Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 4389064, 29 pages

https://doi.org/10.1155/2017/4389064

## Optimization of Production-Distribution Problem in Supply Chain Management under Stochastic and Fuzzy Uncertainties

Department of Industrial Engineering, Kırıkkale University, 71451 Kırıkkale, Turkey

Correspondence should be addressed to Umit Sami Sakalli

Received 4 April 2017; Revised 4 July 2017; Accepted 12 July 2017; Published 17 October 2017

Academic Editor: Anna M. Gil-Lafuente

Copyright © 2017 Umit Sami Sakalli. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Production-Distribution Problem (PDP) in Supply Chain Management (SCM) is an important tactical decision. One of the challenges in this decision is the size and complexity of supply chain system (SCS). On the other side, a tactical operation is a mid-term plan for 6–12 months; therefore, it includes different types of uncertainties, which is the second challenge. In the literature, the uncertain parameters were modeled as stochastic or fuzzy. However, there are a few studies in the literature that handle stochastic and fuzzy uncertainties simultaneously in PDP. In this paper, the modeling and solution approaches of PDP which contain stochastic and fuzzy uncertainties simultaneously are investigated for a SCS that includes multiple suppliers, multiple products, multiple plants, multiple warehouses, multiple retailers, multiple transport paths, and multiple time periods, which, to the best of the author’s knowledge, is not handled in the literature. The PDP contains deterministic, fuzzy, fuzzy random, and random fuzzy parameters. To the best of the author’s knowledge, there is no study in the literature which considers all of them simultaneously in PDP. An analytic solution approach has been developed by using possibilistic programming and chance-constrained programming approaches. The proposed modeling and solution approaches are implemented in a numerical example. The solution has shown that the proposed approaches successfully handled uncertainties and produce robust solutions for PDP.

#### 1. Introduction

The global competition enforces the firms to manage their facilities more effectively and to make right decisions in the market. Supply Chain Management (SCM), which is defined as the integration of key business processes from end user through original suppliers which provides products, services, and information that add value for customers and other stakeholders by the Global Supply Chain Forum (GSCF) [1], is a useful management approach to survive in the global market. SCM includes several processes such as supplier relation management, product development and commercialization, procurement, order fulfillment, manufacturing flow management, demand management, customer relationship management, returns management, and information management [1]. Production-Distribution Problem (PDP) in SCM is an important planning operation that affects several processes such as procurement, order fulfillment, and manufacturing flow management.

PDP starts to plan by determining raw materials provided by suppliers and makes decisions about the production planning and the distribution of final products to customers. The researchers and practitioners have been interested in PDP over the past years. Fahimnia et al. [2] indicated that there might be two main reasons increasing the number of studies on PDP: (1) affecting the profitability and (2) responding to the market changes quickly. The studies on the PDP can be classified into the different clusters according to the different criteria such as complexity of the supply chain system (SCS), decision levels, solution approaches, and structure of parameters.

PDP can be handled at different decision levels such as operational, tactical, and strategical levels. The strategic decisions are long-term plans that have vital effects on surviving in the market. The papers in this cluster focus on supply chain network design. They also consider opening plants, warehouses, and so forth [3–5]. In tactical perspective, PDP can be used to determine the production and transportation quantities for aggregate production planning and distribution planning. Besides, it is useful for capacity and resources planning decisions [6–9]. The PDP in operational level seeks to optimize the SCS by adding operational decisions to the aggregate models such as scheduling problem and routing problem [10–13].

There are several differences between operational, tactical, and strategical decisions such as time period, detail of information, responsibility, and the cost of a wrong decision. One of them is uncertainty which depends on the length of time period. The precision and exactness of information about problem parameters decrease when the time period of decision increases. Therefore, uncertainty is a challenge in PDP. PDP can be classified into four groups according to the structure of parameters. The first group is* deterministic parameters*: these models do not include any uncertainty in their parameters. All of the parameters are exact and are known at the beginning of solution process [12, 14]. The second group is* stochastic parameters*: the parameters include stochastic uncertainty. The probability theory models these parameters [3, 15–17]. The third group is* fuzzy parameters*: the fuzzy set theory is an effective modeling approach when the information on parameters is imprecise or inexact. It enables reflecting the decision maker’s judgements into the problem [18–21]. The fourth group is* fuzzy and stochastic parameters*: in some situations, both fuzzy and stochastic uncertainties can occur in parameters simultaneously such as fuzzy random or random fuzzy parameters [22–24].

On the other hand, the size and complexity of supply chain system (SCS) are big challenges in PDP like uncertainty. Fahimnia et al. [2] classified the studies into seven clusters according to the SCS complexity. These clusters are given as follows:* Cluster 1*: single-product models [25–28];* Cluster 2*: multi-product, single-plant models [29–31];* Cluster 3*: multiple-products, multiple-plants, single- or no-warehouse models [32–34];* Cluster 4*: multiple-products, multiple-plants, multiple-warehouses, single-/no-end-user models [35–37];* Cluster 5*: multiple-products, multiple-plants, multiple-warehouses, multiple end users, single-transport-path models [20, 38, 39];* Cluster 6*: multiple-products, multiple-plants, multiple-warehouses, multiple-end-users, multiple-transport-paths, no time period models [40, 41];* Cluster 7*: multiple-products, multiple-plants, multiple-warehouses, multiple-end-users, multiple-transport-paths, multible-period-models [2]. To the best of the author’s knowledge, a new cluster (Cluster 8) can be added to these classifications:* Cluster 8*: multiple-products, multiple-plants, multiple-warehouses, multiple-end-users, single-transport-path, multiple-periods models [42, 43].

The PDP requires using various techniques for solving this problem because of the properties of the PDP which are discussed above. Fahimnia et al. [2] classified these techniques into four clusters: analytic techniques, heuristic techniques, simulation, and genetic algorithms. For* analytic techniques*, the studies in this cluster use mathematical programming to solve PDP, that is, linear programming, nonlinear programming, mixed integer programming, and Lagrangian relaxation [44–46]. For* heuristic techniques*, since analytic techniques have a limitation on solving large-scale PDP, the researchers developed heuristic techniques that obtain feasible solution close to an optimal solution [16, 35, 47]. For* simulation modeling*, simulation is a very useful tool to analyze the system’s behavior and performance criteria when the considered system is very complex to solve analytically [30, 48]. For* genetic algorithms (GA)*, they are effective algorithms that use direct and stochastic search methods to solve large-scale problems [49, 50].

In this paper, the PDP has been handled from a tactical perspective for a SCS. The SCS includes multiple suppliers, multiple products, multiple plants, multiple warehouses, multiple retailers, multiple transport paths, and multiple time periods, which, to the best of the author’s knowledge, is not handled in the literature. A, 0-1 mixed-integer programming model has been developed for the PDP which includes deterministic, fuzzy, fuzzy random, and random fuzzy parameters. To the best of the author’s knowledge, there is no study in the literature which considers deterministic, fuzzy, fuzzy random, and random fuzzy parameters simultaneously in PDP. An analytic solution approach has been developed for 0-1 mixed-integer programming model by using possibilistic programming and chance-constrained programming approaches.

The paper is organized as follows: the modeling uncertainty is given in Section 2. Section 3 represents mathematical model and uncertain parameters for PDP. The proposed solution approach is given in Section 4. The implementation of the proposed solution approach for a real-life industry case is presented in Section 5. The paper is finalized with concluding remarks in Section 6.

#### 2. Modeling Uncertainty

Let us give the definitions of some uncertainty types such as random, fuzzy, random fuzzy, and fuzzy random variables.

*Definition 1. *If is an experiment having sample space and is a function that assigns a real number to every outcome , then is called a* random variable* [51].

*Definition 2. *Let be a set of all outcomes of a random experiment. A (nonempty) collection of subsets (called events) of is assumed to have the following properties: (a) ; (b) if , then ; and (c) if is a countable sequence of events, then . Such a collection is called a -algebra. For each random event , there is a nonnegative number , called its probability, such that (i) and and (ii) for every countable sequence of mutually disjoint events . The triplet is called a probability space and the function Pr is referred to as a probability measure. A* random variable* on the probability space is a function from to the real line for any Borel set of [52].

Normal distribution is very important in both theory and application of statistics. The notation is often used to indicate that the random variable is normally distributed with mean and variance .

After random variable definition, now we can get fuzzy variable definition and properties. Fuzzy set theory was proposed by L. Zadeh and applications of his theory can be found, for example, in artificial intelligent, computer science, control engineering, operation research, and decision theory [53].

*Definition 3. *Let denote a universal set. Then a fuzzy subset of is defined by its membership function which assigns to each element a real number in the interval , where the value of at represents the grade of membership of in .* A fuzzy variable* is defined as a function from the possibility space to the real line [52].

Triangular fuzzy variable is the most known and used fuzzy variable which is denoted by the triplet and has the shape of a triangle.

The concept of the random fuzzy variable was initialized by Liu and defined as a fuzzy variable taking “random values.”

*Definition 4. *A* random fuzzy variable* is defined as a function from the possibility space to the set of random variables [54].

Assume that are random variables and that are real numbers in such that . Thenis a clearly discrete random fuzzy variable [54].

*Definition 5. *Assume that is* a random fuzzy variable*. Then* the probability * is a* fuzzy variable* for any Borel set of [54].

*Definition 6. *A random fuzzy variable is said to be normal if, for each , is a normally distributed random variable; that is, , with and being fuzzy variables defined on the space such that . A normally distributed random fuzzy variable is usually denoted as , and the fuzziness of random fuzzy variable is said to be characterized by fuzzy vector [55].

Roughly speaking, a fuzzy random variable is a measurable function from a probability space to the set of fuzzy variables. In other words, a fuzzy random variable is a random variable taking fuzzy values.

*Definition 7. **A fuzzy random variable* is a function from a probability space to the set of fuzzy variables such that is a measurable function of for any Borel set of .

A random fuzzy variable is defined as a function that assigns a random value to each fuzzy subset. On the other hand, a fuzzy random variable is defined as a function that assigns a fuzzy subset to each possible output of a random experiment.

#### 3. Mathematical Model and Uncertain Parameters for PDP

##### 3.1. Mathematical Model

The SCS includes multiple suppliers, multiple products, multiple plants, multiple warehouses, multiple retailers, multiple transport paths, and multiple time periods. In the PDP, multiple raw materials are supplied from multiple suppliers and transported to the multiple plants by using multiple transport paths. In plants, multiple products are manufactured by using regular time and overtime and the final products are transported to the multiple warehouses by using multiple transport paths. Multiple warehouses deliver multiple products to the multiple retailers by using multiple transport paths. The customers pick up their products from multiple retailers.

Several assumptions have been made to construct a 0-1 mixed-integer programming model which are given as follows:(i)The quantities of raw materials in suppliers are restricted.(ii)The number and capacity of transport paths between all the components in SCS are restricted.(iii)The starting and ending inventories of product and raw materials in plants, warehouses, and retailers are zero.(iv)The plants have ability to produce several products.(v)The plants have ability to store raw materials and products.(vi)The storage capacities of plants are restricted.(vii)The plants have regular-time and overtime production.(viii)The warehouses have ability to store products.(ix)The storage capacities of warehouses are restricted.(x)The retailers have ability to store products.(xi)The storage capacities of retailers are restricted.(xii)The unsatisfied demands are lost.In this paper, the PDP in SCM has been considered in tactical level. Since it is a mid-term plan, it includes a lot of uncertainties. The uncertainties make the problem more complex compared to the deterministic ones because there are challenges in modeling parameters and obtaining robust solutions. In the literature, the researchers have tried to overcome these challenges by using fuzzy set theory or probability theory.

The fuzzy set theory provides a highly effective means of handling imprecise data. It enables incorporating the decision-maker’s expertise and judgements into the problem. However, it is not a powerful theory like probability theory for modeling and solution. On the other side, the probability theory is an effective tool for modeling uncertainties in the stochastic process. It acts with the past data analysis for the forecasting of future events and does not include decision-makers into the decision-making process.

However, the decision-makers have an impact on the future events by the way of their decisions. In PDP, the unit cost of raw materials may change based on the purchased quantity. The unit production cost directly depends on lot size. Producing in overtime or producing and holding in regular time at previous periods are based on planning manager’s decision. The unit transportation cost is related to path type and transported quantity. The unit price of a product may be changed by making discount, giving an advertisement. The capacity of raw materials supplied from the market is based on the contracts made by the decision-makers. The decision-maker can change the workforce level by hiring and firing; therefore, the production capacity can be changed. The parameters in the model related to the above discussion can be modeled by using triangular fuzzy numbers. The triangular fuzzy numbers are well known and are commonly used in many applications because the decision-maker has opinions about pessimistic, optimistic, and most possible values by using his/her expertise and expectation.

The transportation capacities of all echelons in the SCS are related to the number of the transporters in the portfolio of the decision-makers, transportation quantities, and vehicle routing decisions that make the transportation capacity uncertain. Therefore, transportation capacities can be modeled as triangular fuzzy numbers by using the decision-maker’s expertise and judgements. However, the available transportation capacity can occur in different situations based on the suitability of the transporter in the market which are defined as discrete events. These discrete events are determined as high, medium, and low capacities. It is possible to increase the number of situations; however, it will cause confusion in the categorization process. By analyzing the past data, probability levels can be determined for occurrences of each of the situations. Therefore, the transportation capacities can be modeled as fuzzy random parameters.

The demand of product can be modeled as a probability distribution by analyzing the past data. Since the PDP is a mid-term plan, a sum of identically distributed independent demand variables has a normal distribution according to the central limit theorem. However, the decision-maker can affect the demand quantity by making discount, advertisement, or other strategies. These marketing strategies are based on management decisions. Therefore, the demand quantity is modeled as a random fuzzy variable.

The mathematical model is given in Notations.The objective function, given in (2), maximizes the total profit. Total profit is obtained by total revenue, which is gained from total sales, plus total cost. Total cost includes raw material purchasing cost, fixed costs of using path for transportations between all components of SCS, unit transportation costs between all components of SCS, unit production costs in regular time and overtime, holding costs in plants, warehouses, and retailers, and backorder cost. Equation (3) is a capacity constraint for the supplier that ensures that the total transported quantity from supplier for material at each period will be less than or equal to total capacity. Equations (4) and (5) are constructed to select the transportation path from supplier to plant and not to exceed its capacity where is a big number. Equation (6) is an inventory balance constraint for the raw material in a plant. Equations (7), (8), and (9) are capacity constraints for regular production, overtime production, and inventory level in plants, respectively. Equation (10) is an inventory balance constraint for the product in a plant. Equations (11) and (12) are designed to select the transportation path from a plant to the warehouse and not to exceed its capacity. Equation (13) is an inventory balance constraint in a warehouse. Equation (14) is an inventory capacity constraint for a warehouse. Equations (15) and (16) are designed to select the transportation path from a warehouse to the retailer and not to exceed its capacity. Equation (17) is a balance constraint for inventory and backorder level. Equation (18) is an inventory capacity constraint for a retailer. Equation (19) ensures meeting customer demand. Equation (20) gives the definitions of the decisions variables.

##### 3.2. Uncertain Parameters in PDP

The 0-1 mixed-integer mathematical model given in (2)–(20) includes uncertain parameters in both objective function and constraints which are fuzzy, fuzzy random, and random fuzzy.

###### 3.2.1. Fuzzy Parameters

Fuzzy parameters, which are symbolized by “,” are , , , , , , , , , , , and in objective function and , , and in constraints. All of the fuzzy parameters in objective function and constraints are modeled by using triangular fuzzy numbers. As mentioned above, triangular fuzzy number is denoted by the triplet :(1)The most possible value () that definitely belongs to the set of available values (membership degree = 1 if normalized)(2)The most optimistic value () that has a very low likelihood of belonging to the set of available values (membership degree = 0 if normalized)(3)The most pessimistic value () that has a very low likelihood of belonging to the set of available values (membership = 0 if normalized)

###### 3.2.2. Fuzzy Random Parameters

Fuzzy random parameters, which are symbolized by “,” are , , and in constraints. Fuzzy random parameters are related to the capacities of transportation paths. They can be modeled by using triangular fuzzy number as follows:where , , and represent the probabilities of transportation capacity situations such as high, medium, and low capacities. On the other hand, triangular fuzzy numbers , , and represent the amounts of each of the transportation capacities for probability levels.

###### 3.2.3. Random Fuzzy Parameters

There is one random fuzzy parameter, , which represents the customer demand. The customer demand includes two main parameters: the probability and quantity. Therefore, the demand can be calculated as the sum of multiplication of probability value and quantity which can be referred to as expected value of discrete random variable. The probability and quantity of demand are random fuzzy variables.

The probability of demand is modeled as follows: there are three states about the demand; it may be high, medium, or low. Assume that these three probabilities are represented as , , and . The probability of demand state is affected by three indicators which are (1) political developments, (2) competitors’ strategies, and (3) sectoral expectation. For example, if the competitors perform a strong strategy in the market, the demand quantity will be affected by this situation; most likely it will decrease. These indicators are related to the expertise and expectations. Therefore, it is a very difficult task to model the demand states in deterministic or stochastic case. However, random fuzzy variables enable modeling these situations easier than the remaining ones. It is possible to reflect the decision-maker’s judgements and expectations into the demand state by using random fuzzy variables. The modeling of demand states by using random fuzzy variables can be explained with an example for situations and in Table 1.