Question

Sensitivity analysis (20 marks, 2 marks per subquestion)

A garden store prepares various grades of pine bark for mulch: nuggets (x₁), mini-nuggets (x₂), and chips (x₃). The production process requires pine bark, machine time, and labour time. Each bag of nuggets requires 5 pounds of barks, 2 minutes of machine time, and 2 hours of labour time. To produce one bag of mini-nuggets, 6 pounds of barks, 4 minutes of machine time, and 4 hours of labour time are needed. The production of one bag of chips consumes 3 pounds of barks, 5 minutes of machine time, and 3 labour hours. The store has 600 pounds of barks, 660 minutes of machine time, and 480 hours of labour time available for the next production run. The demand for the three products is at least a total of 150 bags. The unit profit contribution of nuggets, mini-nuggets, and chips are $9, $9, and $6 per bag, respectively. The following linear programming model has been developed:

Decision variables

x₁ = bags of nuggets to produce

x₂ = bags of mini-nuggets to produce

x₃ = bags of chips to produce

Objective function

Maximize        z = 9x₁ + 9x₂ + 6x₃

Constraints:

Bark:               5x₁+ 6x₂ + 3x₃< 600 pounds

Machine:         2x₁+ 4x₂ + 5x₃< 660 minutes

Labour:           2x₁+ 4x₂ + 3x₃< 480 hours

Demand:         x₁+ x₂ + x₃> 150 bags

Non-negativity requirement: x₁, x₂, x₃> 0

The Excel Solver sensitivity report is given on the next page, please answer the following questions based on the report.

NOTE:

1. EACH QUESTION THAT FOLLOWS REFERS TO THE ORIGINAL PROBLEM. THAT IS, EACH QUESTION IS INDEPENDENT OF THE OTHER QUESTIONS.

2. IF IT IS IMPOSSIBLE TO ANSWER THE QUESTION WITHOUT USING THE COMPUTER, YOU MUST STATE THAT IN YOUR ANSWER AND BRIEFLY EXPLAIN WHY.

3. NO MARKS WILL BE GIVEN FOR UNSUPPORTED ANSWERS. CALCULATIONS SHOULD BE DONE IN THE SPACE PROVIDED. CLEARLY PRINT YOUR ANSWER IN THE BOX PROVIDED.

4. ALL EXPLANATIONS MUST BE LIMITED TO 15 WORDS. ONLY THE FIRST 15 WORDS OF YOUR EXPLANATION WILL BE READ BY THE MARKER.

Sensitivity Report

(a) What are the optimal solution and its objective function value?

Answer

x₁ =   x₂ =   x₃ =   

The objective function value is:

Explain where you find this information and show your calculation of the objective function value below (clearly PRINT no more than 15 words):

Calculation

(b) If the manager is able to obtain additional 50 minutes of machine time through better scheduling. What is the impact of this change on the objective function value?

Calculations

The new objective function value is:

Answer (Complete ONE appropriate box ONLY)

The impact on the objective function can be obtained without resolving the problem on the computer. Mark X here:         The new objective function value is: Explain (clearly PRINT no more than 15 words):

The problem must be re-solved on the computer. Mark X here: Explain (clearly PRINT no more than 15 words):

(c) What is the impact on the optimal solution and the objective function value if the unit profit contribution for mini-nuggets increases from $9/bag to $10/bag?

Calculations

Answer (Complete ONE appropriate box ONLY)

Current solution remains optimal. Mark X here:       the profit contribution increases for mini-nuggets is less than reduced cost The new objective function value is:   Explain (clearly PRINT no more than 15 words):

The problem must be re-solved on the computer. Mark X here: Explain (clearly PRINT no more than 15 words):

(d) The manager noticed that a new product, super-chip, is now available at its competitors’ stores. Based on his estimate, super-chips would be able to bring in a profit of $7.00 per bag, requiring 4 pounds of barks, 7 minutes of machine time, and 4 labour hours. Assume that the total demand for all products remains at 150 bags. What is the impact on the optimal solution and objective function value if the store manager decides to introduce this new product?

Calculations

Answer (Complete ONE appropriate box ONLY)

Current solution remains optimal. Mark X here:       The new objective function value is:   Explain (clearly PRINT no more than 15 words):

The problem must be re-solved on the computer. Mark X here: Explain (clearly PRINT no more than 15 words):

(e) After the production process goes on for one week, the manager found that each bag of nuggets needs only 4 pounds of barks rather than 5. What is the impact on the objective function value if this error is corrected?

Calculations

Answer (Complete ONE appropriate box ONLY)

The impact on the objective function can be obtained without resolving the problem on the computer. Mark X here:         The new objective function value is: Explain (clearly PRINT no more than 15 words):

The problem must be re-solved on the computer. Mark X here: Explain (clearly PRINT no more than 15 words):

(f) The manager now realizes that he has limited storage space for the product: he will not be able to stock up anything beyond 160 bags for the total of all products. What is the impact on the optimal solution?

Calculations

Answer (Complete ONE appropriate box ONLY)

Current solution remains optimal. Mark X here:       The new objective function value is:   Explain (clearly PRINT no more than 15 words):

The problem must be re-solved on the computer. Mark X here: Explain (clearly PRINT no more than 15 words):

(g) If the supplier does not have enough barks and, hence, provides only 550 pounds of barks to the manager, what is the impact on the objective function value?

Calculations

Answer (Complete ONE appropriate box ONLY)

The impact on the objective function can be obtained without resolving the problem on the computer. Mark X here:         The new objective function value is:   Explain (clearly PRINT no more than 15 words):

The problem must be re-solved on the computer. Mark X here: Explain (clearly PRINT no more than 15 words):

(h) What is the impact on the optimal solution and the objective function value if the unit profit contributions for nuggets and mini-nuggets both increase by 10%?

Calculations

Answer (Complete ONE appropriate box ONLY)

Current solution remains optimal. Mark X here:       The new objective function value is:   Explain (clearly PRINT no more than 15 words):

The problem must be re-solved on the computer. Mark X here: Explain (clearly PRINT no more than 15 words):

(i) The manager has been notified that he should not worry about the availability of labour hours as another production line has been closed due to the recent economic turmoil. Therefore, there will be unlimited labour hours available (in the context of the size of this particular business) to him without any additional cost. What is the impact on the optimal solution and the objective function value?

Calculations

Answer (Complete ONE appropriate box ONLY)

Current solution remains optimal. Mark X here:       The new objective function value is:   Explain (clearly PRINT no more than 15 words):

The problem must be re-solved on the computer. Mark X here: Explain (clearly PRINT no more than 15 words):

Answer 1

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a)

X1 = 53.6842, X2 = 0 and X3 = 110.526

Objective function value = 9*53.6842+9*0+6*110.526 = 1146.3138

Value of X1,X2and X3 are found from Cells $B$10,$c$10 and $D$10

b)

The impact on the objective function can be obtained without resolving the problem on the computer. X
The new objective function value is: 1146.3138+50*0.157895 = 1154.20855 [ Shadow price of machine time 0.157895 is valid for another 86.6666666667 minutes(Allowable increase)]

c)

Current solution remains optimal. X [ As the profit contribution increase for mini-nuggets is less than reduced cost]   

The new objective function value is same as earlier 1146.3138

d)

The problem must be re-solved on the computer. X

It is not possible to estimate how much share will the new product grab within the existing product mix without resolving on the computer

e)

The problem must be re-solved on the computer. X

It is not possible to estimate the effect on objective function without resolving on the computer

f)

New Constraint x1 + x2 + x3 <= 160 plus the current optimal solution in it

New constraint, current optimal solution doesn't satisfy it.

g)

RHS by 50 pounds, allowable decreas 90

Shadow price $1.737/lb is valid

New OFV = old OFV +

RHS decreases within the Allowable decrease limit. Shadow price is valid

h)

Nugget profit Increase by 1% = increase by 0.9

Mini nugget profit increase by 1% = increase by 0.9

0.9/1 + 0.9/2.052 = 1.34 > 100%

Two simulataneous objective function coefficients, change, 100% rule is violated

i)

Remove the labour hour contraint, as it is non-binding no change

The problem must be re-solved on the computer. Mark X here: X

The new objective function value = 1146,314

Removal of a non-binding constraint and there will be no change.

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