Assignment Problem for Team Performance Promotion under Fuzzy Environment

Abstract

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.

1. Introduction

The assignment problem (AP) is a well-known and practical mathematical model. Given two disjoint 𝑛-sets 𝐼={1,2,…,𝑛}, 𝐽={1,2,…,𝑛} and the set 𝐾={(𝑖,𝑗)∣𝑖∈𝐼 and 𝑗∈𝐽}, the AP can be stated as the following 0-1 programming problem: min^𝑛𝑛_𝑖=1𝑗=1𝑐_𝑖𝑗⋅𝑥_𝑖𝑗𝑐_𝑖𝑗≥0s.t.^𝑛_𝑗=1𝑥_𝑖𝑗=1for𝑖∈𝐼_𝑛𝑖=1𝑥_𝑖𝑗𝑥=1for𝑗∈𝐽_𝑖𝑗∈{0,1}for𝑖,𝑗∈𝐾. (1)

Let 𝑋 denote a feasible solution of (1). In a simplex-type algorithm, 𝑋 is highly degenerate and consists of exactly 𝑛 variables with the value 1; the remaining 𝑛²--𝑛 variables have the value 0. Defining 𝝎={(𝑖,𝑗)∣𝑥_𝑖𝑗=1,𝑥_𝑖𝑗∈𝑋}, 𝝎 constitutes the assignment corresponding to 𝑋. Often (1) is practically interpreted as 𝑛 jobs that must be performed by 𝑛 workers at minimal total cost, and 𝑐_𝑖𝑗 is the job cost associated with worker 𝑖 performing job 𝑗. Here, 𝝎 implies that each job must be assigned to one and only one worker, and vice versa. In the model, all 𝑐_𝑖𝑗 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 𝑘-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.

2. Model Construction

Suppose that 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. 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. 𝑐̃_𝑖𝑗 and 𝑐_𝑇̃ denote the cost of worker 𝑖 performing job 𝑗 and the total cost, respectively.

Let 𝑄_𝑖𝑗(0<𝑄_𝑖𝑗≤1) denote the highest achievable job quality 𝑗 performed by worker 𝑖. The membership function of job cost 𝑐̃_𝑖𝑗 is defined as the monotonically increasing linear function shown in (2). In this function, 𝛼_𝑖𝑗 is the minimum cost for worker 𝑖 performing job𝑗, and 𝛽_𝑖𝑗 is the minimum cost associated with worker 𝑖 performing job𝑗 to reach 𝑄_𝑖𝑗. When the input job cost lies between 𝛼_𝑖𝑗 and 𝛽_𝑖𝑗, the job cost is linearly and positively related to the job quality, that is, the performance of the worker. However, any expense exceeding 𝛽_𝑖𝑗 is inefficient since the quality can no longer be enhanced. Without loss of generality, it is assumed that 0<𝛼_𝑖𝑗<𝛽_𝑖𝑗. If 𝑥_𝑖𝑗=0 in any feasible solution, there is no real expense; therefore 𝑥_𝑖𝑗=1 is added to the following: 𝜇_𝑖𝑗𝑐_𝑖𝑗=𝑞_𝑖𝑗if𝑐_𝑖𝑗≥𝛽_𝑖𝑗,𝑥_𝑖𝑗𝑄=1_𝑖𝑗𝑐_𝑖𝑗−𝛼_𝑖𝑗𝛽_𝑖𝑗−𝛼_𝑖𝑗if𝛼_𝑖𝑗≤𝑐_𝑖𝑗≤𝛽_𝑖𝑗,𝑥_𝑖𝑗=10otherwise (2)⟨𝛼_𝑖𝑗,𝛽_𝑖𝑗⟩ denotes 𝑐̃_𝑖𝑗. Matrix [𝑐̃_𝑖𝑗] is shown as follows: 𝑐̃_𝑖𝑗=𝛼_𝑖𝑗,𝛽_𝑖𝑗. (3) In addition, the 𝛼_𝑖𝑗 values form the matrix [𝛼_𝑖𝑗], the 𝛽_𝑖𝑗 values form the matrix [𝛽_𝑖𝑗], and the 𝑄_𝑖𝑗 values form the matrix [𝑄_𝑖𝑗]. Further, we define 𝛾_𝑖𝑗=𝛽_𝑖𝑗−𝛼_𝑖𝑗𝑄_𝑖𝑗for𝑖,𝑗∈𝐾, (4) where 𝛾_𝑖𝑗 is indeed the slope of the membership function.

The membership function of total cost 𝑐_𝑇̃ is defined as the monotonically decreasing linear function shown in (5). 𝑎 and 𝑏 are the lower and upper bounds of 𝑐_𝑇̃, respectively. Total cost is related to the performance of the manager. When the total cost is lower than 𝑎, his performance is 1; when the total cost is greater than 𝑏, his performance is 0; and when the total cost lies between 𝑎 and 𝑏, the total cost is linearly and negatively related to the performance of the manager. ⟨𝑎,𝑏⟩ denotes the fuzzy interval 𝑐_𝑇̃: 𝜇_𝑇𝑐_𝑇=𝜇_𝑇^𝑛𝑛_𝑖=1𝑗=1𝑐_𝑖𝑗𝑥_𝑖𝑗=1if𝑐_𝑇≤𝑎𝑏−^𝑛_𝑖=1^𝑛_𝑗=1𝑐_𝑖𝑗𝑥_𝑖𝑗=𝑏−𝑎𝑏−𝑐_𝑇𝑏−𝑎if𝑎≤𝑐_𝑇≤𝑏0if𝑐_𝑇≥𝑏. (5) Since the performance of worker 𝑖, denoted as 𝜋_𝑖, is the job quality achieved, then 𝜋_𝑖=𝜇_𝑖𝑗𝑐_𝑖𝑗for𝑥_𝑖𝑗=1,𝑖=1,2,…,𝑛. (6) In addition, the performance of the manager, denoted as 𝜋_𝑚, is as follows: 𝜋_𝑚=𝜇_𝑇^𝑛𝑛_𝑖=1𝑗=1𝑐_𝑖𝑗𝑥_𝑖𝑗. (7)

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_∀𝑖𝜋_𝑖,𝜋_𝑚 (8) or max--min_{𝑥𝑖𝑗=1}𝜇_𝑖𝑗𝑐_𝑖𝑗,𝜇_𝑇^𝑛𝑛_𝑖=1𝑗=1𝑐_𝑖𝑗𝑥_𝑖𝑗, (9) where 𝑥_𝑖𝑗 is an element of a feasible solution X of (1). The FAP model can then be constructed as follows: max--min_{𝑥𝑖𝑗=1}𝜇_𝑖𝑗𝑐_𝑖𝑗,𝜇_𝑇^𝑛𝑛_𝑖=1𝑗=1𝑐_𝑖𝑗𝑥_𝑖𝑗s.t.^𝑛_𝑗=1𝑥_𝑖𝑗=1for𝑖∈𝐼_𝑛𝑖=1𝑥_𝑖𝑗𝑥=1for𝑗∈𝐽_𝑖𝑗∈{0,1}for𝑖,𝑗∈𝐾. (10)

Utilizing the membership functions illustrated in (2) and (5), (10) can be represented as the following equivalent problem: max𝜆s.t.𝜆⋅𝑥_𝑖𝑗≤𝑄_𝑖𝑗𝑐^𝜆_𝑖𝑗−𝛼_𝑖𝑗𝛽_𝑖𝑗−𝛼_𝑖𝑗𝑥_𝑖𝑗∑for𝑖,𝑗∈𝐾𝜆≤𝑏−^𝑛_𝑖=1∑^𝑛_𝑗=1𝑐^𝜆_𝑖𝑗𝑥_𝑖𝑗𝑏−𝑎_𝑛𝑗=1𝑥_𝑖𝑗=1for𝑖∈𝐼_𝑛𝑖=1𝑥_𝑖𝑗=1for𝑗∈𝐽0≤𝜆𝑥_𝑖𝑗≤𝑄_𝑖𝑗𝑥_𝑖𝑗𝑐for𝑖,𝑗∈𝐾_𝜆𝑖𝑗≥0,𝑥_𝑖𝑗∈{0,1}for𝑖,𝑗∈𝐾, (11) where 𝑐^𝜆_𝑖𝑗 denotes the 𝜆-cut of 𝑐̃_𝑖𝑗 and ∑^𝑛_𝑖=1∑^𝑛_𝑗=1𝑐^𝜆_𝑖𝑗𝑥_𝑖𝑗 is the corresponding total cost 𝑐^𝜆_𝑇. Since 𝑥_𝑖𝑗, 𝑐^𝜆_𝑖𝑗, and 𝜆 all are decision variables, (11) can be treated as a mixed nonlinear programming problem.

3. Model Transformation

Let 𝝎 be a specific feasible assignment of (1), that is, of (10). If discussion is confined on 𝝎, (11) can be simplified as follows: 𝑄max𝜆s.t.𝜆≤^𝑖𝑗𝑐^𝜆_𝑖𝑗−𝛼_𝑖𝑗𝛽_𝑖𝑗−𝛼_𝑖𝑗∑for𝑖,𝑗∈𝝎𝜆≤𝑏−_{(𝑖,𝑗)∈𝝎}𝑐^𝜆_𝑖𝑗𝑏−𝑎𝜆≤𝑄_𝑖𝑗𝑐for𝑖,𝑗∈𝝎_𝜆𝑖𝑗≥0,𝜆≥0for𝑖,𝑗∈𝝎. (12)

Theorem 1. Let 𝜆^∗ be the optimal objective value of (12) corresponding to 𝝎, then 𝜆^∗=max--min𝑏−∑_{(𝑖,𝑗)∈𝝎}𝛼_𝑖𝑗𝑏−𝑎+∑_{(𝑖,𝑗)∈𝝎}𝛾_𝑖𝑗,min_{𝑖,𝑗∈𝝎}𝑄_𝑖𝑗. (13)

Proof. Transforming (12) into a linear programming problem, we find the following: max𝜆s.t.−𝑐^𝜆_𝑖𝑗+𝛽_𝑖𝑗−𝛼_𝑖𝑗𝑄_𝑖𝑗⋅𝜆≤−𝛼_𝑖𝑗for𝑖,𝑗∈𝝎_{(𝑖,𝑗)∈𝝎}𝑐^𝜆_𝑖𝑗+(𝑏−𝑎)⋅𝜆≤𝑏𝜆≤min_{𝑖,𝑗∈𝝎}𝑄_𝑖𝑗𝑐^𝜆_𝑖𝑗,𝜆≥0for𝑖,𝑗∈𝝎. (14) The dual problem of (14) can then be obtained as follows: min−_{𝑖,𝑗∈𝝎}𝛼_𝑖𝑗𝑡_𝑖+𝑏𝑡_𝑛+1+min_{𝑖,𝑗∈𝝎}𝑄_𝑖𝑗⋅𝑦s.t.−𝑡_𝑖+𝑡_𝑛+1≥0for𝑖∈𝐼_{(𝑖,𝑗)∈𝝎}𝛽_𝑖𝑗−𝛼_𝑖𝑗𝑄_𝑖𝑗𝑡_𝑖+(𝑏−𝑎)𝑡_𝑛+1𝑡+𝑦≥1_𝑖𝑡≥0,for𝑖∈𝐼_𝑛+1≥0,for𝑦≥0. (15) By (14), 𝑐^𝜆_𝑖𝑗≥𝛼_𝑖𝑗+((𝛽_𝑖𝑗−𝛼_𝑖𝑗)/𝑄_𝑖𝑗)⋅𝜆>0 holds for (𝑖,𝑗)∈𝝎. According to the complementary slackness theorem, −𝑡_𝑖+𝑡_𝑛+1=0for𝑖∈𝐼 (16) or 𝑡_𝑛+1=𝑡₁=𝑡₂=⋯=𝑡_𝑛. (17) Using (17) and (4), (15) can be simplified into the following: min𝑏−_{(𝑖,𝑗)∈𝝎}𝛼_𝑖𝑗𝑡_𝑛+1+min_{𝑖,𝑗∈𝝎}𝑄_𝑖𝑗⋅𝑦s.t.𝑏−𝑎+_{(𝑖,𝑗)∈𝝎}𝛾_𝑖𝑗𝑡_𝑛+1𝑡+𝑦≥1_𝑛+1≥0,𝑦≥0 (18) The dual problem of (18) is as follows: max𝜆s.t.𝑏−𝑎+_{(𝑖,𝑗)∈𝝎}𝛾_𝑖𝑗⋅𝜆≤𝑏−_{(𝑖,𝑗)∈𝝎}𝛼_𝑖𝑗𝜆≤min_{(𝑖,𝑗)∈𝝎}𝑄_𝑖𝑗𝜆≥0 (19) (19) can be rewritten as follows to complete the proof: max𝜆s.t.𝜆≤𝑏−∑_{(𝑖,𝑗)∈𝝎}𝛼_𝑖𝑗𝑏−𝑎+∑_{(𝑖,𝑗)∈𝝎}𝛾_𝑖𝑗𝜆≤min_{𝑖,𝑗∈𝝎}𝑄_𝑖𝑗𝜆≥0. (20)
Obviously, 𝜆^∗ depends on the given 𝝎; that is, 𝜆^∗ is a function of 𝝎. The following corollary is trivial.

Corollary 2. Let 𝝎₁,𝝎₂,…,𝝎_𝑝 be all of the feasible assignments of (1). 𝜆^∗₁,𝜆^∗₂,…,𝜆^∗_𝑝 are the corresponding optimal objective function values in (12). Let 𝜆^∗_𝑖=max(𝜆^∗₁,𝜆^∗₂,…,𝜆^∗_𝑝); then 𝝎_𝑖 (subsequently denoted by 𝝎^∗) is the optimal assignment of (11) and 𝜆^∗_𝑖 (subsequently denoted by 𝑧^∗) is the optimal objective function value in (11).

By Theorem 1 and Corollary 2, (11) can be transformed to (21), which is a general 0-1 fractional programming problem: max--min𝑏−∑^𝑛_𝑖=1∑^𝑛_𝑗=1𝛼_𝑖𝑗𝑥_𝑖𝑗𝑏−𝑎+∑^𝑛_𝑖=1∑^𝑛_𝑗=1𝛾_𝑖𝑗𝑥_𝑖𝑗,min1−1−𝑄_𝑖𝑗𝑥_𝑖𝑗∀𝑖,𝑗∈𝐾s.t.^𝑛_𝑗=1𝑥_𝑖𝑗=1for𝑖∈𝐼_𝑛𝑖=1𝑥_𝑖𝑗𝑥=1for𝑗∈𝐽_𝑖𝑗∈{0,1}for𝑖,𝑗∈𝐾. (21)

4. 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 trade-off 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: max𝑓=𝑏−∑^𝑛_𝑖=1∑^𝑛_𝑗=1𝛼_𝑖𝑗𝑥_𝑖𝑗𝑏−𝑎+∑^𝑛_𝑖=1∑^𝑛_𝑗=1𝛾_𝑖𝑗𝑥_𝑖𝑗s.t.^𝑛_𝑗=1𝑥_𝑖𝑗=1for𝑖∈𝐼_𝑛𝑖=1𝑥_𝑖𝑗𝑥=1for𝑗∈𝐽_𝑖𝑗∈{0,1}for𝑖,𝑗∈𝐾. (22)

Since the constraints of (22) form a totally unimodular matrix and its objective function is a pseudomonotonic model [14], the condition 𝑥_𝑖𝑗∈{0,1} in (22) can be replaced with 𝑥_𝑖𝑗≥0. Furthermore, we can define 𝛼_𝑖𝑗=𝑏/𝑛−𝛼_𝑖𝑗 and 𝛾_𝑖𝑗=(𝑏−𝑎)/𝑛+𝛾_𝑖𝑗. Equation (22) then becomes equivalent to the following: ∑max𝑓=^𝑛𝑖=1∑^𝑛_𝑗=1𝛼_𝑖𝑗𝑥_𝑖𝑗∑^𝑛_𝑖=1∑^𝑛_𝑗=1𝛾_𝑖𝑗𝑥_𝑖𝑗s.t.^𝑛_𝑗=1𝑥_𝑖𝑗=1for𝑖∈𝐼_𝑛𝑖=1𝑥_𝑖𝑗𝑥=1for𝑗∈𝐽_𝑖𝑗≥0for𝑖,𝑗∈𝐾. (23)

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 𝐗⁰ of the maximal-objective-function transportation problem with [𝛼_𝑖𝑗/𝛾_𝑖𝑗] as the profit matrix and 1 as the amount of each supply and demand. 𝐗⁰ 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 𝑓 of 𝐗⁰ in (23) is determined and associates 𝑢_𝑖 and 𝑣_𝑗 with row 𝑖 and column 𝑗, respectively. Set 𝑢₁=0 and identify the other 𝑢_𝑖 and 𝑣_𝑗 using the equation 𝑢_𝑖+𝑣_𝑗=𝛼_𝑖𝑗−𝑓⋅𝛾_𝑖𝑗 according to the basic variables of 𝐗⁰. Define the assignment as 𝝎={(𝑖,𝑗)∣𝑥_𝑖𝑗=1,𝑥_𝑖𝑗∈𝐗⁰}. Remove the degenerated basic variables of 𝐗⁰ from the table.

Figure 1 

The results of the FCMP method.

Figure 2 

Computing values of 𝑢𝑖, 𝑣𝑗, and Δ𝑖𝑗.

Taking 𝝎 as the initial assignment, subprocedures mixed approaches of [5, 11, 18–20] are used repeatedly to improve 𝝎. When Δ_𝑖𝑗=𝑢_𝑖+𝑣_𝑗+𝑓⋅𝛾_𝑖𝑗−𝛼_𝑖𝑗≥0 for each (𝑖,𝑗)∉𝝎, 𝝎 is the optimal assignment of (23). Based on 𝝎, the other condition in (21) is extracted. Let 𝑔=min_{𝑖,𝑗∈𝝎}𝑄_𝑖𝑗; (24) then 𝑧=min(𝑓,𝑔) is the objective function value in (21) corresponding to 𝝎. To find the assignment 𝝎^∗ that maximizes the objective function value in (21), elements of matrix [𝛼_𝑖𝑗] can be repeatedly changed so that 𝑓 decreases or remains constant and 𝑔 increases to improve 𝑧 until the optimal objective function value 𝑧^∗ in (21) is reached. The proposed S&T algorithm consisting of initialization and optimization procedures is summarized as follows.

Phase I. Initialization

Step  1: Let 𝑧^∗→−∞ and 𝝎^∗=∅. Apply the FCMP method to get 𝐗⁰ and 𝑓.

Step  2: Determine all values of 𝑢_𝑖 and 𝑣_𝑗 according to 𝐗⁰.

Step  3: Remove the degenerate basic variables of 𝐗⁰ and obtain 𝝎.

Phase II. Optimization

Step A. Compute Δ_𝑖𝑗, for each (𝑖,𝑗)∉𝝎. If all Δ_𝑖𝑗≥0; then proceed to Step E. Otherwise, choose (𝑘,ℓ) as the pivot cell where Δ_𝑘ℓ=min(Δ_𝑖𝑗)<0.

Step B. Proceed with the following labeling procedure until further labeling is impossible or row 𝑘 has received a label. If row 𝑘 has been labeled, go to Step C; otherwise, go to Step D.

Labeling Procedure

(1)Give label 𝑘 to column ℓ.(2)If column 𝑞 has been labeled, label the row 𝑖 with 𝑞, where 𝑥_𝑖𝑞=1.(3)If row 𝑝 has been labeled, label the nonlabeled column 𝑗 with 𝑝, where Δ_𝑝𝑗=0.

Step C. Form a unique cycle by following the labels in reverse order. In the cycle, every odd cell is 𝑥_𝑖𝑗=0, and every even cell is 𝑥_𝑖𝑗=1. The odd cells are the recipient cells and thus become 𝑥_𝑖𝑗=1; the even cells are the donor cells, which become 𝑥_𝑖𝑗=0. Let 𝑥_𝑖𝑗 remain unchanged if cell (𝑖,𝑗) is not in the cycle. The new assignment is obtained and is taken as the new 𝝎. The objective function value corresponding to the new 𝝎 is taken as the new 𝑓.

By keeping 𝑢_𝑘 unchanged, the new values of 𝑣_ℓ and 𝑣_𝐿(𝑘) can be obtained by the following: 𝑣_ℓ=𝛼_𝑘ℓ−𝑢_𝑘−𝑓⋅𝛾_𝑘ℓ,𝑣_𝐿(𝑘)=𝛼_{(𝑘)𝑘,𝐿}−𝑢_𝑘−𝑓⋅𝛾_{(𝑘)𝑘,𝐿}, (25) where 𝐿(𝑘) is the label of row𝑘.

Other new variables 𝑢_𝑖 and 𝑣_𝑗 related to the cycle can be obtained by keeping Δ_𝑖𝑗=0 in the cycle. If the new values of 𝑢_𝑖 and 𝑣_𝑗 are not related to the cycle, hold 𝑢_𝑖 unchanged and identify 𝑣_𝑗=𝛼_𝑖𝑗−𝑢_𝑖−𝑓⋅𝛾_𝑖𝑗, where (𝑖,𝑗)∈𝝎. Return to Step A.

Step D. Let 𝐽={(𝑖,𝑗)∣Δ_𝑖𝑗>0 and row 𝑖 labeled, but column 𝑗 is not labeled}, 𝜀=min_{(𝑖,𝑗)∈𝐽}{Δ_𝑖𝑗}; or 𝜀=|Δ_𝑘ℓ| if 𝐽 is a null set.

If row 𝑖 is labeled, 𝜀 is subtracted from 𝑢_𝑖; and if column 𝑗 is labeled, 𝜀 is added to 𝑣_𝑗. Recompute Δ_𝑖𝑗 for (𝑖,𝑗)∉𝝎. If all Δ_𝑖𝑗≥0, proceed to Step E. If Δ_𝑘ℓ<0 still holds, take (𝑘,ℓ) as the pivot cell; if Δ_𝑘ℓ≥0, choose (𝑘,ℓ)̂ as the new pivot cell where Δ𝑘ℓ̂=min(Δ_𝑖𝑗)<0. Replace 𝑘 with 𝑘, ℓ with ℓ̂, and return to Step B.

Step E. Determine 𝑔=min_{(𝑖,𝑗)∈𝝎}(𝑄_𝑖𝑗) and 𝑧=min(𝑓,𝑔). If 𝑧≤𝑧^∗, proceed to Step H; otherwise, proceed to Step F.

Step F. Let 𝑧^∗=𝑧 and 𝝎^∗=𝝎. If 𝑧≥𝑓, proceed to Step H; otherwise, proceed to Step G.

Step G. Update [𝛼_𝑖𝑗], where 𝛼_𝑖𝑗⟵𝛼_𝑖𝑗if𝑄_𝑖𝑗>𝑧^∗.−𝑀(𝑀⟶∞)if𝑄_𝑖𝑗≤𝑧^∗. (26)

Revise the objective function value 𝑓 corresponding to 𝝎. To renew 𝑢_𝑖 and 𝑣_𝑗, hold 𝑢_𝑖 unchanged and identify 𝑣_𝑗=𝛼_𝑖𝑗−𝑢_𝑖−𝑓⋅𝛾_𝑖𝑗, where (𝑖,𝑗)∈𝝎. Go back to Step A.

Step H. Terminate the algorithm. The current 𝝎^∗ and 𝑧^∗ are the optimal assignment and optimal value of (21), respectively.

5. Numerical Example

In this section, for demonstration purpose, the S&T algorithm is applied to a numerical example.

Suppose the matrices [𝑐̃_𝑖𝑗] and [𝑄_𝑖𝑗] are 𝑐̃_𝑖𝑗=⟨⟩⟨⟩⟨⟩,𝑄5,22⟩⟨7,16⟩⟨23,395,21⟩⟨21,31⟩⟨16,3114,28⟩⟨23,32⟩⟨5,15_𝑖𝑗=;0.850.60.80.640.80.750.70.720.8 (27) then define matrices 𝛼_𝑖𝑗=,𝛽57235211614235^𝑖𝑗=,𝛾221639213131283215_𝑖𝑗=.2015202512.5202012.512.5 (28) Specify the sum of maximal elements in each row of matrix [𝛽_𝑖𝑗] as 𝑏=102 and the sum of minimal elements in each row of matrix [𝛼_𝑖𝑗] as 𝑎=15. Matrices [𝛼_𝑖𝑗] and [𝛾_𝑖𝑗] can be obtained as follows: 𝛼_𝑖𝑗=,292711291318201129𝛾_𝑖𝑗=4944495441.5494941.541.5. (29) The S&T algorithm is applied to solve (21).

Phase I. Set 𝑧^∗→−∞ and 𝝎^∗=∅. The FCMP method is used to obtain 𝐗⁰, as shown in Figure 1. Its corresponding objective function value 𝑓=170/279 can be obtained from (23).

Figure 2 shows all values of 𝑢_𝑖, 𝑣_𝑗, and 𝝎={(1,2),(2,1),(3,3)}. They are obtained from 𝐗⁰ and 𝑓.

Phase II. Each Δ_𝑖𝑗 for (𝑖,𝑗)∉𝝎 is shown in Figure 2 the cell centers. Since all Δ_𝑖𝑗≥0 for (𝑖,𝑗)∉𝝎, 𝑔=min(0.6,0.64,0.8)=0.6 and 𝑧=min(170/279,0.6)=0.6. Because 𝑧>𝑧^∗, let 𝑧^∗=0.6 and 𝝎^∗={(1,2),(2,1),(3,3)}. Since 𝑧<𝑓, replace 𝛼₁₂ with −𝑀 and update 𝑓=(116−𝑀)/279. The updated 𝑣₁, 𝑣₂, and 𝑣₃ are shown in Figure 3(a), which also shows the recomputed Δ_𝑖𝑗 for each (𝑖,𝑗)∉𝝎. (2, 2) is selected as the pivot cell in the labeling procedure. In Figure 3(a), the labels of row 2 and column 2 are given by the numbers in brackets on the left and top margins.

(a)

(b)

(c)

(d)

Figure 3 

Set of table for Optimization Procedure.

Since row 2 has not received a label in Figure 3(a), 𝐽={(1,1)} and 𝜀=(290+10𝑀/279) are determined. Continuing to follow the steps of Phase II yields Figures 3(b)–3(d). Through this procedure, the optimal assignment 𝝎^∗={(1,2),(2,1),(3,3)} and optimal value 𝑧^∗=0.6 are obtained.

6. 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 the LINGO 11 package. All algorithms are run on an Intel Pentium III-500 CPU.

The sizes of the problems tested range from 3×3 to 16×16. For each problem of 𝑁×𝑁 size, numbers of uniform distribution between 10 and 10+20𝑁 are randomly generated for 𝛼_𝑖𝑗 and numbers between 6𝑁 and 10𝑁 are generated for the difference between 𝛽_𝑖𝑗 and 𝛼_𝑖𝑗. Numbers with a uniform distribution between 0.6 and 1.0 are generated for 𝑄_𝑖𝑗. For each problem size, 30 problems are tested and the average computational times (in seconds) of the three approaches are shown in Table 1. These results show that from the perspective of computational time, the S&T algorithm is greatly superior to the LINGO package. In particular, the S&T algorithm is superior to the index-based B&B algorithm when dealing with an increasing problem size.

Table 1 

Computational results for the general case.

Depending on the random number generated, the manager or one of the workers may be the one who decides the team performance. Now, we move to a discussion of two special cases. In the first case, it is assumed that the company supports the team at any expense. Thus, 𝑏→∞ and the computational results of the three approaches are shown in Table 2. They illustrate that, although a small problem size causes the B&B algorithm to perform faster, the S&T algorithm is still efficient. The elapsed time of the LINGO package is also reduced. However, the variances of the elapsed times of the B&B algorithm and the LINGO package are large. Practically, 𝑏→∞ causes the membership function in (5) to approach 𝜇_𝑇(𝑐_𝑇)=1. Thus, the manager is never the one who determines the team performance. The workers also obtain sufficient expenses and can all reach their highest quality (performance). The team performance is decided by one of the workers. Therefore, 𝑏→∞ simplifies (21) to the following: maxmin1−1−𝑄_𝑖𝑗𝑥_𝑖𝑗;∀𝑖,𝑗∈𝐾s.t.^𝑛_𝑗=1𝑥_𝑖𝑗=1;∀𝑖∈𝐼_𝑛𝑖=1𝑥_𝑖𝑗𝑥=1;∀𝑗∈𝐽_𝑖𝑗∈{0,1};∀𝑖,𝑗∈𝐾. (30) 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 𝑏→∞ is known in advance.

Table 2 

Computational results when 𝑏→∞.

In the second special case, 𝑏 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(1−(1−𝑄_𝑖𝑗)𝑥_𝑖𝑗) for all (𝑖,𝑗)∈𝐾 in the objective function of (21) is redundant. Thus, min𝑏−∑^𝑛_𝑖=1∑^𝑛_𝑗=1𝛼_𝑖𝑗𝑥_𝑖𝑗𝑏−𝑎+∑^𝑛_𝑖=1∑^𝑛_𝑗=1𝛾_𝑖𝑗𝑥_𝑖𝑗,min1−1−𝑄_𝑖𝑗𝑥_𝑖𝑗∀=𝑖,𝑗∈𝐾𝑏−∑^𝑛_𝑖=1∑^𝑛_𝑗=1𝛼_𝑖𝑗𝑥_𝑖𝑗𝑏−𝑎+∑^𝑛_𝑖=1∑^𝑛_𝑗=1𝛾_𝑖𝑗𝑥_𝑖𝑗. (31) Problem (21) can then be simplified to (22) and can be solved with Phase I and Steps A, B, C, and D of Phase II of the proposed S&T algorithm.

Table 3 

Computational results when 𝑏 is sufficiently small.

An overview of Tables 1–3 reveals that, among the three approaches, the S&T algorithm is the most efficient.

7. 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.