the material submodel outlined above in the course of a search for optimal Q'j and Rij values (the production order quantities and finished goods stockpile reorder points). The procedure involves solving the material subproblem for each trial solution of the production problem.

To determine the economic lot size for the production submodel, consider the cost function TC' given by (8). We can rewrite (8) as

TCP = Aij/Qij + BijQ1j + Cal,

where

Ai = C:JXij

B1j = Cij Aij Z aim I 1/Pjim, in I

and

C1j = Ad ajm, A (Tfjifn + Tim+ T+ + CijXij. m I

Differentiating w.r.t. Q'j gives

dTCP/dQij =_-A1j/Q2 + Bij,

d2TCJdQ2% 2Aij/Q3, _ 0, for Qij 2 0.

Hence, the Qj that satisfied dTCP/dQij = 0 minimizes TCP. This Q'j is defined by Qij = NIA11/Bj.

The Fibonnaci search routine defined above for the material submodel is also used to search the neigh- borhood of Qij.

To determine the approximately optimal order trigger points R1j at the Finished Goods Stockpile submodel, we use the results from Cohen, Kleindorfer and Lee [1985a, b]. It can be proven that

(i) Eij is decreasing in Rij, (ii) Iij is increasing in Rij.

Hence, the fill rate, #ij = Qij/(Qij + E1j) is also increasing in Rij. An approximate solution is to find the smallest value of Rij that satisfies the fill rate constraint (11). Possible increases beyond ljj that may lead to cost reduction can be explored via direct search methods.

226 / COHEN AND LEE

It is important to observe that the search for the optimal values of (Qi, Rij) is directed toward mini- mizing the sum of material, production and finished goods stockpile costs. The finished goods service constraint must be satisfied. Finally, recall that the material costs for each candidate choice of (Qi1, Rij) are based on the optimal materials inventory control solution (Qrj, Rrj). As noted earlier, optimization of the distribution subproblem (minimize (15) subject to (14) for all products and locations) is approximated by application of explicit decision rules. Structural analysis and algorithmic development for the overall optimization problem are on-going research activities.

4. An Illustrative Example

For illustrative purposes, we examine a problem that consists of two finished products, three raw materials, one plant, two production lines within the plant, and three distribution centers. The distribution review period was one day and the production planning period consisted of 20 days.

The base case requires fill rate service levels of 0.70, 0.75, and 0.70 for the three raw materials, respectively. It also requires a finished goods service level of 0.95 and distribution center service levels of 0.85 and 0.90

for products 1 and 2, respectively. A fully optimized (cost minimum, service feasible) run based on the heuristics of Section 3 was carried out for alternative service levels of the finished goods stockpile. Table I contains a summary output for the base case.

Figure 4 illustrates the relationships among the stockpile of finished goods service level, the cost of distribution, and the cost of manufacturing. In general, increasing the service level of the finished goods stockpile decreases the distribution costs because the lead-time to replenish the distribution network becomes stochastically smaller. On the other hand, increasing the stockpile service level is costly. Based on these two relationships, we can derive the

Table I Cost Components and Finished Goods

Service Level Stockpile Service Level

Costs 0.35 0.55 0.75 0.95 0.99

Materials 1160.58 1160.68 1160.58 1160.58 1159.09 Production 3008.91 3008.91 3008.91 3008.91 3008.91 Finished goods 901.45 902.27 923.76 1014.29 1088.18 Distribution 2523.62 2527.10 2408.31 2182.72 2102.05

Total 7594.56 7598.86 7501.56 7366.50 7358.23

Distribution Costs vs. Service Level 2.60 - 2.60 -

2.50 -2.50 -

2.40- T-

2.30 - 2.30 -

f 2.20 - 2.20 -

2.10 - 2.10 -

2.00 2.00 -r-- 0.20 0.40 0.60 I 0.80 1.00 2.00 2.20 2.A) 2.60

Finished foods service levce (Mhousands) Distsibutioo costs

Manufacturing Costs vs. Service Level Manufacturing vs. Distribution Costs 5.30 - 5.30 - 5.28 5 5.28 - 5.26 5 5.26- 5.24 -5.24-

2 5.22 S 5.22- 5.20 - 5.20

5114 - 5.14- 5.12 - 5.12- 5.10 _ _5.10 - 5.08 - 5.08 - 5(06 5.06 - 504Hi -r-I-- II----1 '- 5.04 $ g _. 0(20 ().4) 05.6) 0)2() (0) 2A(X) 2.20 2.40 2.60)

i'mi~lshed g(xFs s gevic Cosvel (TIrhoesanaos)

Figure 4. Cost/service tradeoffs.


tradeoffs between investments in manufacturing ver- sus distribution.

The model structure developed in this paper can also be used to evaluate alternative strategic options. For example, suppose that the firm is concerned with improved customer service, due to competitive pres- sures. Improvement could be achieved by increasing inventory in the distribution network, or by improving the finished goods availability at the plant stockpile. The first strategy amounts to defining a higher service level for the distribution network. The second strategy specifies a higher service level at the finished goods stockpile. Both options are evaluated by means of a series of runs that use the test problem data.

For the base case, the customer service level and the finished goods stockpile service level were set at 0.8

and 0.6, respectively. If we keep the same level of service for finished goods, but increase the customer service level to 0.95, then an increase in inventory held at the distribution network is necessary. On the other hand, we can also simultaneously increase the service level at the finished goods stockpile. Increasing the service level of the stockpile leads to an increase in the manufacturing and stockpile costs. However, it also lessens the need to increase inventory levels at the distribution network. The results of the various possible combinations of strategic options are given in Table II. Figure 5 illustrates the impact of the options on manufacturing and distribution costs. It can be seen that, for this example, the best option is to improve the service level at the stockpile.

5. Conclusions

The model formulations described in this paper represent an ambitious departure from the standard analytical methods currently used to analyze supply chain inventories. The key innovation lies in the in- tegration of the entire range of inventory subsystems and the associated linkage of decisions and performance measures. Moreover, each submodel in the model structure uses tractable stochastic models.

A software package to support the model structure has been developed. It is designed to translate the model structure into a usable strategy analysis tool. A highly modular system design was used to facilitate debugging, validation and revisions. As noted in our discussions of the various submodels, many assumptions were made in order to render the computations tractable and accessible. Much work remains to be done in the area of validation and efficiency improvement. Additional enhancements associated with alternative model formulations and/or assumptions are also being developed. These enhancements are required if the models and methods introduced here are to have a wide range of applications. In spite of these shortcomings, we feel that the notion of developing a fully integrated supply chain model was successfully explored and was shown to be both feasible and valuable.

Acknowledgment

This research was partially supported by grants ECS- 8406695 and DMC-8609840 of the National Science Foundation. The model formulation and concept development embodied in this paper were substantially assisted by the participation of Messrs. Tom Jones, Michael Webber, Richard Behling and Steve Griffiths

Table II Analysis of Strategic Options

Strategy No. Performance

0 1 2 3 4

Customer service level 0.8 0.95 0.95 0.95 0.95 Finished goods stockpile 0.6 0.6 0.8 0.95 0.99

service level Manufacturing cost 5074 5074 5099 5182 5226 Distribution cost 2314 2836 2676 2470 2346

Total cost 7388 7910 7775 7653 7602

8.10

Strategy

7.90

Strategy 7.80 2

7.70 Manufacturing + Distribution Costs Strate y

3

7.60 Strategy 4

*z 7.50

5.28 Strategy 4

5.20 Strategy 3

Manufacturing Costs

Strategy

5.00 0,7 0.9

Finished Goods Stockpile Service Level

Figure 5. Strategic analysis.

228 / COHEN AND LEE

of Booz, Allen and Hamilton, Inc. System design and programming efforts were carried out by Christopher Jones. Computational assistance was provided by Manuel Baganha and David Pyke. The authors grate- fully acknowledge the helpful comments by two anon- ymous referees.

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