Answered Essay: Consider the Linear Programming formulation and its associated MS Excel solution below. Carefully examine the

Consider the Linear Programming formulation and its associated MS Excel solution below. Carefully examine the MS Excel output above.

Solver Options Max Time 100 sec, erations 100, Precision 0.000001 Max Subproblems Unlimited, Max Integer Sols Unlimited, Integer Tolerance 5 Min z 38x1 2x2 Objective Cell (Min) Cell Name Original Value Final Value S.t. SB$44. Z 0 82.66666667 X1 6x, 12 5x1 4x2 40 Variable Cells 1x1 2x, 2 12 Cell Name Original Value Final Value nteger 0x1 1x, S 6 $B$42 Solution X1 0 5.333333333 Contin X1, X2 0 SC$42 Solution X2 3.333333333 Contin variable cells Reduced Objective Allowable Allowable Fina Cell Name Value Cost Coefficient Increase Decrease $B$42 Solution X1 5.333333333 SC$42 Solution X2 3.333333333 12 5.6 Constraints Shadow Constraint Allowable Allowable Fina Cell R. H. Side Increase Decrease Name Value Price $B$47 LHS -9.333333333 12 1E+30 21.33333333 $B$48 LH 40 0.666666667 40 12.8 16 12 4.666666667 $B$49 LHS 3.2 3.368421053 12 $B$50 LHS 3.333333333 1E+30 2.666666667

The dual price associated with constraint two is 0.67. If the right hand side of constraint two is increased by 5 units (i.e., 5×1 + 4×2 >=40 becomes 5×1 + 4×2 >=45) what impact does this have on the optimal objective function value of z = 82.67? Specifically, show any calculations to get a revised z-value.

Similarly, suppose that the right hand side of constraint two is decreased by three (3) units. What impact does the aforementioned change have on the objective function value of z = 82.67? Specifically show any calculations to get a revised z-value.

Expert Answer

The shadow price of constraint is an amount by which the objective function value will increase or decrease if a unit of the binding constraint is made available or reduced respectively. But the shadow price is valid till the increase in constraint value is within allowable increase and decrease.

The shadow price or dual price of painting hours is 0.67, it means by increasing the RHS by 1 unit the objective function value will increase by 0.67 per unit increase or the 0.67 per unit decrease.

The allowable increase in the RHS of constraint is 12.8 and allowable decrease is 16 units, the shadow price of the constraint is valid between (40+12.8 =) 54.8 to (40-16 =) 24.

If the RHS of the constraint is increased by 5 units, which is within the allowable increase, the objective function value will increase by 0.67 x 5 = 3.35. thus, revised objective function value = 82.67 + 3.35 = 86.02.

If the RHS of the constraint is decreased by 3 units, which is within the allowable decrease, the objective function value will decrease by 0.67 x 3 = 2.01. thus, revised objective function value = 82.67 – 2.01 = 80.66.

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