Chapter 19 – Linear Programming
6. a. The optimal values of the decision variables are: x1 = 4, x2 = 0, x3 = 18. The optimal value
of the objective function value = Z = 106.
Refer to the Excel Solver solution below:
Formulas used:
Cell
Formula
B13
=(B$4*B8)+(C$4*C8)+(D$4*D8)
B14
=(B$4*B9)+(C$4*C9)+(D$4*D9)
B15
=(B$4*B10)+(C$4*C10)+(D$4*D10)
F4
=(B4*B7)+(C4*C7)+(D4*D7)
Chapter 19 – Linear Programming
19–32
Solver Setup
Chapter 19 – Linear Programming
19–33
b. The optimal values of the decision variables are: x1 = 15, x2 = 10, x3 = 0. The optimal
value of the objective function value = Z = 210.
Refer to the Excel Solver solution below:
Formulas used:
Cell
Formula
B13
=(B$4*B8)+(C$4*C8)+(D$4*D8)
B14
=(B$4*B9)+(C$4*C9)+(D$4*D9)
B15
=(B$4*B10)+(C$4*C10)+(D$4*D10)
F4
=(B4*B7)+(C4*C7)+(D4*D7)
Chapter 19 – Linear Programming
19–34
Solver Setup
19–35
7. a. For Problem 6 Part a, determine the range of feasibility for the three constraints.
To determine the range of feasibility for each constraint, we must use the following values
listed under Constraints in the Sensitivity Report:
R.H. Side, Allowable Increase, and Allowable Decrease.
Lower Limit = Constraint R.H. Side – Allowable Decrease
Upper Limit = Constraint R.H. Side + Allowable Increase
Range of Feasibility for Constraint 1:
25 – 3 and 25 + 1E+30
Range = 22 to +
b. For Problem 6 Part a, determine the range of optimality for the three coefficients of the
objective function.
To determine the range of optimality for each coefficient, we must use the following
values found under Variable Cells in the Sensitivity Report:
Objective Coefficient, Allowable Increase, and Allowable Decrease.
Range of Optimality for Coefficient of x1:
4 – 1.5 and 4 + 11
Range = 2.5 to 15
Range of Optimality for Coefficient of x2:
2 – 1E+30 and 2 + 8.6
Chapter 19 – Linear Programming
19–36
19–37
8. a. For Problem 6 Part b, determine the range of feasibility for the three constraints.
To determine the range of feasibility for each constraint, we must use the following values
listed under Constraints in the Sensitivity Report:
R.H. Side, Allowable Increase, and Allowable Decrease.
Lower Limit = Constraint R.H. Side – Allowable Decrease
Upper Limit = Constraint R.H. Side + Allowable Increase
Range of Feasibility for Constraint 1:
25 – 5 and 25 + 1.667
b. For Problem 6 Part b, determine the range of optimality for the three coefficients of the
objective function.
To determine the range of optimality for each coefficient, we must use the following
values found under Variable Cells in the Sensitivity Report:
Objective Coefficient, Allowable Increase, and Allowable Decrease.
Range of Optimality for Coefficient of x1:
10 – 4 and 10 + 2
Range = 6 to 12
The Sensitivity Report from Excel Solver is shown below:
Chapter 19 – Linear Programming
19–38
Chapter 19 – Linear Programming
Formulas used:
Cell
Formula
B13
=(B$4*B8)+(C$4*C8)+(D$4*D8)
B14
=(B$4*B9)+(C$4*C9)+(D$4*D9)
B15
=(B$4*B10)+(C$4*C10)+(D$4*D10)
F4
=(B4*B7)+(C4*C7)+(D4*D7)
Solver Setup