Question

Kilgore’s Deli is a small delicatessen located near a major university. Kilgore’s does a large walk-in carry-out lunch business. The deli offers two luncheon chili specials, Wimpy and Dial 911. At the beginning of the day, Kilgore needs to decide how much of each special to make (he always sells out of whatever he makes). The profit on one serving of Wimpy is $0.5, on one serving of Dial 911, $0.68. Each serving of Wimpy requires 0.2 pound of beef, 0.2 cup of onions, and 5 ounces of Kilgore’s special sauce. Each serving of Dial 911 requires 0.2 pound of beef, 0.4 cup of onions, 2 ounces of Kilgore’s special sauce, and 5 ounces of hot sauce. Today, Kilgore has 19 pounds of beef, 15 cups of onions, 80 ounces of Kilgore’s special sauce, and 55 ounces of hot sauce on hand.

1. Develop a linear programming model that will tell Kilgore how many servings of Wimpy and Dial 911 to make in order to maximize his profit today. If required, round your answers to two decimal places. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. (Example: -300)
   

   Let	W = number of servings of Wimpy to make
   	D = number of servings of Dial 911 to make

   Max	W	+	D
   s.t.
   	W	+	D			(Beef)
   	W	+	D			(Onions)
   	W	+	D			(Special Sauce)
   	W	+	D			(Hot Sauce)
   			W, D	≥	0

2. Find an optimal solution. If required, round your answers to two decimal places.
   
   W = , D = , Profit = $  
   
3. What is the shadow price for special sauce? If required, round your answers to two decimal places.
   
   $  
   
   Interpret the shadow price.
   
   The input in the box below will not be graded, but may be reviewed and considered by your instructor.
   
   
   
4. Increase the amount of special sauce available by 1 ounce and re-solve. If required, round your answers to two decimal places.
   
   W = , D = , Profit = $  
   
   Does the solution confirm the answer to part (c)?
   
   

Answer 1

Let W = number of servings of Wimpy to make

D = number of servings of Dial 911 to make

Linear Programming Model that will tell Kilgore how many servings of Wimpy and Dial 911 to make in order to maximize his profit today is:

(a) Maximize = 0.5 * W + 0.68* D

Constraints formed from the given information are:

0.2 *W + 0.2*D ≤ 19 (for Beef)

0.2 *W + 0.4*D ≤ 15 (for Onion)

5 *W + 2*D ≤ 80 (for Special Sauce)

5*D ≤ 55 (for Hot Sauce)

W, D ≥ 0

(b) Find an optimal solution. If required, round your answers to two decimal places.

Load the data into Excel.

Consider two decision variable columns and a maximization profit column.

Let us use the Excel function to find the used amount.

Similarly, do for the other columns.

Find the maximum profit for the last step.

Now, go to Data>Solver.

Use the steps as below:

Now click on Solve.

The output will be as follows:

Therefore, to maximize the profit, 11.6 W and 11 W should be made to maximize the profit. The maximum profit will be $13.28.

(c) What is the shadow price for special sauce? If required, round your answers to two decimal places.

$  

Interpret the shadow price.

Go to Data>Solver and select the sensitivity report option to get the output as below:

Thus, dual value for the special sauce is $0.1 which is shadow price.

(d) Increase the amount of special sauce available by 1 ounce and re-solve. If required, round your answers to two decimal places.

Now, the loaded data into Excel is:

The output will be as follows using the same analysis as we did in part (b):

Therefore, to maximize the profit, 11.8 W and 11 W should be made to maximize the profit. The maximum profit will be $13.38.

Does the solution confirm the answer to part (c)?

Yes, because the sensitivity report is still the same and dual value for the special sauce is $0.1 which is shadow price.