Sensitivity analysis report

Question

ABC Company manufactures three types of products and has provided you with the following linear problem:

Max Z=15X1+20X2+14X3 (Total profit)
s.t.
5X1+6X2+4X3<=210 (Total labor hours available)
10X1+8X2+5X3<=200 (Total material in pounds available)
4X1+2X2+5X3<= 170 (Total machine minutes available)

1. Explain what the parameters of each decision variable (x1, x2, & x3) in the objective  function and constraints define? Solve the problem using solver or Tora and provide the sensitivity analysis report.

2. Based on the sensitivity report answer the following questions with a clear description of your answer (No credit will be given if your answer is not based in the sensitivity analysis report):

a) What is the optimal solution and what is its value?

b) By how much would the profit per unit of product 1 have to increase for it to have a non-zero value in the optimal solution?

c) If the profit per unit of product 2 increased by $2, would the optimal values of products 2 and 3 change? Would the optimal value of the objective function change? By how much?

d) If the available labor decreased by 12 hours, would it cause a change in the optimal values of the decision variables? Would anything change?

e) If the available amount of material increased by 10 pounds, how would that affect the optimal value of the objective function?

Answer 1

Part 1:

X1 – number of units of product 1 to be produced
X2 – number of units of product 2 to be produced
X3 – number of units of product 3 to be produced

In the objective function, the coefficients of X1, X2 and X3 are their respective profits. In constraint 1, the coefficients of X1, X2 and X3 are their respective labor hours to produce one unit of each and the right hand side indicates the total labor hours available. In constraint 2, the coefficients of X1, X2 and X3 are their respective pounds of material to produce one unit of each and the right hand side indicates the total material in pounds available. In constraint 3, the coefficients of X1, X2 and X3 are their respective minutes in machine to produce one unit of each and the right hand side indicates the total machine minutes available.

On solving the problem in solver, we get the optimal solution and sensitivity report as attached in the excel file.

Part 2:

(a) The optimal solution is to produce 5 units of product 2 and 32 units of product 3 which gives a total profit of $548.

(b) The profit per unit of product 1 would have to increase by $10.6 (reduced gradient value) for it to have a non-zero value in the optimal solution.

(c) If the profit per unit of product 2 increased by $2, the optimal values of products 2 and 3 would have change since their reduced gradient values are zero. Yes, the optimal value of the objective function would change by 5x, where x is the increase in profit of product 2. In this case, it is $2 and hence the optimal value of the objective function will increase by $10 (5*$2) and the value is $558.

(d) If the available labor decreased by 12 hours, it would not cause a change in the optimal values of the decision variables because the Lagrange multiplier for labor hours is zero. Only the slack will decrease by 12. Other than that, there will be no change in anything.

(e) If the available amount of material is increased by 10 pounds, the optimal value of the objective function will increase by 2.4x (Lagrange multiplier for material usage), where x is the increase in the amount of material available in pounds. In our case, since it is 10 pounds, the optimal value will increase by $24, thus resulting in a value of $572.