A company is considering a blending problem whose linear programming model is as
ID: 381097 • Letter: A
Question
A company is considering a blending problem whose linear programming model is as given below, where X is the quantity of product i produced.
Max Z= 90x1 + 125x2 + 45x3 + 65x4
s.t. 0.10x1 + 0.25x2 + 0.08x3 + 0.21x4 <= 72
3x1 + 3x2 + x3 + x4 <= 1200
36x1 + 48x2 + 25x3 + 35x4 <= 25000
x1 + x2 <= 500
x3 + x4 <= 500
x1,x2,x3,x4 >= 0
In solving the problem in excel, the attached answer and sensitivity reports are produced and correspond to the optimal solution. Use the reports to answer the following questions (note that the questions are independent of one another).
a) What is the impact on profit if the company insists on producing product 4 and by how much?
b) What is the current solution to the problem?
c) What is the impact of increasing resource 2 by 200 units?
d) How much would you be willing to pay for one additional unit of resource 1?
e) What are the values of the slack variables associated with the constraints?
f) For what range of values for the objective coefficient for x1 does the current solution remain valid?
g) How much of resources 1 and 4 are used individually at the optimal solution?
Variable Cells Final Cell SC$19 Var. Values (C19-F19) x1 Reduced Objective Allowable Cost Coefficient Increase 175.5555556 SE$19 Var. Values (C19-F19) x3 SF$19 Var. Values (C19-F19) x4 90 11.92307692 125 13.21428571 1E+30 65 10.33333333 0 0 -10.33333333 45 Constraints Final Value Constraint Allowable R.H. Side Increase Cell $C$26 Constraints x1 $C$27 x1 SC$28 x1 $C$29 x1 SC$30 x1 Name Price 72 233.3333333 72 26.33333333 260 1E+30 E+30 500 185.7142857 1200 22.22222222 0 1200 25000 21593.33333 233.3333333 Microsoft Excel 14.0 Answer Report Worksheet: [Problem 4-final-4-26-16.xlsx)Sheetl Report Created: 4/25/2016 2:13:07 PM Result: Solver found a solution. All Constraints and optimality conditions are satisfied. Solver Engine Engine: Simplex LP Solution Time:0.047 Seconds. Iterations: 3 Subproblems: O Solver Options Max Time Unlimited, Iterations Unlimited, Precision 0.000001 Max Subproblems Unlimited, Max Integer Sols Unlimited, Integer Tolerance 1%. Assume NonNegat, Objective Cell (Max) Cell Name Original Value Final Value $C$24 Profit x1 0 45522.22222 Variable Cells Name Original Value Final Value Integer $C$19 Var. Values (C19-F19) x1 SDS19 Var. Values(C19-F19)x2 SES19 Var. Values (C19-F19) x3 SF$19 Var. Values (C19-F19) x4 0 175.5555556 Contin 500 Contin 0 Contin Constraints Cell Value Formula Status Slack Cell SC$26 Constraints x1 SC$27 x1 SC$28 x1 SC$29 x1 SC$30 x1 Name 72 $C$26s SH$14 Bindin 1200 SCS27Explanation / Answer
a) If product 4 is made, the profit would go down by 10.33 per unit production of Product4. This is because the reduced cost of Product4 is showing -10.33 which means if made the profit goes down by 10.33 or, the cost of product 4 has to go down by 10.33 for it become attractive for production.
b) current solution: to maxime profit from all products units made for each:
x1 = 175.556
x2 = 57.778
x3 = 500
x4 = 0
maximum profit = 45522.22
c) If resource 2 is increase by 200 units the profit would increase by = 200*22.2222 = 4444.44
This is because the shadow price of resource 2 is 22.222 which means that per unit increase in resource would result in increase in profit by 22.222. This increase is valid till 260 units increase in resource 2.
d) shadow price of resource 1 = 233.333
which means that per unit increase in resource 1 results in 233.333 increase in profit. Hence, maximum price to be paid for resource1 should be anything lower than 233.333 such that we make some profit at the end.
e) constraint 1: 0.10x1 + 0.25x2 + 0.08x3 + 0.21x4 <= 72 ; slack =0
constraint 2 : 3x1 + 3x2 + x3 + x4 <= 1200 ; slack = 0
Constraint 3: 36x1 + 48x2 + 25x3 + 35x4 <= 25000 ; slack = 3406.667
Constraint 4: x1 + x2 <= 500 ; slack = 266.667
Constraint 5: x3 + x4 <= 500 ; slack = 0
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