Question 7: 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

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 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):

(j) If the availability of Barks and machine time has been changed to 610 pounds and 620 minutes, respectively, 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):

E F G H A B D 1 Adjustable Cells Final Cell Name Value $B$10 Bags of Nugget 53.6842 $C$10 Bags of Mini-nugget | 0 $D$10 Bags of Chip 110.526 Reduced Objective Allowable Allowable Cost Coefficient Increase Decrease 1 2.166667 -2.05263 9 2.05263158 1E+30 6 16.5 0.6 8 Constraints 10 11 Cell Name $B$17 Bark LHS $B$18 Machine LHS $B$19 Labour LHS $B$20 Demand LHS Final Shadow Value Price 600 1.736842 660 0.157895 438.947 164.211 Constraint Allowable Allowable R.H. Side Increase Decrease 600 195 90 660 86.6666667 135 480 1 E+30 41.05263 150 14.2105263 1E+30 13 14 15 16