the pizza party Company
The Pizza Party manufacture two types of Pizza, small and large.
Delivery Van A can deliver 8 trays of small pizza and 8 trays of large pizzas per trip.
Delivery Van B can deliver 4 trays of small pizza and 24 trays of large pizzas per trip
Each day the Company needs to deliver at least 400 trays of small pizzas and at least 960 trays of large pizzas.
The maintenance crew has told management that the Pizza company delivery vans cannot make more than 90 trips per day.
Fuel costs for Delivery Van A are $8 per trip and delivery Van B $12 per trip
The new fuel costs for Van A are $18 and for van B $9 per trip (Excellence bit)
Calculate in a report the options the company should explore to minimise their fuel costs
How does the increase of fuel cost affect the optimum number of trips of each truck?
Delivery Van A can deliver 8 trays of small pizza and 8 trays of large pizzas per trip.
Delivery Van B can deliver 4 trays of small pizza and 24 trays of large pizzas per trip
Each day the Company needs to deliver at least 400 trays of small pizzas and at least 960 trays of large pizzas.
The maintenance crew has told management that the Pizza company delivery vans cannot make more than 90 trips per day.
Fuel costs for Delivery Van A are $8 per trip and delivery Van B $12 per trip
The new fuel costs for Van A are $18 and for van B $9 per trip (Excellence bit)
Calculate in a report the options the company should explore to minimise their fuel costs
How does the increase of fuel cost affect the optimum number of trips of each truck?
solutions
The Pizza Party manufacture two types of Pizza, small and large.
Delivery Van A can deliver 8 trays of small pizza and 8 trays of large pizzas per trip.
Delivery Van B can deliver 4 trays of small pizza and 24 trays of large pizzas per trip
Each day the Company needs to deliver at least 400 trays of small pizzas and at least 960 trays of large pizzas.
The maintenance crew has told management that the Pizza company delivery vans cannot make more than 90 trips per day.
Fuel costs for Delivery VanA are $8 per trip and delivery Van B $12 per trip
The costs associated with each van are represented by x and y
Therefore, the cost equation is cost = 8x + 12y
Delivery Van A can deliver 8 trays of small pizza and 8 trays of large pizzas per trip.
Delivery Van B can deliver 4 trays of small pizza and 24 trays of large pizzas per trip
Each day the Company needs to deliver at least 400 trays of small pizzas and at least 960 trays of large pizzas.
The maintenance crew has told management that the Pizza company delivery vans cannot make more than 90 trips per day.
Fuel costs for Delivery VanA are $8 per trip and delivery Van B $12 per trip
The costs associated with each van are represented by x and y
Therefore, the cost equation is cost = 8x + 12y
Equations:
x = number of trips for delivery van A
y = number of trips for delivery van B
8x + 4y ≥ 400 representing the quantities of small pizzas
8x + 24y ≥ 960 representing the quantities of large pizzas
y ≤ 90 number of Van A trips
x ≤ 90 number of Van B trips
desmos graphing
label the graph appropriately
costs based on graph points A,B,C and D
Cost = 8x +12y
A (5,90) = 8x5+12x90=$1,120
B (90,90)= 8x90+12x90=$1,800
C (90,10)= 8x90+12x10=$840
D (36,28)= 8x36+12x28=$624
A (5,90) = 8x5+12x90=$1,120
B (90,90)= 8x90+12x90=$1,800
C (90,10)= 8x90+12x10=$840
D (36,28)= 8x36+12x28=$624
The point that minimises fuel costs is D(36,28) [ACHIEVED]
[MERIT]
To minimise the fuel costs, The Pizza Company should run 36 trips a day for delivery van A and 28 trips a day for delivery van B. The daily fuel cost is at a minimum of $624
And for High Merit:
From the increased fuel cost equation worked out above F = 18x + 9y
A(5,90) cost = 18 x 5 + 9 x 90 = 900
B(90,90) cost = 18 x 90 + 9 x 90 = 2,430
C(90,10) cost = 18 x 90 + 9 x 10 = 1,710
D(36,28) cost = 18 x 36 + 9 x 28 = 900
END MERIT
START EXCELLENCE
The company now has two options that give the lowest fuel cost. It can either run 5 trips with delivery van A and 90 trips with delivery vans B each day or 36 trips with delivery van A and 28 trips with delivery van B each day. The minimum cost is $900 an increase of $276 over the last option. That takes it to high Merit.
From the increased fuel cost equation worked out above F = 18x + 9y
A(5,90) cost = 18 x 5 + 9 x 90 = 900
B(90,90) cost = 18 x 90 + 9 x 90 = 2,430
C(90,10) cost = 18 x 90 + 9 x 10 = 1,710
D(36,28) cost = 18 x 36 + 9 x 28 = 900
END MERIT
START EXCELLENCE
The company now has two options that give the lowest fuel cost. It can either run 5 trips with delivery van A and 90 trips with delivery vans B each day or 36 trips with delivery van A and 28 trips with delivery van B each day. The minimum cost is $900 an increase of $276 over the last option. That takes it to high Merit.
Excellence
For Excellence you need to notice the there are two minimum values achieved, at points A and D.
(It isn't always A and D but that happens in most of our activities). The equation for that line is
8x + 4y ≥ 400, now find its gradient by rearranging it.
4y ≥ -8x + 400
y ≥ -2x + 100 so the gradient = -2
The new cost function 18x + 9y has the same gradient as 4y ≥ -8x + 400 above as follows,
18x + 9y = cost - don't worry about c
9y = -18x + c
y = -2x + c so the gradient is -2 as above.
What does this mean?
Any integer (whole) coordinates along the line 4y ≥ -8x + 400 for 5 ≤ x ≤ 36, will minimise the cost. This means the company can run the delivery vans in different combinations, not limited to 36 trips for van A and 28 trips for van B. For example the company could run 6 trips for van A and 88 trips for van B. When they increase Van A’s trip number by 1, Van B’s trip will decrease by 2 (see desmos graph).
(5,90), (6,88), (7,86)……..(35,30),(36,28)
The daily fuel cost will be at minimum of $ 900.
(It isn't always A and D but that happens in most of our activities). The equation for that line is
8x + 4y ≥ 400, now find its gradient by rearranging it.
4y ≥ -8x + 400
y ≥ -2x + 100 so the gradient = -2
The new cost function 18x + 9y has the same gradient as 4y ≥ -8x + 400 above as follows,
18x + 9y = cost - don't worry about c
9y = -18x + c
y = -2x + c so the gradient is -2 as above.
What does this mean?
Any integer (whole) coordinates along the line 4y ≥ -8x + 400 for 5 ≤ x ≤ 36, will minimise the cost. This means the company can run the delivery vans in different combinations, not limited to 36 trips for van A and 28 trips for van B. For example the company could run 6 trips for van A and 88 trips for van B. When they increase Van A’s trip number by 1, Van B’s trip will decrease by 2 (see desmos graph).
(5,90), (6,88), (7,86)……..(35,30),(36,28)
The daily fuel cost will be at minimum of $ 900.