Supply Chain Management (SCM) is the process of planning, implementing and controlling the operations of the Supply Chain (SC) in an efficient way. SCM spans all movements and storage of raw materials, work-in-process inventory, and finished goods from the point-of-origin to the point-of-consumption [1]. Production and distribution operations are two key business functions in SCM. To achieve optimal operational performance in a SC, it is critical to integrate these two functions and schedule them jointly in a coordinated manner. For the most part, these two functions have been studied independently of each other leading to globally sub-optimal decisions [2].

A critical review carried out, measures the relationships between SC integration and performance: a high level of integration has a positive impact on corporate and SC performance. So, in this paper an integrated production-distribution model consisting multi periods and multi products and considering a plant, distribution centers (DCs) and customer zones is presented.

Whereas, SCs are generally complex and are described by numerous activities spread over multiple functions and organizations, which raise interesting challenges for effective SC coordination and as regards SC, is a dynamic network of several business entities that involve a high degree of imprecision. Most studies have focused on traditional analytical and heuristic methods to model the SC network planning problem.

A few studies have attempted to model the problem in an uncertainty environment. Likewise, stochastic models in such conditions may not lead to fully satisfactory results, so using fuzzy models allows us to remove this drawback [3]. However, the majority of existing fuzzy models consider only separate aggregate production planning without taking into account the interdependent nature of production and distribution systems. This limited approach often leads to inadequate results. This is mainly due to its real-world character, where uncertainties in activities extending from the suppliers to the customers make SC imprecise. Therefore fuzzy set theory is a suitable tool to come up with such a complicated system [4].

In this paper, we present a model formulation for application in the consumer goods industry which integrates the production and distribution plans taking into account the fuzziness.

This model is used to optimize the quantity of each product produced in plant and inventory level of products in each DC, transportations between plants, DCs and customer zones when costs are available in fuzzy format.

The rest of this paper is organized as follows. Section 2 presents a literature review about fuzzy applications in production-distribution planning. Section 3 is devoted to problem formulation in deterministic condition and we review briefly fuzzy set theory. The fuzzy mixed-integer programming (FMIP) model is proposed in Section 4. In Section 5, the behavior of the model is evaluated. This is followed by some concluding remarks is Section 6.

The production and distribution systems have been considered the main processes in SC. These systems traditionally were viewed as different functions in the firms and usually optimized seperately or sequentially. In two recent decades, an integrated view of production and distribution systems have received considerable attention regarding its ability to help enterprises to make effective tactical and operational decisions and to compete effectively in the marketplace by optimizing different functions simultaneously. Many researchers, [5-10], provide goog review of the literature related to the integrated analysis of the production and distribution systems in supply chan. In this way, Lee and Kim [11] develop a hybrid simulation-analytic approach to solve an integrated production-distribution planning problem in SC. Review an industrial example and present a mixed integer program model to deal with the integrated production-distribution problem. [12] investigate the differences between the values of coordinting production and distribution planning and those of production scheduling and distribution scheduling solved separately. [13] used a new solution procedure for an integrated production-distribution planning problem consistiong Lagrangian Relaxation and a heuristic algorithm. Lee and Kim [14] and Lee and Kim [11] propose a hybrid approach that combines analytic and simulation method for production–distribution planning in SC.

By seeking the literature of SC, limited number of research that use fuzzy set theory is found. Petrovic et al. [15] developed a fuzzy generative SC model to determine target order-up-to levels of inventories under uncertain demand and external supply of raw materials. Giannoccaro et al. [16] presented a methodology to define a three-stage SC inventory management policy which the echelon stock concept was adopted to manage the SC inventory in an integrated manner, whereas fuzzy set theory was used to properly model the uncertainty associated with both market and inventory costs. Xie et al. [17] presented a new hierarchical, two-level approach to inventory management and control in SCs. They supposed that the SC operates under uncertainty in customer demand, which is described by imprecise terms and modelled by fuzzy sets. Chen et al. [18] present a fuzzy decision-making approach to deal with the supplier selection problem in SC system. Chang et al. [19] propose a fuzzy multiple attribute decision making method based on the fuzzy linguistic quantifier to select SC artners at different phases of product life cycle. [4] presented Five crisp and fuzzy models for supp SC of an automotive manufacturing system. Aliev et al. [3] developed a fuzzy integrated multi-period and multi-product production and distribution model in SC. The model is formulated in terms of fuzzy programming and the solution is provided by genetic optimization. Xu et al. [20] proposed a random fuzzy programming model and a methodology for solving a multi-stage SC design problem of a realistic scale in the random fuzzy environment. Based on the concept of random fuzzy variable. They used probability theory, fuzzy set theory and optimizing theory as their research tool. They presented novel technique called spanning-tree based on genetic algorithm to get the heuristic solutions. Selim et al. [21] asserted that fuzzy goal programming approaches can effectively be used in handling the collaborative production–distribution planning problems in both centralized and decentralized SC structures. Liang and Ceng [22] presented a systematic framework that facilitates fuzzy decision-making for solving the multi-objective manufacturing/distribution planning decision problems with multi-product and multi-time period in SC s under an uncertain environment, enabling the decision maker to adjust the search direction during the solution procedure to obtain a preferred satisfactory solution. Bilgen [23] provided a model formulation for application in the consumer goods industry consisting of multiple manufacturers, multiple production lines and multiple (DCs) which integrates the production and distribution plans taking into account the fuzziness in the available capacity restrictions and the aspiration level of costs. Mula et al. [24] applied a fuzzy approach to a SC production planning problem with lack of knowledge in demand data. They studied an application of known possibilistic programming in a SC planning case study. Liang [22] proposed a fuzzy mathematical programming methodology for solving manufacturing/distribution planning decision integration problems attempting to minimize the total manufacturing and distribution costs by considering the levels of inventory, subcontracting and backordering, the available machine capacity and labor levels at each source, forecast demand and available warehouse space at each destination.

In this section we review some basic concepts in SCM models and fuzzy sets.

Let are triangular fuzzy numbers the model changes as follows:

Min Z:

Subject to The basic idea in this approach is to transform the fuzzy SCM model into a crisp linear programming problem by applying an alternative α-cut approach. This approach assumes that the solution lies in the interval and defines suitable variables for this solution. This method used α-cut approach so it does not retain the uncertainty in formation completely. In other words, each fuzzy number is converted to an interval with the same membership in the entire interval. We propose an alternative fuzziness. The advantage of the new approach is that the uncertainty concept will be kept throughout the computation.

In this part, in order to demonstrate the efficiency of the proposed model a numerical example with 3 (DCs), 2 products, 3 periods and 5 customer zones is generated accordance with the guidelines in the literature. The model is coded in LINGO 8.0 Software on a PC including two Intel® CoreTM2 and 2 GB RAM. Experiment results are indicated in Table 1 and Table 2.

SC design problems have recently raised a lot of interest since the opportunity of an integrated management of the SC can reduce the propagation of undesirable events through the network and can affect decisively the profitability of the members. Conventional SCM models deals with crisp or accurate data, however in real world problems accurate data may not be available. In many situations the data available are imprecise like fuzzy numbers. In the existing models uncertainties of the fuzzy data are effectively ignored through the computation. In this paper costs are considered as triangular fuzzy numbers and this paper proposes an alternative approach to retain fuzziness of the model by maximizing the membership functions of fuzzy data. The value of the objective function is equivalent to α-cut of interval programming problem with maximum possible value of the objective function.

For further studies, it is suggested to explore: 1) reducing the size of the converted (crisp equivalent) problem, and 2) possible linearization of the nonlinear model.

The authors declare no conflict of interest.

All data are included in the text.

This research received no specific grant from funding agencies in the public, commercial, or not-for-profit sectors.