Question

Consider the following linear programming problem

Maximize	$4X1 + $5X2
Subject To	2X1 + 5X2 ≤ 40 hr	Constraint A
	3X1 + 3X2 ≤ 30 hr	Constraint B
	X1, X2 ≥ 0	Constraint C

if A and B are the two binding constraints.

(Round to ONLY two digits after decimal points)

a) What is the range of optimality of the objective function?

  Answer ≤ C1/C2  ≤  Answer

b) Suppose that the unit revenues for X1 and X2 are changed to $100 and $18, respectively. Will the current optimum remain the same?

AnswerYesNO that because the new C1/C2 is Answer which is Answerwithinnot within the range of optimality

c) Suppose that the unit revenue of X1 is fixed $4. What is the associated range for the unit revenue for X2 that will keep the optimum unchanged?

   Answer  ≤ C2 ≤  Answer

d) The Shadow Price for Constraint A is Answer.

e) The Shadow Price for Constraint B is Answer

f) If only the capacity of Constraint A is increased from the present 40 hours to 45 hours, The increase in revenue will be = $Answer

g) A suggestion is made to increase the capacities of Constraint A and B by an hour at the additional cost of $1/hr. Is this advisable?

This is advisable for AnswerConstraint AConstraint BBoth Constraints and the total additional net revenue per hour would be $Answer

Answer 1

a)

A and B are binding constraints. Therefore, ratio of objective function coefficients (C1/C2) must be between the ratio of coefficients of binding constraints. Therefore,

Range of optimality of the objective function is:

2/5 <= C1/C2 <= 3/3

or,

0.4 <= C1/C2 <= 1

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b)

No, that because the new C1/C2 is 100/18 = 5.56 , which is not within the range of optimality.

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c)

The range of optimality for C2 is

4/1 <= C2 <= 4/0.4

or,

4 <= C1 <= 10

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e)

Using Excel Solver

Create Excel model as follows:

Enter Solver Parameters as follows:

Click Solve to generate the solution.

In the next window, check Sensitivity to generate the sensitivity report

and click OK

Result:

Sensitivity report :

d)

Shadow prices are read from the sensitivity report,

The shadow price for constraint A is : 0.333 ,

And the feasibility range is :  40-20 = 20 <= Constraint A <= 40+10 = 50

Feasibility range is determined by adding allowable increase and subtracting allowable decrease from the Constraint R.H. Side

e)

The shadow price for constraint B is : 1.111 ,

And the feasibility range is : 30-6 = 24 <= Constraint A <= 30+30 = 60

Feasibility range is determined by adding allowable increase and subtracting allowable decrease from the Constraint R.H. Side