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

Adjustable Cells
			Final	Reduced	Objective	Allowable	Allowable
	Cell	Name	Value	Cost	Coefficient	Increase	Decrease
	$B$10	Bags of Nugget	53.6842	0	9	1	2.166667
	$C$10	Bags of Mini-nugget	0	-2.05263	9	2.0526316	1E+30
	$D$10	Bags of Chip	110.526	0	6	16.5	0.6
Constraints
			Final	Shadow	Constraint	Allowable	Allowable
	Cell	Name	Value	Price	R.H. Side	Increase	Decrease
	$B$17	Bark LHS	600	1.73684	600	195	90
	$B$18	Machine LHS	660	0.15789	660	86.666667	135
	$B$19	Labour LHS	438.947	0	480	1E+30	41.05263
	$B$20	Demand LHS	164.211	0	150	14.210526	1E+30

(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 PRINTno more than 15 words):

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

Explain (clearly PRINT no more than 15 words):