Fast Food (Q81 P222)

Problem: A fast food restaurant needs to decide how many chicken pieces and chicken nuggets to

cook per hour in peak times.

Fast Food

• They must cook at least 1200 chicken pieces per hour• The total amount of chicken is limited by the cooking

capacity, there are 10 000units of cooking capacity available.

• It takes 2 cooking units for each chicken piece and 1 cooking unit for each chicken nugget.

• The number of pieces of chicken being cooked cannot exceed the quantity of nuggets by more than 800 being cooked at the same time.

Fast Food

• Let c=chicken pieces and let n=nuggetsConstraints• c ≥ 1200• n+ 2c ≤ 10 000• c-n ≤ 800We also need to keep in mind the implied constraints

of this situation i.e. you cannot have a negative value for either c or n so

c ≥ 0 and n ≥ 0

Fast Food

Using y=c and x=n I can plot these

• y=1200• y=(10 000-x)÷2• y=800+x

Then draw if you can not see your functions graphed then change the view so that you are able to.

Fast FoodFinding the feasible region

Identify possible feasible regions and chose a coordinate pair from within each region that has been identified.

(IMPORTANT: the coordinate MUST be inside the region identified)

I have selected these are possible feasible regions because they are bound by all the constraints.

In Region 1 we chose (100, 3000) In Region 2 we chose (3000, 1300)

These coordinates are tidy numbersand are close to a constraint I know the value of.

Fast Food Test each coordinate against the constraints.

If ALL statements are TRUE then we have identified the FEASIBLE REGION.

IMPORTANT: we must take the time to identify the feasible region correctly, our answers from here are based on the region we choose.

Region 1 coordinate (100, 3000) n=100, c=3000• c ≥ 1200 3000 ≥ 1200 True• n+ 2c ≤ 10 000 100+6000 ≤ 10 000 True• c-n ≤ 800 3000-100 ≤ 800 FalseBecause of this false Region 1 is not our feasible region.

Fast Food Test each coordinate against the constraints.

If ALL statements are TRUE then we have identified the FEASIBLE REGION.

IMPORTANT: we must take the time to identify the feasible region correctly, our answers from here are based on the region we choose.

Region 2 coordinate (3000, 1300) n=3000, c=1300• c ≥ 1200 1300 ≥ 1200 True• n+ 2c ≤ 10 0003000+2600 ≤ 10 000 True• c-n ≤ 800 1300-3000 ≤ 800 True

This is our feasible region.

Fast FoodNow that Feasible regionis established and labelled.

this region represents all the possible values that meet the given constraints

We need to find the vertices of this region, where the optimal(maximum or minimum) values can be found.

we do this using the G-slv ISCT function of our calculators, selecting the lines of interest using the replay button

The intercept of constraint 1 and constraint 2 is B= (7600, 1200) ie n=7600 and c=1200

We repeat to find all vertices.

Feasible region

A

BC

Fast FoodIt is easier to tabulate your results from here…

Apply Optimising Function to the vertices to find the optimum that will answer our question.

“The profit on chicken is $0.15 a piece and on nuggets is $0.02 a nugget.” Profit = 0.15c+ 0.02n Work out the coordinates for the greatest profit to be made and the profit per hour.”

Point Nuggets (n, x) Chicken (c, y)

A 2800 3600

B 7600 1200

C 400 1200

Fast FoodIt is easier to tabulate your results from here…

Apply Optimising Function to the vertices to find the optimum that will answer our question.

“The profit on chicken is $0.15 a piece and on nuggets is $0.02 a nugget.” Profit = 0.15c+ 0.02nWork out the coordinates for the greatest profit to be made and the profit per hour.”

Point Nuggets (n, x) Chicken (c, y) Profit = 0.02n + 0.15c

A 2800 3600 =0.02×2800 + 0.15×3600=596

B 7600 1200 =0.02×7600 + 0.15×1200=332

C 400 1200 =0.02×400 + 0.15×1200=188

Fast Food

In order for the Fast food restaurant to make the greatest profit per hour they need to cook 2800 chicken nuggets and 3600 chicken pieces. This should result in a profit of $596 per hour.

• So what have we done to get this solution?– what steps have we taken?

Linear Programming

• Find constraints and graph them

• Identify feasible region– Choose coordinate to test against constraints

• Find coordinates for the vertices (plural for vertex)

• Apply Optimising Function

• Answer question in a succinct statement, in context of course.