# Mathematical Model for Designing Supply Chains

Jairo R. Montoya-Torres (Universidad de La Sabana, Colombia)
DOI: 10.4018/978-1-4666-5202-6.ch135

## Abstract

Supply Chain Management (SCM) is the process of planning, implementing and controlling the operations of the supply chain in an effective way. Part of the planning process consists of finding the best possible chain configuration. For manufacturing firms, a production-distribution problem consists of defining the flow of products from plants to clients (markets) via a set of warehouses. Defining the location of such warehouses is also part of the decision-making problem. This problem is known in the literature as the two-echelon uncapacitated facility location problem (TUFLP). This chapter analyses the performance of a mathematical model that minimizes the total cost of warehouse location and production and distribution. This problem is studied and is solved using optimization methods.
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## Introduction

According to well-known definitions available in the literature (Simchi-Levy, Kaminsky, & Simchi-Levy, 2004; Melo, Nickel, & Saldanha-da-Gama, 2009), Supply Chain Management (SCM) is “the process of planning, implementing and controlling the operations of the supply chain in an effective way. SCM spans all movements and storage of raw material, work-in-process inventory, and finished goods from the point-of-origin to the point-of-consumption.” Part of the planning process in SCM aims to find the best possible configuration of the supply chain. In addition to the generic facility location problem, other areas (e.g., procurement, production, inventory, distribution, routing) has to be considered as well (Cordeau, Pasin, & Solomon, 2006). These areas can be considered either as an integrated decision-making model or as independent, yet interconnected, models. The former approach drives to a very complex model, while the latter is easier to understand and to implement in real practice, yet requiring consistency between solutions of each decision level. When applying a hierarchical decision-making approach, three decision-making levels are usually distinguished depending on the time horizons: strategic, tactical and operational (Vidal & Goetschalckx, 1997; Bender, Hennes, Kalcsics, Melo, & Nickel, 2002). In this chapter, we focus on the strategic level. It deals with decisions that have a long-lasting effect on the firm such as the number, location and capacities of warehouses and manufacturing plants (Simchi-Levi et al., 2004). In strategic decisions due to the large amounts of investments, facilities that are located now are expected to operate for a long-term horizon (Melo et al., 2009). The design of the distribution network and the definition of material flows has become a major challenge for organizations as they simultaneously aim to decrease costs and increase responsiveness in today’s competitive market. In this regard, operational aspects such as lead times and material flows have been identified as issues which are best treated by considering them once the network design is established.

In this chapter, we consider a decision-making problem related to simultaneously define the location of facilities (warehouses) and the flow of products through the supply network. When only two stages (manufacturing plants and warehouses) are taken into account and warehouses are considered to be of infinite capacity, this problem is known in the literature as the Two-Echelon Uncapacitated Facility Location Problem (TUFLP). This problem belongs to the family of multi-item (or multiple products) production-distribution problems, which is one of the most important problems in Facility Location Theory (Daskin, 1995). In a general case, the problem is defined as follows. A firm may have relatively few products and a number of plants. Products are shipped from plants to markets (or customers) via a set of warehouses. The key issues we are concerned with are:

• 1.

How many warehouses to have,

• 2.

Where to locate warehouses, and

• 3.

How the products should flow through the system.

Implicit in the product flow decision are other decisions about which products should be produced at which plants for which markets. Figure 1 is a schematic of such a system.

Figure 1.

Representation of the TUFLP

The problem can be formalized using mathematical programming, but it is known to be NP-hard (Cornuejols, Nemhauser, & Wolsey, 1990), which means that optimal solutions for large-sized instances are not easy to find in reasonable amount of time. This chapter presents a well-known mathematical model based on mixed integer linear programming. The reader must note that the NP-hardness of this problem suggests the use of approximate algorithms to obtain efficient solutions. Some heuristics and meta-heuristics algorithms are also studied in this book (Montoya-Torres, 2013).

## Key Terms in this Chapter

Supply Chain Design: The activity of determine the configuration of a supply system.

Computational Time: Time required by a computer to perform a given set of computations.

Exact Solution Procedure: A formal solution algorithm that explores the complete space of possible solutions for a given problem allowing finding the best (the optimum) one. An exact procedure is mainly based on mathematical programming or branch-and-bound strategies.

Supply Chain: A system of organizations, people, technology, activities, information and resources involved in moving products or services from suppliers to customers or clients. The activities of such a system intends at transforming some materials and components into finished products or services.

Facility Location: A decision making process consisting on determining the optimal, or at least good locations of facilities, such as manufacturing plants, warehouses, distribution centers, etc., in a supply chain or distribution system.

Production-Distribution Problem: A decision making problem in logistics or supply chain management related with the optimal solution of both production (manufacturing) and distribution (transportation), simultaneously.

Approximate Algorithm: A solution procedure (algorithm) that gives a solution to a given problem that is close to optimum. In general, computational time of approximate algorithm is lower than exact solution procedures.

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