Assignment Problem for Team Performance Promotion under Fuzzy Environment

This paper constructs a general fuzzy assignment problem (GFAP) based on a real-world scenario and proposes a solution procedure. Suppose a project team consists of n workers and a manager. The n workers are responsible for performing n jobs and the manager for restraining the total cost. The corresponding cost for a worker to perform his assigned job is not defined deterministically but as a subnormal fuzzy interval with increasing linear membership function. Job quality is then linearly and positively related to the cost of the job and is taken as the performance of the worker. On the other hand, the performance of the manager is negatively related to the total cost and is defined as a fuzzy interval with a decreasing linear membership function. It is common practice for a company to regard the lowest performance among members as the team performance in order to increase overall team performance. Hence, using the max–min criterion, a mixed nonlinear programming model of the GFAP is constructed. The model can be transformed into a general 0-1 fractional programming problem with max–min objective function. An algorithm that combines simplex and trade-off approaches is proposed to solve the problem. A numerical example and the computational results show that the constructed model and the proposed algorithm are useful and efficient.


Introduction
The assignment problem (AP) is a well-known and practical mathematical model.Given two disjoint -sets  = {1, 2, . . ., },  = {1, 2, . . ., } and the set  = {(, ) |  ∈  and  ∈ }, the AP can be stated as the following 0-1 programming problem: (1) Let  denote a feasible solution of (1).In a simplex-type algorithm,  is highly degenerate and consists of exactly  variables with the value 1; the remaining  2 - variables have the value 0. Defining  = {(, ) |   = 1,   ∈ },  constitutes the assignment corresponding to .Often (1) is practically interpreted as  jobs that must be performed by  workers at minimal total cost, and   is the job cost associated with worker  performing job .Here,  implies that each job must be assigned to one and only one worker, and vice versa.In the model, all   values are deterministic.
However, in many real-world applications, job costs are not deterministic.Fuzzy theory has been supported to slash the risk of improper models and solutions which do not reflect the practical problems [1].In recent years, many researchers have investigated the AP and its variants under fuzzy environments.Sakawa et al. [2] dealt with production and workforce assignment problems of a housing material manufacturer.Belacela and Boulasselb [3] developed a multicriteria fuzzy AP (FAP) that they applied to medical diagnosis.Ridwan [4] investigated a fuzzy preference-based traffic assignment problem.Lin and Wen [5] investigated a kind of FAP with an upper bound constraint on total cost.Liu and Li [6] proposed a new fuzzy quadratic AP as well as a genetic algorithm to solve their problem.Feng and Yang [7] developed a two-objective fuzzy -cardinality AP.Liu and Gao [8] studied the fuzzy weighted equilibrium multijob AP.Bashiri and Badri [9] proposed an interactive method for group decision making based on incomplete information using the fuzzy linear assignment method.Mukherjee and Basu [10] introduced an intuitionistic FAP using similarity measures and described the solution procedures.An investigation into sensitivity analysis of FAPs can be found in Lin et al. [11].
In this paper, a general FAP (GFAP) of Lin and Wen [5] is presented and an efficient solution procedure is proposed.The GFAP excludes the upper bound constraint on total cost, which makes the model more complicated and expands the application area.In addition to the team performance in human resource management, the GFAP can be used to situations that equally emphasize the time and cost, the quality and quantity, the humanity and technology, and so forth.The remainder of this paper is structured as follows.In the next section, the GFAP is constructed as a mixed nonlinear programming model.In Section 3, the constructed model is transformed into a general 0-1 fractional programming problem.The proposed algorithm for solving this problem is described in Section 4. A numerical example is illustrated in Section 5.In Section 6, computational results and two special cases are discussed.Finally, conclusions and suggestions for further research are given in Section 7.

Model Construction
Suppose that a project team consists of  workers and a manager.The  workers are responsible for performing  jobs and the manager for restraining the total cost.In many applications, the cost of a job depends not only on the skill of the worker but also on the job quality achieved.In general, the higher the job quality achieved, the greater the job cost required, and since a worker's performance is defined as job quality, the worker desires a job cost in order to raise his personal performance.On the other hand, the manager is required to restrain the total cost as much as possible.Thus, the total cost is negatively related to the performance of the manager.c and c denote the cost of worker  performing job  and the total cost, respectively.
Let   (0 <   ≤ 1) denote the highest achievable job quality  performed by worker .The membership function of job cost c is defined as the monotonically increasing linear function shown in (2).In this function,   is the minimum cost for worker  performing job, and   is the minimum cost associated with worker  performing job to reach   .When the input job cost lies between   and   , the job cost is linearly and positively related to the job quality, that is, the performance of the worker.However, any expense exceeding   is inefficient since the quality can no longer be enhanced.
Without loss of generality, it is assumed that 0 <   <   .If   = 0 in any feasible solution, there is no real expense; therefore   = 1 is added to the following: ⟨  ,   ⟩ denotes c .Matrix [c  ] is shown as follows: In where   is indeed the slope of the membership function.
The membership function of total cost c is defined as the monotonically decreasing linear function shown in (5). and  are the lower and upper bounds of c , respectively.Total cost is related to the performance of the manager.When the total cost is lower than , his performance is 1; when the total cost is greater than , his performance is 0; and when the total cost lies between  and , the total cost is linearly and negatively related to the performance of the manager.⟨, ⟩ denotes the fuzzy interval c : Since the performance of worker , denoted as   , is the job quality achieved, then In addition, the performance of the manager, denoted as   , is as follows: Usually, the company takes the lowest performance among members of the team, including all workers and the manager, as the team performance in order to increase overall performance.To maximize, the performance of each team member has to be equally emphasized.Hence, the Bellman-Zadeh criterion [12], which maximizes the minimum of all membership functions corresponding to that solution, is selected for this problem.Therefore, max-min or max-min where   is an element of a feasible solution X of ( 1).The FAP model can then be constructed as follows: max-min Utilizing the membership functions illustrated in ( 2) and ( 5), ( 10) can be represented as the following equivalent problem: where    denotes the -cut of c and ∑  =1 ∑  =1      is the corresponding total cost    .Since   ,    , and  all are decision variables, (11) can be treated as a mixed nonlinear programming problem.

Model Transformation
Let  be a specific feasible assignment of (1), that is, of (10).If discussion is confined on , (11) Theorem 1.Let  * be the optimal objective value of (12) corresponding to , then Proof.Transforming ( 12) into a linear programming problem, we find the following: The dual problem of ( 14) can then be obtained as follows: By (14),    ≥   + ((  −   )/  ) ⋅  > 0 holds for (, ) ∈ .According to the complementary slackness theorem, −  +  +1 = 0 for  ∈  (16) or Using ( 17) and ( 4), ( 15) can be simplified into the following: The dual problem of ( 18) is as follows: Obviously,  * depends on the given ; that is,  * is a function of .The following corollary is trivial.

Solution Algorithm
Of course, (21) can be solved by branch and bound (B&B) algorithms.However, in this section the S&T algorithm-a hybrid algorithm employing simplex and tradeoff approaches [13]-is proposed to obtain the optimal assignment and optimal objective function value in (21).
There is a tradeoff between the total cost and the jobs quality achieved, that is, between the performance of manager and workers.The S&T algorithm balances the maximum of manager's performance and the minimum of the workers' performances to maximize team performance.Usually, some S&T algorithm procedures must be repeated several times to solve problems with different updated parameters.In each solution, the maximum of manager's performance decreases (or remains unchanged) and the minimum of the workers' performances simultaneously increases until the maximal team performance is reached.
To obtain the maximal manager's performance, ( 22) is extracted from (21) as follows: Since the constraints of (22) form a totally unimodular matrix and its objective function is a pseudomonotonic model [14], the condition   ∈ {0, 1} in (22) can be replaced with   ≥ 0. Furthermore, we can define   = / −   and   = ( − )/ +   .Equation ( 22) then becomes equivalent to the following: Equation ( 23) is a special linear fractional problem, that is, a fractional assignment problem.Some algorithms for solving fractional assignment problems have been developed by Shigeno et al. [15] and Kabadi and Punnen [14] who showed that the optimum of (23) can be found at an extreme point of its solution polyhedron.A simplex-type algorithm can then be employed to solve (23).
The S&T algorithm consists of initialization and optimization procedures.The initialization procedure is first performed in a transportation table, as shown in Figure 1.The FCMP method [16] is employed to obtain a basic feasible solution X 0 of the maximal-objective-function transportation problem with [  /  ] as the profit matrix and 1 as the amount of each supply and demand.X 0 is then used as the initial solution to (23), as shown in Figure 2.This is viewed as the fractional transportation problem [17].Next, the objective function value  of X 0 in (23) is determined and associates   and V  with row  and column , respectively.Set  1 = 0 and identify the other   and V  using the equation   + V  =   −  ⋅   according to the basic variables of X 0 .Define the assignment as  = {(, ) |   = 1,   ∈ X 0 }.Remove the degenerated basic variables of X 0 from the table.
Step 2: Determine all values of   and V  according to X 0 .
Step 3: Remove the degenerate basic variables of X 0 and obtain .
Step B. Proceed with the following labeling procedure until further labeling is impossible or row  has received a label.If row  has been labeled, go to Step C; otherwise, go to Step D.
(2) If column  has been labeled, label the row  with , where   = 1.
Step C. Form a unique cycle by following the labels in reverse order.In the cycle, every odd cell is   = 0, and every even cell is   = 1.The odd cells are the recipient cells and thus become   = 1; the even cells are the donor cells, which become   = 0. Let   remain unchanged if cell (,) is not in the cycle.The new assignment is obtained and is taken as the new .The objective function value corresponding to the new  is taken as the new .
By keeping   unchanged, the new values of V ℓ and V () can be obtained by the following: where () is the label of row.
Other new variables   and V  related to the cycle can be obtained by keeping Δ  = 0 in the cycle.If the new values of   and V  are not related to the cycle, hold   unchanged and identify V  =   −   −  ⋅   , where (, ) ∈ .Return to Step A.
Step G. Update [  ], where Revise the objective function value  corresponding to .To renew   and V  , hold   unchanged and identify V  =   −   −  ⋅   , where (, ) ∈ .Go back to Step A.
Step H. Terminate the algorithm.The current  * and  * are the optimal assignment and optimal value of (21), respectively.

Numerical Example
In this section, for demonstration purpose, the S&T algorithm is applied to a numerical example.
Suppose the matrices [c  ] and [  ] are The S&T algorithm is applied to solve (21).
Phase I. Set  * → −∞ and  * = 0.The FCMP method is used to obtain X 0 , as shown in Figure 1.Its corresponding objective function value  = 170/279 can be obtained from (23).

Computational Experience
The computational test in this section verifies the efficiency of the proposed S&T algorithm.An index-based B&B algorithm is also used to solve (21).Both the S&T algorithm and the index-based B&B algorithm are coded in Visual Basic 6.0.They are compared with the global optimal function in In fact, problem (30) is a variant of the bottleneck AP.Hence, a proper algorithm for the bottleneck AP will improve the efficiency of (21) if  → ∞ is known in advance.
In the second special case,  is small; that is, the budget is insufficient for the team and none of the workers can reach the highest quality (performance) with the provided money.The computational results of the three approaches are shown in Table 3.It is clear that the proposed S&T algorithm is the most efficient.The elapsed time of the LINGO package is shorter than the value in Table 1; however, that of the index-based B&B algorithm is longer.In this case, the team performance is therefore chiefly determined by the manager and min An overview of Tables 1-3 reveals that, among the three approaches, the S&T algorithm is the most efficient.

Conclusion
This study formulated a GFAP to overcome the uncertain environment in practice and developed the S&T algorithm to solve the problem.A numerical example was presented to demonstrate the solution procedure and the computational results of the S&T algorithm were compared with those of the B&B algorithm and the LINGO package.The results show that, when all aspects are considered, the S&T algorithm is the most useful and efficient.In terms of future research, the development of a sensitivity analysis procedure for modifying the membership function could be an interesting field of study with many potential applications.

Figure 1 :
Figure 1: The results of the FCMP method.
addition, the   values form the matrix [  ], the   values form the matrix [  ], and the   values form the matrix [  ].
To find the assignment  * that maximizes the objective function value in (21), elements of matrix [  ] can be repeatedly changed so that  decreases or remains constant and  increases to improve  until the optimal objective function value  * in (21) is reached.

Table 3 :
Computational results when  is sufficiently small.