A Bilevel Programming Model for an Agriculture Production-Distribution System with Fuzzy Random Parameters

Abstract

In this paper, we focus on a production-distribution system consists of a producer and a distribution center. We formulate a bilevel multiobjective programming model for it. The upper model is to maximize the producer’s profit and the lower model is to maximize both the distribution center’s profit and customers’ satisfaction degree. Some decision parameters, including unit production cost, customers’ demand and unit sale price, are assumed as fuzzy random variables due to complex decision environment. Chance-constrained technique are used to tackle the uncertainty of this model. A modified genetic algorithm is be used to solve the problem. A numerical example illustrates the effectiveness and efficiency of the model and the algorithm.

1 Introduction

Production-distribution system is a key issue in ensuring the effectiveness of whole supply chain. The problems of designing production-distribution systems have attracted a lot of researchers [1, 4, 16].

Most of models presented hitherto assume that there is only one decision maker, that is, the production section and distribution section belong to a principal firm, in the production-distribution system, However, in real production-distribution system, the production section and distribution section are distinct and they are non-cooperative for their own benefits. For instance, in a agricultural products supply chain, farmers provide agricultural products to distribution center, and the distribution center sell agricultural products to customers. Both the farmers and the distribution center optimize their objectives respectively. Obviously, their benefits are not completely consistent. Farmers hope sell their product with a good price but distribution center tend to purchase cheaper products to reduce its cost. Classical single level programming models are not appropriate for the situation. It is more likely to be formulated by using bilevel programming model.

A bilevel programming model is a programming model in which a subset of the variables is required to be an optimal solution of a second mathematical programming problem. Bilevel programming models are effective tools to tackle hierarchical structures, which widely exist in many real complex programming problems. Since its rich contents and wide application, bilevel programming is studied by researcher from various aspects [2, 6, 8, 13, 15].

In a production-distribution problem, some parameters, such as demand, can be describe as a fuzzy variable based on the decision maker’s experience and judgement. However, it is not easy to estimate the most possible value exactly. Nevertheless, we can obtain the distribution of the most possible value on the basis of statistics characteristics of the market. So it is reasonable to defined these parameters as fuzzy random variables. The concept of fuzzy random variable was introduced by Kwakernaak [9, 10] to study randomness and fuzziness at the same time. Later on, some variants and extensions were developed by other researchers for different purposes; see for example, Puri and Ralescu [14], López-Diaz and Gil [11], Luhandjula [12], and Gil [7].

The remainder of this paper is organized as follows: Sect. 6.2 states the bilevel decision structure and the motivation of using fuzzy random variables. In Sect. 6.3, we formulate a bilevel multiobjective programming model with fuzzy random parameters for the production-distribution center. Comprehensive explanation of the proposed st-GA is given in Sect. 6.4. Section 6.5 provides an application example to illustrate the performance of the st-GA using actual data obtained from a company in China. The efficacy and the efficiency of this method are demonstrated by comparing its numerical experiment results with those of tradition matrix-based genetic algorithm in Sect. 6.5. Finally, concluding remarks are outlined in Sect. 6.6.

2 Key Problem Statement

The problem considered in this paper is a decision on a production-distribution system. The general characteristics of the system is bilevel multiobjective decision structure and fuzzy random decision environment, which are stated as follows.

2.1 Bilevel Multiobjective Decision Structure

Consider the production-distribution system of a kind of well-sold produce. A farmer yields the raw produce and the distribution center is responsible for purchasing, holding, processing the raw produce and transporting and selling the finished products. The interests of the producer and the distribution center are not the same. The producer wants two sell the produce with a good price, but the distribution center hopes to purchase raw produce with a lower price to reduce its cost. So it is unsuitable for modelling the production-distribution system problem with only one decision maker. It is more likely to be modeled by using bilevel programming, which has been proposed in the literature as an appropriate model for hierarchical decision processes with two different decision makes, the leader at the upper level of the hierarchy and the follower at the lower level. For the problem considered in this study, the distribution center is an agricultural industrialization leading enterprise. It controls the producer’s decision by setting the purchasing price and order quantity. The producer determine output according to purchasing price and order quantity. The output affects the production centers’ adversely. So the leader of the hierarchical production-distribution system is the distribution company and the follower is the farmer.

As to objectives, the discretion center aims to maximize its profit and minimize transportation time. The follower, on receiving the purchasing price and order quantity, decides the output seeking to maximize his profit. As mention above, the production-distribution system can be formulated by a bilevel multiobjective programming model illustrated by Fig. 6.1.

Fig. 6.1

Decision structure

2.2 Fuzzy Random Decision Environment

The need to address uncertainty in production-distribution is widely recognized because uncertainties exist in a variety of system parameters. As a result, the inherent complexity and stochastic uncertainty existing in real-world water resources decision-making have essentially placed them beyond conventional deterministic optimization methods.

Take the customers’ demand as an instance, many studies assumed demand as deterministic for simplifying models. However, demand is influenced by both natural and socials factors. So the demand tends to fluctuate. It is difficult for the decision maker to give a crisp description for this parameter, but he can depict the demand as a fuzzy variable based on his experience and judgement. In this study, a fuzzy variable is assumed as be triangular, i.e., $(a, \rho , b)$. On the basis of statistics characteristics of the market, it is found that the most possible value of demand approximately follows a normal distribution, i.e., $\rho \sim \mathcal {N}(\mu , \sigma ^2)$. In this situation, a fuzzy random variable is applied to depict the demand which combined fuzziness and randomness

3 Model Formulation

In this section, the some relevant assumptions and notations are outlined. Then a brief introduction of fuzzy random variable is presented. After that, a bilevel multiobjective programming model for the production-transportation system is formulated.

Assumptions

• 1. Inventory is not considered.

• 2. All the raw materials are processed.

• 3. The price is inversely proportional to the demand.

• 4. Demands, transportation time from distribution center to customers are fuzzy random variables.

• 5. The distribution center undertakes all the transport task.

• 6. All the produce from the agricultural base is purchased by the distribution center.

• 7. The unit production cost inversely proportional to the square root of the output.

Notations  

$K$ :

potential number of vehicles in the distribution center, indexed by $k$,

$J$ :

potential number of customers, indexed by $j$,

$\tilde{\bar{p}}_j$ :

the unit price to customer $i$,

$\theta _{kj}$ :

loss rate of transportation from the distribution center to customer $j$ by vehicle $k$,

$y$ :

the purchasing quantity,

$v_k$ :

the unit transportation cost from the production base to the distribution center by vehicle $k$,

$w_k$ :

the fixed transportation from the production base to the distribution center by vehicle $k$,

$v_{jk}$ :

the unit transportation cost from the distribution center to customer $j$ by vehicle $k$,

$w_{jk}$ :

the fixed transportation from the distribution center to customer $j$ by vehicle $k$,

$b$ :

the unit proceeding cost in the distribution center,

$\vartheta$ :

the processing loss rate,

$C$ :

the processing capacity of the distribution center,

$\tilde{\bar{h}}_{jk}$ :

transportation time from the distribution center to customer $j$ by vehicle $k$,

$T_j$ :

the aspired time of customer $j$ for the product,

$p'$ :

the unit production cost for production center,

$Y$ :

the production capacity of the agricultural base,

$t_k$ :

the transportation capacity of vehicle $k$,

$x_{jk}$ :

the transportation quantity from the distribution center to customer $j$ by vehicle $k$,

$x$ :

the unit purchasing price,

$x_k$ :

the transportation quantity from the agricultural base to the distribution center by vehicle $k$,

$y$ :

the agricultural base’s output.

 

3.1 Bilevel Model Formulation

We formulate upper level model for the distribution center and lower level model for the farmer. By integrating the lower level model into the upper level model, we formulate the global model for the production-distribution system.

3.1.1 Upper Level Model

The distribution center is an agricultural industrialization enterprise. A brand products of the distribution center is popular with customers. The brand products are made from the materials from the agricultural base. The distribution center is the leader of the production-distribution system, and influence the agricultural base’s decision by determining purchasing price.

For the distribution center, the profit function is formulated by:

$$\begin{aligned} \widetilde{\overline{F}}_1&=\sum \limits _{j=1}^J \tilde{\bar{p}}_j \sum \limits _{k=1}^K(1-\theta _{jk})x_{jk}- xy-\sum \limits _{k=1}^K(x_kv_{k}+\eta (x_k)w_k)\nonumber \\&\quad -\sum \limits _{j=1}^J\sum \limits _{k=1}^K(x_{jk}v_{jk}+ \eta (x_{jk})w_{jk})-b\vartheta \sum \limits _{k=1}^Kx_k, \end{aligned}$$

(6.1)

where the first item represents the sales revenue, the second item is the purchasing cost, the third and the fourth items compose the transportation cost, and the last item is the processing cost. $\eta (x_k)$ takes values of 1 if $x_k$ is positive, i.e., vehicle $k$ is used; otherwise $\eta (x_k)$ is 0. $\eta (x_{jk})$ has the same meaning. It follows from assumption 3 that sale price $p_{j}$ can be formulated by:

$$\begin{aligned} \tilde{\bar{p}}_j=\frac{p_j^0}{\widetilde{\overline{D}}_j}, \end{aligned}$$

(6.2)

where $p_j^0$ are constants, $j=1,2,\cdots ,J$. So (6.1) can be written as:

$$\begin{aligned} \widetilde{\overline{F}}_1&=\sum \limits _{j=1}^J \frac{p_j^0}{\widetilde{\overline{D}}_j} \sum \limits _{k=1}^K(1-\theta _{jk})x_{jk}- xy-\sum \limits _{k=1}^K(x_kv_{k}+\eta (x_k)w_k) \nonumber \\&\quad -\sum \limits _{j=1}^J\sum \limits _{k=1}^K(x_{jk}v_{jk}+ \eta (x_{jk})w_{jk})-b\vartheta \sum \limits _{k=1}^Kx_k. \end{aligned}$$

(6.3)

In a competitive market, customers’ satisfaction degree plays an important role for a company’s development in the long run. Assume the product is well-sold, the customers hope their demands are met as far as possible in a certain period. The customers’ satisfaction degree is measured by the proportion of the actually received products in the whole demand, denoted by:

$$\begin{aligned} \widetilde{\overline{F}}_{2j}=\sum \limits _{k=1}^K(1-\theta _{jk})x_{jk}/\widetilde{\overline{D}}_j,\quad j=1,2,\cdots , J. \end{aligned}$$

(6.4)

By integrating the above objectives, the processing capacity constraint condition of the distribution center, the raw materials constraint condition and time constraints and apply the chance-constrained technique [5], the upper model is formulated as:

$$\begin{aligned} \left\{ \begin{array}{l} \max \limits _{x, x_k,x_{jk}}~\left[ \overline{F}_1,\overline{F}_{21},\overline{F}_{22},\cdots ,\overline{F}_{2J}\right] \\ \text {s.t.}~ \left\{ \begin{array}{l} Ch\left\{ \sum \limits _{j=1}^J \frac{p_j^0}{\widetilde{\overline{D}}_j} \sum \limits _{k=1}^K(1-\theta _{jk})x_{jk}- xy-\sum \limits _{k=1}^K(x_k\tilde{\bar{v}}_{k}+\eta (x_k)w_k)\right. \\ \left. -\sum \limits _{j=1}^J\sum \limits _{k=1}^K(x_{jk}\tilde{\bar{v}}_{jk}+ \eta (x_{jk})w_{jk}) -b\vartheta \sum \limits _{k=1}^Kx_k\ge \overline{F}_1\right\} (\alpha ^{(1)}_1)\ge \beta ^{(1)}_1 \\ Ch\left\{ \sum \limits _{k=1}^K(1-\theta _{jk})y_{jk}/\widetilde{\overline{D}}_j\ge \overline{F}_{2j}\right\} (\alpha ^{(1)}_{2j})\ge \beta ^{(1)}_{2j},\quad j=1,2,\cdots , J\\ \sum \limits _{k=1}^K(1-\theta _{jk})x_{jk}\le C\\ (1-\vartheta )\sum \limits _{j=1}^J\sum \limits _{k=1}^K (1-\theta _{jk})x_{jk}=\sum \limits _{k=1}^K(1-\zeta _k)x_k\\ \sum \limits _{k=1}^K x_k=y\\ Ch\left\{ \sum \limits _{k=1}^K\eta (y_{jk})\tilde{\bar{h}}_{jk}\le T_j\right\} (\gamma ^{(2)}_j)\ge (\delta ^{(2)}_j),\quad j=1,2,\cdots , J\\ x_k\le t_k, \sum \limits _{j=1}^J x_{jk}\le t_k,\quad k=1,2,\cdots , K\\ x\ge 0, x_{k}\ge 0,x_{jk}\ge 0,\quad j=1,2,\cdots , J, \; k=1,2,\cdots , K. \end{array}\right. \end{array}\right. \end{aligned}$$

(6.5)

3.1.2 Lower Level Model

The production base makes its decision of the output, after receiving the price given by the distribution center. The only objective of the production is to maximize its profit:

$$\begin{aligned} f=(x-p')y, \end{aligned}$$

(6.6)

the level model for the distribution center can formulated as:

$$\begin{aligned} \left\{ \begin{array}{l} \max \limits _{y} f=\left( x-\frac{p'_0}{\sqrt{y}}\right) y\\ \text {s.t.}~ \left\{ \begin{array}{l} y\le Y\\ y\ge 0.\\ \end{array}\right. \end{array}\right. \end{aligned}$$

(6.7)

3.1.3 Overall Model

As mentioned above, the problem considered in this paper is formulated as a bilevel multiobjective programming model. As the production center makes decision first, the production center is the leader, while the contractor is the follower. By embedding lower level model (6.7) in upper level model (6.5), we develop the global model as:

$$\begin{aligned} \left\{ \begin{array}{l} \max \limits _{x, x_k,x_{jk}}~\left[ \overline{F}_1,\overline{F}_{21},\overline{F}_{22},\cdots ,\overline{F}_{2J}\right] \\ \text {s.t.} \left\{ \begin{array}{l} Ch\left\{ \sum \limits _{j=1}^J \frac{p_j^0}{\widetilde{\overline{D}}_j} \sum \limits _{k=1}^K(1-\theta _{jk})x_{jk}- xy-\sum \limits _{k=1}^K(x_kv_{k}+\eta (x_k)w_k)\right. \\ \left. -\sum \limits _{j=1}^J\sum \limits _{k=1}^K(x_{jk}v_{jk}+ \eta (x_{jk})w_{jk}) -b\vartheta \sum \limits _{k=1}^Kx_k\ge \overline{F}_1\right\} (\alpha ^{(1)}_1)\ge \beta ^{(1)}_1 \\ Ch\left\{ \sum \limits _{k=1}^K(1-\theta _{jk})y_{jk}/\widetilde{\overline{D}}_j\ge \overline{F}_{2j}\right\} (\alpha ^{(1)}_{2j})\ge \beta ^{(1)}_{2j},\quad j=1,2,\cdots , J\\ \sum \limits _{k=1}^K(1-\theta _{jk})x_{jk}\le C\\ (1-\vartheta )\sum \limits _{j=1}^J\sum \limits _{k=1}^K (1-\theta _{jk})x_{jk}=\sum \limits _{k=1}^K(1-\zeta _k)x_k\\ \sum \limits _{k=1}^K x_k=y\\ Ch\left\{ \sum \limits _{k=1}^K\eta (y_{jk})\tilde{\bar{h}}_{jk}\le T_j\right\} (\gamma ^{(2)}_j)\ge (\delta ^{(2)}_j), \quad j=1,2,\cdots , J\\ x_k\le t_k, \sum \limits _{j=1}^J x_{jk}\le t_k,\quad k=1,2,\cdots , K\\ x\ge 0, x_{k}\ge 0,x_{jk}\ge 0,\quad j=1,2,\cdots ,\; J,~ k=1,2,\cdots , K, \end{array}\right. \\ \text {where~y~solves}\\ \quad \left\{ \begin{array}{l} \max \limits _{y} f=\left( x-\frac{p'_0}{\sqrt{y}}\right) y\\ \text {s.t.}~ \left\{ \begin{array}{l} y\le Y\\ y\ge 0.\\ \end{array}\right. \end{array}\right. \end{array}\right. \end{aligned}$$

(6.8)

4 Solution Procedure

The main solution procedure for bilevel model (6.8) model consists of two steps: transforming the bilevel model into a single level model and designing a hybrid genetic algorithm.

4.1 KKT Transformation

Although simplest bilevel linear programming is a NP-hard problem [3], the lower level model of (6.8) is so simple that can be transformed by KKT condition. In order to use the KKT condition, we rewrite the lower level model as the following standard form:

$$\begin{aligned} \left\{ \begin{array}{l} \min \limits _{y} -f=\sqrt{y}p'_0-xy \\ \text {s.t.}~ \left\{ \begin{array}{l} g_1(y)=y-Y\le 0\\ g_2(y)=-y\le 0,\\ \end{array}\right. \end{array}\right. \end{aligned}$$

(6.9)

$$\begin{aligned} \left\{ \begin{array}{l} \frac{p'_0}{2\sqrt{y}}-x+\lambda _1-\lambda _2=0 \\ \lambda _1(y-Y)=0\\ -\lambda _2 y=0\\ \lambda _1, \lambda _2\ge 0. \end{array}\right. \end{aligned}$$

(6.10)

In a real production-distribution system, the output $y$ must be positive. It follows from $-\lambda _2 y=0$ that $\lambda _2=0$. If $\lambda _1=0$, $y=\frac{p'^2_0}{4x^2}$, and $\lambda _1>0$, $y=Y$. Here we discuss the two cases respectively.

Case 6.1

$\lambda _1=0, y=\frac{p'^2_0}{4x^2}$. For this case,

$$f=\left( x-\frac{p'_0}{\sqrt{y}}\right) y=-\frac{p'^2_0}{4x}<0$$

implies a contradiction with reality, i.e., no one will do a business at a loss.

Case 6.2

$\lambda _1>0, y=Y$. For this case, $x=\frac{p'_0}{2\sqrt{Y}}+\lambda _1$. It follows from $\lambda _1=x-\frac{p'_0}{2\sqrt{Y}}>0$ that the lower level model can be transformed into the inequality constraint $x>\frac{p'_0}{2\sqrt{Y}}.$

4.2 Fuzzy Random Simulation-based Genetic Algorithm

After transforming (6.8) into a single-level model, we design the so-called fuzzy random simulation-based genetic algorithm to solve it. Simulation is an imprecise technique which provides only statistical estimates rather than exact results and is also a slow and costly way to study problems. However, it is indeed a powerful tool dealing with complex problems without analytic techniques. Fuzzy random simulation is a twofold simulation of fuzzy simulation and stochastic simulation. The fuzzy random simulation-based genetic algorithm is to embed the fuzzy random simulation technique into genetic algorithm.

The overall procedure of the fuzzy random simulation-based genetic algorithm is illustrated by Fig. 6.2.

Fig. 6.2

The flowchart of the fuzzy random simulation-based genetic algorithm

5 Practical Application

The problem between a organic agricultural products base in Mianyang and a distribution center in Chengdu, is used as a practical application example.

Table 6.1 Fixed transportation costs, $w_k, k=1,2,3$ (yuan)

Table 6.2 Unit transportation costs, $v_k, k=1,2,3$ (yuan/ton)

5.1 Presentation of Case Problem

The base in Mianyang produces a kind of organic agricultural product-mini jujube, which is very popular with the market. The distribution center has the functions of processing, inventory and distribution. There are 6 customers in all from Guangzhou, Wuhan, Changsha, Nanjing, Hangzhou and Nanchang. The production capacity of this base is 100,000 tons and budget is 200,000,000 yuan. The aspired delivery time of the 6 customers are 36, 54, 48, 36, 60 and 54 h, respectively. There are 3 transport fleet belong to the distribution center with capacities of 20,000, 30,000 and 50,000 tons, respectively. For the base, the unit production cost can be presented by the following fuzzy random variable: $\tilde{\bar{e}}=(\bar{e}, 500, 100)_{LR},\overline{e}\sim \mathcal {N}(60{,}000, 100).$

Table 6.3 Loss rates during transportation (%)

Since the road conditions, distances and status of the products are different between base-distribution center and distribution center-customers, the fixed transportation cost and the unit transportation cost differs for the same fleet. The fixed transportation costs and the unit transportation costs of the three fleets are listed in Tables 6.1 and 6.2, respectively.

Table 6.4 Customers’ demands (ton)

Table 6.5 Unit prices, (yuan/ton)

Table 6.6 Optimal transport scheme as $\alpha =0.6$

Table 6.7 Optimal transport scheme as $\alpha =0.7$

Table 6.8 Optimal transport scheme as $\alpha =0.8$

A larger fleet has higher fixed cost but lower unit cost due to the economies of scale. On the contrary, a smaller fleet has lower fixed cost but higher unit cost.

The loss rates during transportation are listed in Table 6.3.

The inventory capacity of the distribution center is 96,000 tons and unit storage cost is 10 yuan/ton. The customers’ demands are listed in Table 6.4. The unit prices to different customers are listed in Table 6.5.

5.2 Results and Discussion

To show the practicality and efficiency of the optimization method for the benefit trade-off problem presented in this paper, the rough simulation and interactive fuzzy programming combined with hGA is conducted and ran on MATLAB. First, the rough simulation is conducted 30 times of running the rough simulation is finished, and the mean value of the outcomes is.

As shown from these tables, the optimal values of the objective function get worse with the confidence levels increasing since the feasible religion narrows. Higher confidence levels means lower risk degree but lower return. How to set confidence levels depends on decision maker’s attitude towards risk (Tables 6.6, 6.7 and 6.8).

6 Conclusions

In this paper, we developed a novel multiobjective programming model with rough interval parameters, and proposed a rough interval goal programming approach and spanning tree-based genetic algorithm to tackle the model. The model and algorithm are applied to a practical problem. The experimental results and the comparison analysis illustrate the effectiveness of our model and algorithm.

Several extensions of the proposed approach are worthwhile further investigating. Further study on rough intervals, such as other operators and order relations, can be discussed as basis of rough interval programming. More rough interval programming models should be investigated. Also, improving algorithm quality and more practical applications are important issues for future research.