Question

In a new restaurant Jake’s offers three different burgers that is open 40 hours per week. Each
product requires the following processing times (in minutes) in each of three station.
Burger Max Burger Special Mega Burger
Grill 2 2 1
Preparation 3 4 6
Packaging 4 6 5
Each station must be run by one of 19 cross-trained workers who are each available 35 hours per week. The plant has 10 grills, 6 preparation stations, and 8 packaging machines available. The three products contribute $0.90, $1.20, and $1.50, respectively, in marginal profits per unit produced.
a) (4 Points) Formulate an LP model for this problem.
b) (1 Point) Can this problem be solved graphically? Argue your point.
c) (8 Points) What is the optimal solution?
d) (2 Points) Assume that the number of available workers is fluctuating. Describe two ways
how it can be tested whether these fluctuations lead to changes in the production plan.

Answer 1

Consider the provided details of Inline Electronics in the question 16 at the end of chapter 3 to solve the subparts. The screenshot of the data into Excel spreadsheet is shown below:

The plant is open 40 hours per week to produce three type of different products, and 10 types of machine 1 available, 6 type of 2 machines are available and 8 type of 3 machines are available. Thus calculate the hours used and hours required in the column F and G, and then calculate the hours used to produce the required products in the column E by using the formula. The screenshot of the formulas used is shown below:

Now introduce the changing cell for quantity required to make the electronic products in the cell B2 to D2 and fill the initial values as 0. And, calculate the total profit in the cell F3 by using the formula. The screenshot of the formulas used and the changing cells is shown below:

The values will be calculated, the screenshot of the calculated values and the changing cells is shown below:

Now click on the “Data” tab in the menu bar and select the “Solver option”, a new dialog box will appear. Set the objective as cell $F$3 and maximize it, select the changing cell as $B$2:$D$2. And then click on the “Add” button, a new dialog box will appear, select the non-negative constraint for the changing cell as shown below:

Now click on the “Add” button in the above dialog box then add the next constraint for hours used to produce the three type of products should not be greater the hours available. The screenshot is shown below:

Press “OK” option in the above dialog box, the screenshot of the solver parameters is shown below:

Select the solving method as “Simplex Linear programming” and then press solve option in the above dialog box; a new dialog box will appear select the option “Keep solver solution and select the “Sensitivity” option into reports box” and then press “OK” option to get the results. The screenshot of the obtained results is shown below:

And the screenshot of the sensitivity reports is shown below:

a. No, the solution is not degenerate because the solution is finite by using the simplex method and all the basic variables takes non-zero values.

b. According to the results obtained, the shadow price for the machine 1 is $0 which indicate that if the Inline Electronics add more units of this machine will not increase the profit.

c. According to the results obtained, the shadow price for the total labor hours used is $5 which indicate that if the Inline Electronics add more cross-trained workers then it will increase the profit by $5 per hour to above the current labor cost.

d. According to the results obtained, the allowable decreases for the quality product 2 is $0 which indicate that if calculate the marginal profit for each product from using estimated cost then it will not affect optimal solution.

e. According to the results obtained, all the 19 workers are present and works 35 hours per week and company have total hours used are 665. The company wants to calculate the total profit for the total available labor hours varies from 600 to 665 hours, Thus use the “PsiOptParam(600,665,665)” formula in the cell F9 to calculate the total profit for the total available labor hours varies from 600 to 665 hours. The screenshot of the used formulas is shown below:

Now click on the “Analytic solver platform” option, then select the model and click on the run option. The screenshot is shown below:

A new dialog box will appear, select the “continue” option and press “OK” option. The screenshot of the obtained results is shown below:

According to the above results, the hours used to produce 3 products by machine 1 is 80, machine 2 is 240 and machine 3 is 280.