Geometric Programming with Discrete Variables Subject to Max-Product Fuzzy Relation Constraints

1School of Mathematics and Information Science, Guangzhou University, Guangzhou, Guangdong 510006, China 2Guangzhou Vocational College of Science and Technology, Guangzhou University, Guangzhou, Guangdong 510550, China 3Department of Industrial and System Engineering, North Carolina State University, Raleigh, NC 27695, USA 4Department of Mathematics and Statistics, Hanshan Normal University, Chaozhou, Guangdong 521041, China


Introduction
Since fuzzy relation equations with max-min composition were firstly introduced by Sanchez [1][2][3], they have attracted much research attention.As an extension, fuzzy relation inequalities associated with max--norm were also studied.As demonstrated in [4], the complete solution set of max-norm fuzzy relation equations can be completely determined by a unique maximum solution and a finite number of minimal solutions.It is easy to compute the maximum solution, but finding all the minimal solutions is an NP-hard problem [4][5][6].It is worth mentioning that Li and Fang [5] provided a complete survey and detailed discussion on fuzzy relational equations.They studied fuzzy relational equations in a general lattice-theoretic framework and introduced classification of basic fuzzy relational equations.
Meanwhile, optimization problems subject to fuzzy relation equations or inequalities were introduced and studied.Fang and Li [7] firstly investigated a linear optimization problem with a consistent system of max-min equations.They converted it into a 0-1 integer programming problem and solved this by the branch-and-bound method.Then it became a unified framework to deal with the resolution of linear optimization problem subject to a system of fuzzy relational equations with max--norm composition.And linear programming problem with fuzzy relation constraints became a hot topic for further research [8][9][10].Guo and Xia [11] proposed a method to accelerate the resolution of this problem.Zhang et al. [12] studied a linear objective optimization problem with max-min fuzzy relation inequalities.
For nonlinear programming with fuzzy relation constraints, many achievements were gained.Wang et al. [13] first studied latticized linear programming subject to maxmin fuzzy relation inequalities.In [14] Li and Fang made a further study on the latticized linear optimization (LLO) problem and its variant, which are a special class of optimization problems constrained by fuzzy relational equations or inequalities.Yang and Cao [15] and Wu [16] considered geometric optimization problems with single-term exponents under fuzzy relation equation constraints with max-min composition, where the objective function is  = ⋁  =1 (  ∧     ).Zhou and Ahat [17] investigated similar problem where the composition was replaced by max-product.Yang et al. [18] investigated min-max programming problem subject to fuzzy relation inequality constraints with a special composition of addition-min.Yang et al. [19] studied the single-variable term semi-latticized geometric programming subject to maxproduct fuzzy relation equations.As important nonlinear programming, geometric programming with fuzzy relational constraints attracted some researchers' attention.Yang and Cao [20] proposed a monomial geometric programming subject to max-min fuzzy relation equations with the objective function being  = ∏  =1     .Shivanian and Khorram [21] considered monomial geometric programming subject to fuzzy relation inequalities with max-product composition.Zhou et al. [22] investigated a special posynomial geometric programming problem subject to max-min fuzzy relation equations, in which the exponents of each variable are all nonpositive or nonnegative real numbers.Aliannezhadi et al. [23] investigated a monomial geometric programming objective function subject to bipolar max-product fuzzy relation constraints.All the objective functions of these optimization problems are special geometric functions.Because of the nonconvexity and complexity, problems with a general geometric objective function have not been studied.At the same time, the data in the real world could often be discrete instead of continuous.In particular, statistics data are often discrete numbers associated with real facts.Therefore, in this paper we study geometric programming subject to maxproduct fuzzy relational constraints, in which the objective function is a general geometric function and all the variables are of discrete values.
The rest of the paper is organized as follows.In Section 2, we introduce some basic concepts.A method to transform the original problem into a linear mix-integer programming model is proposed in Section 3. Numerical experiments are given to illustrate the effectiveness of the proposed solution method in Section 4. A simple conclusion is arranged in Section 5.
We analyze the three kinds of max-product fuzzy relation constraints with discrete variables first.
(ii) For an equation constraint: It can be written as an upper-bound constraint and a lowerbound constraint, that is, From (i) we know that the upper-bound constraint can be included in a new set of discrete values, and the lowerbound constraint can be grouped into other lower-bound constraints.Consequently, the constraints can all be transformed into max-product lower-bound constraints with discrete variables.Without loss of generality, in this paper we consider the following geometric programming problem subject to max-product fuzzy relational inequalities with discrete variables: where The major difficulty in solving problem (8) comes from the nonconvexity caused by the cross-product terms in the objective function and the max operation in the fuzzy inequalities.We will introduce methods to transform the original problem into an equivalent 0-1 linear mixed integer programming problem and adopt the branch-and-bound scheme to find an optimal solution.
It is important to know that the ℎ binary variables of {   },  ∈  and  ∈ , induced by ( 9), may cause heavy computational burden when  and ℎ become large.To avoid this burden, Li et al. [24] proposed a logarithmic approach which needs only ⌈log 2 ℎ⌉ binary variables (⌈⋅⌉ denotes the ceiling function), ℎ nonnegative variables, and ⌈log 2 ℎ⌉ linear equations to represent the  discrete variables with ℎ values by considering where  = {1, 2, . . ., ⌈log 1 for  ∈ ; then we have Moreover, we denote Then the objective function can be expressed as In this way we can express the objective function of problem (8) where    is defined in (9), and  , is a sufficient large positive value such that Proof.Assume that   =    * for a particular  * .From (9), we have    * = 1 and other    = 0.
Theorem 4 (see [26,27]).Let  ∘  ≥  be a system of maxproduct fuzzy relation inequalities; then it is consistent if and only if x = (1, 1, . . ., 1) ∈ (, , ≥).Moreover, if the system is consistent, the solution set (, , ≥) can be fully determined by one maximum solution and a finite number of minimal solutions, that is, where X(, , ≥) is the set of all minimal solutions of (22).
The feasible domain restricted by the max-product fuzzy relation inequalities ( 22) can be fully characterised by the solution set (, , ≥).Actually, if we know all the minimal solutions, then the solution set (, , ≥) can be expressed.The main problem we face is computing all the minimal solutions of system (22).However this is an NP-hard problem [5,26].Hence we introduce a method to find all the potential minimal solutions instead of minimal solutions.
Definition 5 (see [26]).Matrix  = ( , ) × is called a discrimination matrix of (22) with Theorem 6 (see [26]).Let matrix  be the discrimination matrix of the system (22).The system is consistent if and only if, for any  ∈ , there exists at least one   ∈  such that  ,  ̸ = 0.
We call  a solution matrix if, for any  ∈ , there exists a unique   ∈  such that  ,  ̸ = 0.
From Theorem 8, we know that, for any minimal solution x of ( 22), the solution matrix  and its corresponding   can be formed such that   = x .Then the feasible domain restricted by the max-product fuzzy relation inequalities constraints (22) can be expressed as where () denotes the set of all the solution matrix determined by discrimination matrix  of system (22), and   is a solution corresponding to the solution matrix .
Now we illustrate this model by a simple example.

Computational Experiments
To illustrate the effectiveness of the proposed method, we conduct some computational experiments.All the experiments have been run on a PC equipped with the Intel Core I7-6700HQ CPU, 8 GB RAM, and Windows 10 (64 bit) operating system.GUROBI (2016) is the chosen MIP solver for solving all the instances.To ensure the existence of an optimal solution in each case, we apply the following rules to randomly generate its coefficients: (1)  , are generated randomly by the uniform distribution over (0, 1], and   are taken randomly from [−5, 5] for  ∈  and  ∈.
The computational results are shown in Table 1.In the Table , V-Bin denotes the number of 0-1 variables, V-Con denotes the number of continuous variables, L-Eq denotes the number of linear equation constraints, L-Ine denotes the number of linear inequality constraints,  ave () denotes the average CPU time in seconds of solving 20 randomly generated instances of each case, and  max () means the longest CPU time in seconds we meet in each case.
From Table 1, we can see that there are hundreds of binary variables and thousands of linear inequality constraints in each case.All the problem can be solved in 4 minutes in average.Even in the worst condition, we can also find the optimal solution in 1 hour.Therefore, our computational experiments support the efficiency of the proposed algorithm.
Because of the nonconvexity of the geometric function and fuzzy relation equations (inequalities), the optimization problem of geometric programming with fuzzy relational constraints is difficult to solve.To make problem easier to solve, researchers chose special geometric functions as objective functions, such as monomials in [20,21,23] and posynomial function with special indexes in [17].In our problem, the objective function is a general geometric function.At the same time, we add the restriction of discrete values for The next step is to reformulate the objective function.Remembering that    ∈ {0, 1} and ∑ ∈    = 1, we denote  ,1 = 2 ℎ⌉} is an index set;    ∈ {0, 1} is a binary variable;    ≥ 0 is now a nonnegative variable, for  ∈ ,  ∈  and  ∈ ; and  , are binary variables obtained by solving the equations of 1 + ∑ ∈ 2 −1  , =  for  ∈ .