Genetic algorithm based hybrid approach to solve fuzzy multiobjective assignment problem using exponential membership function
 Jayesh M. Dhodiya^{1}Email author and
 Anita Ravi Tailor^{1}
Received: 7 June 2016
Accepted: 14 November 2016
Published: 28 November 2016
Abstract
This paper presents a genetic algorithm based hybrid approach for solving a fuzzy multiobjective assignment problem (FMOAP) by using an exponential membership function in which the coefficient of the objective function is described by a triangular possibility distribution. Moreover, in this study, fuzzy judgment was classified using αlevel sets for the decision maker (DM) to simultaneously optimize the optimistic, most likely, and pessimistic scenarios of fuzzy objective functions. To demonstrate the effectiveness of the proposed approach, a numerical example is provided with a data set from a realistic situation. This paper concludes that the developed hybrid approach can manage FMOAP efficiently and effectively with an effective output to enable the DM to take a decision.
Keywords
Background
The assignment problem (AP) has been extensively used in manufacturing and developing service systems, to optimally resolve the problem of assigning N duties to N employees to optimize the total resources. Furthermore, in AP, N employees must be assigned N number of duties, where in each employee must complete their individual assigned duty. However, because of personal ability or other reasons, each employee may spend a different amount of resources to complete various duties. The objective is to assign each duty to the appropriate employee to optimize the total utilization of resources and to complete all duties. Two types of objectives are generally measured in AP: maximization and minimization. Minimization refers to minimizing aspects such as duty cost and total duration, whereas maximization refers to maximizing aspects such as the overall manufacturing profit and overall sale of manufacture.
Responsible parameters that are considered for determining the assignment plan in a realworld scenario should not be precise but should be exaggerated by indistinctness and imprecision represented by linguistic terms which expressed by the DM. In such a scenario, the AP is converted into a fuzzy assignment problem (FAP). The concept of fuzzy set theory was introduced by Zadeh (1965), which providesa highly effective method for handling imprecise data. In the decisionmaking real world problems, AP is more advantageous by fuzzy theory, subjective preference of DM. The fuzzy models of AP have been described in detail in several papers (Biswas and Pramanik 2011; Lin and Wen 2004; Lin et al. 2011; Li et al. 2012; Tanaka et al. 1984; Kagade and Bajaj 2009, 2010; Kumar and Gupta 2011; Liu and Gao 2009; Gupta and Mehlawat 2013; Mukherjee and Basu 2010; Feng and Yang 2006).
For handling objective functions and\or constraints with fuzzy coefficients and fuzzy information of realworld decisionmaking problems, possibilistic decisionmaking models play a vital role. Possibility distribution converts the fuzzy objectives and\or constraints into crisp objectives and\or constraints with respect to three scenarios, optimistic, mostlikely, and pessimistic. In addition, possibility distribution is used to maintain the uncertainty of the problem until the solution is obtained (Gupta and Mehlawat 2014). Several studies in literature have employed possibility distribution to solve fuzzy objective function and\or constraintbased optimization problems (Tanaka et al. 1984; Luhandjula 1987; Rommelfanger et al. 1989; Rommelfanger 1989; Lai and Hwang 1992).
There are several studies on FMOAP available in literature. To the best of our knowledge, Yang and Liu (2005) obtained a solution for a fuzzy multiobjective assignment problem (FMOAP) through the dependentchance goal programming model by using the tabu search algorithm based on fuzzy simulation. Gupta and Mehlawat (2014) proposed a possibilistic programming approach for FMOAPs to obtain the most favorable, most likely, and least favorable scenarios by using the linear membership function. Li et al. (2012) proposed two solution models for FAP by combining agenetic algorithm (GA) and the APs to express an actual execution strategy. Tapkan et al. (2013) provided a direct approach to solving FMOAPs by using fuzzy ranking methods to rank the objective function values and to determine the feasibility of the constraints within the bees algorithm (metaheuristic search algorithm). Tailor and Dhodiya (2016a) developed a hybrid approach to solving FMOAPs by using a GA and exponential membership function. The Yager’s ranking method was proposed in Biswas and Pramanik (2011) to solve FMOAPs by transforming the MOAP into an equivalent single objective AP problem. Pramanik and Biswas (2012) developed a prioritybased fuzzy goal programming method for generalized trapezoidal FMOAPs. Thorani and Shankar (2013) developed a linear programming model for FMOAPs by employing various linear and nonlinear functions with L–R fuzzy numbers by using Yager’s ranking method. Esmaieli et al. (2011) solved the fuzzy multijob and multicompany employee AP with penalty by using a GA.
The aforementioned approaches present the solution for FMOAP by using various techniques, such as the possibilistic approach by using a linear membership function, Yager’s ranking method, prioritybased fuzzy goal programming method, and bees algorithm, etc. However, in realworld problems, the decision parameters are affected by various imprecise and vague factors that cannot be precisely calculated. Moreover, the DM’s review of the estimates may be based on partial knowledge about the task itself, which may affect the decision of task allocations to a particular employee. Under such circumstances, the task allocation decision becomes a one ofchoice from a fuzzy set of subjective interpretations. Therefore, in this paper, we propose a GAbased hybrid approach to solving FMOAP by using a fuzzy exponential membership function in which the FMOAP is converted into a single objective nonlinear optimization problem with some realistic constraints, and it is considered as a “NPhard” problem. GA is an appropriate technique to solve such “NPhard” problems (Gupta and Mehlawat 2013; Papadimitriou and Steiglitz 1982). It is a wellknown random search and global optimization method, considering the aspects of evolution and natural selection, and an appropriate method for solving largescale nonlinear, discrete, and nonconvex optimization problems because it searches for optimal solutions by simulating the natural evolution process (Eiben and Smith 2003; Holland 1992; Mendes et al. 2009; Gupta and Mehlawat 2013). GAs are highly efficient in the resolution of various NPhard problems, including resource allocation.
Fuzzy multiobjective assignment problem formulation
The main characteristics and assumptions of the FMOAP are as follows: (1) Each duty is completed by only one employee, and if an employee accepts more than one duty, all the duties must be completed. (2) It is not compulsory to assigned any duty to some employees. (3) The number of employees who have been assigned duties must be specified to balance the amount of work between the employees. (4) In the decisionmaking method, each employee’s working ability is considered. We assume that each employee is assigned the number of duties in a certain range.
Fuzzy multiobjective assignment model
Formulation of objective functions
Model constraints
Decision problem
Some preliminaries
Possibilistic programming approach
The collection data on realworld problems generally involve some type of uncertainty. In fact, many pieces of information cannot be quantified because of their nature and hence are represented using fuzzy numbers. These types of fuzzy numbers are modeled using possibility distribution (Hsu and Wang 2001; Buckley 1988; Gupta and Mehlawat 2014; Wang and Liang 2005; Lai and Hwang 1992). Possibilistic distribution has been used in many crucial applications to solve fuzzy optimization models with imprecise coefficients in the objective function. Thus, we converted the FMOAP model into an auxiliary crisp multiobjective optimization model by using the possibilistic approach (Gupta and Mehlawat 2014).
Triangular possibilistic distribution (TPD)
Because of the imprecise nature of the uncertain parameters, the triangular possibilistic distribution (TPD) is commonly used due to its simplicity and computational effectiveness in obtaining data.
αLevel sets
An αlevel set is the most essential theory to establish an association between traditional and fuzzy set theories, which was introduced by Zadeh (1965). The αlevel reflects the confidence of the DM regarding his fuzzy judgment; it can also be termed as the confidence level. The smallest αvalue yields an interval judgment with a large spared, which indicates a high level of pessimism and uncertainty. The largest αvalue yields a smaller but more optimistic judgment in which the upper and lower bounds have a greater degree of membership in the initial fuzzy sets. Several researchers (Tanaka et al. 1984; Luhandjula 1987; Rommelfanger et al. 1989; Rommelfanger 1989; Tailor and Dhodiya 2016a; Lai and Hwang 1992) have used this αlevel set concept to find the solutions for fuzzy optimizationrelated problems; therefore, we also used this concept in the present study to determine the confidence of the DM with respect to his fuzzy judgment.
Formulation of multiobjective 0–1 programming model
Using the αlevel sets concepts (\(0\le \alpha \le 1\)), each \(c_{ij}\) can be stated as \(\left( c_{ij} \right) _{\alpha }=\left( \left( c_{ij} \right) _{\alpha }^{o},\left( c_{ij}\right) _{\alpha }^{m},\left( c_{ij} \right) _{\alpha }^{p} \right)\), where \(\left( c_{ij} \right) _{\alpha }^{o} =c_{ij}^{o} +\alpha \left( c_{ij}^{m} c_{ij}^{o} \right) , \left( c_{ij} \right) _{\alpha }^{m} =c_{ij}^{m}, \left( c_{ij}\right) _{\alpha }^{p} =c_{ij}^{p} \alpha \left( c_{ij}^{p}c_{ij}^{m} \right).\)
Auxiliary multiobjective 0–1 programming model
Solution method for auxiliary model
To characterize the indistinct aspiration level of the DM, fuzzy membership functions such as linear, piecewise linear, exponential, and tangent are used. Out of these, the linear membership function is most commonly used because it is defined by two fixed points, the upper bound and lower bound of the objective, and also considered only a violent calculation of realworld circumstances. In addition, membership functions are used for describing the behavior of uncertain values, fuzzy data use, and preference etc. In such situation, the nonlinear membership function provides a more efficient representation than others to reflect the reality as the marginal rate of increasing membership values as a function of model parameter, which is not constant (Gupta and Mehlawat 2013).
GA is one of the most adaptive optimization search methodologies, which is based on natural genetics, natural selection, and survival of the fittest in a biological system. It mimics the evaluating principle and chromosome processing of natural genetics (Eiben and Smith 2003; Esmaieli et al. 2011; Gen et al. 1995; Holland 1992; Li et al. 1997; Mendes et al. 2009; Gupta and Mehlawat 2013; Sivanandam and Deepa 2007; Tailor and Dhodiya 2016a, b). To determine the solution of a single optimization FMAOP through GA, the chromosomes are first encoded according to the problem and a fitness function is defined for measuring the chromosomes. Subsequently, three operators, selection, crossover, and mutation, are applied to generate the new population. The selection process involves the formation of a parent population for creating the next generation. The crossover process involves the selection of two parent chromosomes to produce a new offspring chromosome. Mutation refers to randomly altering the selected positions in a selected chromosome (Gupta and Mehlawat 2013; Tailor and Dhodiya 2016b). Thus, the new population is generated by replacing some chromosomes in the parent population with those of the children population to determine effective solutions for FMOAPs (Tailor and Dhodiya 2016b).
This section presents a GAbased hybrid approach to determining the most efficient solution for an FMOAP by using the exponential membership function to characterize the indistinct aspiration levels of the DM. In addition, this approach provides greater flexibility to solve multiobjective optimization problems by considering the various choices of aspiration level for each objective function. This approach optimizes each objective by maximizing the degree of satisfaction with respect to cost, time, and quality to provide more effective assignment plans.
Steps for find the solution of FMOAP using genetic algorithm based approach
 Step1 :

Formulate the model1 of FMOAP, using appropriate triangular possibilities distribution.
 Step2 :

According to confidence level α, define the crisp objective function model (model2).
 Step3 :

Findout the positive ideal solution (PIS) and negative ideal solution (NIS) (Gupta and Mehlawat 2014) for each objective function of the model2.
 Step4 :

Find fuzzy exponential membership value for \(z_{ij} ({\rm i}=1, 2, 3; {\rm j}=1, 2, 3)\).where, \(\psi _{ij} \left( x\right) =\frac{z_{ij} z_{ij}^{{\rm PIS}} }{z_{ij}^{{\rm NIS}} z_{ij}^{{\rm PIS}}}\) and S is nonzero shape parameter given by DM that \(0\le \mu _{z_{ij}} \left( x\right) \, \le 1\,\). For \(S>0\, (S<0)\), the membership function is strictly concave (convex) in [\(z_{ij}^{{\rm PIS}}, z_{ij}^{{\rm NIS}}\)]. The value of this fuzzy membership function allows us to model the grades of precision in corresponding objective function (Gupta and Mehlawat 2013).$$\begin{aligned} \mu _{z_{ij}}^{E} \left( x\right) =\left\{ \begin{array}{ll} 1;&{}\quad {\rm if}\, \, z_{ij} \le z_{ij}^{{\rm PIS}} \\ \frac{e^{S\psi _{ij} \left( x\right) } e^{S}}{1e^{S}};&{}\quad {\rm if}\, \, z_{ij}^{{\rm PIS}}<z_{ij} <z_{ij}^{{\rm NIS}}\\ 0;&{}\quad {\rm if}\, \, z_{ij} \ge z_{ij}^{{\rm NIS}} \end{array}\right. \end{aligned}$$(12)
 Step5 :

Fuzzy membership functions are comprehensive by using the product operator. Thus, FMOAP can be written in the singleobjective optimizationproblem (SOP) as follows:$$\begin{aligned}&{\mathbf{(Model}}{\text{}}{{\bf 3)}} \\&\max \, W = \prod \limits _{i = 1}^3 {\prod \limits _{j =1}^3 {{\mu _{{z_{ij}}}}}} \\&{\rm Subject}\,{\rm to}{:}\,{\rm Constraints}\,(1{}5) \end{aligned}$$$${\mu _{{z_{ij}}}}\left( x \right)  \overline{{\mu _{{z_{ij}}}}} \left( x \right) \ge 0;\quad i = 1, 2, 3;\quad j = 1, 2, 3$$(13)$$\mu _{z_{2j}} \left( x\right) \overline{\mu _{z_{2j}} }\left( x\right) \ge 0;\quad j=1, 2, 3$$(14)where \(\overline{\mu }_{z_{ij}} \left( x\right) ; i=1, 2, 3; j=1, 2,\, 3\,\) is the desired aspiration level of fuzzy goals corresponding to each objective. The above model can be solved for varying aspiration levels of the DM regarding the achievement of various fuzzy membership functions (Gupta and Mehlawat 2013).$$\mu _{z_{3j}} \left( x\right) \overline{\mu _{z_{3j}} }\left( x\right) \ge 0;\quad j=1, 2, 3$$(15)
 Step6 :

To solve the singleobjective optimization problem model3 of the FMOAP, GA is used with various choices of the shape parameter.

Chromosome encoding
To generate a solution for the FMOAP, the data structure of chromosomes must be considered, which represents the solution to the problem in the encoding space. In the encoding space, we set all 0’s to all \(n\times n\) genes on a chromosome, and then for a randomly selected gene on the chromosome, we set 1’s in each column exactly one and those in each row less or equal to \(l_{i}\) that satisfies constraints (1)–(5) of model3. Each component in the string (chromosome) can be uniquely expressed as \(2^r\); where r is real value varying from 0 to \({\rm n}1\).

Fitness function evaluation
In the GA, the fitness function is the major parameter for solving the FMOAP. The objective function of model3 that satisfies constraints (1)–(5) and (13)–(15) is evaluated.

Selection
The selection operator is used to determine which chromosome from the current population will be used to reproduce a new child with higher fitness for the next population. It is carefully formulated to select the chromosome in the population with the highest fitness for mutation/next generation. This operator improves the average quality of the chromosomes in the population for the next generation by providing the chromosomes with the highest quality a higher chance to get copied into Gupta and Mehlawat (2013) and Tailor and Dhodiya (2016a, b).
In this study, we used tournament selection for determining the solution for the FMOAP because of its efficiency and easy implementation. In tournament selection, N chromosomes are randomly selected from the population and compared with each other. The chromosome with the highest fitness (winner) is selected for the next generation and others are disqualified. This selection is continued until the number of winners is equal to the population size.

Crossover
After successful completion of tournament selection, the crossover operator is used to produce a new offspring for the next generation. The principle underlying crossover is that the offspring may exhibit a higher level of fitness than both parents if it inherits highquality characteristics from each parent.
To generate a solution for the FMOAP, we used the twopoint crossover operator to generate a new offspring. In a twopoint crossover, the gene values are exchanged between two random crossover points on the two selected parent chromosomes to generate the new offspring (Sivanandam and Deepa 2007; Tailor and Dhodiya 2016a).

Threshold construction
To maintain the population diversity after crossover, a threshold is constructed to generate the FMOAP’s solution. In this step, from the set of parenthood and childhood population some are selected for the new iteration.
For constructing the threshold, one method of selecting the population may be to sort the entire population in an ascending order of their objective function values and selecting predetermined individual strings from each category. The population is divided into four categories on the basis of their objective function values: values above \(\mu +3*\sigma\), values between \(\mu +3*\sigma\) and \(\mu\), values between \(\mu\) and \(\mu 3*\sigma\), and values less then \(\mu 3*\sigma\). Thus, the most efficient string cannot be missed (Sivanandam and Deepa 2007; Tailor and Dhodiya 2016a, b).

Mutation
For recovering the lost genetic materials and for randomly disturbing genetic information, the mutation operator is applied. In this study, we applied the swap mutation (Gupta and Mehlawat 2013; Tailor and Dhodiya 2016a, b) out of the numerous mutation operators available. In a swap mutation, two random spots are selected in a string and the corresponding values are swapped between the positions.
If we swap the string \(\langle 1, 2, 3, 4, 5\rangle\) at second and fourth position then the new mutated string become \(\langle 1, 4, 3, 2, 5\rangle\).

Termination criteria
When the algorithm completes a given number of iterations, it stops and provides the optimal solution as the output. The iteration process is repeated until a termination condition is reached.
If the obtained solution is accepted by the DM, then it is considered as the ideal compromise solution and the iteration is stopped, else the value ofis changed and steps 2–5 are repeated untila satisfactory solution is achieved.
Algorithm
Flowchart
Convergence criteria
A GA usually converges when no significant improvement is observed in the fitness values of the population from one generation to the next. GA converging at a global optima for an NPhard problem is impossible, unless the optimum solution for a test data set is already known. In GA, the convergence criteria also depends on the problem size. In this study, we considered a problem with size \(6 \times 6\) size problem. For this problem, we set the experimental parameters as follows: size = 4500, iterations = 90 at different values \(\alpha =0.1, \alpha =0.5\) and \(\alpha =0.9\). The experiment is presented in the following section.
Numerical illustration
Costtimequality matrix
Worker (i)  Job (j)  

Job1  Job2  Job3  Job4  Job5  Job6  
Worker1  
\(c_{ij}\)  (4, 6, 8)  (3, 4, 6)  (4, 5, 8)  (6, 8, 11)  (7, 10, 14)  (4, 6, 7) 
\(t_{ij}\)  (2, 4, 5)  (16, 20, 24)  (7, 9, 12)  (2, 3, 5)  (5, 8, 10)  (7, 9, 12) 
\(q_{ij}\)  (0, 1, 3)  (1, 3, 5)  (0, 1, 3)  (0, 1, 3)  (0, 1, 3)  (3, 5, 7) 
Worker2  
\(c_{ij}\)  (4, 6, 7)  (4, 5, 7)  (5, 6, 9)  (3, 5, 7)  (6, 9, 11)  (6, 8, 11) 
\(t_{ij}\)  (4, 6, 9)  (15, 18, 22)  (6, 8, 12)  (5, 7, 10)  (14, 17, 20)  (6, 8, 10) 
\(q_{ij}\)  (1, 3, 5)  (3, 5, 7)  (1, 3, 5)  (3, 5, 7)  (5, 7, 9)  (3, 5, 7) 
Worker3  
\(c_{ij}\)  (8, 11, 14)  (5, 7, 9)  (2, 4, 6)  (5, 8, 12)  (2, 3, 4)  (3, 4, 6) 
\(t_{ij}\)  (2, 3, 4)  (6, 8, 10)  (17, 20, 24)  (5, 7, 10)  (12, 15, 18)  (5, 7, 10) 
\(q_{ij}\)  (0, 1, 3)  (5, 7, 9)  (3, 5, 7)  (1, 3, 5)  (3, 5, 7)  (5, 7, 9) 
Worker4  
\(c_{ij}\)  (7, 9, 12)  (7, 10, 12)  (6, 8, 11)  (4, 6, 8)  (8, 10, 12)  (3, 4, 6) 
\(t_{ij}\)  (10, 12, 16)  (10, 13, 16)  (12, 14, 18)  (4, 6, 9)  (7, 9, 12)  (8, 10, 14) 
\(q_{ij}\)  (3, 5, 7)  (7, 9, 10)  (1, 3, 5)  (3, 5, 7)  (1, 3, 5)  (1, 3, 5) 
Worker5  
\(c_{ij}\)  (3, 4, 6)  (4, 6, 8)  (5, 7, 10)  (7, 9, 12)  (6, 8, 12)  (5, 7, 10) 
\(t_{ij}\)  (7, 9, 12)  (5, 8, 11)  (5, 7, 10)  (11, 14, 18)  (3, 5, 8)  (7, 9, 12) 
\(q_{ij}\)  (1, 3, 5)  (7, 9, 10)  (5, 7, 9)  (3, 5, 7)  (1, 3, 5)  (1, 3, 5) 
Worker6  
\(c_{ij}\)  (2, 3, 4)  (4, 5, 7)  (8, 11, 15)  (8, 10, 13)  (9, 12, 15)  (6, 8, 12) 
\(t_{ij}\)  (14, 17, 21)  (10, 13, 16)  (2, 3, 5)  (3, 5, 8)  (10, 13, 17)  (5, 7, 10) 
\(q_{ij}\)  (1, 3, 5)  (1, 3, 5)  (3, 5, 7)  (5, 7, 9)  (3, 5, 7)  (5, 7, 9) 
PIS and NIS for fuzzy objective functions
αLevel  Solutions  Objectives  

\(z_{11}\)  \(z_{12}\)  \(z_{13}\)  \(z_{21}\)  \(z_{22}\)  \(z_{23}\)  \(z_{31}\)  \(z_{32}\)  \(z_{33}\)  
\(\alpha =0.1\)  PIS  15.8  23  32  20  29  40.7  3.9  12  22.8 
NIS  46.6  61  77.2  81.8  98  118.7  31.2  42  51.9  
\(\alpha =0.5\)  PIS  19  23  28  24  29  35.5  7.5  12  18 
NIS  53  61  70  89  98  109.5  36  42  47.5  
\(\alpha =0.9\)  PIS  22.2  23  24  28  29  30.3  11.1  12  13.2 
NIS  59.4  61  62.8  96.2  98  100.3  40.8  42  43.1 
According to triangular possibility distribution, the assignment plans for FMOAP are reported in below tables with different values of the shape parameters and aspiration levels which specified by the DM. We use here different values of for \(\alpha =0.1,\alpha =0.5\) and \(\alpha =0.9\) to reflect the different scenario of DM’s confidence about fuzzy decision.
Different values of shape parameters and aspiration level
Case  Shape parameter \(\left( K_{1}, K_{2}, K_{3} \right)\)  Aspiration level \(\left( \bar{\mu }_{Z_{1j}} (x),\bar{\mu }_{Z_{2j}} (x), \bar{\mu }_{Z_{3j}} (x)\right)\) 

Case1  (−5, −1, −2)  0.7, 0.8, 0.9 
Case2  (−5, −1, −2)  0.8, 0.85, 0.7 
Case3  (−5, −1, −2)  0.9, 0.7, 0.8 
Case4  (−1, −2, −5)  0.7, 0.8, 0.85 
Case5  (−1, −2, −5)  0.8, 0.7, 0.75 
Case6  (−2, −5, −1)  0.8, 0.85, 0.7 
Case7  (−2, −5, −1)  0.9, 0.75, 0.8 
Summary results for \(\alpha =0.1, \alpha =0.5\) and \(\alpha =0.9\)
α  Case  W  Membership values \(\left( \mu _{Z_{1j}}, \mu _{Z_{2j}}, \mu _{Z_{3j}} \right)\)  Objective values \(Z_{1}, Z_{2}, Z_{3}\)  Optimum allocations 

\(\alpha =0.1\)  1  0.8954  (0.9111, 0.9241, 0.8954)  (32.1, 42, 57.3)  \(x_{11}, x_{14}, x_{46}, x_{55}, x_{62}, x_{63}\) 
(0.9185, 0.9189, 0.9136)  (28.1, 38, 51.5)  
(0.9601, 0.9522, 0.9505)  (7, 16, 26.8)  
2  0.8527  (0.9499, 0.9640, 0.9488)  (28.9, 37, 51.4)  \(x_{11}, x_{14}, x_{23}, x_{46}, x_{55}, x_{62}\)  
(0.8730, 0.8691, 0.8527)  (32.2, 43, 58.3)  
(0.9869, 0.9777, 0.9769)  (5, 14, 24.8)  
3  0.8611  (0.9127, 0.9343, 0.9070)  (32, 41, 56.3)  \(x_{13}, x_{14}, x_{31}, x_{46}, x_{55}, x_{62}\)  
(0.8626, 0.8691, 0.8611)  (33.1, 43, 57.4)  
(1, 1, 1)  (3.9, 12, 22.8)  
4  0.9115  (0.9115, 0.9419, 0.9265)  (22.7, 29, 40.7)  \(x_{13}, x_{14}, x_{35}, x_{46}, x_{51}, x_{62}\)  
(0.9450, 0.9471, 0.9449)  (47.3, 59, 75.2)  
(0.9300, 0.9170, 0.9142)  (7, 16, 26.8)  
5  0.8667  (0.9254, 0.9419, 0.9172)  (21.8, 29, 41.6)  \(x_{11}, x_{34}, x_{35}, x_{46}, x_{51}, x_{62}\)  
(0.9274, 0.9271, 0.9220)  (50.4, 63, 80.1)  
(0.9032, 0.8711, 0.8667)  (8.1, 18, 28.8)  
6  0.7799  (0.8000, 0.8248, 0.7799)  (24.9, 33, 46.5)  \(x_{11}, x_{13}, x_{24}, x_{46}, x_{55}, x_{62}\)  
(0.8913, 0.8850, 0.8771)  (36.3, 48, 63.3)  
(0.9948, 0.9936, 0.9933)  (7, 16, 26.8)  
7  0.8240  (0.8974, 0.9182, 0.9055)  (20.8, 28, 38.8)  \(x_{11}, x_{13}, x_{24}, x_{35}, x_{36}, x_{62}\)  
(0.8334, 0.8240, 0.8298)  (42.4, 55, 69.4)  
(0.9709, 0.9709, 0.9690)  (13, 22, 32.8)  
\(\alpha =0.5\)  1  0.9080  (0.9178, 0.9241, 0.9080)  (36.5, 42, 50.5)  \(x_{11}, x_{14}, x_{46}, x_{55}, x_{62}, x_{63}\) 
(0.9178, 0.9189, 0.9158)  (32.5, 38, 45.5)  
(0.9564, 0.9522, 0.9512)  (11, 16, 22)  
2  0.8595  (0.9574, 0.9640, 0.9555)  (32.5, 37, 45)  \(x_{11}, x_{14}, x_{23}, x_{46}, x_{55}, x_{62}\)  
(0.8711, 0.8691, 0.8595)  (37, 43, 51.5)  
(0.9826, 0.9777, 0.9773)  (9, 14, 20)  
3  0.8644  (0.9241, 0.9343, 0.9191)  (36, 41, 49.5)  \(x_{13}, x_{14}, x_{31}, x_{46}, x_{55}, x_{62}\)  
(0.8657, 0.8691, 0.8644)  (37.5, 43, 51)  
(1, 1, 1)  (7.5, 12, 18)  
4  0.9155  (0.9271, 0.9419, 0.9328)  (25.5, 29, 35)  \(x_{13}, x_{14}, x_{35}, x_{46}, x_{51}, x_{62}\)  
(0.9460, 0.9471, 0.9458)  (52.5, 59, 68)  
(0.9240, 0.9170, 0.9155)  (11, 16, 22)  
5  0.8687  (0.9388, 0.9419, 0.9274)  (25, 29, 36)  \(x_{13}, x_{34}, x_{35}, x_{46}, x_{51}, x_{62}\)  
(0.9273, 0.9271, 0.9241)  (56, 63, 72.5)  
(0.8884, 0.8711, 0.8687)  (12.5, 18, 24)  
6  0.7983  (0.8124, 0.8248, 0.7983)  (28.5, 33, 40.5)  \(x_{11}, x_{13}, x_{24}, x_{36}, x_{55}, x_{62}\)  
(0.9120, 0.9076, 0.9053)  (38.5, 45, 53)  
(0.9815, 0.9810, 0.9805)  (15, 20, 26)  
7  0.8385  (0.8877, 0.9005, 0.8945)  (25, 29, 35)  \(x_{11}, x_{13}, x_{35}, x_{36}, x_{44}, x_{62}\)  
(0.8389, 0.8335, 0.8362)  (47, 54, 62)  
(0.9709, 0.9709, 0.9698)  (17, 22, 28)  
\(\alpha =0.9\)  1  0.9183  (0.9236, 0.9241, 0.9209)  (40.9, 42, 43.7)  \(x_{11}, x_{14}, x_{46}, x_{55}, x_{62}, x_{65}\) 
(0.9189, 0.9189, 0.9183)  (36.9, 38, 39.5)  
(0.9530, 0.9522, 0.9520)  (15, 16, 17.2)  
2  0.8671  (0.9628, 0.9640, 0.9623)  (36.1, 37, 38.6)  \(x_{11}, x_{14}, x_{23}, x_{46}, x_{55}, x_{62}\)  
(0.8695, 0.8691, 0.8671)  (41.8, 43, 44.7)  
(0.9786, 0.9777, 0.9776)  (13, 14, 15.2)  
3  0.8681  (0.9326, 0.9343, 0.9313)  (40, 41, 42.7)  \(x_{13}, x_{14}, x_{31}, x_{46}, x_{55}, x_{62}\)  
(0.8684, 0.8691, 0.8681)  (41.9, 43, 44.6)  
(1, 1, 1)  (11.1, 12, 13.2)  
4  0.8719  (0.8738, 0.8773, 0.8719)  (33.2, 34, 35.6)  \(x_{13}, x_{14}, x_{46}, x_{51}, x_{55}, x_{62}\)  
(0.9778, 0.9779, 0.9774)  (47.8, 49, 50.8)  
(0.9616, 0.9599, 0.9774)  (13, 14, 15.2)  
5  0.8707  (0.9159, 0.9181, 0.9139)  (30.2, 31, 32.5)  \(x_{13}, x_{24}, x_{46}, x_{51}, x_{55}, x_{62}\)  
(0.9682, 0.9682, 0.9675)  (51.7, 53, 54.9)  
(0.8754, 0.8711, 0.8707)  (16.9, 18, 19.2)  
6  0.8191  (0.8226, 0.8248, 0.8191)  (32.1, 33, 34.5)  \(x_{11}, x_{13}, x_{24}, x_{36}, x_{55}, x_{62}\)  
(0.9085, 0.9076, 0.9072)  (43.7, 45, 46.6)  
(0.9811, 0.9810, 0.9809)  (19, 20, 21.2)  
7  0.8335  (0.8981, 0.9005, 0.8992)  (28.2, 29, 30.2)  \(x_{11}, x_{13}, x_{35}, x_{36}, x_{44}, x_{62}\)  
(0.8345, 0.8335, 0.8340)  (52.6, 54, 55.6)  
(0.9709, 0.9709, 0.9707)  (21, 22, 23.2) 
The convergence rate of GA for FMOAP
To defuzzify the fuzzy number, Lai and Hwang (1992) provided the concept of most likely values to verify the efficiency of outputs. They determined crisp values for each objective corresponding to the triangular number. If cost \(\tilde{C}=\left( C^{o}, C^{m}, C^{p} \right)\) is a triangular fuzzy number, then the crisp value of cost objective is given as \(\tilde{C}=\left( \frac{C^{o} +4C^{m} +C^{p}}{6} \right)\) which provided the most likely value of the objective function.
Furthermore, if the DM is not satisfied with the obtained compromise solution, then the desired objective function can be improved as per the preference of the DM. For example, in an AP with fuzzy cost, time, and quality objectives, if the DM prioritizes the cost objective in determining the period of allocation plan, the solution that satisfies the cost objective function most favorably than others is selected by the DM. However, this can result in poor degrees of satisfaction level because the performance of one objective may be compensated by the efficient performance of others. Hence, the DM can select different solutions in different situations, according to his/her requirements. Therefore, to generate a new membership function, the upper bound of the selected objective function is modified using the DM’s preference. The model is resolved using new parameters and the iterations are continued until the DM terminates the process (Gupta and Mehlawat 2013, 2014; Tailor and Dhodiya 2016a).
Compromised solutions with respect to improvement desired in various objective at confidence level \(\alpha =0.1\) with different shape parameter
Case  Obj. function  Bounds  \(\lambda\)  Objective values  Solution variables 

(\(Z_{1}, Z_{2}, Z_{3}\))  \(x_{ij}\)  
Shape parameter: (−5, −1, −2)  
Aspiration level: (0.8, 0.85, 0.7)  
1  Cost  \(15.8 \le z_{11} \le 32,\)  0.8025  (26.9, 35, 48.5),  \(x_{11}, x_{23}, x_{36}, x_{44}, x_{55}\),\(x_{62}\) 
\(23 \le z_{12} \le 41,\)  (31.3, 43, 58.3),  
\(32 \le z_{13}\le 56.3\)  (12.1, 22, 32.8)  
2  Quality  \(3.9\le z_{31}\le 7,\)  0.8611  (32, 41, 56.3),  \(x_{13}, x_{14}, x_{31}, x_{46}, x_{55}\),\(x_{62}\) 
\(12 \le z_{32} \le 16,\)  (33.1, 43, 57.4),  
\(22.8 \le z_{33} \le 26.8\)  (3.9, 12, 22)  
Aspiration level: (0.9, 0.7, 0.8)  
1  Cost  \(15.8 \le z_{11}\le 28.9,\)  0.7104  (22.8, 30, 40.8),  \(x_{11}, x_{23}, x_{35}, x_{44}, x_{46}\),\(x_{62}\) 
\(23 \le z_{12} \le 41,\)  (43.4, 56, 72.2),  
\(37 \le z_{13} \le 51.4\)  (10.1, 20, 30.8)  
Aspiration level: (0.7, 0.8, 0.9)  
1  Cost  \(15.8 \le z_{11} \le 32.9\)  0.8143  (25.9, 34, 47.5),  \(x_{11}, x_{13}, x_{44}, x_{46}, x_{55}\),\(x_{62}\) 
\(23 \le z_{12}\le 41,\)  (35.3, 47, 62.3),  
\(32 \le z_{13} \le 56.3\)  (7, 26, 26.8)  
Shape parameter: (−2, −5, −1)  
Aspiration level: (0.8, 0.85, 0.7)  
1  Time  \(20 \le z_{21} \le 47.3,\)  0.8159  (24.9, 33, 46.5),  \(x_{13}, x_{14}, x_{46}, x_{51}, x_{55}, x_{62}\) 
\(29 \le z_{22} \le 59,\)  (33.3, 45, 59.4),  
\(40.7 \le z_{23} \le 75.2\)  (11, 20, 30.8)  
Shape parameter: (−1, −2, −5)  
Aspiration level: (0.8, 0.7, 0.75)  
1  Quality  \(3.9 \le z_{31}\le 13.1\),  0.7784  (23.7, 30, 41.7),  \(x_{12}, x_{15}, x_{23}, x_{34}, x_{46}, x_{61}\) 
\(12 \le z_{32} \le 22\),  (46.3, 58, 75.1),  
\(22.8\le z_{33} \le 32.8\)  (8.1, 18, 28.8) 
Compromised solutions with respect to improvement desired in various objective at confidence level \(\alpha =0.5\) with different shape parameter
Case  Obj. function  Bounds  \(\lambda\)  Objective values  Solution variables 

(\(Z_{1}, Z_{2}, Z_{3}\))  \(x_{ij}\)  
Shape parameter: (−5, −1, −2)  
Aspiration level: (0.8, 0.85, 0.7)  
1  Cost  \(19 \le z_{11} \le 36.5\),  0.8255  (30.5, 35, 42.5),  \(x_{11}, x_{23}, x_{36}, x_{44}, x_{55}, x_{62}\) 
\(23 \le z_{12} \le 42\),  (36.5, 43, 51.5),  
\(28 \le z_{13} \le 50.5\)  (16.5, 22, 28)  
2  Quality  \(7.5 \le z_{31} \le 11\),  0.7311  (32.5, 37, 45)  \(x_{11}, x_{14}, x_{23}, x_{46}, x_{55}, x_{62}\) 
\(12 \le z_{32} \le 16\),  (37, 43, 51.5),  
\(18 \le z_{33} \le 22\)  (9, 14, 20)  
3  Quality  \(7.5 \le z_{31} \le 9\),  0.8644  (36, 41, 49.5)  \(x_{13}, x_{14}, x_{31}, x_{46}, x_{55}, x_{62}\) 
\(12 \le z_{32} \le 14\),  (37.5, 43, 51),  
\(18 \le z_{33} \le 20\)  (7.5, 12, 18)  
Aspiration level: (0.9, 0.7, 0.8)  
1  Cost  \(19 \le z_{11} \le 32.5\),  0.7550  (26, 30, 36),  \(x_{11}, x_{23}, x_{35}, x_{36}, x_{44}, x_{62}\) 
\(23 \le z_{12} \le 37\),  (46, 53, 61.5),  
\(28 \le z_{13} \le 45\)  (18.5, 24, 30)  
2  Quality  \(7.5 \le z_{31} \le 9\),  0.8644  (36, 41, 49.5)  \(x_{13}, x_{14}, x_{31}, x_{46}, x_{55}, x_{62}\) 
\(12 \le z_{32} \le 14\),  (37.5, 43, 51),  
\(18 \le z_{33} \le 20\)  (7.5, 12, 18)  
Aspiration level: (0.7, 0.8, 0.9)  
1  Cost  \(19 \le z_{11} \le 36\),  0.8142  (29.5, 34, 41.5),  \(x_{11}, x_{23}, x_{24}, x_{46}, x_{55}, x_{62}\) 
\(23 \le z_{12} \le 41,\)  (40.5, 47, 56),  
\(28 \le z_{13} \le 49.5\)  (12.5, 18, 24)  
Shape parameter: (−2, −5, −1)  
Aspiration level: (0.8, 0.85, 0.7)  
1  Cost  \(19 \le z_{11} \le 25.5\),  0.8187  (21.5, 25, 30.5),  \(x_{13}, x_{24}, x_{35}, x_{46}, x_{61}, x_{62}\) 
\(23 \le z_{12} \le 29\),  (63.5, 71, 81),  
\(28 \le z_{13} \le 35.5\)  (14.5, 20, 26)  
2  Time  \(24 \le z_{21} \le 51.5\),  0.8048  (32.5, 37, 45),  \(x_{11}, x_{14}, x_{23}, x_{46}, x_{55}, x_{62}\) 
\(29 \le z_{22} \le 58\),  (37, 43, 51.5),  
\(35.5 \le z_{23} \le 67.5\)  (9, 14, 20)  
Shape parameter: (−1, −2, −5)  
Aspiration level: (0.7, 0.8, 0.85)  
1  Quality  \(7.5 \le z_{31} \le 15\),  0.8124  (28.5, 33, 40.5),  \(x_{11}, x_{13}, x_{24}, x_{46}, x_{55}, x_{62}\) 
\(12 \le z_{32} \le 20\),  (41.5, 48, 56.6),  
\(18 \le z_{33} \le 26\)  (9, 14, 20)  
2  Quality  \(7.5 \le z_{31} \le 11\),  0.7199  (31, 36, 44.5),  \(x_{11}, x_{13}, x_{34}, x_{46}, x_{55}, x_{62}\) 
\(12 \le z_{32} \le 16\),  (41.5, 48, 56.6),  
\(18 \le z_{33} \le 22\)  (9, 14, 20)  
Aspiration level: (0.8, 0.85, 0.7)  
1  Quality  \(7.5 \le z_{31} \le 17\),  0.7848  (26.5, 30, 36.5),  \(x_{14}, x_{23}, x_{35}, x_{46}, x_{51}, x_{62}\) 
\(12 \le z_{32} \le 22\),  (51.5, 58, 67.5),  
\(18 \le z_{33} \le 28\)  (12.5, 18, 24) 
Compromised solutions with respect to improvement desired in various objective at confidence level \(\alpha =0.9\) with different shape parameter
Case  Obj. function  Bounds  \(\lambda\)  Objective values  Solution variables 

\(Z_{1}, Z_{2}, Z_{3}\)  \(x_{ij}\)  
Shape parameter: (−5, −1, −2)  
Aspiration level: (0.8, 0.85, 0.7)  
1  Cost  \(22.2 \le z_{11} \le 40.9\),  0.8578  (33.1, 34, 35.5),  \(x_{11}, x_{13}, x_{36}, x_{44}, x_{55}, x_{62}\) 
\(23 \le z_{12} \le 42\),  (42.7, 44, 45.6),  
\(24 \le z_{13} \le 43.7\)  (19, 20, 21.2)  
2  Time  \(28 \le z_{21} \le 36.9\),  0.7574  (46.7, 48, 49.8),  \(x_{14}, x_{31}, x_{36}, x_{52}, x_{55}, x_{62}\) 
\(29 \le z_{22} \le 38\),  (28, 29, 30.4),  
\(30.3 \le z_{23} \le 39.5\)  (25, 26, 27.1)  
3  Quality  \(11.1 \le z_{31} \le 15\),  0.8681  (40, 41, 42.7),  \(x_{13}, x_{14}, x_{31}, x_{46}, x_{55}, x_{62}\) 
\(12 \le z_{32} \le 16\),  (41.9, 43, 44.6),  
\(13.2 \le z_{33} \le 17.2\)  (11.1, 12, 13.2)  
Aspiration level: (0.9, 0.7, 0.8)  
1  Cost  \(22.2 \le z_{11} \le 36.1\),  0.7322  (29.3, 30, 31.3),  \(x_{14}, x_{23}, x_{35}, x_{36}, x_{51}, x_{62}\) 
\(23 \le z_{12} \le 37\),  (53.7, 55, 56.8),  
\(24 \le z_{13} \le 38.6\)  (20.9, 22, 23.2)  
2  Time  \(28 \le z_{21} \le 41.8\),  0.8029  (41.8, 43, 44.7),  \(x_{14}, x_{31}, x_{36}, x_{52}, x_{55}, x_{62}\) 
\(29 \le z_{22} \le 43\),  (31.9, 33, 34.5),  
\(30.3 \le z_{23} \le 44.7\)  (21, 22, 23.1)  
3  Quality  \(11.1 \le z_{31} \le 13\),  0.8681  (40, 41, 42.7),  \(x_{13}, x_{14}, x_{31}, x_{46}, x_{55}, x_{62}\) 
\(12 \le z_{32} \le 14\),  (41.9, 43, 44.6),  
\(13.2 \le z_{33} \le 15.2\)  (11.1, 12, 13.2)  
Aspiration level: (0.7, 0.8, 0.9)  
1  Cost  \(22.2 \le z_{11} \le 40\),  0.8009  (33.1, 34, 35.4),  \(x_{13}, x_{21}, x_{44}, x_{46}, x_{55}, x_{62}\) 
\(23 \le z_{12} \le 41\),  (47.7, 49, 50.9),  
\(24 \le z_{13} \le 42.7\)  (16.9, 18, 19.2)  
Shape parameter: (−2, −5, −1)  
Aspiration level: (0.8, 0.85, 0.7)  
1  Cost  \(22.2 \le z_{11} \le 33.2\),  0.8215  (25.3, 26, 27.2),  \(x_{13}, x_{24}, x_{35}, x_{46}, x_{51}, x_{62}\) 
\(23 \le z_{12} \le 34\),  (61.6, 63, 64.9),  
\(24 \le z_{13} \le 35.6\)  (18.9, 20, 21.2)  
2  Time  \(28 \le z_{21} \le 47.8\),  0.7126  (37, 38, 39.5),  \(x_{14}, x_{21}, x_{36}, x_{53}, x_{55}, x_{62}\) 
\(29 \le z_{22} \le 49\),  (39.8, 41, 42.7),  
\(30.3 \le z_{23} \le 50.8\)  (22.9, 24, 25.2)  
Shape parameter: (−1, −2, −5)  
Aspiration level: (0.7, 0.8, 0.85)  
1  Quality  \(11.1 \le z_{31} \le 19,\)  0.7972  (33.2, 34, 35.6),  \(x_{13}, x_{14}, x_{46}, x_{51}, x_{46}, x_{62}\) 
\(12 \le z_{32} \le 20\),  (47.8, 49, 50.8),  
\(13.2 \le z_{33} \le 21.2\)  (13, 14, 15.2)  
Aspiration level: (0.8, 0.7, 0.75)  
2  Quality  \(11.1 \le z_{31} \le 21\),  0.8191  (32.1, 33, 34.5),  \(x_{11}, x_{13}, x_{24}, x_{46}, x_{55}, x_{62}\) 
\(12 \le z_{32} \le 22\),  (46.7, 48, 49.7),  
\(13.2 \le z_{33} \le 23.2\)  (15, 16, 17.2) 
Tables 5, 6 and 7 report the preferred compromise solutions obtained by modifying the upper bounds of various objectives with differing values of confidence level and shape parameters, and differing estimates of aspiration levels for various confidence levels α. As shown in the above table, the GAbased hybrid approach helps to improve the TPD by modifying the upper bound of each objective function for particular values of α (Gupta and Mehlawat 2014). If the DM is not satisfied with the obtained assignment plans, more assignment plans can be generated by integrating the preference of the DM for various objectives and also altering the various shape parameters.
The GAbased hybrid approach provides flexibility and facilitates the collection of large amounts of information in terms of altering the α level and shape parameters in the exponential membership function and providing various scenario analyses to the DM for fuzzy allocation strategy.
Sensitivity analysis with respect to the number of workers and jobs
Post optimality analysis with respect to the number of workers and jobs is discussed in this section to measure, how the proposed solution method handle FMOAP effectively when new workers and jobs are involve. In this paper, sensitivity analysis is considered by adding the new data of jobs and keeping the workers fixed as given in the article of Gupta and Mehlawat (2014). For \(\alpha =0.1\), and the fuzzy input data \(l_{i} =3\) and s = 4, solution and assignment plans of FMOAP with extra nine jobs (Job7 to Job15) and same six workers are shown in Table 9 with its triangular possibilistic distributions for each objectives.
The computational results are shown in the Table 9 for same six worker and extra nine jobs (6workers and 9jobs, 6workers and 11jobs, 6workers and 13jobs), respectively by taking different estimation of the aspiration levels for each combination of the shape parameters.
Different values of shape parameters and aspiration level
Case  Shape parameter \(\left( K_{1}, K_{2}, K_{3} \right)\)  Aspiration level \(\left( \bar{\mu }_{Z_{1j}} (x),\bar{\mu }_{Z_{2j}} (x), \bar{\mu }_{Z_{3j}} (x)\right)\) 

Case1:  (−5, −1, −2)  0.7, 0.8, 0.9 
Case2:  (−5, −1, −2)  0.8, 0.85, 0.7 
Case3:  (−5, −1, −2)  0.9, 0.7, 0.8 
Case4:  (−1, −2, −5)  0.7, 0.8, 0.85 
Case5:  (−1, −2, −5)  0.8, 0.7, 0.75 
Case6:  (−2, −5, −1)  0.8, 0.85, 0.7 
Results summery of sensitivity analysis w.r.t number of jobs at \(\alpha = 0.1\)
No. of jobs  Case  W  Membership values \(\left( \mu _{Z_{1j}}, \mu _{Z_{2j}}, \mu _{Z_{3j}} \right)\)  Objective values \(\left( Z_{1}, Z_{2}, Z_{3} \right)\)  Optimum allocations  

\(x_{1j}\)  \(x_{2j}\)  \(x_{3j}\)  \(x_{4j}\)  \(x_{5j}\)  \(x_{6j}\)  
Jobs9  1  0.8235  (0.9574, 0.9567, 0.9287) (0.8549, 0.8367, 0.8235) (0.9184, 0.9124, 0.9152)  (49.8, 66, 90.3) (47.9, 65, 85.7) (14.4, 27, 42.3)  \({x_{11}}\) \({x_{13}}\) \({x_{14}}\)  \(x_{37}\)  \({x_{48}}\) \({x_{49}}\)  \({x_{52}}\) \({x_{55}}\) \({x_{56}}\)  
2  0.8716  (0.9148, 0.9170, 0.8824) (0.8899, 0.8757, 0.8716) (0.9270, 0.9124, 0.9066)  (55.9, 73, 96.4) (43.8, 60, 78.9) (13.5, 27, 43.2)  \({x_{11}}\) \({x_{14}}\) \({x_{19}}\)  \(x_{38}\)  \({x_{55}}\) \({x_{56}}\)  \({x_{62}}\) \({x_{63}}\) \({x_{67}}\)  
3  0.8436  (0.9626, 0.9644, 0.9431) (0.8713, 0.8447, 0.8436) (0.8882, 0.8936, 0.8957)  (48.7, 64, 87.4) (46, 64, 82.9) (17.3, 29, 44.3)  \({x_{11}}\) \({x_{13}}\) \({x_{14}}\)  \(x_{36}\)  \({x_{48}}\) \({x_{49}}\)  \({x_{52}}\) \({x_{55}}\)  \(x_{67}\)  
4  0.8036  (0.8671, 0.8829, 0.8684) (0.8233, 0.8243, 0.8036) (0.9515, 0.9442, 0.9423)  (40.4, 53, 71) (60.8, 77, 100.4) (24.6, 39, 54.3)  \({x_{11}}\) \({x_{14}}\) \({x_{17}}\)  \(x_{23}\)  \({x_{32}}\) \({x_{35}}\) \({x_{38}}\)  \({x_{46}}\) \({x_{49}}\)  
5  0.7452  (0.7856, 0.7832, 0.7452) (0.9174, 0.8998, 0.8813) (0.9740, 0.9709, 0.9709)  (45.7, 61, 81.7) (47, 65, 87.5) (19.5, 33, 48.3)  \({x_{11}}\) \({x_{13}}\)  \(x_{38}\)  \({x_{44}}\) \({x_{46}}\) \({x_{49}}\)  \({x_{52}}\) \({x_{55}}\)  \(x_{67}\)  
6  0.8434  (0.8763, 0.8807, 0.8434) (0.9665, 0.9623, 0.9527) (0.8902, 0.8552, 0.8466)  (44.6, 59, 80.6) (59, 77, 101.3) (12.6, 27, 43.2)  \(x_{13}\)  \({x_{21}}\) \({x_{24}}\)  \(x_{37}\)  \({x_{46}}\) \({x_{48}}\) \({x_{49}}\)  \(x_{55}\)  \(x_{62}\)  
Jobs11  1  0.8528  (0.9490, 0.9456, 0.9283) (0.8735, 0.8685, 0.8528) (0.9241, 0.9203, 0.9201)  (63.3, 84, 111) (57.2, 77, 102.2) (17.7, 33, 51.9)  \({x_{11}}\) \({x_{13}}\) \({x_{14}}\)  \(x_{38}\)  \({x_{46}}\) \({x_{411}}\)  \({x_{52}}\) \({x_{55}}\) \({x_{59}}\)  \({x_{67}}\) \({x_{610}}\)  
2  0.8516  (0.9503, 0.9567, 0.9406) (0.8655, 0.8552, 0.8516) (0.9241, 0.9203, 0.9201)  (63, 81, 108) (58.3, 79, 102.4) (17.7, 33, 51.9)  \({x_{13}}\) \({x_{14}}\) \({x_{111}}\)  \(x_{210}\)  \(x_{31}\)  \({x_{46}}\) \({x_{48}}\) \({x_{49}}\)  \({x_{52}}\) \({x_{55}}\)  \(x_{67}\)  
3  0.8706  (0.9296, 0.9456, 0.9359) (0.8885, 0.8881, 0.8706) (0.8995, 0.8908, 0.8883)  (66.9, 84, 109.2) (55.1, 74, 99.2) (20.8, 37, 55.9)  \({x_{13}}\) \({x_{14}}\) \({x_{111}}\)  \(x_{210}\)  \(x_{31}\)  \({x_{46}}\) \({x_{47}}\)  \({x_{55}}\) \({x_{58}}\)  \({x_{62}}\) \({x_{69}}\)  
4  0.8304  (0.8498, 0.8519, 0.8304) (0.8409, 0.8402, 0.8461) (0.9651, 0.9609, 0.9587)  (50.9, 68, 90.5) (72.3, 93, 116.4) (27.9, 45, 63.9)  \({x_{13}}\) \({x_{111}}\)  \(x_{21}\)  \({x_{36}}\) \({x_{37}}\) \({x_{38}}\)  \({x_{44}}\) \({x_{45}}\) \({x_{49}}\)  \({x_{62}}\) \({x_{610}}\)  
5  0.8105  (0.9074, 0.9081, 0.8874) (0.8210, 0.8221, 0.8105) (0.9438, 0.9354, 0.9336)  (45.8, 62, 83.6) (75.3, 96, 123) (33, 51, 69)  \({x_{13}}\) \({x_{111}}\)  \(x_{24}\)  \(x_{35}\)  \({x_{46}}\) \({x_{49}}\)  \({x_{51}}\) \({x_{52}}\) \({x_{58}}\)  \({x_{67}}\) \({x_{610}}\)  
6  0.8093  (0.9096, 0.9167, 0.9066) (0.9590, 0.9615, 0.9575) (0.8577, 0.8236, 0.8093)  (50.8, 67, 88.6) (76.2, 96, 123) (19, 37, 56.8)  \({x_{14}}\) \({x_{111}}\)  \({x_{21}}\) \({x_{23}}\)  \({x_{35}}\) \({x_{37}}\)  \({x_{46}}\) \({x_{48}}\) \({x_{49}}\)  \({x_{62}}\) \({x_{610}}\)  
Jobs13  1  0.8118  (0.9098, 0.9265, 0.9127) (0.8518, 0.8248, 0.8118) (0.9182, 0.9110, 0.9014)  (83.4, 105, 137.4) (71, 98, 128.6) (24.1, 43, 66.4)  \({x_{13}}\) \({x_{113}}\)  \({x_{210}}\) \({x_{211}}\)  \({x_{31}}\) \({x_{36}}\)  \({x_{44}}\) \({x_{48}}\) \({x_{49}}\)  \({x_{55}}\) \({x_{512}}\)  \({x_{62}}\) \({x_{67}}\) 
2  0.8636  (0.8734, 0.8705, 0.8837) (0.9086, 0.9056, 0.9031) (0.8824, 0.9713, 0.8636)  (87.9, 114, 142.8) (61.4, 83, 110) (29.2, 49, 71.5)  \({x_{13}}\) \({x_{14}}\) \({x_{113}}\)  \({x_{21}}\) \({x_{26}}\) \({x_{29}}\)  \({x_{38}}\) \({x_{312}}\)  \({x_{45}}\) \({x_{47}}\) \({x_{411}}\)  \(x_{52}\)  \(x_{610}\)  
3  0.8170  (0.9598, 0.9672, 0.9550) (0.8469, 0.8305, 0.8170) (0.9109, 0.8985, 0.8872)  (73.2, 93, 125.4) (71.8, 97, 127.6) (25.2, 45, 68.4)  \({x_{11}}\) \({x_{14}}\) \({x_{111}}\)  \({x_{23}}\) \({x_{210}}\)  \({x_{36}}\) \({x_{37}}\)  \({x_{48}}\) \({x_{412}}\)  \({x_{55}}\) \({x_{513}}\)  \({x_{62}}\) \({x_{69}}\)  
4  0.8260  (0.8260, 0.8322, 0.8364) (0.8727, 0.8786, 0.8586) (0.9708, 0.9609, 0.9542)  (52.9, 70, 89.8) (67.1, 86, 113.9) (26.1, 45, 64.8)  \({x_{13}}\) \({x_{111}}\)  \(x_{21}\)  \({x_{36}}\) \({x_{37}}\) \({x_{38}}\)  \({x_{44}}\) \({x_{45}}\) \({x_{49}}\)  \({x_{62}}\) \({x_{610}}\)  
5  0.8296  (0.8380, 0.8421, 0.8296) (0.8778, 0.8786, 0.8630) (0.9830, 0.9773, 0.9740)  (51.9, 69, 90.6) (66.2, 86, 113) (21, 39, 58.8)  \({x_{13}}\) \({x_{111}}\)  \(x_{21}\)  \(x_{38}\)  \({x_{46}}\) \({x_{47}}\) \({x_{49}}\)  \(x_{55}\)  \({x_{62}}\) \({x_{67}}\) \({x_{610}}\)  
6  0.8186  (0.8438, 0.8540, 0.8352) (0.9749, 0.9726, 0.9684) (08619, 0.8347, 0.8186)  (74.3, 95, 125.6) (82.8, 108, 139.5) (30.3, 51, 73.5)  \({x_{15}}\) \({x_{111}}\) \({x_{113}}\)  \({x_{23}}\) \({x_{24}}\)  \(x_{37}\)  \({x_{46}}\) \({x_{48}}\)  \({x_{51}}\) \({x_{512}}\)  \({x_{62}}\) \({x_{69}}\) \({x_{610}}\) 
The computational results and its corresponding assignment plans of sensitivity analysis (15jobs and 6workers) are presented in the Table 9 at \(\alpha =0.1\) for different shape parameter and aspiration level.
Moreover, this paper presents a sensitivity analysis by adding the new data of workers (Worker7 to Worker9) and keeping the jobs (Job1 to Job15) as given in the article of Gupta and Mehlawat (2014). For \(\alpha =0.1\), and the fuzzy input data \(l_{i} =3\) and s = 4, solution and assignment plans of FMOAP with extra workers (worker7 to worker9) and fifteen jobs are shown in Table 11 with its triangular possibilistic distributions for each objectives.
The computational results are shown in the Table 11 or additional worker and 15 jobs (7workers and 15jobs, 8workers and 15jobs, 9workers and 15jobs) respectively by taking different estimation of the aspiration levels for each combination of the shape parameters.
Different values of shape parameters and aspiration level
Case  Shape parameter \(\left( K_{1}, K_{2}, K_{3} \right)\)  Aspiration level \(\left( \bar{\mu }_{Z_{1j}} (x),\bar{\mu }_{Z_{2j}} (x), \bar{\mu }_{Z_{3j}} (x)\right)\) 

Case1  (−5, −1, −2)  0.8, 0.85, 0.7 
Case2  (−5, −1, −2)  0.9, 0.7, 0.8 
Case3  (−1, −2, −5)  0.7, 0.8, 0.85 
Case4  (−1, −2, −5)  0.8, 0.7, 0.75 
Case5  (−2, −5, −1)  0.8, 0.85, 0.7 
Results summery of sensitivity analysis w.r.t number of workers at \(\alpha = 0.1\)
No. of worker  Case  \(\lambda\)  \(\mu _{ij}\)  Objective values  Optimum allocations \(x_{ij}\) 

Workers7  1  0.7480  (0.8986, 0.9066, 0.8964) (0.9182, 0.9036, 0.8840) (0.8606, 0.8387, 0.7480)  (105, 132, 168.9) (68, 95, 131.9) (36.7, 61, 87.1)  \({x_{13}, x_{112}, x_{115}, x_{24}, x_{210},}\) \({x_{31}, x_{36}, x_{38}, x_{411}, x_{413},}\) \({x_{55}, x_{59}, x_{62}, x_{67}, x_{614}}\) 
2  0.8067  (0.9198, 0.9280, 0.9414) (0.8768, 0.8616, 0.8491) (0.9052, 0.8875, 0.8067)  (100.9, 127, 156.7) (76.1, 104, 140) (29.6, 53, 79.1)  \({x_{11}, x_{14}, x_{111}, x_{23}, x_{210},}\) \({x_{215}, x_{38}, x_{312}, x_{45}, x_{52},}\) \({ x_{56}, x_{513}, x_{69}, x_{614}, x_{77}}\)  
3  0.7303  (0.7716, 0.7433, 0.7303) (0.8944, 0.8933, 0.8805) (0.9648, 0.9577, 0.9162)  (85.2, 114, 148.2) (84.9, 111, 148.8) (40.7, 65, 91.1)  \({x_{12}, x_{18}, x_{112}, x_{23}, x_{24},}\) \({x_{36}, x_{37}, x_{314}, x_{49}, x_{411},}\) \({x_{413}, x_{52}, x_{610}, x_{615}, x_{75}}\)  
4  0.8131  (0.8131, 0.8202, 0.8304) (0.8362, 0.8263, 0.8287) (0.9597, 0.9523, 0.9070)  (80.6, 104, 133.7) (98.3, 128, 163.1) (42.7, 67, 93.1)  \({x_{13}, x_{19}, x_{111}, x_{21}, x_{213},}\) \({x_{32}, x_{35}, x_{314}, x_{44}, x_{46},}\) \({x_{47}, x_{58}, x_{512}, x_{515}, x_{610}}\)  
5  0.7539  (0.8179, 0.8122, 0.8251) (0.9353, 0.9320, 0.9332) (0.8937, 0.8652, 0.7539)  (90, 117, 147.6) (113.4, 144, 180.9) (23.6, 47, 74)  \({x_{19}, x_{111}, x_{215}, x_{33}, x_{34},}\) \({x_{314}, x_{45}, x_{48}, x_{51}, x_{512},}\) \({x_{513}, x_{62}, x_{610}, x_{76}, x_{77}}\)  
Workers8  1  0.7932  (0.9092, 0.9083, 0.9110) (0.8911, 0.8836, 0.8578) (0.8482, 0.8187, 0.7932)  (108.2, 137, 170.3) (71.8, 97, 134.8) (37.8, 63, 90)  \({x_{111}, x_{114}, x_{24}, x_{215}, x_{38},}\) \({x_{46}, x_{47}, x_{51}, x_{63}, x_{69},}\) \({x_{710}, x_{712}, x_{713}, x_{82}, x_{85}}\) 
2  0.7752  (0.9492, 0.9511, 0.9385) (0.8113, 0.7849, 0.7752) (0.9389, 0.9266, 0.9225)  (97.9, 124, 161.8) (87.4, 118, 154) (21.4, 43, 69.1)  \({x_{11}, x_{13}, x_{112}, x_{210}, x_{34},}\) \({x_{314}, x_{46}, x_{47}, x_{411}, x_{58},}\) \({x_{513}, x_{62}, x_{615}, x_{85}, x_{89}}\)  
3  0.7408  (0.7504, 0.7408, 0.7626) (0.8687, 0.8702, 0.8712) (0.9754, 0.9685, 0.9638)  (91, 118, 146.8) (90.8, 116, 149.3) (34.7, 59, 85.1)  \({x_{111}, x_{112}, x_{21}, x_{23}, x_{215},}\) \({x_{314}, x_{45}, x_{48}, x_{49}, x_{52},}\) \({x_{59}, x_{610}, x_{84}, x_{87}, x_{813}}\)  
4  0.8069  (0.8112, 0.8181, 0.8096) (0.8719, 0.8666, 0.8621) (0.9698, 0.9604, 0.9537)  (83.6, 107, 139.4) (90, 117, 152.1) (37.8, 63, 89.1)  \({x_{14}, x_{111}, x_{115}, x_{29}, x_{38},}\) \({x_{314}, x_{46}, x_{412}, x_{55}, x_{513},}\) \({x_{62}, x_{610}, x_{77}, x_{83}}\)  
5  0.8168  (0.8434, 0.8352, 0.8423) (0.9572, 0.9560, 0.9540) (0.8645, 0.8347, 0.8168)  (90, 117, 147.6) (104.2, 132, 168.9) (25.6, 49, 76)  \({x_{14}, x_{111}, x_{215}, x_{314}, x_{46},}\) \({x_{48}, x_{53}, x_{55}, x_{59}, x_{61},}\) \({x_{610}, x_{612}, x_{77}, x_{82}, x_{813}}\)  
Workers9  1  0.7975  (0.9192, 0.8991, 0.8994) (0.9197, 0.9064, 0.8898) (0.8401, 0.8158, 0.7975)  (108.8, 143, 177.2) (65.7, 90, 125.1) (37.6, 61, 87.1)  \({x_{16}, x_{111}, x_{113}, x_{28}, x_{29},}\) \({x_{215}, x_{31}, x_{37}, x_{44}, x_{52},}\) \({x_{73}, x_{85}, x_{812}, x_{910}, x_{914}}\) 
2  0.8208  (0.9371, 0.9380, 0.9279) (0.8636, 0.8265, 0.8208) (0.9149, 0.8936, 0.8810)  (104.1, 132, 168.9) (77.4, 108, 142.2) (23.6, 47, 74)  \({x_{113}, x_{21}, x_{210}, x_{312}, x_{44},}\) \({x_{48}, x_{411}, x_{55}, x_{62}, x_{67},}\) \({x_{714}, x_{89}, x_{93}, x_{96}, x_{915}}\)  
3  0.7298  (0.7346, 0.7429, 0.7298) (0.9093, 0.8960, 0.8809) (0.9524, 0.9381, 0.9258)  (94.8, 120, 154.2) (80, 107, 144.8) (43.8, 69, 95.1)  \({x_{14}, x_{112}, x_{115}, x_{23}, x_{29},}\) \({x_{211}, x_{38}, x_{314}, x_{46}, x_{51},}\) \({x_{62}, x_{75}, x_{713}, x_{97}, x_{910}}\)  
4  0.7653  (0.8351, 0.8271, 0.8083) (0.8056, 0.7922, 0.7653) (0.9626, 0.9497, 0.9435)  (81.8, 107, 141.2) (105.2, 134, 176.3) (39.8, 65, 90.2)  \({x_{17}, x_{18}, x_{111}, x_{23}, x_{26},}\) \({x_{34}, x_{314}, x_{45}, x_{49}, x_{413},}\) \({x_{52}, x_{512}, x_{61}, x_{910}, x_{915}}\)  
5  0.7615  (0.8495, 0.8517, 0.8607) (0.9718, 0.9657, 0.9585) (0.8275, 0.7847, 0.7615)  (90.8, 111.6, 145.7) (94.3, 124, 164.5) (28.7, 53, 80)  \({x_{111}, x_{38}, x_{314}, x_{44}, x_{46},}\) \({x_{413}, x_{55}, x_{61}, x_{610}, x_{77},}\) \({x_{89}, x_{812}, x_{815}, x_{92}, x_{93}}\) 
The computational results and the corresponding assignment plans obtained through sensitivity analysis (15 jobs and 9 employees) are presented in Table 11 at \(\alpha =0.1\) for various shape parameters and estimated aspiration levels. Moreover, as previously discussed, if the DM is not satisfied with the obtained compromise solution, more solutions can be obtained by improving an individual objective function as per the DM’s preference.
Thus, the sensitivity analysis reveals that the developed solution approach can handle the FMOAP successfully and proficiently when an additional employee and \or job is considered. In addition, if the DM is not satisfied with the obtained assignment plans, more assignment plans can be generated by changing the values of the shape parameters in the exponential membership function (Gupta and Mehlawat 2013, 2014; Tailor and Dhodiya 2016a).
Comparison
comparison between obtained solutions by GA based hybrid approach using exponential membership function with different approaches at \(\alpha = 0.1\)
Shape parameter  Aspiration level  Hannan (1981)  Lai and Hwang (1992)  Yager (1981)  Gupta and Mehlawat (2014)  Proposed hybrid approach 

Max \(\lambda\)  Max \(\lambda\)  Max \(\lambda\)  Max \(\lambda\)  Max \(\lambda\)  
(−5, −1, −2)  0.8, 0.85, 0.7  0.67342  0.53334  0.642109  0.695652  0.8954 
(−5, −1, −2)  0.9, 0.7, 0.8  0.8527  
(−5, −1, −2)  0.7, 0.8, 0.9  0.8611  
(−2, −5, −1)  0.8, 0.85, 0.7  0.9115  
(−2, −5, −1)  0.9, 0.75, 0.8  0.8667  
(−1, −2, −5)  0.7, 0.8, 0.85  0.7799  
(−1, −2, −5)  0.8, 0.7, 0.75  0.8240 
Conclusion
The GAbased hybrid approach provided the solution for the FMOAP by using the fuzzy exponential membership function with some realistic constraints to optimize the optimistic, most likely, and pessimistic scenarios of fuzzy objective functions with TPD. Moreover, the developed hybrid approach provided flexibility for the DM in terms of the various choices in aspiration levels, shape parameters, upper bound improvement and also provided more effective assignment plans.
Declarations
Authors' contributions
This work was carried out in collaboration between all authors. The authors AT and JMD participated actively to each phase of this work, as designed the algorithm, developing the solution method. The author AT designed the study, managed the literature searches, wrote the first draft of the manuscript the conduction experimental results. The author JMD managed the analyses of the study, designed the sensitivity and revised the manuscript critically for important intellectual content. Both authors read and approved the final manuscript.
Acknowledgements
The authors are thankful to Applied mathematics and humanities department of S. V. National Institute of Technology, Surat for the encouragement and facilities.
Competing interests
The authors declare that they have no competing interests regrading the publication of this paper.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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