Fuzzy Multi-Choice Goal Programming and Artificial Bee Colony Algorithm for Triangular and Trapezoidal Membership Functions

Multi-choice goal programming (MCGP) has been widely used to find satisfying solutions for multiple criteria/objective decision problems in which the target value of “the more, the better,” or “the less, the better” can easily be obtained. This paper proposes two new models for representing the triangular and trapezoidal membership functions, which improve the efficacy of fuzzy MCGP (FMCGP). Two real-world applications are provided in this study to demonstrate the usefulness of the proposed models. Furthermore, the same problems are resolved by using the proposed nature-inspired optimization method (NIOM) to find the differences between them. While the artificial bee colony (ABC) algorithm is a well-known NIOM technique, studies have shown that it has an excellent performance with high-quality solutions. Thus, this study initially uses the ABC algorithm to find the differences between MCGP and ABC. Finally, some insightful information is obtained from the comparison to contribute to the NIOM and MCGP fields and their respective applications.


I. INTRODUCTION
In recent years, fuzzy multiple objective decision making (FMODM) has become more important for helping companies make an appropriate decision under environmental uncertainty. Chang [1]- [3] proposed a series of multi-choice goal programming (MCGP) methods that effectively solve multi-aspiration level problems to contribute to the field of multiple objective decision making (MODM). In the MCGP, the multi-aspiration level can be represented as discrete, vector, and utility functions (UF) according to decisionmakers' (DM) needs. By using MCGP to solve MODM problems, both qualitative and quantitative issues can be considered simultaneously. Moreover, MCGP has been widely applied to solve many real-world decision-making problems such as supplier selection [4], [5], house selection [6], The associate editor coordinating the review of this manuscript and approving it for publication was Stavros Souravlas .
Membership function (MF) is usually used to quantify linguistic terms and represent a fuzzy set graphically. For example, an MF for a fuzzy set on the universe of discourse is defined as µ A : x → [0, 1]. Further, the applications of MF can be seen in automatic control, decision-making capacity, and fuzzy logic. Chang [3] was the first to add two popular UFs (linear and S-shaped UFs) to MCGP in dealing with both qualitative and quantitative issues simultaneously. This technique enriches the fields of MCGP and MCDM. Chang [3] first introduced the fuzzy method to MCGP, called the FMCGP, to formulate triangular MF and enrich MCGP for qualitative decision-making issues. However, there are two limitations on their method: (1) it only deals with triangular MF, which is less flexible in resolving practical problems, and (2) it only maximizes the values of MF (i.e., qualitative issue) in the objective function; while it may be necessary to consider both qualitative and quantitative issues in practice.
Additionally, Hocine et al. [17] proposed a WA-FMCGP method to formulate triangular and trapezoidal MFs. However, there is a major restriction on their method: 2n binary variables were required in their model to formulate n triangular MFs, which becomes time-consuming when the problem size increases. Li et al. [18] also proposed a large-scale group decision-making method to manage crisp, interval and triangular numbers, and [19] a generalized fuzzy number to express decision makers' preferences. On the other hand, Zheng and Chang [20] proposed an MGCP model with a trapezoidal utility function to measure senior citizens' satisfaction in topology design. Zhang et al. [21] derived a consensus-reaching method for social network group decision making by considering leadership and bounded confidence. Zhang et al. [22] proposed a consensus reaching method for group decision making with multi-granular unbalanced linguistic information. Patro et al. [23] also presented an MCGP model using Vandermonde's interpolating polynomial, binary variables, and a least square approximation method.
Moreover, MCGP with triangular and trapezoidal MFs has been rarely studied in the past. To fill this gap, this study proposes a novel method with two contributions: (1) it can be easily used to formulate triangular and trapezoidal MFs, which will improve the usefulness of MCGP in solving real-world problems, and (2) only log 2 2n binary variables are required to formulate n triangular MFs. In addition, given the lack of literature, this study aims at understanding the relationship between MCGP and the nature-inspired optimization method to understand the related technologies of MODM better. Therefore, the new FMCGP and the nature-inspired optimization method (NIOM) are proposed to improve the usefulness of MCGP in the field of MODM. The relationships between the MCGP family members are shown in Figure 1. Although Lingo 12 [24] or related software can be used to solve the FMCGP model, Lingo cannot guarantee to find the 95268 VOLUME 9, 2021 global optimal solution when it faces a non-linear problem. Thus, many meta-heuristics, including nature-inspired optimization methods, are proposed as global optimization search algorithms to solve FMCGP problems.
The artificial bee colony (ABC) algorithm is a well-known technique, and many studies have shown that it has an excellent performance with high-quality solutions. Thus, this study first uses the ABC algorithm to solve FMCGP problems and finds out the differences between MCGP and ABC methods. This study also fully applies the ''equation constraint'' feature in the proposed NIOM to reduce the number of required decision variables for the original model, dramatically reducing a problem's complexity. Furthermore, the role of this study is to explore a new area that enriches the field of MODM. This paper proposes two new methods to formulate popular triangular and trapezoidal MFs. In addition, a NIOM method is also provided to enrich the related fields of MCGP. To prove the novelty of the NIOM and the proposed FMCGP methods, these same methods are used to solve the same set of MODM problems with triangular and trapezoidal MFs. A comparison of the accuracy of the solutions of both methods is further provided. To briefly demonstrate the comparison in the study, a flowchart is shown in Figure 2. First, it defines the triangular and trapezoidal MFs. Second, it establishes the MCGP and NIOM models for triangular and trapezoidal MFs. Third, it solves the MCGP and NIOM models and then compares the results. If the NIOM result is better than the MCGP, the findings are obtained, and the procedure is halted. Otherwise, the NIOM algorithm would be revised and compared again. Furthermore, this study provides more insightful information regarding the NIOM algorithm, which contributes to the MCGP and NIOM fields and their respective applications.
The remainder of this paper is organized as follows: Section 2 presents model formulations for the proposed model with triangular and trapezoidal MFs. Section 3 presents the artificial bee colony algorithm for solving triangular and trapezoidal MFs. In Section 5, we conclude our findings and recommendations for future research are provided.

II. MODEL FORMULATIONS A. PREVIOUS MODELS
The MCGP with UF µ i (y i ) proposed by Chang [3] allows DM to set their preferences mapping with linear UF for an MODM problem. The linear UFs can be expressed as in (1) and (2) as shown in Figure 3.
where g i,max and g i,min are lower and upper bounds for the ith goal, respectively. Further, the MCGP with utility functions proposed by Chang [3] was expressed, as follows: , i = 1, 2, . . . , n, for LLUF , i = 1, 2, . . . , n, for RLUF (4) x ∈ F, (F is a feasible set, while x is unrestricted in sign), where w i and β i are weights attached to deviational variables d + i , d − i and f − i ; λ i is the utility value. As described in (6), the highest possible value of the L(R)LUF (from (3) and (4)) is 1. Other variables are defined as in MCGP (see Chang [1]).

B. PROPOSED MODEL FOR TRIANGULAR MF FORMULATION
A triangular MF is widely used to solve many uncertainty problems, which can be expressed as:   Particularly, an example of two triangular MFs is demonstrated in Figure 4, which can be represented by M i1 = µ i1 ∪ µ i2 and M i2 = µ i3 ∪ µ i4 , where µ ij is a sub-triangular MF. To improve the usefulness of MCGP, a new triangular MF should be added. Therefore, Figure 4 can be intuitively formulated, as follows: X ∈ F, (F is a feasible set), where d + i and d − i are positive and negative deviational variables attached to |f i (y) −g i |; α i and β i are the weights attached to d + i and d − i ; w ij is the weight attached to e − ij ; f i (y) is the i th objective function; z ij is the binary variable; µ ij is the MF ofg ij ; e − ij is the negative deviational variable used to force the value of µ ij approaching 1 (i.e., the maximum value of MF). g max ij (g min ij ) is upper (lower) bound of y ij ; y i1 ∈ [g min i1 ,g i1 ], y i2 ∈ [g i1 , g max i1 ], y i3 ∈ [g min i2 ,g i2 ], and y i4 ∈ [g i2 , g max i2 ] are additional variables. Proposition 1: P1 and Figure 4 are equivalent in the sense that they have the same optimal solutions.
By referring to Chang [25], the quadratic binary term z i1 z i2 in (10) can be linearized, as follows: , where x i satisfy the following inequalities.
The management implication of P1 is that two MFs (M1 and M2) represent the aspiration levels on the right-hand side of (10) as ''choosing one from them.'' For example, the two MFs can be used to map two types of customer satisfaction in the MODM model. This provides an optimal choice of MFs for MODM problems. As seen in P1, plus 4 sign constraints (from (18)) are required to deal with 2 triangular MFs, 2 binary variables (from (10)), and 16 auxiliary constraints (from (15)-(17)). Accordingly, to deal with the number of n triangular MFs, log 2 2n binary variables, 8n auxiliary constraints, plus 2n sign constraints are required. This handling is better than the model of Hocine et al. [17], where 2n binary variables are needed in their model to deal with the number of n triangular MFs. In order to reduce the auxiliary constraints in P1, Proposition 2 is introduced, as follows: Proposition 2: Given the two infinite sets K = {1, . . . , n} and J = {1, . . . , log 2 n }, extra continuous variables are denoted as c k (k ∈ K ), which can be constructed by adding a logarithmic number of binary variables b j (j ∈ J ) and the following constraints. where Example 1: A company is manufacturing two products, y 1 and y 2 . For product y 1 , there are two types of customers, namely general customers (GC) and VIP customers (VIPC), with ''approximate'' demands of 30 and 50, respectively. These demands can be represented by different MFs, as shown in Figure 4. The maximum allowable negative and positive deviations for GC and VIPC from their goals are set as 4 and 5, respectively. Further, (G1) is denoted as y 1 ∼ = 30 or 50 and (G2) as y 2 ∼ = 15 or 30. The selling profit for the product y 1 (y 2 ) is 10 (12) dollars. The information about these two products is shown in Table 1. However, due to limitations such as political ones, the company must select only one of its customers for each product. A profit of at least 450 dollars  from the product is expected. Weights w 1 and w 2 are given as 0.4 and 0.3, respectively. The available amount of resources for products y 1 and y 2 is shown in Table 2. The objective is to find the best degree of achievement of the two fuzzy goals.
Based on the proposed FMCGP and P1 models, the problem can be formulated as follows.
Based on Proposition 2, (22) and (25) can be replaced, as follows: where c i , ∀i are continous variables, and z i , ∀i are binary variables. This problem is solved again using Lingo 12 [24] to obtain the same solutions.

C. PROPOSED MODEL FOR TRAPEZOIDAL MF FORMULATION
A trapezoidal MF can be expressed as (37).
An example of two trapezoidal MFs is demonstrated in Figure 5, which can be represented by To improve the usefulness of MCGP, a new trapezoidal MF should be added. Therefore, Figure 5 can be intuitively formulated, as follows: where d + i and d − i are positive and negative deviational variables attached to |f i (y i ) −g i |; α i and β i are the weights attached to d + i and d − i ; w ij is the weight attached to e − ij ; f i (y i ) is the objective function; z ij is the binary variable; µ ij is the MF; e − ij is the negative deviational variable used to force µ ij and R ij approaching 1 (i.e., maximum value of MF). g max are additional variables. Proposition 3: P2 and Figure 5 are equivalent in the sense that they have the same optimal solutions.
Example 2: Eating disorders or imbalanced diets have recently become a serious health problem worldwide. Many chronic diseases such as atherosclerosis, heart disease, and stroke are associated with imbalanced diets. Basal metabolic rate (BMR) is the rate at which the body uses energy while at rest to maintain vital functions such as breathing and body temperature. In this example, the revised Harris-Benedict equation from the American college of sports medicine is used to determine a person's daily energy expenditure in calories. The BMR is calculated as follows: BMR(male) = (13.397 × weight(kg)) +(4.799 × height(cm)) − (5.677 × age) + 66 BMR(female) = (9.247 × weight(kg)) + (3.098 × height(cm)) − (4.33 × age) + 447.593 The basic summation of calories to be burned off is BMR plus 400 Kcal, to account for daily activities such as walking, thinking, breathing, and other important functions.
A 26-year-old man, Sam, is selected as the analytical target whose height and weight are 168cm and 77kg, respectively. Sam's basic consumption of calories is calculated as follows: The acceptable distribution range of nutrients is shown in Table 3. On the other hand, the range of target values for carbohydrate, protein, and fat is depicted in Figures 6-8. Twenty-seven types of food have been chosen from four categories (grains, meat, vegetable, egg), where each food has nine types of nutrients, as shown in Table 4. The objective is to find the best degree of carbohydrate, protein, and fat achievement for Sam.

1) GOALS
Maximize the degree of achievement of carbohydrate, protein, and fat  total intake amount of carbohydrate (Kcal) x 2 : total intake amount of protein (Kcal) x 3 : total intake amount of fat (Kcal) x 4 : total intake amount of vitamin A (µg) x 5 : total intake amount of vitamin C (mg) x 6 : total intake amount of vitamin B1 (mg) x 7 : total intake amount of vitamin B2 (mg) x 8 : total intake amount of calcium (mg) x 9 : total intake amount of iron (mg) x U j : upper bound of total intake of the nutrient j x L j : lower bound of total intake of the nutrient j . . , 27) and j th nutrients (j = 1, . . . , 9) The objective function maximizes the degree of achievement of carbohydrate, protein, and fat, as follows: Max M 1 (y) + M 2 (y) + M 3 (y) Furthermore, the constraints of the problem are described as follows: 1. Nutrients x 1 , . . . , x 9 must fall within a certain interval, as shown in Table 4. Therefore, the constraint x L j ≤ x j ≤ x U j (j = 1, . . . , 9) should be obeyed. Assuming his personal needs, his nutrition may have some non-linear limitations as in 2x 2 6 + 3x 2 7 ≤ 27 and x 6 x 7 x 9 ≤ 120.  2. The amount of nutrients is made up of selected foods. Therefore, it can be expressed as: 1, . . . , 9).

III. SOLVING M1 AND M2 BY ARTIFICIAL BEE COLONY ALGORITHM
The ABC algorithm is used to solve M1 and M2 and demonstrate the usefulness of the proposed methods. The ABC algorithm is a swarm-based metaheuristic algorithm introduced by Karaboga [26] for optimizing numerical problems. Satisfactory ABC-based algorithm results have been reported when applied to the numerical test functions of optimization problems [27], [28]. Moreover, a honeybee swarm inspires the emergence of ABC algorithm by its intelligent foraging behavior, and has been transformed into artificial bees postulated in three groups: employed bees, onlooker bees, and scout bees. An employed bee exploits a food source; an onlooker bee decides to search a food source; and a scout bee performs a random search for a new food source. When an employed bee abandons a food source, it becomes a scout bee. Meanwhile, as employed bees and onlooker bees perform the exploitation process, the scout bees control the exploration process.
The proposed ABC algorithm uses four variables as parameters: (1) SN refers to the number of food sources, (2) limit denotes a predefined number that if the food source cannot be improved in successive limit iterations, the employed bee becomes a scout bee to explore a new solution, (3) G max is the number of maximal allowable iterations, and (4) Penalty is the penalty of a solution if equations are violated. Both numbers of employed bees and onlooker bees are set as SN .
The pseudo-code of the proposed ABC algorithm is shown in Figure 9. It begins with a population of randomly generated food sources (initialize each solution X i in the first population by random, i = 2, . . . ., SN ). In order to obtain a good initial solution, we set the initialize solution X 1 in the first population using the solution obtained by FMCGP. Assume that the food source is X i (X i = [x i,1 , x i,2 , . . . , x i,D ]) and j ∈ {1, . . . , D}, where D is the number of decision variables in the solution; x i,j = x min,j + rand(0, 1)(x max,j − x min,j ), where x min,j andx max,j are the minimum and maximum values of the x i,j , and rand(0, 1) gives a random real value between 0 and 1. For simplicity, we set x min,j and x max,j to be 0 and 1, respectively, so all variables will range from 0 to 1.
Furthermore, the calculation of the fitness of food sources is shown in Figure 8 (later explained). A typical iteration of ABC proceeds as follows. While the employed bees and onlooker bees are both placed in their food sources, the latter is concerned with their nectar volume. On the other hand, scout bees sent to the search area to discover new food sources X i do not change in successive limits (limit). An onlooker's selection is contingent on the probability associated with a certain food source: p i = fit i / Finally, the best food source (X best ) found is recorded to the memory. This process repeats until it satisfies the termination condition (the number of iterationGequals the maximum number of iteration G max ) [29], [30].
To apply the ABC algorithm in solving an FMCGP problem, a solution (food source) is represented by D continuous values, where D is the number of the necessary decision variables in the model. The continuous values range from 0.0 to 1.0 (x min,j and x max,j set to be 0 and 1, respectively). For each continuous variable in the model, its value can be determined by v lower + (v upper − v lower ) · [ * ], where v lower and v upper are the lower and upper bound of the variable and [ * ] is the value obtained for the variable from the proposed ABC algorithm. For each binary variable in the model, if the value obtained from the proposed ABC algorithm is smaller than 0.5, the binary variable is set to 0; otherwise, the binary variable is set to 1. Although many decision variables exist in the FMCGP, many of them are deducible when parts of the decision variables are given (many of the equations equal (=) one) which significantly reduce the complexity of the problem.
As shown in Figure 10, after values of variables of the FMCGP model are obtained, the OFV can then be calculated directly. For each equation, if the violation happens, a hefty penalty is added to the OFV, computed by a constant plus the degree of violation. Further, the fitness value of a solution is computed as 1.0/OFV -the larger the value, the better the solution.
Moreover, the proposed algorithm is implemented using Visual C++ 2015 under Windows 10 operating system. The parameter SN and limit are set to 20 and 100, respectively, to be consistent with the default as in Karaboga and Basturk [31]. The penalty is set to 100,000. In addition, the number of iterations is set to 500,000 to balance the computing time and solution quality.

A. SOLVING M1 BY ABC ALGORITHM
In solving the M1 through the proposed ABC algorithm, a solution has the following 14 variables: x 1 , x 2 , . . . , x 8 , y 1 , and y 2 are non-negative variables with given ranges, while z 1 , z 2 , z 3 , and z 4 are binary variables. If the values of the 14 decision variables are given, the values The best solution is taken among 10 calculations of the M1 by the proposed ABC algorithm. The detailed best solutions obtained by Lingo 12 [24] and the proposed ABC algorithm for M1 are shown in Table 5. A prominent result is shown by the ABC (OFV = 0.9150181) compared to that of Lingo 12 (OFV = 15.2080426). Furthermore, the solution obtained by the ABC is (y 1 , y 2 , µ 1 , µ 2 , µ 3 , µ 4 , z 1 , z 2 , z 3 , z 4 ) = (28.0010294, 14.2849002, 1.0, 0.5002614, 1.0, 1.0, 1, 0, 1, 1) and the degree of achievement of two fuzzy goalsg y1 andg y2 is 50.026143% and 100.00%  In solving the M2 by the proposed ABC algorithm, a solution has the following 42 variables: y 1 to y 9 are continuous decision variables, while z 1 to z 6 , G 1 to G 6 , M 7 to M 15 , V 16 to V 24 , and E 25 to E 27 are binary decision variables. If the value of the above decision variables are given, the value of x 1 to x 9 , R 1 to R 2 , u 1 to u 6 , and e 1 to e 9 can be obtained by the (54)-(66) in the model. After all values of variables of the FMCGP model are obtained, the OFV can be calculated directly. A penalty is added to avoid the violation of (67)-(68) and greatly enlarge the OFV resulting in a smaller fitness value of the solution. As shown in Table 6, the OFV obtained by Lingo 12 [24] and the proposed ABC algorithm is 2.7870707 and 2.6809091, respectively. Consequently,  we can imply that the solution obtained by the ABC algorithm for M2 is better than that of Lingo 12. The solution obtained by the ABC is (x 1 , x 2 , x 3 , µ 1 , R 1 , µ 2 , µ 3 , R 2 , µ 4 , µ 5 , R 3 , µ 6 , z 1 , z 2 , z 3 , z 4 , z 5 , z 6 , M 1 (y), M 2 (y), M 3 (y)) = (1307.9, 485.2, 234. 75, 1, 1, 1, 0.6044444, 0, 1, 0.7146465, 0, 1, 1, 0, 0). This means that the demand for carbohydrates is fully met, although the degree of achievement of protein and fat is 60.444444% and 71.464646%, respectively. Further, the degree of protein achievement is much better, while the degree of achievement of fat is closer to that of Lingo. To show the convergence trend of the proposed ABC algorithm for M1 and M2, the OFV obtained with the ''number of iterations'' is displayed in Figures 11 and 12, respectively. As shown, the solution improvement rate decreases over an increasing number of iterations, and there is no more improvement in the best solution obtained after a certain number of iterations. Thus, using more iterations may not enhance the solution quality. In addition, the ABC algorithm can obtain more precise solutions than that obtained by the MCGP method.
According to the results of M1 and M2, the proposed ABC algorithm may obtain better solutions than the immediate solutions from Lingo 12 [24]. The reason the proposed ABC algorithm acquires better solutions may be as follows: (1) the ABC algorithm is a global optimization search algorithm; (2) many equations equal (=) one in the fuzzy MCGP and after giving the values of parts of the decision variables in these equations, the remaining variables could be deduced, which significantly reduces the complexity of the problem.
Moreover, this fills in the gap between the MCGP and NIOM methods. In the future, we can integrate MCGP and NIOM as a new method to enrich the field of MCDM techniques for solving qualitative and quantitative issues.

IV. CONCLUSION AND FUTURE RESEARCH
With the current status of available knowledge, this study is the first to investigate the relationship between MCGP and NIOM, aiming with two contributions between them and further improve the usefulness of MCGP. Therefore, new models are proposed to enrich FMCGP with triangular and trapezoidal MFs and improve the usefulness of MCGP for solving more MODM problems. NIOM is also provided to solve the same problem and know the differences between them. In addition, a comparison of the two methods is exhibited to understand their differences better and enrich the knowledge of both MCGP and NIOM. Furthermore, the new algorithms provided improve the efficacy of NIOM in solving fuzzy decision/management problems. Meanwhile, the proposed ABC algorithm uses many equal (''='') equations in the fuzzy multi-choice model. After giving the values of parts of the decision variables in these equations, the remaining variables could be deduced in the NIOM, which significantly reduces the complexity of the fuzzy MCGP model. In addition, the proposed methods could easily be used to solve multiple objective problems while considering qualitative and quantitative issues at the same time [20].
Moreover, several possible future directions exist for this research. In the future, other NIOMs, such as particle swarm optimization, firefly algorithm, bacterial foraginginspired algorithm, genetic algorithm, and cuckoo search algorithm, can also be used to solve the FMCGP problems with non-linear constraints and MFs. Another exciting potential research direction is applying the proposed NIOM algorithm in solving more complex FMCGP problems with non-linear constraints and MFs in real applications. Finally, the proposed NIOM algorithm and new MCGP methods can be used to solve multi-attribute decision-making problems [32] while considering qualitative and quantitative issues simultaneously.