# Linear Programming Case Study - Case Problem

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1. Mar 29, 2016

### mpoli

1. The problem statement, all variables and given/known data
Linear Programming Case Study - Case Problem ( Page # 109 Decision making methods) “The Possibility” Restaurant?
In the case problem, Angela and Zooey wanted to develop a linear programming model to help determine the number of beef and fish meals they should prepare each night. Solve Zooey and Angela’s linear programming model by using QM.
Angela and Zooey are considering investing in some advertising to increase the maximum number of meals they serve. They estimate that if they spend $30 per day on a newspaper ad, it will increase the maximum number of meals they serve per day from 60 to 70. Should they make the investment? Zooey and Angela are concerned about the reliability of some of their kitchen staff. They estimate that on some evenings they could have a staff reduction of as much as 5 hours. How would this affect their profit level? The final question they would like to explore is raising the price of the fish dinner. Angela believes the price for a fish dinner is a little low and that it could be closer to the price of a beef dinner without affecting customer demand. However, Zooey has noted that Pierre has already made plans based on the number of dinners recommended by the linear programming solution. Angela has suggested a price increase that will increase profit for the fish dinner to$14. Would this be acceptable to Pierre, and how much additional profit would be realizes?

2. Relevant equations
Resource Availability 1200 minutes of labor per day
60 maximum meals each night
Decision variables
X1 = number of fish meal
X2 = number of beef meal
Objective function Maximize
Z = $12X1 +$16X2
Where Z = total profit per day
$12X1 = profit from fish meals$16X2 = profit from beef meals
Resource Constraints
X1 + X2 ≤ 60 (the maximum estimated meals each night)
15X1 + 30X2 ≤ 1200 (labor in minutes to prepare meals)
2X1 - 3X2 ≥ 0 (they'll sell at least 3fish/2beef meal)
X1 - 9X2 ≤ 0 (at least10% of customers will order beef meal)
Non-Negativity Constrains
X1 ≥ 0 ; X2 ≥ 0
3. The attempt at a solution

2. Mar 29, 2016

### Staff: Mentor

Since there are only two variables, x1 and x2, one method of solving this system is to graph all of the constraint inequalities. This will give you the feasible region. The objective function takes on its maximum value at a corner point of the feasible region.

3. Mar 29, 2016

### Ray Vickson

Google "linear programming"; you will receive links to many web pages that go into such problems in great detail. Chapter 1 of any introductory textbook on Operations Research will do numerous examples like this one. This Forum is not the right place for answering questions like yours, because the solution involves a lot of two-dimensional graphs and plots, which are a bit tricky to include here and which would require US to spend a lot of our own time constructing the diagrams, etc. Lots of other people have already done that and have posted the results; you just need to look.

Last edited: Mar 29, 2016