A VaR-based optimization model for crop production planning under imprecise uncertainty

Abstract

This paper introduces the fuzzy Value-at-Risk (VaR) into crop production planning and proposes a risk-based decision-making approach to the problem in imprecise or fuzzy parameters environments. In the proposed fuzzy crop production planning VaR model, the profit coefficients are imprecise uncertain and assumed to be fuzzy variables with known possibility distributions. Due to the fuzzy variable parameters with infinite supports, the VaR model is inherently an infinite-dimensional optimization problem that can rarely be solved directly via conventional mathematical programming methods. Therefore, algorithm procedures for solving this optimization problem must rely on approximation schemes and heuristic computing. The paper presents a heuristic algorithm, which integrates approximation approach (AA), neural network (NN) and genetic algorithm (GA), to solve the fuzzy crop production planning VaR model. Finally, a practical example is given to show the feasibility and effectiveness of the proposed model and heuristic algorithm.

Keywords

Fuzzy programming Value-at-Risk (VaR)production planning approximation approach heuristic algorithm

1 Introduction

The amount of crops produced by farms may directly influence market prices of crops as well as the farms’ income, hence, the crop production planning is extremely important from both economic and inventory management points of view. Moreover, how to integrating crop production planning decisions by effective mathematical methods is a key issue in ensuring the overall production profit.

Mathematical programming models for crop production planning problems have been studied and used widely since Heady [1] demonstrated the use of linear programming (LP) for land allocation to crop production planning problems. For example, Bender et al. [2] studied energy crop evaluation by LP. Jolayemi and Olaomi [3] proposed a mathematical programming procedure for selecting crops for mixed-cropping schemes. Jolayemi and Olorunniwo [4] dealt with production quantities in a multi-plant, multi-warehouse environment with extensible capacities via deterministic mathematical programming model. Reis and Leal [5] presented a novel deterministic mathematical model, which is applied into a stochastic optimization model for the soybean supply chain in Brazil.

Although LP has been applied broadly to agricultural production planning problems, the main weakness of conventional LP formulation is that all the parameters of the LP problem need to be specified precisely in the practical crop production planning environment. Nevertheless, in most of the practical production decision problems, uncertainty plays an important role. Considering these uncertainty results in more realistic crop production planning models, and the uncertainties are normally modelled as randomness and/or fuzziness. Clearly, the conventional LP methods and algorithms described above cannot solve all crop production planning problems in uncertain environments.

In order to handle probabilistic uncertainty in the production decision systems, some stochastic crop production planning models and algorithms have been studied in the literature. Huirne et al. [6] introduced a stochastic dynamic programming model to determine the economic optimal replacement policy in swine breeding herds. Detlefsen and Jensen [7] described a decision support system to support the farmer when selecting a winter wheat variety and built a simple stochastic optimization model. Flaten and Lien [8] presented a discrete stochastic two-stage utility-efficient programming model of organic dairy farms. Kouedeu et al. [9] investigated a production system consisting of two parallel machines with production-dependent failure rates. Susanne and Jutta [10] studied supply planning for processors of agricultural raw materials by two-stage stochastic programming.

Meanwhile, a number of fuzzy crop production planning problems have been considered by many researchers. On the basis of fuzzy set theory and possibility theory [11 –14], Sher and Amir [15] presented an agricultural production planning model with fuzzy constraints. Miller et al. [16] provided an LP formulation to determine the production schedule for a fresh tomato packinghouse, and the corresponding elements were fuzzified into a fuzzy model which was solved by using an auxiliary model. Animesh and Bijay [17] presented how fuzzy goal programming can be efficiently used for modelling and solving land-use planning problems in agricultural systems for optimal production of several seasonal crops in a planning year. Donya et al. [18] studied a two-stage real world capacitated production system with lead time and setup decisions in which some parameters such as production costs and customer demand are uncertain.

In the real world, most farmers focus only on the final crop production profit. Therefore, the risk of crop production optimization decision-making becomes more important. Value-at-Risk (VaR) [19] is a broadly used risk metric from finance to industry sectors, which are original proposed in the framework of probability theory in 1994. Peng [20] presented the average value at risk method for fuzzy risk analysis. Huang et al. [21] used conditional value-at-risk measures to model risks associated with the decisions in a stochastic environment. Moussa et al. [22] focused on the portfolio value-at-risk and expected shortfall computation when the underlying risk factors exhibit some imprecision and vagueness and have heavy-tailed distributions. Zhang et al. [23] investigated the dynamic portfolio selection problem based on a benchmark process coupled with a dynamic value-at-risk constraint. In order to measure more effectively the risk with imprecise uncertainties. Wang et al. [24] discussed the VaR metrics defined by possibility, necessity [14] and credibility measures [25 –28], and showed that the self-duality of credibility is indispensable to measuring risk in fuzzy environments. An extension of fuzzy VaR to fuzzy random cases can be found in Wang and Watada [29]. However in contrast to previous stochastic and fuzzy VaR problem, credibility theory and VaR optimization method to crop production planning problems have not been studied extensively in the literature.

Making use of the fuzzy VaR for crop production planning, this paper presents a new class of fuzzy crop production planning VaR model, in which profit coefficients are uncertain and assumed to be fuzzy variables with known possibility distributions, and we will adopt VaR optimization method in the objective of the fuzzy crop production planning model. Based on the proposed model, our aim is motivated by the economic and ecological relevance of crop production of providing consistent decision rules to control the amount and quality of crops. At the same time, since the proposed fuzzy crop production planning problem includes fuzzy parameters which in general are defined through possibility distributions with infinite supports, it is inherently an infinite-dimensional optimization problem that cannot be solved directly. To overcome this difficulty, this paper applies the approximation approach (AA) [30] to estimate the credibility function of fuzzy crop production planning VaR problem. Using this method, we can convert an infinite-dimensional crop production planning problem into an approximate finite-dimensional one. Finally, in order to solve the approximate fuzzy crop production planning problem, we develop a heuristic algorithm by integrating the AA, neural network (NN) and genetic algorithm (GA), in which the NN is used for acceleration.

The remaining of this paper is organized as follows. Section 2 recalls some basic concepts in credibility theory. Section 3 presents a formulation of fuzzy crop production planning problem with VaR criteria. In Section 4, we apply the proposed AA to determine the credibility function in fuzzy crop production planning VaR problem, and solve it via a heuristic algorithm by integrating the AA, NN and GA. The intent of Section 5 is to demonstrate the feasibility and effectiveness of the designed heuristic algorithm through a practical example. Finally, Section 6 gives some brief conclusions.

2 Preliminaries

Given a universe $Γ, P (Γ)$ is the power set of Γ and Pos is a set function defined on $P (Γ)$ . Let ξ be a fuzzy variable with membership function μ (x) and r a real number. Then the possibility measure of a fuzzy event {ξ ≤ r} is defined as $Pos {ξ \leq r} = sup_{x \leq r} μ (x)$ for any real number r.

The credibility measure [25] of the fuzzy event {ξ ≤ r} is defined as $Cr {ξ \leq r} = \frac{1}{2} (1 + sup_{x \leq r} μ (x) - sup_{x > r} μ (x))$ for any real number r.

If we assume that the triplet $(Γ, P (Γ), Cr)$ is a credibility space and ξ = (ξ₁, ξ₂, ⋯ , ξ_n) is a function from Γ to the space $R^{n}$ , then ξ is called a fuzzy vector. As n = 1, ξ is called a fuzzy variable.

The fuzzy variables ξ₁, ξ₂, ⋯ , ξ_n defined on a credibility space $(Γ, P (Γ), Cr)$ are said to be mutually independent if $\begin{matrix} Cr {γ ∣ ξ_{1} (γ) \in B_{1}, ξ_{2} (γ) \in B_{2}, \dots, ξ_{n} (γ) \in B_{n}} \\ = min_{1 \leq i \leq n} Cr {γ ∣ ξ_{i} (γ) \in B_{i}} \end{matrix}$ for any subsets B₁, B₂, ⋯ , B_n of $R$ .

If ξ be an n-ary fuzzy vector defined on credibility space $(Γ, P (Γ), Cr)$ , then its joint membership function is derived from the credibility measure by $\begin{matrix} μ_{ξ} (t_{1}, t_{2}, \dots, t_{n}) \\ = min {2 Cr {γ ∣ ξ_{1} (γ) = t_{1}, \\ ξ_{2} (γ) & = t_{2}, \dots, ξ_{n} (γ) = t_{n}}, 1} \end{matrix}$ for any $(t_{1}, t_{2}, \dots, t_{n}) \in R^{n}$ .

Let ξ be an m-ary fuzzy vector. The support of the fuzzy vector ξ, denoted Ξ, is defined as the closure of the subset ${(t_{1}, t_{2}, \dots, t_{m}) \in R^{m} ∣ μ_{ξ} (t_{1}, t_{2}, \dots, t_{m}) > 0}$ of $R^{m}$ , it is the smallest closed subset of $R^{m}$ such that $Cr {γ ∣ ξ (γ) \in Ξ} = 1 .$

An m-ary fuzzy vector ξ is said to be bounded if its support Ξ is a bounded subset of $R^{m}$ .

3 Problem formulation

Based on credibility theory and fuzzy VaR optimization method, we will develop a new type of crop production planning VaR model under fuzzy environment in this section.

In addition, we will adopt the following indices and parameters in the rest of the paper:

Indices:

i : index of producible kinds of crops, i = 1, 2, ⋯ , N;

t : index of producible periods of crops, t = 1, 2, ⋯ , T;

f : index of types of fertilizers, f = 1, 2, ⋯ , F.

Parameters:

c_it (γ) : the fuzzy unit profit at crop i in period t;

m_it : the work time for growing crop i at the unit area in period t;

w_it : the supply level of water for growing crop i at the unit area in period t;

f_it : the supply level of fertilizer for growing crop i at the unit area in period t.

Decision variables:

x_it : the cultivation area for crop i in period t.

Objective function:

The VaR of production profit $\sum_{i = 1}^{N} \sum_{t = 1}^{T} c_{it} (γ) x_{it}$ is $\begin{matrix} {VaR}_{1 - α} (x_{it}) \\ = sup {λ | Cr {\sum_{i = 1}^{N} \sum_{t = 1}^{T} c_{it} (γ) x_{it} \geq λ} \geq α} . \end{matrix}$ which denotes the potentially largest profit to the credibility level (confidence level) 1 - α. And the constant value α is predetermined by production decision maker.

Constraints:

C1 : Since the land of a farm is limited, the total area of farm’s land $\sum_{i = 1}^{N} \sum_{t = 1}^{T} x_{it}$ has to be less than or equal to a certain L (Land Constraint): $\sum_{i = 1}^{N} \sum_{t = 1}^{T} x_{it} \leq L .$

C2 : The total length of working time of a farm is limited, $\sum_{i = 1}^{N} \sum_{t = 1}^{T} m_{it} x_{it}$ has to be less than or equal to a certain M (Labor Constraint):

$\sum_{i = 1}^{N} \sum_{t = 1}^{T} m_{it} x_{it} \leq M .$

C3 : Water is an essential input for growing the crops in each period. Therefore, $\sum_{t = 1}^{T} w_{it} x_{it}$ has to be more than or equal to a certain W_i for growing crop i (Water Constraint):

$\sum_{t = 1}^{T} w_{it} x_{it} \geq W_{i} .$

C4 : In order to maintain the productivity of the soil, different kinds of fertilizers are to be used in different periods. Therefore, $\sum_{i = 1}^{N} \sum_{t = 1}^{T} f_{it} x_{it}$ has to be more than or equal to a certain P_f for each fertilizer (Fertilizer Constraint):

$\sum_{i = 1}^{N} \sum_{t = 1}^{T} f_{it} x_{it} \geq P_{f} .$

C5 : The cultivation area for crop i in period t cannot be negative. $x_{it} \geq 0; i = 1, 2, \dots, N; t = 1, 2, \dots, T .$

Using the above indices, parameters and constraints, we will construct a new type of fuzzy crop production planning VaR model with fuzzy coefficients in this paper. Then, the following N-crop T-period production planning problem is considered: $\begin{matrix} max & {VaR}_{1 - α} (x_{it}) \\ s . t . & \sum_{i = 1}^{N} \sum_{t = 1}^{T} x_{it} \leq L, \\ \sum_{i = 1}^{N} \sum_{t = 1}^{T} m_{it} x_{it} \leq M, \\ \sum_{t = 1}^{T} w_{it} x_{it} \geq W_{i}, \\ \sum_{i = 1}^{N} \sum_{t = 1}^{T} f_{it} x_{it} \geq P_{f}, \\ x_{it} \geq 0, λ \geq 0, \\ i = 1, 2, \dots, N; t = 1, 2, \dots, T, \\ f = 1, 2, \dots, F . \end{matrix}$ (1) where $\begin{matrix} {VaR}_{1 - α} (x_{it}) \\ = sup {λ | Cr {\sum_{i = 1}^{N} \sum_{t = 1}^{T} c_{it} (γ) x_{it} \geq λ} \geq α} . \end{matrix}$

The profit c_it (γ) of crop i in period t is uncertain and characterized by a fuzzy variable with known possibility distribution. In problem (1), we assume that the fuzzy vector ξ′ (γ) = (c₁₁ (γ) , c₁₂ (γ) , ⋯ , c_NT (γ)) is defined on a credibility space $(Γ, P (Γ), Cr)$ .

4 Solution procedure

The proposed fuzzy crop production planning VaR problem (1) is an infinite-dimensional optimization problem in Section 3, so we cannot usually solve it by conventional optimization algorithms. Based on this, we will depend on heuristic algorithm to solve it. There are several kinds of heuristic algorithms [31 –34] have been proposed such as GA (Genetic Algorithm), PSO (Particle Swarm Optimization), ABC (Artificial Bee Colony), BFO (Bacterial Feeding Optimization), and so on. These algorithms have been successfully applied in different optimization problems. GA is one of the most popular ones, and is adopted in this paper. GA has numerous advantages over other classical optimization methods. First, GA does not require the specific mathematical analysis of optimization problems. Second, GA that uses recombination operators is able to mix good characteristics from different solutions. Finally, GA can work with the solution space in multiple directions or parallel at once. Therefore, we will design a heuristic algorithm to solve the proposed model by integrating the AA, the trained NN and GA in this paper. In the following, the first is an AA of credibility function, and the second is a heuristic algorithm combining AA, NN and GA.

4.1 Evaluating credibility function by AA

Since the proposed fuzzy crop production planning VaR problem (1) often includes fuzzy coefficients defined through possibility distributions with infinite supports, it is inherently an infinite-dimensional optimization problem that cannot be solved directly. Thus, the intent of this section is to discuss the approximation of fuzzy coefficients with infinite supports in credibility function of problem (1). In order to solve the proposed fuzzy production planning problem (1), it is required to evaluate the following credibility function ${VaR}_{1 - α} : x \to \sup {λ | Cr {ξ^{'} (γ) x \geq λ} \geq α}$ (2) where we assume that the fuzzy vector ξ (γ) is obtained by piecing together the fuzzy profit coefficients c_it (γ) (1 ≤ i ≤ N, 1 ≤ t ≤ T) involved in the problem (1). And we can compute the value of the credibility function (2) according to the following method. On the other hand, the decision vector x = (x₁₁, x₁₂, ⋯ , x_NT)′ in the credibility function (2).

For convenience, we assume that the fuzzy vector $\begin{matrix} ξ^{'} (γ) & = & (c_{11} (γ), c_{12} (γ), \dots, c_{NT} (γ)) \\ = & (ξ_{1}, ξ_{2}, \dots, ξ_{N \cdot T}) \end{matrix}$ in the original fuzzy crop production planning VaR problem (1) is a continuous fuzzy vector with the infinite support $Ξ = \prod_{j = 1}^{N \cdot T} [a_{j}, b_{j}]$ in this paper, where [a_j, b_j] is the support of ξ_j, then we will apply AA to approximate the possibility distribution of ξ by a sequence of possibility distributions of primitive fuzzy vectors {ζ_m}, where fuzzy vector $ζ_{m}^{'} = (ζ_{m, 1}, ζ_{m, 2}, \dots, ζ_{m, N \cdot T}) .$

For each j ∈ {1, 2, ⋯ , N · T}, define fuzzy variable ζ_m,j = g_m,j (ξ_j) for m = 1, 2, ⋯ , where the function g_m,j is

$\begin{matrix} g_{m, j} (u_{j}) & = & sup {\frac{k_{j}}{m} | k_{j} \in Z, s . t . \frac{k_{j}}{m} \leq u_{j}}, \\ u_{j} \in [a_{j}, b_{j}] \end{matrix}$ and Z is the set of integers.

Moreover, for each j, 1 ≤ j ≤ N · T, by the definition of ζ_m,j, as ξ_j takes its values in [a_j, b_j], the fuzzy variable ζ_m,j takes its values in the set

${\frac{k_{j}}{m} ∣ k_{j} = [{ma}_{j}], [{ma}_{j}] + 1, \dots, K_{j}},$ where [r] is the maximal integer such that [r] ≤ r, and K_j = mb_j - 1 or [mb_j] according as mb_j is an integer or not an integer. What’s more, for each integer k_j, the fuzzy variable ζ_m,j takes the value $\frac{k_{j}}{m}$ as ξ_j takes its values in the interval $[\frac{k_{j}}{m}, \frac{k_{j} + 1}{m})$ . Finally, the possibility distribution of the fuzzy variable ζ_m,j, denoted $V_{m, j}$ is

$V_{m, j} (\frac{k_{j}}{m}) = Pos {γ ∣ \frac{k_{j}}{m} \leq ξ_{j} (γ) < \frac{k_{j} + 1}{m}}$ for k_j = [ma_j] , [ma_j] +1, ⋯ , K_j.

From the construction of ζ_m,j, for each γ ∈ Γ, we have

$| ξ_{j} (γ) - ζ_{m, j} (γ) | < \frac{1}{m} .$

Note that ξ (γ) and ζ_m (γ) are N · T-ary fuzzy vectors, and ξ_j (γ) and ζ_m,j (γ) are their jth components, respectively. Therefore, we have $\begin{matrix} ∥ ζ_{m} (γ) - ξ (γ) ∥ \\ = \sqrt{\sum_{j = 1}^{N \cdot T} (ζ_{m, j} (γ) - ξ_{j} (γ))^{2}} \leq \frac{\sqrt{N \cdot T}}{m} \end{matrix}$ for every γ ∈ Γ, which implies that the sequence {ζ_m (γ)} of fuzzy vectors converges to fuzzy vector ξ (γ) uniformly.

We now provide an example to show the AA described above.

Example 1. In the proposed fuzzy crop production planning problem (1), we suppose N = 1, T = 2, ξ₁ = c₁₁ (γ) and ξ₂ = c₁₂ (γ) are mutually independent triangular fuzzy variables, their possibility distribution functions are $μ_{ξ_{1}} (r) = {\begin{matrix} r - 1, & if 1 \leq r < 2 \\ 3 - r, & if 2 \leq r \leq 3 \\ 0, & otherwise, \end{matrix}$ and $μ_{ξ_{2}} (r) = {\begin{matrix} r - 2, & if 2 \leq r < 3 \\ 4 - r, & if 3 \leq r \leq 4 \\ 0, & otherwise, \end{matrix}$ respectively. Then we denote that $ξ^{'} = (ξ_{1}, ξ_{2})$ . Determine the possibility distributions of the discrete fuzzy vectors $ζ_{m}^{'} = (ζ_{m, 1}, ζ_{m, 2}), m = 1, 2, \dots,$ where the fuzzy variables ζ_m,j = g_m,j (ξ_j) , j = 1, 2, with $\begin{matrix} g_{m, 1} (u_{1}) & = & sup {\frac{k_{1}}{m} | k_{1} \in Z, s . t . \frac{k_{1}}{m} \leq u_{1}}, \\ u_{1} \in [1, 3] \end{matrix}$ and $\begin{matrix} g_{m, 2} (u_{2}) & = & sup {\frac{k_{2}}{m} | k_{2} \in Z, s . t . \frac{k_{2}}{m} \leq u_{2}}, \\ u_{2} \in [2, 4] \end{matrix}$

In the following, we first deduce the possibility distributions $V_{m, 1}$ of fuzzy variables ζ_m,1, m = 1, 2, ⋯ .

Let m = 1 . Then fuzzy variable ζ_1,1 takes the values 1 and 2 as ξ₁ takes its values in the intervals [1,2) and 2,3 , 1,2) and 2,3], respectively. Therefore, we have $\begin{matrix} V_{1, 1} (1) = Pos {ζ_{1, 1} = 1} = Pos {1 \leq ξ_{1} < 2} = 1, \\ V_{1, 1} (2) = Pos {ζ_{1, 1} = 2} = Pos {2 \leq ξ_{1} \leq 3} = 1, \end{matrix}$ i.e., the fuzzy variable ζ_1,1 takes the values 1 and 2 with possibility 1 each.

Let m = 2 . Then fuzzy variable ζ_2,1 takes the values 1, 1.5, 2 and 2.5 as ξ₂ takes its values in the intervals [1,1.5), 1.5,2), 2,2.5) and 2.5,3], respectively. Therefore, we have $\begin{matrix} V_{2, 1} (1) & = & Pos {ζ_{2, 1} = 1} \\ = & Pos {1 \leq ξ_{1} < 1.5} = 0.5, \\ V_{2, 1} (1.5) & = & Pos {ζ_{2, 1} = 1.5} \\ = & Pos {1.5 \leq ξ_{1} < 2} = 1, \\ V_{2, 1} (2) & = & Pos {ζ_{2, 1} = 2} \\ = & Pos {2 \leq ξ_{1} < 2.5} = 1, \\ V_{2, 1} (2.5) & = & Pos {ζ_{2, 1} = 2.5} \\ = & Pos {2.5 \leq ξ_{1} \leq 3} = 0.5, \end{matrix}$ i.e., the fuzzy variable ζ_2,1 takes the values 1, 1.5, 2 and 2.5 with possibility 0.5, 1, 1 and 0.5, respectively.

Generally, the fuzzy variable ζ_m,1 takes on values $\frac{k_{1}}{m}, k_{1} = m, m - 1, \dots, 3 m - 1,$ and the possibility that ζ_m,1 takes the value $\frac{k_{1}}{m}$ is calculated as follows.

For k₁ = m, m - 1, ⋯ , 2m - 1, we have $\begin{matrix} V_{m, 1} (\frac{k_{1}}{m}) & = & Pos {ζ_{m, 1} = \frac{k_{1}}{m}} \\ = & Pos {\frac{k_{1}}{m} \leq ξ_{1} \leq \frac{k_{1} + 1}{m}} \\ = & μ_{ξ_{1}} (\frac{k_{1} + 1}{m}) \\ = & \frac{k_{1} + 1}{m} - 1 . \end{matrix}$

On the other hand, for k₁ = 2m, 2m + 1, ⋯ , 3m - 1, we have $\begin{matrix} V_{m, 1} (\frac{k_{1}}{m}) & = & Pos {ζ_{m, 1} = \frac{k_{1}}{m}} \\ = & Pos {\frac{k_{1}}{m} \leq ξ_{1} \leq \frac{k_{1} + 1}{m}} \\ = & μ_{ξ_{1}} (\frac{k_{1}}{m}) = 3 - \frac{k_{1}}{m} . \end{matrix}$

Combining the above gives the possibility distribution $V_{m, 1}$ of ζ_m,1 as follows $V_{m, 1} (\frac{k_{1}}{m}) = {\begin{matrix} \frac{k_{1} + 1}{m} - 1, & if m \leq k_{1} < 2 m \\ 3 - \frac{k_{1}}{m}, & if 2 m \leq k_{1} \leq 3 m - 1 \\ 0, & otherwise . \end{matrix}$ (3)

Also, by the definition of ζ_m,1, we have $ξ_{1} - \frac{1}{m} < ζ_{m, 1} \leq ξ_{1}, m = 1, 2, \dots .$ (4)

Using the similar method, we can obtain the possibility distribution of fuzzy variables ζ_m,2 $V_{m, 2} (\frac{k_{2}}{m}) = {\begin{matrix} \frac{k_{2} + 1}{m} - 2, & if 2 m \leq k_{2} < 3 m \\ 4 - \frac{k_{2}}{m}, & if 3 m \leq k_{2} \leq 4 m - 1 \\ 0, & otherwise . \end{matrix}$ (5) and the link between ζ_m,2 and ξ₂ $ξ_{2} - \frac{1}{m} < ζ_{m, 2} \leq ξ_{2}, m = 1, 2, \dots .$ (6)

By (3) and (5), the possibility distribution of fuzzy vector $ζ_{m}^{'} = (ζ_{m, 1}, ζ_{m, 2}),$ denoted μ_m, is obtained as follows:

$\begin{matrix} μ_{m} (\frac{k_{1}}{m}, \frac{k_{2}}{m}) \\ = {\begin{matrix} min {\frac{k_{1} + 1}{m} - 1, \frac{k_{2} + 1}{m} - 2}, & if m \leq k_{1} < 2 m, \\ 2 m \leq k_{2} < 3 m \\ min {\frac{k_{1} + 1}{m} - 1, 4 - \frac{k_{2}}{m}}, & if m \leq k_{1} < 2 m, \\ 3 m \leq k_{2} < 4 m - 1 \\ min {3 - \frac{k_{1}}{m}, \frac{k_{2} + 1}{m} - 2}, & if 2 m \leq k_{1} < 3 m - 1, \\ 2 m \leq k_{2} < 3 m \\ min {3 - \frac{k_{1}}{m}, 4 - \frac{k_{2}}{m}}, & if 2 m \leq k_{1} < 3 m - 1, \\ 3 m \leq k_{2} < 4 m - 1 \\ 0, & otherwise . \end{matrix} \end{matrix}$

In addition, it follows from (4) and (6) that $∥ ζ_{m} - ξ ∥ = \sqrt{(ζ_{m, 1} - ξ_{1})^{2} + (ζ_{m, 2} - ξ_{2})^{2}} < \frac{\sqrt{2}}{m},$ which implies that the sequence {ζ_m} of the discrete fuzzy vectors converges uniformly to the continuous fuzzy vector ξ.

We now replace the possibility distribution of ξ by that of ζ_m, and approximate the credibility function VaR_1-α (x) in (2) by the approximating credibility function ${({\hat{VaR}}_{1 - α})}_{m} (x) = sup {λ | Cr {{\hat{ζ}}_{m}^{'} (γ) x \geq λ} \geq α}$ where fuzzy vector $\begin{matrix} {\hat{ζ}}_{m} (γ) & = & {({\hat{ζ}}_{m, 1}, {\hat{ζ}}_{m, 2}, \dots, {\hat{ζ}}_{m, N \cdot T})}^{'} \\ = & {({\hat{c}}_{11} (γ), {\hat{c}}_{12} (γ), \dots, {\hat{c}}_{NT} (γ))}^{'} \end{matrix}$ and $\begin{matrix} {({\hat{c}}_{11} (γ), {\hat{c}}_{12} (γ), \dots, {\hat{c}}_{NT} (γ))}^{'} \\ = {(g_{m, 1} (ξ_{1}), g_{m, 2} (ξ_{2}), \dots, g_{m, N \cdot T} (ξ_{N \cdot T}))}^{'} . \end{matrix}$

Letting $λ_{k} = ({\hat{ζ}}_{m}^{k})' (γ) x$ , then ${({\hat{VaR}}_{1 - α})}_{m} (x)$ can be computed by ${({\hat{VaR}}_{1 - α})}_{m} (x) = max {λ_{k} | c_{k} \geq α}$ (7) where

$\begin{matrix} c_{k} & = & \frac{1}{2} (1 + max_{1 \leq j \leq k} {V_{j} | λ_{j} \geq λ_{k}} \\ - max_{1 \leq j \leq k} {V_{j} | λ_{j} < λ_{k}}) \end{matrix}$ (8)

The process to estimate the credibility function of fuzzy crop production planning problem (1) is summarized as follows.

[Approximation approach]

Generate K points ${\hat{ζ}}_{m}^{k} = {({\hat{ζ}}_{m, 1}^{k}, {\hat{ζ}}_{m, 2}^{k}, \dots, {\hat{ζ}}_{m, N \cdot T}^{k})}^{'}$ for k = 1, 2, ⋯ , K uniformly from the support Ξ of ξ;

Calculate $\begin{matrix} {\hat{ζ}}_{m} = g_{m} (ξ) \\ = {(g_{m, 1} (ξ_{1}), g_{m, 2} (ξ_{2}), \dots, g_{m, N \cdot T} (ξ_{N \cdot T}))}^{'}; \end{matrix}$

Set $\begin{matrix} V_{k} & = & V_{m, 1} ({\hat{ζ}}_{m, 1}^{k}) \land V_{m, 2} ({\hat{ζ}}_{m, 2}^{k}) \\ \land \dots \land V_{m, N \cdot T} ({\hat{ζ}}_{m, N \cdot T}^{k}) \end{matrix}$ for k = 1, 2, ⋯ , K;

Compute $c_{k} = Cr {({\hat{ζ}}_{m}^{k})' (γ) x \geq λ_{k}}$ for k = 1, 2, ⋯ , K according to formula (8);

Return the approximating value ${({\hat{VaR}}_{1 - α})}_{m} (x)$ of VaR_1-α (x) via the estimation formula (7).

Using the approximation approach described in the above section, we can formally construct the following approximating fuzzy crop production planning optimization problem $\begin{matrix} max & {({\hat{VaR}}_{1 - α})}_{m} (x) \\ s . t . & \sum_{i = 1}^{N} \sum_{t = 1}^{T} x_{it} \leq L, \\ \sum_{i = 1}^{N} \sum_{t = 1}^{T} m_{it} x_{it} \leq M, \\ \sum_{t = 1}^{T} w_{it} x_{it} \geq W_{i}, \\ \sum_{i = 1}^{N} \sum_{t = 1}^{T} f_{it} x_{it} \geq P_{f}, \\ x_{it} \geq 0, λ \geq 0, \\ i = 1, 2, \dots, N; t = 1, 2, \dots, T, \\ f = 1, 2, \dots, F . \end{matrix}$ (9) where ${({\hat{VaR}}_{1 - α})}_{m} (x) = sup {λ | Cr {{\hat{ζ}}_{m}^{'} (γ) x \geq λ} \geq α} .$

The problem (9) is referred to as the approximating fuzzy crop production planning VaR problem (1) and the following fuzzy vector ${\hat{ζ}}_{m} (γ) = {({\hat{ζ}}_{m, 1} (γ), {\hat{ζ}}_{m, 2} (γ), \dots, {\hat{ζ}}_{m, N \cdot T} (γ))}^{'}$ is also defined on a credibility space $(Γ, P (Γ), Cr)$ , since fuzzy vector ${\hat{ζ}}_{m} (γ)$ is the function of fuzzy vector ξ (γ), ${\hat{ζ}}_{m} (γ) = g_{m} (ξ (γ))$ .

The readers who are interested in the detailed discussion about the convergence of the optimal value of the approximating crop production planning problem (9) may consult the reference [30].

4.2 A heuristic algorithm

Till now, we have discussed the approximation of fuzzy crop production planning VaR optimization problem (1) so that an infinite-dimensional optimization problem can be approximated by a finite-dimensional optimization one. On the other hand, to speed up the solution process of heuristic algorithm, we desire to replace the credibility function VaR_1-α (x) by an NN since a trained NN has the ability to approximate functions. In this section, we will employ the fast BP algorithm to train a feedforward NN to approximate the credibility function VaR_1-α (x). And we only consider the NN with input layer, one hidden layer and output layer connected in a feedforward way, in which there are N · T input neurons in the input layer representing the input value of decision vector x, p neurons in the hidden layer and 1 neurons in the output layer representing the value of the credibility function VaR_1-α (x). In the following, we will incorporate AA, NN and GA to produce a heuristic algorithm for solving the N-crop T-period fuzzy production planning VaR problem (1). The reader who want to learn detailed discussion about GA may also refer to [24] and [35 –37] for some other heuristics in fuzzy optimization problems. The solution process of heuristic algorithm is summarized as follows.

[A Heuristic Algorithm]

Input the parameters pop _ size, P_c, P_m and a;

Generate a set of input-output data for the function ${VaR}_{1 - α} : x \to \sup {λ | Cr {ξ^{'} (γ) x \geq λ} \geq α}$ by AA;

Train an NN to approximate the function VaR_1-α (x) by the generated data;

Initialize pop _ size chromosomes $\begin{matrix} x_{k} & = & (x_{11}^{k}, x_{12}^{k}, \dots, x_{NT}^{k}), \\ k & = & 1, 2, \dots, pop_size \end{matrix}$ at random from the potential region ${x | \begin{matrix} \sum_{i = 1}^{N} \sum_{t = 1}^{T} x_{it} \leq L, \\ \sum_{i = 1}^{N} \sum_{t = 1}^{T} m_{it} x_{it} \leq M, \\ \sum_{t = 1}^{T} w_{it} x_{it} \geq W_{i}, \\ \sum_{i = 1}^{N} \sum_{t = 1}^{T} f_{it} x_{it} \geq P_{f}, \\ x_{it} \geq 0, λ \geq 0, i = 1, 2, \dots, N, \\ t = 1, 2, \dots, T; f = 1, 2, \dots, F . \end{matrix}}$ uniformly and check of the feasibility of chromosomes by the trained NN;

Update the chromosomes $x_{k} (k = 1, 2, \dots, pop_size)$ by crossover operations. To select the parents for crossover operation, we repeat the following process from k = 1 to pop _ size. First, randomly generate a real number r from the interval [0, 1], the chromosome x_k is selected as a parent when the real number r is less than P_c, where the parameter P_c is the probability of crossover. Second, assume that the selected parents $x_{1}^{'}, x_{2}^{'}, \dots, x_{pop_size}^{'}$ are divided into $(x_{1}^{'}, x_{2}^{'}), (x_{3}^{'}, x_{4}^{'}), \dots, (x_{pop_size - 1}^{'}, x_{pop_size}^{'}) .$ Next, let us show the crossover operator on each pair $(x_{k}^{'}, x_{k + 1}^{'}) (k = 1, 2, \dots, pop_size - 1) .$ Third, randomly generate a number λ (λ ∈ (0, 1)) . Then the crossover operator on $x_{k}^{'}$ and $x_{k + 1}^{'}$ will produce two children $\begin{matrix} X & = & λ \cdot x_{k}^{'} + (1 - λ) \cdot x_{k + 1}^{'}, \\ Y & = & (1 - λ) \cdot x_{k}^{'} + λ \cdot x_{k + 1}^{'} . \end{matrix}$ If both children are feasible, then the parents are we replaced by them. If not, we keep the feasible one if it exists, and then repeat the crossover process by generating a new number λ until two feasible children are obtained. Finally, we will obtain pop _ size new chromosomes $x_{k}^{'} (k = 1, 2, \dots, pop_size);$

Update the chromosomes $x_{k} (k = 1, 2, \dots, pop_size)$ by mutation operations. Similar to crossover operation, the chromosomes x_k are selected as parents when the randomly generated real number r (r ∈ [0, 1]) is less than P_m at the kth selection, where k = 1, 2, ⋯ , pop _ size and the parameter P_m is the probability of mutation. For each selected parent x′ = (x₁₁, x₁₂, ⋯ , x_NT), we choose a mutation direction d in $R^{NT}$ randomly. If X = x′ + M · d is feasible for the constraints, then replace the parent x′ with it. Otherwise, we set M as a random number between 0 and M until it is feasible, where M is an appropriate large positive number;

Calculate the objective values ${VaR}_{1 - α}^{k}$ for all chromosomes x_k (k = 1, 2, ⋯ , pop _ size) by the trained NN;

Compute the fitness of each chromosome according to the objective values ${VaR}_{1 - α}^{k}$ . In this paper, we assume that the parameter a ∈ (0, 1). The rank-based evaluation function is defined as $\begin{matrix} eval (x_{k}) = a (1 - a)^{k - 1}, \\ k = 1, 2, \dots, pop_size, \end{matrix}$ where x₁, x₂, ⋯ , x_{pop_size} are the chromosomes which are rearranged from good to bad according to their objective values ${VaR}_{1 - α}^{k}$ ;

Select the chromosomes by spinning the roulette wheel;

Repeat Step 4 to Step 8 for a given number of cycles;

Report the best chromosome x^∗ as the optimal solution.

5 Practical example

To show the feasibility and effectiveness of the proposed optimization model and heuristic algorithm, we will consider the following fuzzy crop production planning problem with N = 5, T = 4 and the profit coefficients are assumed to be triangular and trapezoidal fuzzy variables with known possibility distributions. Also, the fuzzy variables involved in this problem are assumed to be mutually independent. The farm will grow carrot, cucumber, cabbage, potatoes and watermelon in four periods. Let those cultivation areas be x₁₁, x₁₂, ⋯ , x₅₄ (unit:10 m²) during four periods, respectively. The required data set for this crop production system is given in Table 1. Moreover, the possibility distributions of fuzzy profit coefficients in this production planning VaR problem are provided in Table 1. On the other hand, if a decision maker takes the certain values α = 0.95, L = 10000 (unit:10m²), M = 100000 (unit:hour), W₁ = 5000 m³, W₂ = 6000 m³, W₃ = 5000 m³, W₄ = 8000 m³, W₅ = 5000 m³, P₁ = 100 kg (Nitrogen fertilizer (N)), P₂ = 80 kg (Phosphate fertilizer (P)), P₃ = 70 kg (Potash fertilizer (K)) and P₄ = 150 m³ (Organic fertilizer (O)), respectively, then the fuzzy crop production planning VaR problem is as follows

$\begin{matrix} max & {VaR}_{0.05} (x_{it}) \\ s . t . & \sum_{i = 1}^{5} \sum_{t = 1}^{4} x_{it} \leq 10000, \\ \sum_{i = 1}^{5} \sum_{t = 1}^{4} m_{it} x_{it} \leq 100000, \\ 500 x_{11} + 300 x_{12} + 400 x_{13} \geq 5000, \\ 800 x_{21} + 600 x_{22} + 450 x_{23} + 1500 x_{24} \geq 6000, \\ 600 x_{31} + 500 x_{32} + 1000 x_{34} \geq 5000, \\ 500 x_{41} + 300 x_{42} + 500 x_{43} + 800 x_{44} \geq 8000, \\ 1800 x_{51} + 550 x_{53} + 2000 x_{54} \geq 5000, \\ 1.5 x_{11} + x_{12} + x_{13} + 2 x_{21} + 2 x_{22} + x_{23} \\ + 1.5 x_{24} + 2 x_{31} + 2 x_{32} + 1.5 x_{34} + x_{41} \\ + 1.5 x_{42} + x_{43} + x_{44} + 2.5 x_{51} + 2 x_{53} \\ + 3 x_{54} \geq 100, \\ x_{11} + 0.5 x_{12} + 0.5 x_{13} + 1.5 x_{21} + 1.5 x_{22} \\ + 0.8 x_{23} + x_{24} + 1.5 x_{31} + 1.5 x_{32} + x_{34} \\ + 0.5 x_{41} + x_{42} + 0.6 x_{43} + 0.6 x_{44} + 2 x_{51} \\ + 1.7 x_{53} + 2 x_{54} \geq 80, \\ 0.5 x_{11} + 0.3 x_{12} + 0.3 x_{13} + x_{21} + x_{22} \\ + 0.6 x_{23} + 0.5 x_{24} + x_{31} + x_{32} + 0.5 x_{34} \\ + 0.5 x_{41} + x_{42} + 0.4 x_{43} + 0.5 x_{44} + 1.5 x_{51} \\ + 1.5 x_{53} + 2 x_{54} \geq 70, \\ 2 x_{11} + x_{12} + x_{13} + 3 x_{21} + 3 x_{22} + x_{23} + 2 x_{24} \\ + 3 x_{31} + 3 x_{32} + 2 x_{34} + 2 x_{41} + 2 x_{42} + 2 x_{43} \\ + 2 x_{44} + 4 x_{51} + 3 x_{53} + 3 x_{54} \geq 150, \\ x_{it} \geq 0, λ \geq 0, \\ i = 1, 2, 3, 4, 5; t = 1, 2, 3, 4 . \end{matrix}$ (10)

where

$\begin{matrix} {VaR}_{0.05} (x_{it}) \\ = sup {λ | Cr {\sum_{i = 1}^{5} \sum_{t = 1}^{4} c_{it} (γ) x_{it} \geq λ} \geq 0.95} . \end{matrix}$

Table 1

The data set for fuzzy crop production planning problem

The profit (c_it (γ)) (unit:$)
Periods	1	2	3	4
Crops
Carrot	(25,30,35,40)	(22,26,30)	(20,25,30)	—
Cucumber	(22,25,28)	(28,35,40)	(20,23,26)	(80,85,90,95)
Cabbage	(70,72,75,80)	(30,35,40)	—	(80,90,95)
Potatoes	(30,35,40,45)	(20,25,30)	(15,20,25)	(20,25,30,35)
Watermelon	(50,55,60)	—	(60,70,80)	(85,90,93,95)

The work time (m_it)
Periods	1	2	3	4
Crops
Carrot	95h	80h	90h	—
Cucumber	90h	90h	85h	120h
Cabbage	95h	95h	—	120h
Potatoes	50h	55h	40h	45h
Watermelon	100h	—	90h	120h

The water
consumption (w_it)
Periods	1	2	3	4
Crops
Carrot	500m³	300m³	400m³	—
Cucumber	800m³	600m³	450m³	1500m³
Cabbage	600m³	500m³	—	1000m³
Potatoes	500m³	300m³	500m³	800m³
Watermelon	1800m³	—	550m³	2000m³

The fertilizer requirement (f_it)
Periods	1	2	3	4
Crops	N P K O	N P K O	N P K O	N P K O
Carrot	1.5 1 0.5 2	1 0.5 0.3 1	1 0.5 0.3 1	—————
Cucumber	2 1.5 1 3	2 1.5 1 3	1 0.8 0.6 1	1.5 1 0.5 2
Cabbage	2 1.5 1 3	2 1.5 1 3	—————	1.5 1 0.5 2
Potatoes	1 0.5 0.5 2	1.5 1 1 2	1 0.6 0.4 2	1 0.6 0.5 2
Watermelon	2.5 2 1.5 4	—————	2 1.7 1.5 3	3 2 2 3

In order to solve the above proposed fuzzy crop production planning VaR model (10), for any feasible solution

$x = (x_{11}, x_{12}, \dots, x_{54})',$ we will generate 10000 sample points ${\hat{ζ}}_{m}^{k}$ via AA as described in Section 4 to estimate the value of the following the function

$\begin{matrix} {({\hat{VaR}}_{0.05})}_{m} : (x_{it}) \\ \to sup {λ_{k} | Cr {\sum_{i = 1}^{5} \sum_{t = 1}^{4} {\hat{ζ}}_{m, it}^{k} (γ) x_{it} \geq λ_{k}} \geq 0.95} \end{matrix}$ for k = 1, 2, ⋯ , 10000.

To speed up the solution process of the fuzzy programming problem (10), we first generate a set of 2000 input-output data for objective function VaR_0.05 (x_it). Then we train an NN (20 input neurons representing the value of cultivation area variables x₁₁, x₁₂, ⋯ , x₅₄, 11 hidden neurons, and 1 output neuron representing the output value of the objective function VaR_0.05 (x_it)) via the input-output data to approximate the objective function VaR_0.05 (x_it). After the NN is well trained, it is embeded into GA to produce a heuristic algorithm to search for the optimal solutions of the fuzzy crop production planning problem (10).

If the parameters in the implementation of GA are set as follows: the population size pop _ size is 30, the probability of crossover P_c = 0.3, the probability of mutation P_m = 0.2 and the parameter a = 0.05 in the rank-based evaluation function, then a run of the heuristic algorithm with 10000 generations gives the optimal solutions in Table 2 and objective value of the fuzzy crop production planning VaR problem (10) is 52786.8614.

Table 2

The optimal solutions for fuzzy crop production planning problem

Decision Variables	$x_{11}^{*}$	$x_{12}^{*}$	$x_{13}^{*}$	$x_{14}^{*}$
Cultivation Levels	62.1072	21.8744	41.9369	—
Decision Variables	$x_{21}^{*}$	$x_{22}^{*}$	$x_{23}^{*}$	$x_{24}^{*}$
Cultivation Levels	0.0072	41.4085	81.3842	99.9999
Decision Variables	$x_{31}^{*}$	$x_{32}^{*}$	$x_{33}^{*}$	$x_{34}^{*}$
Cultivation Levels	28.9701	89.1537	—	100.0000
Decision Variables	$x_{41}^{*}$	$x_{42}^{*}$	$x_{43}^{*}$	$x_{44}^{*}$
Cultivation Levels	87.9481	82.7617	39.0753	99.9762
Decision Variables	$x_{51}^{*}$	$x_{52}^{*}$	$x_{53}^{*}$	$x_{54}^{*}$
Cultivation Levels	66.9749	—	100.0000	100.0000

In view of identification of parameters’s influence on solution quality, we compare solutions by careful variation of parameters of GA with the same generations in GA as a stopping rule. The optimal solutions corresponding to various parameters are given in Tables 3, 4, 5 and 6, respectively. The computational results are collected in Table 7, where the parameters of GA are given from the first column to the fourth column, ‘gen’ in the fifth column is the generation in GA, and objective values are provided in the sixth column. In addition, the parameter ‘relative error’ in the last column is defined as (optimal value-objective value)/optimal value × 100%, where the optimal value is the most one of the five objective values in the sixth column. From Table 7, we can see that the relative error does not exceed 1.391% when various parameters of GA are selected, which implies the heuristic algorithm is robust to the parameters settings and effective to solve this fuzzy crop production planning problem with VaR criteria.

Table 3

The optimal solutions (pop _ size = 30, P_c = 0.3, P_m = 0.3 and a = 0.05)

Decision Variables	$x_{11}^{*}$	$x_{12}^{*}$	$x_{13}^{*}$	$x_{21}^{*}$	$x_{22}^{*}$
Cultivation Levels	41.8713	0.2826	47.9924	0.7317	68.9944
Decision Variables	$x_{23}^{*}$	$x_{24}^{*}$	$x_{31}^{*}$	$x_{32}^{*}$	$x_{34}^{*}$
Cultivation Levels	89.9628	100.0000	15.9401	86.7840	99.9999
Decision Variables	$x_{41}^{*}$	$x_{42}^{*}$	$x_{43}^{*}$	$x_{44}^{*}$	$x_{51}^{*}$
Cultivation Levels	98.8458	92.5880	55.3858	99.9984	62.4935
Decision Variables	$x_{53}^{*}$	$x_{54}^{*}$
Cultivation Levels	100.0000	100.0000

Table 4

The optimal solutions (pop _ size = 40, P_c = 0.3, P_m = 0.3 and a = 0.04)

Decision Variables	$x_{11}^{*}$	$x_{12}^{*}$	$x_{13}^{*}$	$x_{21}^{*}$	$x_{22}^{*}$
Cultivation Levels	36.7123	5.0817	47.0607	0.0003	65.8166
Decision Variables	$x_{23}^{*}$	$x_{24}^{*}$	$x_{31}^{*}$	$x_{32}^{*}$	$x_{34}^{*}$
Cultivation Levels	89.2307	99.9999	44.8671	87.6298	100.0000
Decision Variables	$x_{41}^{*}$	$x_{42}^{*}$	$x_{43}^{*}$	$x_{44}^{*}$	$x_{51}^{*}$
Cultivation Levels	83.5018	86.0104	28.5232	99.6523	62.4407
Decision Variables	$x_{53}^{*}$	$x_{54}^{*}$
Cultivation Levels	99.9999	100.0000

Table 5

The optimal solutions (pop _ size = 35, P_c = 0.3, P_m = 0.2 and a = 0.05)

Decision Variables	$x_{11}^{*}$	$x_{12}^{*}$	$x_{13}^{*}$	$x_{21}^{*}$	$x_{22}^{*}$
Cultivation Levels	48.9725	8.9370	54.7600	0.0046	44.2716
Decision Variables	$x_{23}^{*}$	$x_{24}^{*}$	$x_{31}^{*}$	$x_{32}^{*}$	$x_{34}^{*}$
Cultivation Levels	80.0649	100.0000	47.4839	88.3445	100.0000
Decision Variables	$x_{41}^{*}$	$x_{42}^{*}$	$x_{43}^{*}$	$x_{44}^{*}$	$x_{51}^{*}$
Cultivation Levels	83.9175	73.9215	54.6116	91.5110	64.4617
Decision Variables	$x_{53}^{*}$	$x_{54}^{*}$
Cultivation Levels	100.0000	99.9999

Table 6

The optimal solutions (pop _ size = 30, P_c = 0.2, P_m = 0.3 and a = 0.06)

Decision Variables	$x_{11}^{*}$	$x_{12}^{*}$	$x_{13}^{*}$	$x_{21}^{*}$	$x_{22}^{*}$
Cultivation Levels	62.4599	14.5840	48.6876	1.9024	54.6189
Decision Variables	$x_{23}^{*}$	$x_{24}^{*}$	$x_{31}^{*}$	$x_{32}^{*}$	$x_{34}^{*}$
Cultivation Levels	86.9197	99.9906	40.5760	92.5715	99.6207
Decision Variables	$x_{41}^{*}$	$x_{42}^{*}$	$x_{43}^{*}$	$x_{44}^{*}$	$x_{51}^{*}$
Cultivation Levels	87.5208	63.7633	35.5569	90.6163	50.6501
Decision Variables	$x_{53}^{*}$	$x_{54}^{*}$
Cultivation Levels	99.8886	100.0000

Table 7

Comparison solutions of fuzzy crop production planning VaR problem

pop _ size	P _c	P _m	a	gen	Objective value	Error (%)
30	0.3	0.2	0.05	10000	52786.8614	0.964
30	0.3	0.3	0.05	10000	52559.2396	1.391
40	0.3	0.3	0.04	10000	53229.5474	0.133
35	0.3	0.2	0.05	10000	53300.6656	0.000
30	0.2	0.3	0.06	10000	52807.9947	0.924

On the other hand, the confidence level α is set to be 0.9, 0.95 and 0.99, respectively. We run the heuristic algorithm with 10000 generations, and compare solutions presented in Table 8, by considering different confidence levels of VaR and different parameters. Figure 1 illustrates the convergence of VaR for different confidence levels according to the iterations of the heuristic algorithm. And Fig. 2 provides a more detailed sensitivity of the VaR with respected to the confidence level α.

Table 8

Comparison solutions of fuzzy crop production planning VaR problem with different confidence levels

pop _ size	P _c	P _m	a	gen	Objective value	confidence
30	0.3	0.3	0.05	10000	55748.7299	0.90
40	0.3	0.3	0.04	10000	54764.0139	0.90
35	0.3	0.2	0.05	10000	53972.8505	0.90
30	0.2	0.3	0.06	10000	54824.9550	0.90
30	0.3	0.3	0.05	10000	52559.2396	0.95
40	0.3	0.3	0.04	10000	53229.5474	0.95
35	0.3	0.2	0.05	10000	53300.6656	0.95
30	0.2	0.3	0.06	10000	52807.9947	0.95
30	0.3	0.3	0.05	10000	53364.8332	0.99
40	0.3	0.3	0.04	10000	52479.7013	0.99
35	0.3	0.2	0.05	10000	52530.1792	0.99
30	0.2	0.3	0.06	10000	52640.8669	0.99

Fig.1

The convergence of VaR at different confidence levels.

Fig.2

The sensitivity of VaR to the changes of confidence level.

6 Conclusions

In this paper, we have presented a new fuzzy crop production planning problem with VaR criteria. Since the proposed production planning problem contains fuzzy parameters with infinite supports, it is an infinite-dimensional optimization problem. To solve the problem, we employed an AA to approximate the original fuzzy crop production planning VaR problem and converted it to an computable finite-dimensional problem. After that, the paper designed a heuristic algorithm which combines AA, NN and GA. The feasibility and effectiveness of the proposed fuzzy crop production planning model and heuristic algorithm are illustrated via a practical example.

Although this paper presented a novel fuzzy crop production planning problem, it can not solve every situation in real production environment. For example, the proposed production problem can also be modelled by stochastic programming methods, which contain goal programming (GP), chance-constrained programming (CCP) or two-stage programming, and so on. In our future research, we will study other formulation about crop production planning problems as well as their solution methods. Another concern is that we will study some special cases of fuzzy crop production planning problems and design some precise algorithms to solve them.

Footnotes

Acknowledgments

The author thanks the reviewers for their valuable comments that have been incorporated into the final version of this paper. This work was supported by the National Natural Science Foundation of China (No. 61374184), the Natural Science Foundation of Hebei Province (No. A2013410011), the Natural Science Youth Foundation Project of Hebei Province College (No. QN2016226) and High-level Talent Funding Project of Hebei Province (No. A2016001124).

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