The following mathematical formulation describes a problem of allocating three r
ID: 3145672 • Letter: T
Question
The following mathematical formulation describes a problem of allocating three resources to the annual production of three commodities by a manufacturing firm. The amounts of the three products to be produced are denoted by xj, x2, and x3. The objective function reflects the dollar contribution to profit of these products.
Maximize 10x 15x2 5x subject to 2x x2 S 6000 3x1 + 3x2 + x3 9000 1 + 212 + 2x3 S 4000 , x2, x3 0. a. Without using the simplex method, verify that the optimal basis consists of the slack variable of the first constraint, x, and x2 Make use of the information in Part (a) to write the optimal tableau. The Research and Development Department proposes a new product whose production coefficients are represented by [2,4,2 If the contribution to profit is S15 per unit of this new product, should this product be produced? If so, what is the new optimal solution? b. c· d. What is the minimal contribution to profit that should be expected before production of this new product would actually increase the value of the objective function?Explanation / Answer
a and b) The problem is converted to canonical form by adding slack, surplus and artificial variables as appropiate
1. As the constraint 1 is of type '' we should add slack variable S1
2. As the constraint 2 is of type '' we should add slack variable S2
3. As the constraint 3 is of type '' we should add slack variable S3
After introducing slack variables
Max Z = 10 x1 + 15 x2 + 5 x3 + 0 S1 + 0 S2 + 0 S3
subject to
2 x1 + x2 + S1 = 6000
3 x1 + 3 x2 + x3 + S2 = 9000
x1 + 2 x2 + 2 x3 + S3 = 4000
and x1,x2,x3,S1,S2,S30
Iteration-1 Cj 10 15 5 0 0 0
B CB XB x1 x2 x3 S1 S2 S3
S1 0 6000 2 1 0 1 0 0
S2 0 9000 3 3 1 0 1 0
S3 0 4000 1 2 2 0 0 1
Z=0 0=
Zj=CBXB Zj Zj=CBxj 0 0=0×2+0×3+0×1
Zj=CBx1 0 0=0×1+0×3+0×2
Zj=CBx2 0 0=0×0+0×1+0×2
Zj=CBx3 0 0=0×1+0×0+0×0
Zj=CBS1 0 0=0×0+0×1+0×0
Zj=CBS2 0 0=0×0+0×0+0×1
Zj=CBS3
Cj-Zj 10 10=10-0 15 15=15-0 5 5=5-0 0 0=0-0 0 0=0-0 0 0=0-0
Ratio --- --- --- --- --- ---
Since all Cj-Zj0 and all XBi0 thus the current solution is the optimal solution.
Hence, optimal solution is arrived with value of variables as :
x1=0,x2=0,x3=0
Max Z=0
c and d) yes, new product should be produced.
Max Z = 2 x1 + 4 x2 + 2 x3
subject to
2 x1 + x2 6000
3 x1 + 3 x2 + x3 9000
x1 + 2 x2 + 2 x3 4000
and x1,x2,x30;
The problem is converted to canonical form by adding slack, surplus and artificial variables as appropiate
1. As the constraint 1 is of type '' we should add slack variable S1
2. As the constraint 2 is of type '' we should add slack variable S2
3. As the constraint 3 is of type '' we should add slack variable S3
After introducing slack variables
Max Z = 2 x1 + 4 x2 + 2 x3 + 0 S1 + 0 S2 + 0 S3
subject to
2 x1 + x2 + S1 = 6000
3 x1 + 3 x2 + x3 + S2 = 9000
x1 + 2 x2 + 2 x3 + S3 = 4000
and x1,x2,x3,S1,S2,S30
Iteration-1 Cj 2 4 2 0 0 0
B CB XB x1 x2 x3 S1 S2 S3
S1 0 6000 2 1 0 1 0 0
S2 0 9000 3 3 1 0 1 0
S3 0 4000 1 2 2 0 0 1
Z=0 0=
Zj=CBXB Zj Zj=CBxj 0 0=0×2+0×3+0×1
Zj=CBx1 0 0=0×1+0×3+0×2
Zj=CBx2 0 0=0×0+0×1+0×2
Zj=CBx3 0 0=0×1+0×0+0×0
Zj=CBS1 0 0=0×0+0×1+0×0
Zj=CBS2 0 0=0×0+0×0+0×1
Zj=CBS3
Cj-Zj 2 2=2-0 4 4=4-0 2 2=2-0 0 0=0-0 0 0=0-0 0 0=0-0
Ratio --- --- --- --- --- ---
Since all Cj-Zj0 and all XBi0 thus the current solution is the optimal solution.
Hence, optimal solution is arrived with value of variables as :
x1=0,x2=0,x3=0
Max Z=0
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