Applying fuzzy mathematical programming approach to optimize a multiple supply network in uncertain condition with comparative analysis
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► Here, we propose a fuzzy mathematical programming model for a supply chain which considers multiple depots, multiple vehicles, multiple products, multiple customers, and different time periods. ► In this work not only demand and cost but also decision variables are considered to be fuzzy. ► We apply two ranking functions for solving the model.
Introduction
Over the last decade or so, supply chain management has emerged as a key area of research among the practitioners of operations research. In today's increasingly global and competitive market, it is imperative that enterprises work together to achieve common goals such as minimizing the delay of deliveries, the holding and the transportation costs [1]. A supply chain can be defined as a network consisting of suppliers, manufacturers, wholesales, distributors, retailers, and customers through which material and products are acquired, transformed, and delivered to consumers in markets [2]. Thus, more and more companies adopt and explore better supply chain management (SCM) to improve the overall efficiency. A successful SCM requires a change from managing individual functions to integrating activities into key supply chain processes.
Owing to the high complexity and uncertainty of the supply chain in industry, a traditional centralized decisional system seems unable to manage easily all the information flows and actions. The decision delay in the supply chain prolongs the process time and causes a company to lose competence. In order to reduce this delay, the supply chain member needs to give quick response. Thus, a supply chain can be characterized as a logistic network of partially autonomous decision-makers. Supply chain management has to do with the coordination of decisions within the network. In the supply chain, ordering decision and inventory decision are two critical decisions supply chain managers have to face. The orders are usually made based on the forecasted customer demand without considering the uncertain factors in industry. Mathematical programming models have proven their usefulness as analytical tools to optimize complex decision-making problems such as those encountered in supply chain planning. After that, a diversity of deterministic mathematical programming models dealing with the design of supply chain networks can be found in the literature. See for example Goetschalckx et al. [3], Geoffrion and Powers [4], Yan et al. [5], and Amiri [6].
Under the most circumstances, the critical design parameters for the supply chain, such as customer demands, prices and resource capacities are generally uncertain. Uncertain supply chain design has been one of the promising subjects. A big amount of stochastic programming models have been proposed for strategic and tactical planning. See for example Owen and Daskin [7], Cheung and Powell [8], Guillén et al. [9], Min and Zhou [10], and Landeghen and Vanmaele [11]. However, in certain situations, the assumption of precise parameters of probability distributions is seriously questioned. The parameters are fixed statistically estimated using past demand information, while demand does not stay ‘static’ in fact. When the conditioning variables, such as the technologic innovations and preferences of consumers, considerably change, the mean and variance of the demand distribution are possible to change. Besides, it is almost impossible to specify exactly the true values to the parameters, especially in the absence of abundant information as in the case of demand of new products. Thus, based on expert experience fuzzy variables are considered to describe them. In this case, random variables with imprecise parameters are random fuzzy variables.
By random fuzzy programming we mean the optimization theory in random fuzzy environment. Random fuzzy programming makes the supply chain design plan more flexible when the parameters of the coefficients’ distribution are uncertain. In practice, the values of the fuzzy parameters can be obtained according to the expert experience. Different numbers of the fuzzy variables reflect different conditions which affect the parameters of the probability distribution. The concept of random fuzzy variables was provided by Liu [12], which is different from the definition used by Nattier [13]. To the best of our knowledge, there is a little research for the programming and solving supply chain design problems in random fuzzy environment.
The remainder of our work is organized as follows. Next, the proposed problem is fully explained and justified. The mathematical formulation is developed considering fuzzy concepts and models in Section 3. The numerical experiments to illustrate the effectiveness of the proposed methodology are given in Section 4. A comparative analysis using regression meta-modelling is presented in Section 5. We conclude in Section 6.
Section snippets
Problem description
The proposed problem of this paper considers different customers that should be serviced with one supplier. The supplier provides various products and keeps them in different depots. The initial problem is choosing the appropriate depots among a set of candidate depots. Each depot uses different types of vehicles to satisfy the orders. All of the depots already stationed at the related locations.
Here, we propose a supply chain which considers multiple depots, multiple vehicles, multiple
The mathematical model
Here, we present a primal framework of the fuzzy variable linear programming (FVLP) to obtain the solution of the linear programming problem with fuzzy cost coefficients. Consider the FVLP problem,
The proposed fuzzy parameters are supposed as trapezoidal fuzzy numbers. A mathematical notation and a configuration of a trapezoidal fuzzy number are shown in Fig. 2.
Numerical experiments
Here, we propose a numerical example to indicate the effectiveness of the proposed fuzzy mathematical model. The number of customers is three, number of products is three, number of candidate depots is seven, and number of vehicles is two. Because of the return of selected vehicles at each period, we must consider additional period in which no demand exists. Then, period four is supposed as additional period. Both of the fuzzy orders and principal orders in different time periods for different
Comparative analysis
Here, we investigate the variation of our decision variables in different iterations obtained from all three methods. First, we consider the quantity of product obtained from Yager, MOM, and the Principal approaches. We depicted the quantity of products for different iterations in Fig. 3. Clearly, the quantity in Yager and Principal methods are closer than MOM.
Here, we survey the number of vehicles obtained as output of the proposed mathematical model. We see in Fig. 4 that the number of
Conclusions
In order to handle the complexity of real life environment and existing constraints to attain useful data, presentation of a deterministic mathematical program is not sufficient in SCM. Therefore, we proposed a fuzzy mathematical program and compared their solutions with each other. In this way, we used ranking function approach to solve such problems. The results of these problems express flexibility of the proposed model to real life environment. Also, comparative analysis was investigated on
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