Let us consider a product-mix selection problem . Suppose that the Knox Mix company has the option of using one or more of four different types of production processes. The first and second processes yield items of product A, and the third and fourth yield items of product B. The inputs for each process are labor measured in man-weeks, pounds of material Y, and boxes of material Z. Since each process varies in its input requirements, the 81 profitabilities of the process differ, even for processes producing the same item. The manufacturer's decision on a week's production schedule is limited in the range of possibilities by the available amounts of manpower and both kinds of raw materials.
With production levels in processes 1, 2, 3, 4 as x1, x2, x3, x4 respectively. The problem can then be formulated as
z = 4x1 + 5x2 + 9x3 + 11x4 (Profit)
Max z subject to constraints
g1(x) = x1 + x2 + x3 + x4 < 15 (Man Weeks)
g2(x) = 7x1 + 5x2 + 3x3 + 2x4 < 120 (Material Y)
g3(x) = 3x1 + 5x2 + 10x3 + 15x4 < 100 (Material Z)
We then solve this linear programming problem by use of the Simplex Method - (using Tora Software) . The optimal solution is:
x* = (50 / 7, 0, 55 / 7, 0) = (7.14, 0, 7.86, 0) and z* = $ 695 / 7 = $99.29. The actual resources used are 15, 73.57 and 100 units for manweeks, material Y and material Z, respectively
Let us assume that the available constraints in the above problem are imprecise with a tolerance of 25%. Then the membership function of fuzzy constraints are
1 if g1(x) < 15
μ1(x) = 1 - [ (g1(x) - 15) / 4] if 15 < g1(x) < 19
0 if g1(x) > 19
1 if g2(x) < 120
μ2(x) = 1 - [ (g2(x) - 120) / 30] if 120 < g2(x) < 150
0 if g2(x) > 150
1 if g3(x) < 100
μ3(x) = 1 - [ (g3(x) - 100) / 25] if 100 < g3(x) < 125
0 if g3(x) > 125
Thus we have the following problem to solve:
z = 4x1 + 5x2 + 9x3 + 11x4
Max z subject to constraints
μ1(x) > 𝜶
μ2(x) > 𝜶
μ3(x) > 𝜶
𝜶 ∈ [0, 1] and x1, x2, x3, x4 > 0
which is nothing but:
z = 4x1 + 5x2 + 9x3 + 11x4
Max z subject to constraints
g1(x) = x1 + x2 + x3 + x4 < 15 + 4(1 - 𝜶) (Man Weeks)
g2(x) = 7x1 + 5x2 + 3x3 + 2x4 < 120 + 30(1 - 𝜶) (Material Y)
g3(x) = 3x1 + 5x2 + 10x3 + 15x4 < 100 + 25(1 - 𝜶) (Material Z)
𝜶 ∈ [0, 1] and x1, x2, x3, x4 > 0
Set θ = 1 - 𝜶, we get the following parametric programming problem
z = 4x1 + 5x2 + 9x3 + 11x4
Max z subject to constraints
g1(x) = x1 + x2 + x3 + x4 < 15 + 4θ (Man Weeks)
g2(x) = 7x1 + 5x2 + 3x3 + 2x4 < 120 + 30θ (Material Y)
g3(x) = 3x1 + 5x2 + 10x3 + 15x4 < 100 + 25θ (Material Z)
where θ ∈ [0, 1] is a parameter. By use of the parametric technique and the final table of the simplex method shown in Table above, we can obtain the following parametric results: (15θ/7, 66θ/7, 13θ/7)
The final simplex table is shown below. Since the RHS ( 50/7 + 15θ/7 ), (325/7 + 66θ/7) and (55/7 + 13θ/7), for θ ∈ [0,1], are always greater than zero, the optimal solution is then:
x* = (7.14 + 2.14θ, 0, 7.86 + 1.85θ, 0) and z* = $ (99.29 + 13.43θ).
The final parametric tableau of the simplex method for the Knox selection problem:
The solutions obtained for parametric programming problem are-
- Fuzzy Mathematical Programming (Young-Jou Lai,Ching-Lai Hwang)
- Tora Software (Hamdy A. Taha)
- Lai, Y.J. and C.L. Hwang, Interactive fuzzy linear programming, Fuzzy Sets and Systems 45 (1992) 169-183.
- Negoita, C.V. and D. Ralescu, Simulation, Knowledge-Based, Computing, and Fuzzy Statistics (Van Nostrand Reinhold, New York, 1987).