Simplex Method: Goal Programming

In the previous section, we provided an example of Goal Programming. Another example is given below to enhance your knowledge.

We will use the simplex method to solve this problem.

exampleExample 2: Simplex Method of Goal Programming

Minimize z = P1d1− + 2P2d2− + P2d3− + P3d1+

subject to

x1 + x2 + d1− - d1+ = 350
x1 + d2− = 200
x2 + d3− = 300

x1, x2, d1−, d2−, d3−, d1+ ≥ 0

Solution.

Substituting x1 = 0, x2 = 0, d1+ = 0
Therefore, d1− = 350, d2− = 200, d3− = 300

Table 1

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  cj 0 0 P1 2P2 P2 P3  
cB Basic variables
B 		x1 x2 d1− d2− d3− d1+ Solution values
b (=XB)
P1 d1− 1 1 1 0 0 -1 350
2P2 d2− 1 0 0 1 0 0 200
P2 d3− 0 1 0 0 1 0 300
zj-cj P3 0 0 0 0 0 -1 0
P2 2 1 0 0 0 0 700
P1 1 1 0 0 0 -1 350

Key column = x1 column
Minimum positive value = Min(350/1, 200/1) = 200
So, d2− row is the key row.
Therefore, d2− departs & x1 enters

Table 2

  cj 0 0 P1 2P2 P2 P3  
cB Basic variables
B 		x1 x2 d1− d2− d3− d1+ Solution values
b (=XB)
P1 d1− 0 1 1 -1 0 -1 150
0 x1 1 0 0 1 0 0 200
P2 d3− 0 1 0 0 1 0 300
zj-cj P3 0 0 0 0 0 -1 0
P2 0 1 0 -2 0 0 300
P1 0 1 0 -1 0 -1 150

Table 3

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  cj 0 0 P1 2P2 P2 P3  
cB Basic variables
B 		x1 x2 d1− d2− d3− d1+ Solution values
b (=XB)
0 x2 0 1 1 -1 0 -1 150
0 x1 1 0 0 1 0 0 200
P2 d3− 0 0 -1 1 1 1 150
zj-cj P3 0 0 0 0 0 -1 0
P2 0 0 -1 -1 0 1 150
P1 0 0 -1 0 0 0 0

Fina Table: Simplex Method

  cj 0 0 P1 2P2 P2 P3  
cB Basic variables
B 		x1 x2 d1− d2− d3− d1+ Solution values
b (=XB)
0 x2 0 1 0 0 1 0 300
0 x1 1 0 0 1 0 0 200
P3 d1+ 0 0 -1 1 1 1 150
zj-cj P3 0 0 -1 1 1 0 150
P2 0 0 0 -2 -1 0 0
P1 0 0 -1 0 0 0 0

The optimal solution is:
x1 = 200, x2 = 300, d1− = 0, d2− = 0, d3− = 0, d1+ = 150.

Goal programming is a powerful tool to tackle multiple and incompatible goals of an enterprise. It is becoming popular because of its applicability in solving a wide range of problems in diversified areas. The most important advantage of goal programming is its great flexibility, which allows model simulation with numerous variation of constraints and goal priorities. This chapter discussed the solution of a goal programming problem by using the graphical method and simplex method.

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Goal programming Linear programming Simplex Method Assignment Problem