Using Linear Programming to Optimize a New Restaurant

[aspirare21]

Linear Programming "Possibility Restaurant"

Homework Statement

Angela Fox and Zooey Caulfield were food and nutrition majors at State University, as well as close friends and roommates. Upon graduation Angela and Zooey decided to open a French restaurant in Draperton, the small town where the university was located. There were no other French restaurants in Draperton, and the possibility of doing something new and somewhat risky intrigued the two friends. They purchased an old Victorian home just off Main Street for their new restaurant, which they named "The Possibility".

Angela and Zooey knew in advance that at least initially they could not offer a full varied menu of dishes. They had no idea what their local customers' taste in French cuisine would be, so they decided to serve only two full-course meals each night, one with beef and the other with fish. Their chief, Pierre, was confident he could make each dish so exciting and unique that two meals would be sufficient, at least until they could assess which menu items were most popular. Pierre indicated that with each meal he could experiment with different appetizers, soups, salads, vegetable dishes, and desserts until they were able to identify a full selection of menu items.

The next problem for Angela and Zooey was to determine how many meals to prepare for each night so they could shop for ingredients and set up the work schedule. They could not afford too much waste. They estimated that they would sell a maximum of 60 meals each night, Each fish dinner, including all accompaniments, requires 15 minutes to prepare, and each beef dinner takes twice as long. There is a total of 20 hours of kitchen staff labor available each day. Angela and Zooey believe that because of the health consciousness of their potential clientele they will sell at least 3 fish dinners for every 2 beef dinners. However, they also believe that at least 10% of their customers will order beef dinners. The profit from each fish dinner will be approximately $12, and the profit from a beef dinner will be about $16. A) If Angela and Zooey increased the menu price on the fish dinners so that the profit for both dinners was the same, what effect would that have on their solution? Suppose Angela and Zooey reconsidered the demand for beef dinners and decided that at least 20% of their customers would purchase beef dinners. What effect would this have on their meal preparation plan?

Homework 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

The Attempt at a Solution

Complete Linear Programming Model
Maximize Z = $12X1 + $16X2
Subject to X1 + X2 ≤ 60
15X1 + 30X2 ≤ 1200
2X1 - 3X2 ≥ 0
X1 - 9X2 ≤ 0
X1 , X2 ≥ 0
Possibilities

X1 X2 RHS Equation form
Maximize 12 16 Max 12X1 + 16X2
Constraint 1 1 1

What will be the effect if the price is the same and also if the beef is atleast 20% now?

 

[.Scott]

The trickiest part of this problem is identifying what needs to be ignored.
When the problem says "the next problem ... was to determine how many meals to prepare", that is the only problem that they are sharing with us. So the problem of maximizing profit ($12fish+$16beef) is off out menu - so to speak.

Since this problem is 12 years old - I won't be shy about proving the solution out-right.

In order to prepare for each night meals, they simply need to determine the maximum number of beef and fish meals they might need.

A couple of notes on the problem as posted:
-- I would have hired a "Chef", not a "Chief", but to each her own.
-- Aside from the number of meals always being non-negative, they will also always be integers. As it turns out, this will not be a factor.

So let's take each constraint:
No more that a total of 60 meals (and non-negative):
$X_{fish} + X_{beef} ≤ 60$
$0 + X_{beef} = X_{beef} ≤ 60$
$X_{fish} + 0 = X_{fish} ≤ 60$

There's only 1200 staff-minutes available (and non-negative):
$15X_{fish} + 30X_{beef} ≤ 1200$
$0 + 30X_{beef} ≤ 1200$
$X_{beef} ≤ 40$
$15X_{fish} + 0 ≤ 1200$
$X_{fish} ≤ 80$

At least 60% of the orders will be for fish; total order max is 60:
$X_{beef} ≤ 0.4(X_{fish}+X_{beef})$
$X_{beef} ≤ 0.4(60)$
$X_{beef} ≤ 24$

At least 10% of the orders will be for beef; total order max is 60:
$X_{fish} ≤ 0.9(X_{fish}+X_{beef})$
$X_{fish} ≤ 0.9(60)$
$X_{fish} ≤ 54$

So, applying the most restrictive constraints:
$X_{fish} ≤ 54$
Prepare for 54 fish plates. Any more, and you will exceed the maximum portion (90%) of the maximum meal count (60).
$X_{beef} ≤ 24$
Prepare for 54 beef plates. Any more, and you will exceed the maximum portion (40%) of the maximum meal count (60).Follow-on Questions:

Question: If Angela and Zooey increased the menu price on the fish dinners so that the profit for both dinners was the same, what effect would that have on their solution?

Answer: Not at all. The problem we are attacking is meal preparation; not profit.Suppose Angela and Zooey reconsidered the demand for beef dinners and decided that at least 20% of their customers would purchase beef dinners. What effect would this have on their meal preparation plan?

Answer: This would affect the limit on fish as follows:
At least 20% of the orders will be for beef; total order max is 60:
$X_{fish} ≤ 0.8(X_{fish}+X_{beef})$
$X_{fish} ≤ 0.8(60)$
$X_{fish} ≤ 48$
They just save themselves 12 fish plates that would never have been used.

 

[pbuk]

The trickiest part of this problem is identifying what needs to be ignored.
When the problem says "the next problem ... was to determine how many meals to prepare", that is the only problem that they are sharing with us. So the problem of maximizing profit ($12fish+$16beef) is off out menu - so to speak.

I don't agree. Although the problem is not stated very well, I think this ignores

They could not afford too much waste.

In other words, I think it was intendend that no more meals should be prepared than they can serve, giving the constraint $f + b \le 60$.

You have actually written this constraint in your workings:

$X_{fish} + X_{beef} ≤ 60$

but your answer

$X_{fish} ≤ 54$
Prepare for 54 fish plates. Any more, and you will exceed the maximum portion (90%) of the maximum meal count (60).
$X_{beef} ≤ 24$
Prepare for 5424 beef plates. Any more, and you will exceed the maximum portion (40%) of the maximum meal count (60).

violates this (54 + 24 = 78).

Instead, I think the OP had the constraints and the objective function exactly right:

Maximize Z = $12X1 + $16X2
Subject to X1 + X2 ≤ 60
15X1 + 30X2 ≤ 1200
2X1 - 3X2 ≥ 0
X1 - 9X2 ≤ 0
X1 , X2 ≥ 0

This can be solved by graphing or other methods, and it can be seen that

Spoiler

the objective function has its maximum value of $800 at $(x_f, x_b) = (40, 20)$, constrained by $15x_f + 30x_b = 600 + 600 = 1200$ and $x_f + x_b = 40 + 20 = 60$

I am fairly sure that this is the answer the question setter had in mind. For the second part of the question:

Spoiler

making the profits equal increases the profit to a maximum of $960, and now any solution between (40, 20) and (54, 6) is optimal. If the minimum demand for beef is 20% then the upper limit of the range for $x_f$ is reduced to (48,12).

 

[.Scott]

You have actually written this constraint in your workings:
but your answer
violates this (54 + 24 = 78).

First, my answer does not violate anything. If a<=54 and b<=24, it is still possible for a+b<=60.

As to your interpretation of the problem, I don't agree at all.

There is no explicit statement that we address the profitability. That said, "They could not afford too much waste" doesn't mean that they are willing to order less of a meal type than they predict they will likely need - something that is likely to be more costly. Attempting to sway patrons to go for the only item left in the kitchen is not the kind of common sense business model that we should presume.

They had no idea what their local customers' taste in French cuisine would be, so they decided to serve only two full-course meals each night, one with beef and the other with fish. Their chief, Pierre, was confident he could make each dish so exciting and unique that two meals would be sufficient...

Moreover, engineering students should be encouraged to identify explicitly stated requirements and hold off on "implied requirements" until they have been discussed with the stakeholders and made explicit. Since they seem to hold "Chief Pierre"s opinion in high esteem, we should probably invite his assessment on this matter.One of the earmarks of a rookie engineer is one who has an endless list of features they want to implement. But low lying fruit is useless when no one is hungry.