A manufacturing supply chain optimization model for distilling process

doi:10.1016/j.amc.2005.01.041

Applied Mathematics and Computation

Volume 171, Issue 1, 1 December 2005, Pages 464-485

https://doi.org/10.1016/j.amc.2005.01.041 Get rights and content

Abstract

In this paper, the model of an integrated manufacturing supply chain where multiple products are manufactured across multiple manufacturing plants with distilling process is considered. This kind of supply chain often arises in such manufacturing scenarios where the products are distilled from one raw material. To solve the problem, we reformulate it as a minimum cost flow problem plus several bounded variables. Based on this reformulation, we show that the basis of the reformulated problem is closely related with the minimum cost flow problem and design a kind of network simplex method to get the integrated optimal solution of the problem. The efficiency of the method is also tested by our numerical experiments.

Introduction

A supply chain is an integrated network of suppliers, manufacturing plants and distributing channels which are organized to acquire raw materials, to convert those raw materials to final products, and then to distribute those products to customers. In the supply chain, there are several organizations which are responsible for purchasing and manufacturing the raw materials, and distributing the final production, respectively. Although these organizations are owned by a single firm, they always operate independently and sometimes they may conflict. That is, there may not be a single and integrated plan for the firm. Based on this observation, we attempt to present an integrated model of production–distribution system with distilling process in this paper.

To our knowledge, a large number of manufacturing models have been proposed for the design and planning supply chain network, see [1], [2], [3], [4] and the papers cited therein. In [5], Cohen and Lee developed a comprehensive modelling framework which links material management activities throughout the material production–distribution supply chain, in which the framework consists of four stochastic sub-models and the optimal solution for each sub-model is solved individually under certain assumptions. However, it would be hard to find an optimal solution if all sub-models are integrated. Later, Cohen et al. [6] considered the operation of a network of supplier, producers and markets, which includes material requirement balance constraints. Cohen and Lee [7] presented a simplified version of the model in [5].

Recently, more complicated manufacturing network is considered, such as the synthesis of different materials to one product and/or the distilling of one material to many different products. To model this manufacturing scenarios, Fang and Qi [9] described a generalized network model which consists of the operation of a network of suppliers, producers and customers. In this model, the producer has two production modes: distillation and combination. By using a simplified version of the model which is called distribution network, the authors of [9] proposed a network simplex method to solve the problem.

In this paper, we consider a manufacturing supply chain model with distilling process. The model consists of several sub-networks which deal with the raw material and productions separately. Motivated by the work of Ahuja et al. [10] and Calvete [11], we will give a new method to solve the model so that the optimal solution of integrated all sub-networks can be found efficiently. Using a similar technique, we reformulate the model as a minimum cost problem plus several bounded variables. A modified network simplex method which exploits the special structure of basis is presented and we can perform the computations on the network.

The rest of this paper is organized as follows. In Section 2, we would give descriptions of the manufacturing supply chain with distilling process and present its mathematics formulation; in Section 3, we reformulate the problem as a minimum cost flow problem plus several bounded variables and characterizes some the properties; in Section 4, we discuss the structure of basis; in Section 5, we would give a network simplex method to solve this supply chain model, and in the last section, we would give a numerical experiment to the proposed algorithm.

Section snippets

A manufacturing supply chain

We now consider the following manufacturing scenarios: A firm has p manufacture plants that produces q products. The production process starts with the supply of raw material, the raw material are distilled into q products in the next step of the process, and distribution and sales are done in the final step. Our aim is to make an integrated decision which minimizes the total cost of raw material, production and distribution.

For simplify, in network $G$ with node set $N$ and arc set $A$ , the set of

Problem reformulation and properties

In this section, we first give a reformulation of the problem (3) and then present a result which plays an important role in our algorithm.

For simplicity, we give some notation. Without loss of generality, we assume that $M = {1, 2, \dots, p}$ , and $N_{k}$ , the node set of network $G_{k}$ , contains nodes {n_k + 1, … , n_k+1}, k = 0, 1, … , q and 0 = n₀ < p < n₁ < ⋯ < n_q+1 = n.

For l = 1, 2, … , p, if $i \in N_{0}$ , let $β_{i}^{l} = \{\begin{matrix} 1, & if i \in M and i = l, \\ - α_{l}^{k}, & if i = l_{k} and k = 1, 2, \dots, q, \\ 0, & otherwise . \end{matrix}$ It follows that $\sum_{i \in N_{0}} β_{i}^{l} = 1, \sum_{i \in N_{k}} β_{i}^{l} = - α_{l}^{k} l = 1, \dots, p and k = 1, \dots, q,$ and from (5), (6), we obtain $\sum$

Structure of the basis

According to the theorem in last section, problem (8) is a linear programming problem with bounded variables whose coefficient matrix has rank n − q. Therefore, basic feasible solutions are composed by n − q basic variables and the rest of the variables are fixed at their lower or upper bound.

Let us consider that v variables in {x₁, … , x_p} are basic, 1 ⩽ v ⩽ p. For simplicity, we assume that these variables are x₁, … , x_v. Then, to get a basic solution, we should select n − q − v variables x_ij, $(i, j) \in \tilde{A}$ , whose

Network simplex algorithm

In this section, we first give a detailed description of the main steps and then present the network simplex algorithm for (8).

Numerical experiments

In order to illustrate Algorithm 1, let us consider the example presented in Fig. 1 with p = 3 and q = 2. The number adjacent to any node i denotes b_i. The arc costs c_ij and capacities u_ij are shown next to the arcs. Suppose that $α_{1}^{1} = α_{2}^{1} = α_{3}^{1} = 0.7, α_{1}^{2} = α_{2}^{2} = α_{3}^{2} = 0.3$ .

In the network, $N_{0} = {1, \dots, 10}, N_{1} = {11, \dots, 15}, N_{2} = {16, \dots, 20}$ and $A_{0} = {(4, 5), (4, 8), (5, 6), (5, 8), (5, 10), (6, 9), (6, 10), (7, 6), (7, 10), (8, 1), (9, 2), (10, 3)}, A_{1} = {(11, 12), (11, 13), (11, 15), (12, 15), (13, 14), (14, 11), (14, 12), (15, 14)}, A_{2} = {(16, 17), (16, 18), (16, 19), (17, 18$

Concluding remarks

In this paper, we present a manufacturing supply chain model for distillation processing, and we extend previous work on a general equal flow problem with additional side constraints requiring the flow of arcs in some given sets of arcs to take on the some value. The proposed approach is a network-based approach which allows us to benefit from the practical computational advantages of network models. Based on the new reformulation of the problem, the bases of the problem are characterized as

References (11)

H.I. Calvete
Network simplex algorithm for the general equal flow problem
European Journal of Operational Research
(2003)
R.G. Askin et al.
Modeling and analysis of manufacturing system
(1993)
A. Gunasekaren et al.
Modeling and analysis of supply chain management systems: an editorial overview
Journal of Operational Research Society
(2000)
V.A. Marbert et al.
Special research focus on supply chain linkages: challenges for design and management in the 21st Century
Decision Science
(1998)
S.S. Erengue et al.
Integrated production distribution planning in supply chain: an invited review
European Journal of Operation Research
(1999)

There are more references available in the full text version of this article.

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In a companion paper [Dabbene F; Gay P; Sacco N (2008). Optimisation of Fresh-Food Supply Chains in Uncertain Environments, Part II: a Case Study. Biosystems Engineering, accepted], this new methodology is applied to a real-world example concerning meat refrigeration.
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^☆: Supported by the Research Grant Council of Hong Kong, Australia Research Council, and the Natural Science Foundation of Guangxi (No. 0135004).

¹: This work was done when the author visited Department of Mathematics and Statistics, Curtin University of Technology, Perth, Australia.

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A manufacturing supply chain optimization model for distilling process☆

Abstract

Introduction

Section snippets

A manufacturing supply chain

Problem reformulation and properties

Structure of the basis

Network simplex algorithm

Numerical experiments

Concluding remarks

European Journal of Operational Research

Modeling and analysis of manufacturing system

Modeling and analysis of supply chain management systems: an editorial overview

Journal of Operational Research Society

Special research focus on supply chain linkages: challenges for design and management in the 21st Century

Decision Science

Integrated production distribution planning in supply chain: an invited review

European Journal of Operation Research