# Assignment Problem for Team Performance Promotion under Fuzzy Environment

Authors: Chi-Jen Lin
Journal: Mathematical Problems in Engineering
 (2013)

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## 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 ofn 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.


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## Body

## 1. Introduction

The assignment problem (AP) is a well-known and practical mathematical model. Given two disjointn-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:
(1)min∑i=1n∑j=1ncij·xij(cij≥0)s.t.∑j=1nxij=1fori∈I∑i=1nxij=1forj∈Jxij∈{0,1}for(i,j)∈K.LetX 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)∣xij=1,xij∈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 cij 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 cij 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 ofn 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. c~ij and c~T denote the cost of worker i performing job j and the total cost, respectively.LetQij(0<Qij≤1) denote the highest achievable job quality j performed by worker i. The membership function of job cost c~ij is defined as the monotonically increasing linear function shown in (2). In this function, αij is the minimum cost for worker i performing jobj, and βij is the minimum cost associated with worker i performing jobj to reach Qij. When the input job cost lies between αij and βij, the job cost is linearly and positively related to the job quality, that is, the performance of the worker. However, any expense exceeding βij is inefficient since the quality can no longer be enhanced. Without loss of generality, it is assumed that 0<αij<βij. If xij=0 in any feasible solution, there is no real expense; therefore xij=1 is added to the following:
(2)μij(cij)={qijifcij≥βij,xij=1Qij(cij-αij)βij-αijifαij≤cij≤βij,xij=10otherwise〈αij,βij〉 denotes c~ij. Matrix [c~ij] is shown as follows:
(3)[c~ij]=[〈αij,βij〉].
In addition, the αij values form the matrix [αij], the βij values form the matrix [βij], and the Qij values form the matrix [Qij]. Further, we define
(4)[γij]=[βij-αijQij]for(i,j)∈K,
where γij is indeed the slope of the membership function.The membership function of total costc~T is defined as the monotonically decreasing linear function shown in (5). a and b are the lower and upper bounds of c~T, 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~T:
(5)μT(cT)=μT(∑i=1n∑j=1ncijxij)={1ifcT≤ab-∑i=1n∑j=1ncijxijb-a=b-cTb-aifa≤cT≤b0ifcT≥b.
Since the performance of worker i, denoted as πi, is the job quality achieved, then
(6)πi=μij(cij)forxij=1,i=1,2,…,n.
In addition, the performance of the manager, denoted as πm, is as follows:
(7)πm=μT(∑i=1n∑j=1ncijxij).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,
(8)max–min∀i(πi,πm)
or
(9)max–minxij=1(μij(cij),μT(∑i=1n∑j=1ncijxij)),
where xij is an element of a feasible solution X of (1). The FAP model can then be constructed as follows:
(10)max–minxij=1(μij(cij),μT(∑i=1n∑j=1ncijxij))s.t.∑j=1nxij=1fori∈I∑i=1nxij=1forj∈Jxij∈{0,1}for(i,j)∈K.Utilizing the membership functions illustrated in (2) and (5), (10) can be represented as the following equivalent problem:
(11)maxλs.t.λ·xij≤Qij(cijλ-αij)βij-αijxijfor(i,j)∈Kλ≤b-∑i=1n∑j=1ncijλxijb-a∑j=1nxij=1fori∈I∑i=1nxij=1forj∈J0≤λxij≤Qijxijfor(i,j)∈Kcijλ≥0,xij∈{0,1}for(i,j)∈K,
where cijλ denotes the λ-cut of c~ij and ∑i=1n∑j=1ncijλxij is the corresponding total cost cTλ. Since xij, cijλ, and λ all are decision variables, (11) can be treated as a mixed nonlinear programming problem.


## 3. Model Transformation

Letω be a specific feasible assignment of (1), that is, of (10). If discussion is confined on ω, (11) can be simplified as follows:
(12)maxλs.t.λ≤Qij(cijλ-αij)βij-αijfor(i,j)∈ωλ≤b-∑(i,j)∈ωcijλb-aλ≤Qijfor(i,j)∈ωcijλ≥0,λ≥0for(i,j)∈ω.Theorem 1.
Letλ* be the optimal objective value of (12) corresponding to ω, then
(13)λ*=max–min(b-∑(i,j)∈ωαij(b-a+∑(i,j)∈ωγij),min(i,j)∈ω(Qij)).Proof.
Transforming (12) into a linear programming problem, we find the following:
(14)maxλs.t.-cijλ+(βij-αij)Qij·λ≤-αijfor(i,j)∈ω∑(i,j)∈ωcijλ+(b-a)·λ≤bλ≤min(i,j)∈ω(Qij)cijλ,λ≥0for(i,j)∈ω.
The dual problem of (14) can then be obtained as follows:
(15)min-∑(i,j)∈ω(αijti)+btn+1+min(i,j)∈ω(Qij)·ys.t.-ti+tn+1≥0fori∈I∑(i,j)∈ω(βij-αij)Qijti+(b-a)tn+1+y≥1ti≥0,fori∈Itn+1≥0,fory≥0.
By (14), cijλ≥αij+((βij-αij)/Qij)·λ>0 holds for (i,j)∈ω. According to the complementary slackness theorem,
(16)-ti+tn+1=0fori∈I
or
(17)tn+1=t1=t2=⋯=tn.
Using (17) and (4), (15) can be simplified into the following:
(18)min(b-∑(i,j)∈ωαij)tn+1+min(i,j)∈ω(Qij)·ys.t.(b-a+∑(i,j)∈ωγij)tn+1+y≥1tn+1≥0,y≥0
The dual problem of (18) is as follows:
(19)maxλs.t.(b-a+∑(i,j)∈ωγij)·λ≤b-∑(i,j)∈ωαijλ≤min(i,j)∈ω(Qij)λ≥0
(19) can be rewritten as follows to complete the proof:
(20)maxλs.t.λ≤b-∑(i,j)∈ωαij(b-a+∑(i,j)∈ωγij)λ≤min(i,j)∈ω(Qij)λ≥0.
Obviously,λ* depends on the given ω; that is, λ* is a function of ω. The following corollary is trivial.Corollary 2.
Letω1,ω2,…,ωp be all of the feasible assignments of (1). λ1*,λ2*,…,λp* are the corresponding optimal objective function values in (12). Let λi*=max(λ1*,λ2*,…,λp*); then ωi (subsequently denoted by ω*) is the optimal assignment of (11) and λi* (subsequently denoted by z*) is the optimal objective function value in (11).By Theorem1 and Corollary 2, (11) can be transformed to (21), which is a general 0-1 fractional programming problem:
(21)max–min(b-∑i=1n∑j=1n(αijxij)(b-a+∑i=1n∑j=1n(γijxij)),max-minmin(1-(1-Qij)xij)∀(i,j)∈Kb-∑i=1n∑j=1n(αijxij)(b-a+∑i=1n∑j=1n(γijxij)))s.t.∑j=1nxij=1fori∈I∑i=1nxij=1forj∈Jxij∈{0,1}for(i,j)∈K.


## 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:
(22)maxf=b-∑i=1n∑j=1n(αijxij)(b-a+∑i=1n∑j=1n(γijxij))s.t.∑j=1nxij=1fori∈I∑i=1nxij=1forj∈Jxij∈{0,1}for(i,j)∈K.Since the constraints of (22) form a totally unimodular matrix and its objective function is a pseudomonotonic model [14], the condition xij∈{0,1} in (22) can be replaced with xij≥0. Furthermore, we can define α-ij=b/n-αij and  γ-ij=(b-a)/n+γij. Equation (22) then becomes equivalent to the following:
(23)maxf=∑i=1n∑j=1n(α-ijxij)∑i=1n∑j=1n(γ-ijxij)s.t.∑j=1nxij=1fori∈I∑i=1nxij=1forj∈Jxij≥0for(i,j)∈K.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 Figure1. 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 ui and vj with row i and column j, respectively. Set u1=0 and identify the other ui and vj using the equation ui+vj=α-ij-f·γ-ij according to the basic variables of X0. Define the assignment as ω={(i,j)∣xij=1,xij∈X0}. Remove the degenerated basic variables of X0 from the table.Figure 1
The results of the FCMP method.Figure 2
Computing values ofui, vj, and Δij.Takingω as the initial assignment, subprocedures mixed approaches of [5, 11, 18–20] are used repeatedly to improve ω. When Δij=ui+vj+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
(24)g=min(i,j)∈ω(Qij);
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. InitializationStep  1: Let z*→-∞ and ω*=∅. Apply the FCMP method to get X0 and f.Step  2: Determine all values of ui and vj according to X0.Step  3: Remove the degenerate basic variables of X0 and obtain ω.Phase II. OptimizationStep A. Compute Δij, for each (i,j)∉ω. If all Δij≥0; then proceed to Step E. Otherwise, choose (k,ℓ) as the pivot cell where Δkℓ=min(Δij)<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
i
q
=
1
.
(3)
If row
p
has been labeled, label the nonlabeled column
j
with
p
, where
Δ
p
j
=
0
.Step C. Form a unique cycle by following the labels in reverse order. In the cycle, every odd cell is xij=0, and every even cell is xij=1. The odd cells are the recipient cells and thus become xij=1; the even cells are the donor cells, which become xij=0. Let xij 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 keepinguk unchanged, the new values of vℓ and vL(k) can be obtained by the following:
(25)vℓ=α-kℓ-uk-f·γ-kℓ,vL(k)=α-k,L(k)-uk-f·γ-k,L(k),
where L(k) is the label of rowk.Other new variablesui and vj related to the cycle can be obtained by keeping Δij=0 in the cycle. If the new values of ui and vj are not related to the cycle, hold ui unchanged and identify vj=α-ij-ui-f·γ-ij, where (i,j)∈ω. Return to Step A.Step D. Let J-={(i,j)∣Δij>0 and row i labeled, but column j is not labeled}, ε=min(i,j)∈J-{Δij}; or ε=|Δkℓ| if J- is a null set.If rowi is labeled, ε is subtracted from ui; and if column j is labeled, ε is added to vj. Recompute Δij for (i,j)∉ω. If all Δij≥0, proceed to Step E. If Δkℓ<0 still holds, take (k,ℓ) as the pivot cell; if Δkℓ≥0, choose (k^,ℓ^) as the new pivot cell where Δk^ℓ^=min(Δij)<0. Replace k with k^, ℓ with ℓ^, and return to Step B.Step E. Determine g=min(i,j)∈ω(Qij) 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
(26)α-ij⟵{α-ijifQij>z*.-M(M⟶∞)ifQij≤z*.Revise the objective function valuef corresponding to ω. To renew ui and vj, hold ui unchanged and identify vj=α-ij-ui-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 [c~ij] and [Qij] are
(27)[c~ij]=[〈5,22〉〈7,16〉〈23,39〉〈5,21〉〈21,31〉〈16,31〉〈14,28〉〈23,32〉〈5,15〉],[Qij]=[0.850.60.80.640.80.750.70.720.8];
then define matrices
(28)[αij]=[57235211614235],[βij]=[221639213131283215],[γij]=[2015202512.5202012.512.5].
Specify the sum of maximal elements in each row of matrix [βij] as b=102 and the sum of minimal elements in each row of matrix [αij] as a=15. Matrices [α-ij] and [γ-ij] can be obtained as follows:
(29)[α-ij]=[292711291318201129],[γ-ij]=[4944495441.5494941.541.5].
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).Figure2 shows all values of ui, vj, and ω={(1,2),(2,1),(3,3)}. They are obtained from X0 and f.Phase II. Each Δij for (i,j)∉ω is shown in Figure 2 the cell centers. Since all Δij≥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 α-12 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 Δij 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.Set of table for Optimization Procedure.
(a)
(b)
(c)
(d)Since row 2 has not received a label in Figure3(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 from3×3 to 16×16. For each problem of N×N size, numbers of uniform distribution between 10 and 10+20N are randomly generated for αij and numbers between 6N and 10N are generated for the difference between βij and αij. Numbers with a uniform distribution between 0.6 and 1.0 are generated for Qij. 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.
Problem size
B&B algorithm
S&T algorithm
LINGO
Average time (sec.)
Average time (sec.)
Average time (sec.)
3
×
3
—
3.3333
×
10
-
4
1.90667
4
×
4
6.6667
×
10
-
4
1.3333
×
10
-
3
10.545
5
×
5
1.3333
×
10
-
3
4.333
×
10
-
3
28.7183
6
×
6
4.6667
×
10
-
3
7.666
×
10
-
3
34.0763
7
×
7
6.7
×
10
-
3
1.0
×
10
-
2
34.9223
8
×
8
2.203
×
10
-
2
1.4333
×
10
-
2
61.9323
9
×
9
7.41
×
10
-
2
2.303
×
10
-
2
85.8511
10
×
10
0.2236
3.67
×
10
-
2
131.4427
11
×
11
0.6369
5.5067
×
10
-
2
145.2227
12
×
12
3.4179
8.2433
×
10
-
2
216.4897
1
3
×
1
3
7.9434
0.1225
374.1963
1
4
×
1
4
42.0261
0.17357
466.2703
1
5
×
1
5
128.2167
0.24067
656.5563
1
6
×
1
6
516.42
0.3388
870.1540Depending 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 μT(cT)=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:
(30)maxmin((1-(1-Qij)xij);∀(i,j)∈K)s.t.∑j=1nxij=1;∀i∈I∑i=1nxij=1;∀j∈Jxij∈{0,1};∀(i,j)∈K.
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 whenb→∞.
Problem size
B&B algorithm
S&T algorithm
LINGO
Average time (sec.)
Average time (sec.)
Average time (sec.)
3
×
3
—
1.3333
×
10
-
3
13.141
4
×
4
3.3333
×
10
-
4
5
×
10
-
3
26.9487
5
×
5
6.6667
×
10
-
4
8.6667
×
10
-
3
62.3437
6
×
6
1.7
×
10
-
3
1.47
×
10
-
2
33.1177
7
×
7
3
×
10
-
3
2.4367
×
10
-
2
41.2426
8
×
8
6.33
×
10
-
3
4.1733
×
10
-
2
40.42
9
×
9
5.33
×
10
-
3
7.1766
×
10
-
2
49.6247
10
×
10
4.27
×
10
-
2
0.11916
59.917
11
×
11
2.003
×
10
-
2
0.19493
64.9183
12
×
12
3.873
×
10
-
2
0.29843
84.1493
1
3
×
1
3
0.2944
0.4453
94.3677
1
4
×
1
4
1.603
0.63356
119.5947
1
5
×
1
5
0.3962
0.89496
145.8433
16
×
16
39.8416
1.24846
173.3940In 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-Qij)xij) for all (i,j)∈K in the objective function of (21) is redundant. Thus,
(31)min(b-∑i=1n∑j=1n(αijxij)(b-a+∑i=1n∑j=1n(γijxij)),min(1-(1-Qij)xij)∀(i,j)∈Kb-∑i=1n∑j=1n(αijxij)(b-a+∑i=1n∑j=1n(γijxij)))=b-∑i=1n∑j=1n(αijxij)(b-a+∑i=1n∑j=1n(γijxij)).
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 whenb is sufficiently small.
Problem size
Index-based B&Balgorithm
S&T algorithm
LINGO
Average time (sec.)
Average time (sec.)
Average time (sec.)
3
×
3
3.3333
×
10
-
4
6.667
×
10
-
4
0.405
4
×
4
6.6667
×
10
-
4
1.6667
×
10
-
3
0.8493
5
×
5
1.6667
×
10
-
3
3
×
10
-
3
1.6643
6
×
6
4.6667
×
10
-
3
6
×
10
-
3
2.9837
7
×
7
7
×
10
-
3
7.6667
×
10
-
3
5.0093
8
×
8
2.27
×
10
-
2
1.0033
×
10
-
2
7.962
9
×
9
7.6133
×
10
-
2
1.2367
×
10
-
2
13.798
10
×
10
0.2226
1.67
×
10
-
2
18.0663
11
×
11
0.9057
2.27
×
10
-
2
26.4913
12
×
12
3.725
2.8367
×
10
-
2
39.366
1
3
×
1
3
11.3644
3.94
×
10
-
2
84.0933
1
4
×
1
4
54.5097
5.2767
×
10
-
2
90.954
1
5
×
1
5
179.3896
7.1433
×
10
-
2
116.3357
16
×
16
1143.4145
9.3467
×
10
-
2
143.4327An overview of Tables1–3 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.



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*Source: 791415-2013-06-12.xml*