Abstract
For years, researchers and practitioners have primarily investigated the various processes within manufacturing supply chains individually. Recently, however, there has been increasing attention placed on the performance, design, and analysis of the supply chain as a whole. This attention is largely a result of the rising costs of manufacturing, the shrinking resources of manufacturing bases, shortened product life cycles, the leveling of the playing field within manufacturing, and the globalization of market economies. The objectives of this paper are to: (1) provide a focused review of literature in multi-stage supply chain modeling and (2) define a research agenda for future research in this area.
Keywords
supply chain, production, distribution, logistics
1 Introduction
A supply chain may be defined as an integrated process wherein a number of various business entities (i.e., suppliers, manufacturers, distributors, and retailers) work together in an effort to: (1) acquire raw materials, (2) convert these raw materials into specified final products, and (3) deliver these final products to retailers. This chain is traditionally characterized by a forward flow of materials and a backward flow of information. For years, researchers and practitioners have primarily investigated the various processes of the supply chain individually. Recently, however, there has been increasing attention placed on the performance, design, and analysis of the supply chain as a whole. From a practical standpoint, the supply chain concept arose from a number of changes in the manufacturing environment, including the rising costs of manufacturing, the shrinking resources of manufacturing bases, shortened product life cycles, the leveling of the playing field within manufacturing, and the globalization of market economies. The current interest has sought to extend the traditional supply chain to include reverse logistics, to include product recovery for the purposes of recycling, re-manufacturing, and re-use. Within manufacturing research, the supply chain concept grew largely out of two-stage multi-echelon inventory models, and it is important to note that considerable progress has been made in the design and analysis of two-echelon systems. Most of the research in this area is based on the classic work of Clark and Scarf (1960) and Clark and Scarf (1962). The interested reader is referred to Federgruen (1993) and Bhatnagar, et. al. (1993) for comprehensive reviews of models of this type. More recent discussions of two-echelon models may be found in Diks, et. al. (1996) and van Houtum, et. al. (1996). The objectives of this paper are to: (1) provide a focused review of literature in the area of multi-stage supply chain design and analysis, and (2) develop a research agenda that may serve as a basis for future supply chain research.
2 The Supply Chain Defined
As mentioned above, a supply chain is an integrated manufacturing process wherein raw materials are converted into final products, then delivered to customers. At its highest level, a supply chain is comprised of two basic, integrated processes: (1) the Production Planning and Inventory Control Process, and (2) the Distribution and Logistics Process. These Processes, illustrated below in Figure 1, provide the basic framework for the conversion and movement of raw materials into final products.

and Inventory Control
Figure 1. The Supply Chain Process
The Production Planning and Inventory Control Process encompasses the manufacturing and storage sub-processes, and their interface(s). More specifically, production planning describes the design and management of the entire manufacturing process (including raw material scheduling and acquisition, manufacturing process design and scheduling, and material handling design and control). Inventory control describes the design and management of the storage policies and procedures for raw materials, work-in-process inventories, and usually, final products.
The Distribution and Logistics Process determines how products are retrieved and transported from the warehouse to retailers. These products may be transported to retailers directly, or may first be moved to distribution facilities, which, in turn, transport products to retailers. This process includes the management of inventory retrieval, transportation, and final product delivery.
These processes interact with one another to produce an integrated supply chain. The design and management of these processes determine the extent to which the supply chain works as a unit to meet required performance objectives.
3 Literature Review
The supply chain in Figure 1 consists of five stages. Generally, multi-stage models for supply chain design and analysis can be divided into four categories, by modelling approach. In the cases included here, the modelling approach is driven by the nature of the inputs and the objective of the study. The four categories are: (1) deterministic analytical models, in which the variables are known and specified (2) stochastic analytical models, where at least one of the variables is unknown, and is assumed to follow a particular probability distribution, (3) economic models, and (4) simulation models.
3.1 Deterministic Analytical Models
Williams (1981) presents seven heuristic algorithms for scheduling production and distribution operations in an assembly supply chain network (i.e., each station has at most one immediate successor, but any number of immediate predecessors). The objective of each heuristic is to determine a minimum-cost production and/or product distribution schedule that satisfies final product demand. The total cost is a sum of average inventory holding and fixed (ordering, delivery, or set-up) costs. Finally, the performance of each heuristic is compared using a wide range of empirical experiments, and recommendations are made on the bases of solution quality and network structure.
Williams (1983) develops a dynamic programming algorithm for simultaneously determining the production and distribution batch sizes at each node within a supply chain network. As in Williams (1981), it is assumed that the production process is an assembly process. The objective of the heuristic is to minimize the average cost per period over an infinite horizon, where the average cost is a function of processing costs and inventory holding costs for each node in the network.
Ishii, et. al (1988) develop a deterministic model for determining the base stock levels and lead times associated with the lowest cost solution for an integrated supply chain on a finite horizon. The stock levels and lead times are determined in such a way as to prevent stockout, and to minimize the amount of obsolete (dead) inventory at each stock point. Their model utilizes a pull-type ordering system which is driven by, in this case, linear (and known) demand processes.
Cohen and Lee (1989) present a deterministic, mixed integer, non-linear mathematical programming model, based on economic order quantity (EOQ) techniques, to develop what the authors refer to as a global resource deployment policy. More specifically, the objective function used in their model maximizes the total after-tax profit for the manufacturing facilities and distribution centers (total revenue less total before-tax costs less taxes due). This objective function is subject to a number of constraints, including managerial constraints (resource and production constraints) and logical consistency constraints (feasibility, availability, demand limits, and variable non-negativity). The outputs resulting from their model include [12]:
• Assignments for finished products and subassemblies to manufacturing plants, vendors to distribution centers, distribution centers to market regions.
• Amounts of components, subassemblies, and final products to be shipped among the vendors, manufacturing facilities, and distribution centers.
• Amounts of components, subassemblies, and final products to be manufactured at the manufacturing facilities.
Moreover, this model develops material requirements and assignments for all products, while maximizing after-tax profits.
Cohen and Moon (1990) extend Cohen and Lee (1989) by developing a constrained optimization model, called PILOT, to investigate the effects of various parameters on supply chain cost, and consider the additional problem of determining which manufacturing facilities and distribution centers should be open. More specifically, the authors consider a supply chain consisting of raw material suppliers, manufacturing facilities, distribution centers, and retailers. This system produces final products and intermediate products, using various types of raw materials. Using this particular system, the PILOT model accepts as input various production and transportation costs, and consequently outputs:
• Which of the available manufacturing facilities and distribution centers should be open.
• Raw material and intermediate order quantities for vendors and manufacturing facilities.
• Production quantities by product by manufacturing facility.
• Product-specific shipping quantities from manufacturing facility to distribution center to customer.
The objective function of the PILOT model is a cost function, consisting of fixed and variable production and transportation costs, subject to supply, capacity, assignment, demand, and raw material requirement constraints. Based on the results of their example supply chain system, the authors conclude that there are a number of factors that may dominate supply chain costs under a variety of situations, and that transportation costs play a significant role in the overall costs of supply chain operations.
Newhart, et. al. (1993) design an optimal supply chain using a two-phase approach. The first phase is a combination mathematical program and heuristic model, with the objective of minimizing the number of distinct product types held in inventory throughout the supply chain. This is accomplished by consolidating substitutable product types into single SKUs. The second phase is a spreadsheet-based inventory model, which determines the minimum amount of safety stock required to absorb demand and lead time fluctuations. The authors considered four facility location alternatives for the placement of the various facilities within the supply chain. The next step is to calculate the amount of inventory investment under each alternative, given a set of demand requirements, and then select the minimum cost alternative.
Arntzen, et. al. (1995) develop a mixed integer programming model, called GSCM
( Global Supply Chain Model), that can accommodate multiple products, facilities, stages
( echelons), time periods, and transportation modes. More specifically, the GSCM minimizes a composite function of: (1) activity days and (2) total (fixed and variable) cost of production, inventory, material handling, overhead, and transportation costs. The model requires, as input, bills of materials, demand volumes, costs and taxes, and activity day requirements and provides, as output: (1) the number and location of distribution centers, (2) the customer-distribution center assignment, (3) the number of echelons (amount of vertical integration), and (4) the product-plant assignment.
Voudouris (1996) develops a mathematical model designed to improve efficiency and responsiveness in a supply chain. The model maximizes system flexibility, as measured by the time-based sum of instantaneous differences between the capacities and utilizations of two types of resources: inventory resources and activity resources. Inventory resources are resources directly associated with the amount of inventory held; activity resources, then, are resources that are required to maintain material flow. The model requires, as input, product-based resource consumption data and bill-of-material information, and generates, as output: (1) a production, shipping, and delivery schedule for each product and (2) target inventory levels for each product.
Camm, et. al. (1997) develop an integer programming model, based on an uncapacitated facility location formulation, for Procter and Gamble Company. The purpose of the model is to: (1) determine the location of distribution centers (DCs) and (2) assign those selected DCs to customer zones. The objective function of the model minimizes the total cost of the DC location selection and the DC-customer assignment, subject to constraints governing DC-customer assignments and the maximum number of DCs allowed.
3.2 Stochastic Analytical Models
Cohen and Lee (1988) develop a model for establishing a material requirements policy for all materials for every stage in the supply chain production system. In this work, the authors use four different cost-based sub-models (there is one stochastic sub-model for each production stage considered). Each of these sub-models is listed and described below [12]:
(1) Material Control: Establishes material ordering quantities, reorder intervals, and estimated response times for all supply chain facilities, given lead times, fill rates, bills of material, cost data, and production requirements.
(2) Production Control : Determines production lot sizes and lead times for each product, given material response times.
(3) Finished Goods Stockpile (Warehouse): Determines the economic order size and quantity for each product, using cost data, fill rate objectives, production lead times, and demand data.
(4) Distribution: Establishes inventory ordering policies for each distribution facility, based on transportation time requirements, demand data, cost data, network data, and fill rate objectives.
Each of these sub-models is based on a minimum-cost objective. In the final computational step, the authors determine approximate optimal ordering policies using a mathematical program, which minimizes the total sum of the costs for each of the four sub-models.
Svoronos and Zipkin (1991) consider multi-echelon, distribution-type supply chain systems
( i.e., each facility has at most one direct predecessor, but any number of direct successors). In this research, the authors assume a base stock, one-for-one ( S-1, S ) replenishment policy for each facility, and that demands for each facility follow an independent Poisson process. The authors obtain steady-state approximations for the average inventory level and average number of outstanding backorders at each location for any choice of base stock level. Finally, using these approximations, the authors propose the construction of an optimization model that determines the minimum-cost base stock level.
Lee and Billington (1993) develop a heuristic stochastic model for managing material flows on a site-by-site basis. Specifically, the authors model a pull-type, periodic, orderup-to inventory system, and determine the review period (by product type) and the orderup-to quantity (by product type) as model outputs. The authors develop a model which will either: (1) determine the material ordering policy by calculating the required stock levels to achieve a given target service level for each product at each facility or (2) determine the service level for each product at each facility, given a material ordering policy.
Lee, et. al. (1993), develop a stochastic, periodic-review, order-up-to inventory model to develop a procedure for process localization in the supply chain. That is, the authors propose an approach to operational and delivery processes that consider differences in target market structures (e.g., differences in language, environment, or governments). Thus, the objective of this research is to design the product and production processes that are suitable for different market segments that result in the lowest cost and highest customer service levels overall.
Pyke and Cohen (1993) develop a mathematical programming model for an integrated supply chain, using stochastic sub-models to calculate the values of the included random variables included in the mathematical program. The authors consider a three-level supply chain, consisting of one product, one manufacturing facility, one warehousing facility, and one retailer. The model minimizes total cost, subject to a service level constraint, and holds the set-up times, processing times, and replenishment lead times constant. The model yields the approximate economic (minimum cost) reorder interval, replenishment batch sizes, and the order-up-to product levels (for the retailer) for a particular production network.
Pyke and Cohen (1994) follow the Pyke and Cohen (1993) research by including a more complicated production network. In Pyke and Cohen (1994), the authors again consider an integrated supply chain with one manufacturing facility, one warehouse, and one retailer, but now consider multiple product types. The new model yields similar outputs; however, it determines the key decision variables for each product type. More specifically, this model yields the approximate economic (minimum cost) reorder interval (for each product type), replenishment batch sizes (for each product type), and the orderup-to product levels (for the retailer, for each product type) for a particular supply chain network.
Tzafestas and Kapsiotis (1994) utilize a deterministic mathematical programming approach to optimize a supply chain, then use simulation techniques to analyze a numerical example of their optimization model. In this work, the authors perform the optimization under three different scenarios [39]:
(1) Manufacturing Facility Optimization: Under this scenario, the objective is to minimize the total cost incurred by the manufacturing facility only; the costs experienced by other facilities is ignored.
(2) Global Supply Chain Optimization: This scenario assumes a cooperative relationship among all stages of the supply chain, and therefore minimizes the total operational costs of the chain as a whole.
(3) Decentralized Optimization: This scenario optimizes each of the supply chain components individually, and thus minimizes the cost experienced by each level.
The authors observe that for their chosen example, the differences in total costs among the three scenarios are very close.
Towill and Del Vecchio (1994) consider the application of filter theory and simulation to the study of supply chains. In their research, the authors compare filter characteristics of supply chains to analyze various supply chain responses to randomness in the demand pattern. These responses are then compared using simulation, in order to specify the minimum safety stock requirements that achieve a particular desired service level.
Lee and Feitzinger (1995) develop an analytical model to analyze product configuration for postponement (i.e., determining the optimal production step for product differentiation), assuming stochastic product demands. The authors assume a manufacturing process with I production steps that may be performed at a factory or at one of the M distribution centers (DCs). The problem is to determine a step P such that steps 1 through P will be performed at the factory and steps (P+1) to I will be performed at the DCs. The authors solve this problem by calculating an expected cost for the various product configurations, as a sum of inventory, freight, customs, setup, and processing costs. The optimal value of P is the one that minimizes the sum of these costs.
Altiok and Ranjan (1995) consider a generalized production/inventory system with: M (M > 1) stages (j = 1,M) , one type of final product, random processing times (FIFO, for all stages) and set-up times, and intermediate buffers. The system experiences demand for finished products according to a compound Poisson process, and the inventory levels for inventories (intermediate buffers and finished goods) are controlled according to a continuous review (R,r) inventory policy, and backorders are allowed. The authors develop an iterative procedure wherein each of the two-node sub-systems are analyzed individually; the procedure terminates once the estimate average throughput for each sub-system are all approximately equal. Once the termination condition is met, the procedure allows for calculation of approximate values for the two performance measures: (1) the inventory levels in each buffer j, and (2) the backorder probability. The authors conclude that their approximation is acceptable as long as the P(backorder) does not exceed 0.30, in which cases the system is failing to effectively accommodate demand volumes.
Finally, Lee, et. al. (1997) develop stochastic mathematical models describing The Bullwhip Effect, which is defined as the phenomenon in which the variance of buyer demand becomes increasingly amplified and distorted at each echelon upwards throughout the supply chain. That is, the actual variance and magnitude of the orders at each echelon is increasingly higher than the variance and magnitude of the sales, and that this phenomenon propagates upstream within the chain. In this research, the authors develop stochastic analytical models describing the four causes of the bullwhip effect (demand signal processing, rationing game, order batching, and price variations), and show how these causes contribute to the effect.