Question

On the tab labeled "Original Problem", I have provided a problem statement and the modeling of a Linear Programming Model Problem. Carefully read the problem statement. Reference the Wyndor example from the text for an example of the integration of a linear model to Excel.

The first requirement is to fill in the formulas in the GREEN cells for the Objective function and the constraints. There is no need to change the values in the RED cells.

Next, open Solver and fill in the 'Set Target Cell', 'By Changing Cells', and 'Subject to the Constraints' boxes.

Once you completed the Solver inputs and solved the problem, you are required to obtain the sensitivity analysis. If you do this correctly, the 'Sensitivity Report' will show up in a new tab in the bottom of the sheet.

From there, proceed to the Solutions tab where you will find five questions. To answer each question, you will be required to consult the sensitivity analysis and locate the correct information from the analysis to answer each question. All answers should come from solving the model and obtaining and consulting the sensitivity data, not by resolving the model multiple times.

In your response to each question, you should begin with a written sentence(s) that explains where specifically in the sensitivity analysis you found the information you need to answer each question followed by a sentence answering each respective question. Each question will be evaluated on explaining where in the sensitivity analysis you found the information necessary to answer the question (mechanical component) and for a clear, concise, grammatically correct answer to the question (written component). Questions should be answered on the Solutions tab and typed in unbolded print right after the statement of the question.

Good luck, and HAVE FUN!

Problem Statement:

The Charger Club of America sponsors driver education events that provide high-performance driving instruction on actual race tracks. Because safety is a primary consideration at such events, many owners elect to install roll bars in their cars. Wilson Industries manufactures two types of roll bars for Chargers, Model 1 and Model 2. Model 1 requires 20 pounds of a special high alloy steel, 50 minutes of manufacturing time, and 60 minutes of assembly time. Model 2 requires 25 pounds of the special high alloy steel, 100 minutes of manufacturing time, and 50 minutes of assembly time.   Wilson’s steel supplier indicated that at most 40,000 pounds of the high-alloy steel will be available next quarter. In addition, Wilson estimates that 120,000 minutes of manufacturing time and 96,000 minutes of assembly time will be available next quarter. The profit contributions are $200 per unit for Model 1 and $280 per unit for Model 2. The linear programming model for this problem is as follows:

Variables:

Model1 = Number of Model 1 to produce

Model2 = Number of Model 2 to produce

Objective:

Max 200 Model1 + 280 Model2

Constraints subject to:

20 Model1 + 25   Model2 <= 40,000 Steel available

50 Model1 + 100 Model2 <= 120,000 Manufacturing minutes

60 Model1 + 50   Model2 <= 96,000 Assembly minutes

Model1, Model2 >= 0

Question 1) What are the optimal solution and the total profit?

Question 2) If the unit profit of the Model 2 decreases to $250 per unit, how would the optimal solution change?

Question 3): How much should the company be willing to pay to acquire additional capacity in the 'Assembly minutes' area? Explain in detail.

Question 4) How much should the company be willing to pay to acquire additional 'Steel'? Explain in detail.

Question 5) How much will the objective function, 'Total Profit', change if there was one additiional hour of Manufacturing capacity available?

Answer 1

ANSWER:-

Objective Cell (Max)

CELL	NAME	ORIGINAL VALUE	FINAL VALUE
$D$6	Total points	0	397714.29

Variable Cells

CELL	NAME	ORIGINAL VALUE	FINAL VALUE	INTEGER
$B$5	Number to Make Model 1	0	1028.57	Contin
$C$5	Number to Make Model 2	0	685.71	Contin

Constraints

CELL	NAME	CELL VALUE	FORMULA	STATUS	SLACK
$D$9	Steel Available Used	37714.28571	$D$9<=$E$9	Not Binding	2285.714286
$D$10	Manufacturing Minutes Used	120000	$D$10<=$E$10	Binding	0
$D$11	Assembly Minutes Used	96000	$D$11<=$E$11	Binding	0

Sensitivity Report

Variable Cells

CELL	NAME	FINAL VALUE	REDUCED COST	OBJECTIVE COEFFICIENT	ALLOWABLE INCREASE	ALLOWABLE DECREASE
$B$5	Number to Make Model 1	1028.571429	0	200	136	60
$C$5	Number to Make Model 2	685.7142857	0	280	120	113.34

Constraints

CELL	NAME	FINAL VALUE	SHADOW PRICE	CONSTRAINT R.H. SIDE	ALLOWABLE INCREASE	ALLOWABLE DECREASE
$D$9	Steel Available Used	37714.28571	0	40000	1E+30	2285.714286
$D$10	Manufacturing Minutes Used	120000	1.942857143	120000	16000	2285.714286
$D$11	Assembly Minutes Used	96000	1.714285714	96000	10666.67	36000

(1)

Optimal solution: Model 1 = 1028.57

Model 2 = 685.71

Total profit = $397,714.29

(2)

The proposed decrease is 280 - 250 = 30

The proposed decrease is =30

The allowable decrease is 113.33.

So, there will be no change in the optimality.

(3)

Shadow price for assembly minutes = $1.714.

one additional assembly minute will generate an additional profit of $1.714.

the willingness to pay should be any amount less than $1.714 per minute.

(4)

Shadow price for steel = 0. This is a non-binding constraint.

one additional unit of steel will not generate any additional profit.

So, the willingness to pay should be zero.

(5)

Shadow price for manufacturing units = 1.943. So, one additional manufacturing unit will generate an additional profit of $1.943.

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