Kevin Rivas

Executive Summary

Angela and Zooey don’t have the necessary funds to have a full menu. To overcome this

challenge they decide to serve two full course meals of either fish or beef. They are trying to decide how

many of dishes of each should they make to add up to 60 meals because that is there maximum

constraint. Angela and Zooey are also trying to decide on whether to increase the price of fish from $12

and whether or not this change in price would affect the current model they have for profit

maximization. They are also trying to decide on whether spending $30 on advertisement which leads to

the amount of meals sold to increase by 10. Also with an unreliable source of labor they want to know

whether or not the reduction in hours would have an effect on the profit level.

Since Angela and Zooey can only make two dishes the profit maximization point between the

fish and beef dishes would be when they make 40 fish meals and 20 beefs meals. At this ratio of dishes

they profit $800. They should also increase the price of the fish meal from $12 to $16 because it would

not affect the number of customers they get since the allowable increase is $4. By charging the same for

fish and beef the profit is $960. The $30 investment in advertisement should also be spent because the

10 added meals results in an $80 profit increase. Lastly we recommend that Angela and Zooey hire a

more reliable labor so that their reduction in hours is less than 5 so that their profit level won’t be

affected.

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Introduction:

This case involves Angela and Zooey who want to open a French restaurant in Draperton. They

will start off only serving two main dishes. One of them is fish and the other will be beef. The reason

why they only have two dishes is because initially they have no idea the tastes of the locals regarding

French cuisine. They originally expect to be able to sell a maximum of 60 meals each night. Each fish

dinner takes 15 minutes to prepare while each beef dinner takes 30 minutes to prepare. They have a

total of 20 hours of labor in the kitchen each day. They believe that they will sell at a fish to beef ration

of 3 fish for every 2 beef. No matter what, they expect to sell at least 6 beef dinners. Angela and Zooey

expect to make $12 from each fish dinner and $16 from each beef dinner. The analysis asks us to create

a programming model to help estimate the amount of meals they should prepare. The analysis also asks

what effect would an increase in the price of the fish dinner from $12 to $16. Finally, it asks to analyze

the effects on committing 12 beef dinners for each night. The second half of the problem asks us to

solve their problem using the computer rather than by hand. In the second half of the problem we need

to analyze whether or not a $30 cost per day is worth it to sell 70 meals. It also asks to analyze a labor

reduction of five hours effect on the profit level. The final question that needs to be solved is whether or

not increasing the price of the fish dinner to $14 will be best option.

Case Analysis:

A. Key Challenges

Angela and Zooey are opening a restaurant and do not have the funds to have a full menu. They

decide they are going to serve two full course meals with beef and fish. The key challenges for Angela

and Zooey are how many meals they should prepare each night and whether or not they should charge

the same amount of money for each meal. Later on they debate on investing in placing and

advertisement in the newspaper and how much staff they really need to have on board. Lastly, they

debate on if they should raise the price of fish, and wonder if it will affect the demand. They are not sure if

their chef will allow this to happen and how it will affect their profit. Long term they should purchase some

advertising, have less staff, and change the price of fish without affecting the chef’s plans.

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B. Causes of the Problem

The decision variables in this problem are how many meals of fish and beef to prepare. The quantities to be produced are representing as:

● x1 = Fish

● x2 = Beef

The linear programming model formulation used is maximize Z = $12x1+16x2. This formula will be used as an objective function that will maximize the number of meals Angela and Zooey should prepare each night.

● Z = total profit per day ● $12x1 = profit from fish

● 16x2 = profit from beef

There are numerous model constraints:

Resource Constraints

● x1 + x2 ≤ 60 (maximum meals each night)

● 15x1 + 30x2 ≤ 1200 (labor in minutes to prepare meals) OR

o .25x1 + .50x2 ≤ 20

● 2x1 - 3x2 ≥ 0 (will sell at least 3 fish/2 beef meal) OR

o x1 /x2≥ 3/2

● .90x2 - .10x1 ≥ 0 (at least 10% will order beef) OR

o x2 /(x1+x2) ≤ .10

Non-Negativity

● x1 ≥ 0

● x2 ≥ 0

To solve for the maximum point we will use the graphical solution approach. First we need to plot the

constraints as equations. The lines plotted do not represent the entire constraint; it also includes the

values that are less or equal to the line. The shaded in area on the graph is the feasible solution area

because all the points in this area satisfy all constraints. From the shaded area we can find the optimal

point (B) which is the best feasible solution.

Point B:

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-20 ≤ -2.86 ≤ Δ ≤ 12.72 ≤ 20

-2.86 ≤ Δ ≤ 12.72

q1 = 60 + Δ Δ+q1 – 60

-26 ≤ q1 -60 ≤12.72

57.14 ≤ q1 ≤ 72.72

● x1 = 40

● x2 = 20

● Z = $800 Point A:

● x1 = 34.3

● x2 = 22.8

● Z = $776.23 Point C:

● x1 = 6

● x2 = 54

● Z = $744

If we were to change the objective function to Z = $16x1 + 16x2 there would be two optimal solutions (B and C). For both solutions the profit would be $960.

● 16(40) + 16(20) = $960

● 16(6) + 16(54) = $960

If we changed the constraint from 90x2 - .10x1 ≥ 0 to 80x2 - .20x1 ≥ 0 it would have no effect on the

solution because it would not change the optimal point on the graph.

In order to determine if Angela and Zooey should invest in an advertisement we need to look at the sensitivity range for q1.

S1:

● 40 - 2Δ ≥0

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If they added 10 meals it would not affect the

shadow price for the mean demand ($800). With

these added 10 meals they would see an increase in

profit of $80. $8 * $10 = $80

The $80 is more than the $30 of advertising so

therefore the Ad should be purchased.

● 7Δ ≥ -40

● Δ ≥ -20 S2:

● 20 – Δ ≥ 0

● Δ ≥ -20

● Δ ≤ 20 S3:

● 20 + 7Δ ≥ 0

● 7Δ ≥-20

● Δ ≥ -2.86

S4:

● 14 – 1.1Δ ≥ 0

● 1.1Δ ≥ -14

● Δ ≤ 20

If they have a night where they have a staff reduction as much as 5 hours it would affect their profit level in a negative way. Based on the Sensitivity Report we can see that the shadow price is 16 and therefore anything less than that would be unacceptable.

Based on the Sensitivity Report we can see that Angela and Zooey could increase their price by as much as $4. So raising the price from $12 to $14 would be acceptable to Pierre and there would be and $80 profit per night.

Original profit: $800

New profit: (14 * 40)+ (16*20) = $880 $880 - $800 = $80

C. Evaluation: Decision criteria and Alternative solutions

The decisions in which Angela and Zooey have to choose from for the chapter 2 problem are whether or

not making the profit from both dinners the same would have an effect on their solution and what

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effect would allocating 20% of the dinners to beef rather than 10% have on their meal preparation plan.

The solutions for the first half of the problem regarding the profit for both dinners would to keep the

price the same and have a profit of $800 or change the prices to both be $16 and have a profit of $960.

The solutions for the second half of the problem are to keep 10% allocated for beef or to allocate 20%

for beef. It turns out that both of the solutions are the same.

The decision in which Angela and Zooey have to choose from for chapter 3 are whether or not to

advertise, whether or not a reduction in 5 hours of labor will affect their profit level, and whether or not

raising the price of the fish dinner to $14 would be acceptable to Pierre (their Chef) and how much profit

it would earn them. The solutions to whether or not they should advertise are not advertising and make

what they are making right now, or to advertise and gain 10 meals which would increase profit by

$50($80-$30). The solutions to whether or not a decrease in labor available by 5 hours would affect

profit is to keep doing what they are doing or reduce labor. Reducing their labor to anything less than 16

would result in losing profit. Finally, the solutions regarding whether or not to change the price of fish to

$14 are to keep the price at $12 or change it to $14. Keeping the price at $12 would yield profit of $800

while changing the price would yield a profit of $880.

Recommended Solution

Angela and Zooey should serve 40 fish meals and 20 beef meals per night, this is the quantity at which

profit is maximized (when price of fish=12 and price of beef=16, profit=800). We also recommend

increasing the price of fish to $16. By charging the same amount for both meals profit rises from $800 to

$960 ( 40*16 + 20*16 = 960).

Advertising will add 10 more meals per night and increase profit by $80, this increased profit is larger

than the $30 cost of advertising so the newspaper ad should be purchased. More reliable staff should be

hired in the long run. If labor is reduced by 5 hours it will fall to 15 hours, this is below the point at which

profit will be negatively affected (16 hours). In order to keep their profit where it is they cannot afford to

reduce labor. Changing the price of fish to $14 will increase profit by $80, which is acceptable to the

chef. But we recommend increasing the price to $16, this will grant an additional $80 of profit.