A multi-objective linear programming problem is made from fuzzy linear programming problem. It is due the fact that it is used fuzzy programming method during the solution. The Multi objective linear programming problem can be converted into the single objective function by various method s as Chandra Sen’s method, weighted sum method, ranking function method, statistical averaging method. In this paper, Chandra Sen’s method and statistical averaging method both are used here for making single objective function from multi-objective function. Two multi-objective programming problems are solved to verify the result. One is numerical example and the other is real life example. Then the problems are solved by ordinary simplex method and fuzzy programming method. It can be seen that fuzzy programming method gives better optimal values than the ordinary simplex method.
Fuzzy decision-making technique is used repeatedly for fuzzy multi-objective linear programming problem. It has been investigated for more than decades by many researchers. The idea of fuzzy set was first proposed by Zadeh [
The idea of fuzzy decision was mentioned by Bellman and Zadeh in 1970. Multi-Objective Linear Programming problems (MOLPP) with fuzzy goals were taken into account by Zimmerman [
First formulation of fuzzy linear programming (FLP) was proposed by Zimmermann. A new fuzzy primal and dual simplex algorithm is for solving FLP by Mahdavi-Amiri et al. [
Nasseri and Alizadeh [
In this paper, we solve MOLPP by using fuzzy programming method. For making single objective from multi-objective, we apply Chandra Sen’s method and statistical averaging method by Nahar and Alim [
Fuzzy number
Let X be a set of objects for fuzzy statement. The set of ordered pairs A ˜ = { ( x , μ A ( x ) ) | x ∈ X } where μ A : X → [ 0 , 1 ] is a fuzzy set in X. The evaluation function μ A ( x ) is called the membership function.
A fuzzy number A = (a, b, c) is said to be a triangular fuzzy number if its membership function is given by
μ A ( x ) = { x − a b − a , a ≤ x ≤ b 1 , x = b x − b b − c , b ≤ x ≤ c
Mathematical model of multi-objective linear programming problem:
max Z s ( x ) = ∑ j = 1 n c j x j , s = 1 , 2 , ⋯ , s
Subject to
∑ j = 1 n a i j x j ( ≤ , = , ≥ ) b i , i = 1 , 2 , ⋯ , m x j ≥ 0 , j = 1 , 2 , ⋯ , n (1)
To verify the result, two examples are shown. One is numerical example (Hypothetical) and the other is real life example (An investigation for the risk reduction along the coastal area).
Chandra Sen’s method [
In this paper for making single objective from MOLPP, Chandra Sen’s method and statistical averaging methods are used. Later on, the single objective function is solved by the ordinary simplex method and fuzzy programming method and hence the results are compared.
Fuzzy Programming MethodStep 1: Solve MOLPP by using simplex algorithm, considering only one of the objectives at a time and ignoring all others. Repeat the process s times for s different objective functions.
Step 2: Using all solutions in step 1, construct a pay off matrix of size s by s. Then from the pay off matrix estimate the lower bound (Ls) and the upper bound (Us) for the kth objective function zs as:
L s ≤ z s ≤ U s , s = 1 , 2 , ⋯ , s
Step 3: Define a fuzzy linear membership function μ ( z s ( x ) ) for the sth objective function z s , s = 1 , 2 , ⋯ , s
μ ( z s ( x ) ) = { 0 , if z s ≤ L s 1 − U s − z s U s − L s , if L s ≤ z s ≤ U s 1 , if z s ≥ U s (2)
Step 4: Using membership functions we can get a crisp model by introducing an augmented variable λ
max : 1 − U s − z s U s − L s , s = 1 , 2 , ⋯ , s
min : U s − z s U s − L s ; s = 1 , 2 , ⋯ , s
Subject to (1) can be further written as
Minimize λ
Subject to
∑ j = 1 n c j x i + ( U s − L s ) λ ≥ U s , s = 1 , 2 , ⋯ , s ∑ j = 1 n a i j x j ( ≤ , = , ≥ ) b i , i = 1 , 2 , ⋯ , m λ ≥ 0 , x j ≥ 0 , j = 1 , 2 , ⋯ , n (3)
Step 5: Solve the crisp model by simplex algorithm and find optimal solution.
max z 1 = 9 x 0 + 4 x 1 + 5 x 2 max z 2 = 3 x 0 + x 1 + 5 x 2 max z 3 = x 0 + 2 x 1 + 3 x 2 (4)
Subject to
4 x 0 + 2 x 1 + 3 x 2 ≤ 5 5 x 0 + 3 x 1 + 2 x 2 ≤ 9 3 x 0 + 2 x 1 + 7 x 2 ≤ 7 x 0 , x 1 , x 2 ≥ 0 (5)
For the first objective function in Equation (4) with constraints in equation (5), by applying simplex algorithm we get z1 = 11.25 with (5/4, 0, 0). The value of the objective function is obtained using the steps of simplex method.
Similarly for second objective function in equation (4) with same constraints in equation (5) we get z2 = 5.63 with (14/19, 0, 13/19).
And for last objective function in Equation (4) with same constraints in equation (5) we get z3 = 5 with (0, 7/4, 1/2).
A Pay-off matrix is formulated as in “
For the Pay-off matrix, lower and upper bound are established as:
9.5 ≤ z 1 ≤ 11.25 3.75 ≤ z 2 ≤ 5.67 1.25 ≤ z 3 ≤ 5 (6)
Using Equation (3) we can write as
min λ
Subject to
9 x 0 + 4 x 1 + 5 x 2 + 1.75 λ ≥ 11.25 3 x 0 + x 1 + 5 x 2 + 1.92 λ ≥ 5.67 x 0 + 2 x 1 + 3 x 2 + 3.75 λ ≥ 5
4 x 0 + 2 x 1 + 3 x 2 ≤ 5 5 x 0 + 3 x 1 + 2 x 2 ≤ 9 3 x 0 + 2 x 1 + 7 x 2 ≤ 7 x 0 , x 1 , x 2 , λ ≥ 0 (7)
| (x1, x2, x3) | Z1 | Z2 | Z3 |
|---|---|---|---|
| (1.25, 0, 0) | 11.25 | 3.75 | 1.25 |
| (0.74, 0, 0.68) | 10.06 | 5.67 | 2.78 |
| (0, 1.75, 0.5) | 9.5 | 4.25 | 5 |
Using simplex algorithm, we get the result x 0 = 0.69 , x 1 = 0.51 , x 2 = 0.41 , λ = 0.55 .
Finally, the crisp model is solved to get the result φ 1 = 10.3 , φ 2 = 4.63 , φ 3 = 2.94 .
Thus, for simplex method we get the optimal values for z 1 = 11.25 , z 2 = 5.63 , z 3 = 5 and for fuzzy programming method we get φ 1 = 10.3 , φ 2 = 4.63 , φ 3 = 2.94 .
For making single objective from multi-objective applying Chandra Sen’s method by Sen
z = z 1 φ 1 + z 2 φ 2 + z 3 φ 3 = 1 11.25 ( 9 x 0 + 4 x 1 + 5 x 2 ) + 1 5.63 ( 3 x 0 + x 1 + 5 x 2 ) + 1 5 ( x 0 + 2 x 1 + 3 x 2 ) = x 0 ( 0.8 + 0.53 + 0.2 ) + x 1 ( 0.35 + 0.18 + 0.4 ) + x 2 ( 0.44 + 0.89 + 0.6 ) = 1.53 x 0 + 0.93 x 1 + 1.93 x 2
Thus, the single objective becomes
max z = 1.53 x 0 + 0.93 x 1 + 1.93 x 2 (8)
From the single objective function in Equation (8) with same constraints (5) by using simplex algorithm we get z = 2.5925 with (0, 7/4, 1/2).
For φ 1 = 11.25 , φ 2 = 5.63 , φ 3 = 5 using statistical averaging method such as arithmetic mean (A.M), geometric mean (G.M) and harmonic mean (H.M) we get
A . M = 11.25 + 5.63 + 5 3 = 7.29 G . M = 11.25 × 5.63 × 5 3 = 6.687 H . M = 3 1 11.25 + 1 5.63 + 1 5 = 6.42
Applying arithmetic mean formula by Nahar and Alim we can write
max z = 1 A . M ( z 1 + z 2 + z 3 ) = 1 7.29 ( 9 x 0 + 4 x 1 + 5 x 2 + 3 x 0 + x 1 + 5 x 2 + x 0 + 2 x 1 + 3 x 2 ) = 1 7.29 [ x 0 ( 9 + 3 + 1 ) + x 1 ( 4 + 1 + 2 ) + x 2 ( 5 + 5 + 3 ) ] = 1 7.29 ( 13 x 0 + 7 x 1 + 13 x 2 ) = 1.783 x 0 + 0.96 x 1 + 1.783 x 2
Thus, the single objective becomes
max z = 1.783 x 0 + 0.96 x 1 + 1.783 x 2 (9)
From the single objective function in Equation (9) with same constraints (5) by using simplex algorithm we get
z = 2.5715 with (0, 7/4, 1/2)
Applying geometric mean formula by Nahar and Alim we can write
max z = 1 G . M ( z 1 + z 2 + z 3 ) = 1 6.687 ( 13 x 0 + 7 x 1 + 13 x 2 ) = 1.94 x 0 + 1.05 x 1 + 1.94 x 2
Thus, the single objective is
max z = 1.94 x 0 + 1.05 x 1 + 1.94 x 2 (10)
From the single objective function in equation (10) with same constraints (5) by using simplex algorithm we get z = 2.8075 with (0, 7/4, 1/2).
Applying Harmonic mean formula by Nahar and Alim (2017) we can write
max z = 1 H . M ( z 1 + z 2 + z 3 ) = 1 6.42 ( 13 x 0 + 7 x 1 + 13 x 2 ) = 2.02 x 0 + 1.09 x 1 + 2.024 x 2
Thus, the single objective is
max z = 2.02 x 0 + 1.09 x 1 + 2.024 x 2 (11)
From the single objective function in Equation (11) with same constraints (5) by using simplex algorithm we get z = 2.9175 with (0, 7/4, 1/2).
Using the value of fuzzy programming method applying Chanda Sen’s method by Sen
max z = z 1 φ 1 + z 2 φ 2 + z 3 φ 3 = z 1 10.3 + z 2 4.63 + z 3 2.94 = 1 10.3 ( 9 x 0 + 4 x 1 + 5 x 2 ) + 1 4.63 ( 3 x 0 + x 1 + 5 x 2 ) + 1 2.94 ( x 0 + 2 x 1 + 3 x 2 ) = x 0 ( 0.874 + 0.65 + 0.34 ) + x 1 ( 0.38 + 0.216 + 0.68 ) + x 2 ( 0.48 + 1.08 + 1.02 ) = 1.864 x 0 + 1.276 x 1 + 2.58 x 2
Thus, the single objective is
max z = 1.864 x 0 + 1.276 x 1 + 2.58 x 2 (12)
From the single objective function in Equation (12) with same constraints (5) by using simplex algorithm we get z = 3.523 with (0, 7/4, 1/2).
For φ 1 = 10.3 , φ 2 = 4.63 , φ 3 = 2.94 using statistical averaging method such as arithmetic mean (A.M), geometric mean (G.M) and harmonic mean (H.M) we get
A . M = 11.25 + 5.63 + 5 3 = 7.29 G . M = 10.3 × 4.63 × 2.94 3 = 5.11 H . M = 3 1 10.3 + 1 4.63 + 1 2.94 = 4.594
Applying arithmetic mean formula by Nahar and Alim (2017) we can write
max z = 1 A . M ( z 1 + z 2 + z 3 ) = 1 5.957 ( 9 x 0 + 4 x 1 + 5 x 2 + 3 x 0 + x 1 + 5 x 2 + x 0 + 2 x 1 + 3 x 2 ) = 1 5.957 [ x 0 ( 9 + 3 + 1 ) + x 1 ( 4 + 1 + 2 ) + x 2 ( 5 + 5 + 3 ) ] = 1 5.957 ( 13 x 0 + 7 x 1 + 13 x 2 ) = 2.182 x 0 + 1.175 x 1 + 2.182 x 2
max z = 2.182 x 0 + 1.175 x 1 + 2.182 x 2 (13)
From the single objective function in Equation (13) with same constraints (5) by using simplex algorithm we get z = 3.1472 with (0, 7/4, 1/2).
Applying geometric mean formula by Nahar and Alim (2017) we can write
max z = 1 G . M ( z 1 + z 2 + z 3 ) = 1 5.11 ( 13 x 0 + 7 x 1 + 13 x 2 ) = 2.544 x 0 + 1.369 x 1 + 2.544 x 2
Thus, single objective is
max z = 2.544 x 0 + 1.369 x 1 + 2.544 x 2 (14)
From the single objective function in Equation (14) with same constraints (5) by using simplex algorithm we get z = 3.65 with (0, 7/4, 1/2).
Applying harmonic mean formula by Nahar and Alim (2017) we can write
max z = 1 H . M ( z 1 + z 2 + z 3 ) = 1 4.594 ( 13 x 0 + 7 x 1 + 13 x 2 ) = 2.83 x 0 + 1.525 x 1 + 2.83 x 2
Thus, the single objective is
max z = 2.83 x 0 + 1.525 x 1 + 2.83 x 2 (15)
From the single objective function in Equation (15) with same constraints (5) by using simplex algorithm we get z = 4.083 with (0, 7/4, 1/2). All these results can be shown in “
“
| Methods | Chandra Sen’s method | Arithmetic mean | Geometric mean | Harmonic mean |
|---|---|---|---|---|
| Ordinary Simplex Method | 2.5925 | 2.5715 | 2.8075 | 2.9175 |
| Fuzzy programming method | 3.523 | 3.1472 | 3.65 | 4.083 |
For the real life example, an example is taken from Nahar, et al. [
The objective function of fuzzy linear programming problem is by Nahar, et al.
max Z = ( 0 , 0.00335 , 0.067 ) x 1 + ( 0 , 0.36 , 0.72 ) x 2 + ( − 0.072 , − 0.036 , 0 ) x 3 + ( − 0.1413 , − 0.07065 , 0 ) x 4
With fuzzy constraints
( 6.0877 , 26 , 45.9123 ) x 1 + ( 41.7423 , 70 , 98.2577 ) x 2 + ( − 5.1243 , 7 , 19.1243 ) x 3 + ( 3.7907 , 13.75 , 23.7093 ) x 4 ≤ ( 90.8769 , 95 , 99.1231 )
Thus, the multiple objective functions become as like by Nahar, et al.
max Z 1 = 0 x 1 + 0 x 2 − 0.072 x 3 − 0.1413 x 4 max Z 2 = 0.00335 x 1 + 0.36 x 2 − 0.036 x 3 − 0.07065 x 4 max Z 3 = 0.067 x 1 + 0.72 x 2 + 0 x 3 + 0 x 4 (16)
subject to
6.0877 x 1 + 41.7423 x 2 + ( − 5.1243 ) x 3 + 3.7907 x 4 ≤ 90.8769 ( − 19.9123 ) x 1 + ( − 28.2577 ) x 2 + ( − 12.1243 ) x 3 + ( − 9.9593 ) x 4 ≤ − 4.1231 52 x 1 + 140 x 2 + 14 x 3 + 27.5 x 4 ≤ 190 (17)
For the first objective function in Equation (16) with same constraints in Equation (17), by applying simplex algorithm we get
ϕ 1 = 0 with (0, 0.1459, 0, 0)
For the second objective function in Equation (16) with same constraints in Equation (17), by applying simplex algorithm we get
ϕ 2 = 0.4886 with (0, 1.3571, 0, 0)
Similarly for third objective function, in equation (16) with same constraints in Equation (17), we get
ϕ 3 = 0.9771 with (0, 1.3571, 0, 0)
Applying Chandra Sen’s method for making single objective function from multi objective functions
max z = z 1 ϕ 1 + z 2 ϕ 2 + z 3 ϕ 3 = 0.0754178 x 1 + 1.47366 x 2 − 0.0736812 x 3 − 0.144599 x 4
Thus, the single objective function becomes
max Z = 0.0754178 x 1 + 1.47366 x 2 − 0.0736812 x 3 − 0.144599 x 4 (18)
For this objective function in Equation (18) with same constraints in Equation (17) we get the result 2 with (0, 1.3571, 0, 0).
Applying arithmetic mean, geometric mean and harmonic mean among ϕ 1 , ϕ 2 , ϕ 3
A . M = 0.48857 G . M = 0 H . M = 0.97716
Arithmetic averaging method:
max z = 1 A . M ( z 1 + z 2 + z 3 ) = 0.154364 x 1 + 3.016272 x 2 − 0.150809 x 3 − 0.295963 x 4
Thus, the single objective function becomes
max Z = 0.154364 x 1 + 3.016272 x 2 − 0.150809 x 3 − 0.295963 x 4 (19)
For this objective function in Equation (19) with same constraints in equation (17) we get the result 4.0935 with (0, 1.3571, 0, 0).
Harmonic averaging method:
max z = 1 H . M ( z 1 + z 2 + z 3 ) = 0.07718 x 1 + 1.508099 x 2 − 0.0754031 x 3 − 0.147978 x 4
Thus, the single objective becomes
max Z = 0.07718 x 1 + 1.508099 x 2 − 0.0754031 x 3 − 0.147978 x 4 (20)
For this objective function in Equation (20) with same constraints in equation (17) we get the result 2.0467 with (0, 1.3571, 0, 0).
It can be seen from “
Choosing minimum from the optimal values of maximum type in Chandra Sen’s method we get m = 0.4886
max z = 1 0.4886 ( z 1 + z 2 + z 3 ) = 2.0467 ( 0.075417 x 1 + 1.47366 x 2 − 0.0736812 x 3 − 0.144599 x 4 ) = 0.15435 x 1 + 3.01614 x 2 − 0.1508 x 3 − 0..29595 x 4
Thus, the single objective becomes
max Z = 0.15435 x 1 + 3.01614 x 2 − 0.1508 x 3 − 0.29595 x 4 (21)
For this objective function in Equation (21) with same constraints in equation (17) we get the result 4.0933 with (0, 1.3571, 0, 0).
It can be seen from “
| Chandra Sen’s method | Arithmetic averaging method | Harmonic averaging method |
|---|---|---|
| Zmax = 2 with (0, 1.3571, 0, 0) | Zmax = 4.0935 with (0, 1.3571, 0, 0) | Zmax = 2.0467 with (0, 1.3571, 0, 0) |
| Chandra Sen’s Method | Statistical Averaging Method (SAM) | New Statistical Averaging Method (NSAM) |
|---|---|---|
| Zmax = 2 with (0, 1.3571, 0, 0) | Zmax = 4.0935 with (0, 1.3571, 0, 0) | Zmax = 4.0933 with (0, 1.3571, 0, 0) |
max Z 1 = 0 x 1 + 0 x 2 − 0.072 x 3 − 0.1413 x 4 max Z 2 = 0.00335 x 1 + 0.36 x 2 − 0.036 x 3 − 0.07065 x 4 max Z 3 = 0.067 x 1 + 0.72 x 2 + 0 x 3 + 0 x 4 (22)
subject to
6.0877 x 1 + 41.7423 x 2 + ( − 5.1243 ) x 3 + 3.7907 x 4 ≤ 90.8769 ( − 19.9123 ) x 1 + ( − 28.2577 ) x 2 + ( − 12.1243 ) x 3 + ( − 9.9593 ) x 4 ≤ − 4.1231 52 x 1 + 140 x 2 + 14 x 3 + 27.5 x 4 ≤ 190 (23)
For the first objective function in Equation (22) with constraints in equation (23) by using simplex algorithm we get z1 = 0 with (0, 0.1459, 0, 0).
Similarly for second objective function in equation (22) with same constraints in Equation (23) by using simplex algorithm we get z2 = 1.224 with (3.6538, 0, 0, 0).
And for the last objective function in Equation (22) with same constraints in Equation (23) by using simplex algorithm we get z3 = 0.9771 with (0, 1.3571, 0, 0).
A pay-off matrix is formulated as in “
For the Pay-off matrix, lower and upper bound are established as:
0 ≤ z 1 ≤ 0 0.0525 ≤ z 2 ≤ 1.224 0.10505 ≤ z 3 ≤ 0.9771 (24)
Using the membership functions as defined and introducing augmented variable λ a crisp model is formulated as
min λ
Subject to
0 x 1 + 0 x 2 + ( − 0.072 ) x 3 − 0.1413 x 4 ≥ 0 0.00335 x 1 + 0.36 x 2 + ( − 0.036 ) x 3 + ( − 0.07065 ) x 4 + 1.1715 λ ≥ 1.224 52 x 1 + 140 x 2 + 14 x 3 + 27.5 x 4 + 0.872 λ ≥ 0.9771190
6.0877 x 1 + 41.7423 x 2 + ( − 5.1243 ) x 3 + 3.7907 x 4 ≤ 90.8769 ( − 19.9123 ) x 1 + ( − 28.2577 ) x 2 + ( − 12.1243 ) x 3 + ( − 9.9593 ) x 4 ≤ − 4.1231 52 x 1 + 140 x 2 + 14 x 3 + 27.5 x 4 ≤ 190 x 1 , x 2 , x 3 , x 4 , λ ≥ 0 (25)
| (x1, x2, x3) | Z1 | Z2 | Z3 |
|---|---|---|---|
| (0, 0.1459, 0, 0) | 0 | 0.0525 | 0.10505 |
| (3.6538, 0, 0, 0) | 0 | 1.224 | 0.245 |
| (0, 1.3571, 0, 0) | 0 | 0.488 | 0.9771 |
Using simplex algorithm, we get the result x 1 = 0 , x 2 = 1.3571 , x 3 = 0 , x 4 = 0 , λ = 0.6278
Finally, the crisp model is solved to get the result ϕ 1 = 0 , ϕ 2 = 0.4885 , ϕ 3 = 0.977
Using the value of Fuzzy Programming method applying Chanda Sen’s method
max z = 0.0753 x 1 + 1.47382 x 2 − 0.0737 x 3 − 0.1446 x 4 (26)
For the single objective function in Equation (26) with constraints in Equation (23), by using simplex algorithm we get z = 2.002 with (0, 1.3571, 0, 0).
Applying arithmetic mean formula by Nahar and Alim (2017) we get the single objective function
max z = 0.096 x 1 + 1.474 x 2 − 0.1474 x 3 − 0.289 x 4 (27)
For the single objective function in Equation (27) with constraints in equation (23), by using simplex algorithm we get z = 2.004 with (0, 1.3571, 0, 0).
Applying harmonic mean formula by Nahar and Alim (2017) we get the single objective function
max z = 0.108 x 1 + 1.658 x 2 − 0.166 x 3 − 0.325 x 4 (28)
For the single objective function in Equation (28) with constraints in Equation (23), by using simplex algorithm we get z = 2.2501 with (0, 1.3571, 0, 0).
Choosing minimum from the optimal values of maximum type in Chandra Sen’s method we get m = 0.4885
max z = 1 0.2442 ( z 1 + z 2 + z 3 ) = 4.094 ( 0.07035 x 1 + 1.08 x 2 − 0.108 x 3 − 0.21195 x 4 ) = 0.288 x 1 + 4.4215 x 2 − 0.442 x 3 − 0.8677 x 4 (29)
For this objective function in Equation (29) with same constraints in equation (23) we get the result 6.006 with (0, 1.3571, 0, 0).
It can be seen from “
This table can be described in the following bar diagram in “
| Methods | Chandra Sen’s Method | Statistical Averaging Method (SAM) | New Statistical Averaging Method (NSAM) |
|---|---|---|---|
| Ordinary simplex method | Zmax = 2 with (0, 1.3571, 0, 0) | Zmax = 4.0935 with (0, 1.3571, 0, 0) | Zmax = 4.0933with (0, 1.3571, 0, 0) |
| Fuzzy programming method | Zmax = 2 with (0, 1.3571, 0, 0) | Zmax = 2.2501 with (0, 1.3571, 0, 0) | Zmax = 6.006 with (0, 1.3571, 0, 0) |
For finding optimal values here we apply simplex method and fuzzy programming method. Fuzzy programming method gives better optimal values than simplex method. Our next stage is making single objective function from multi-objective functions. For this we apply statistical averaging method and new statistical method. It can be seen that statistical averaging method gives better result than Chandra Sen’s method in both ordinary simplex method and in fuzzy programming method. To reduce risk in the coastal region fuzzy programming method is used here. It can be seen that risk reduction capacity is maximized in fuzzy programming method.
The authors declare no conflicts of interest regarding the publication of this paper.
Nahar, S., Akter, M. and Alim, Md.A. (2023) Solving Multi-Objective Linear Programming Problem by Statistical Averaging Method with the Help of Fuzzy Programming Method. American Journal of Operations Research, 13, 19-32. https://doi.org/10.4236/ajor.2023.132002