Question

A manufacturer produces four types of vehicles: compact, economy, mid-size, and suv. The manufacturer has designed their supply chain in such a way so that the final assembly of their vehicles entails putting together different quantities of the same parts. We will assume that the costs for parts unique to the model have already been accounted for in the reported per-unit profits, and that inventory levels for these other parts do not change the optimal allocation. e parts, and their current inventory levels, are shown in the table below would like to know how many of each model it should manufacture so as to maximize their total profit. However, if inventory of the raw materials is left over after sales within the planning period, then the manufacturer is charged a per-unit fee for each part, which is also shown in the table below. Use this information to construct a linear program and to answer the questions below. Current Part Inventories IndexPart Name Current Inventory Unit Inventory Holding Cost Framing Tires Paint 50000 units 100000 units 45000 units 60000 units 45000 units $2 $1.50 3Steering Wheel 4 $2.50 $4.25 Windows Per-Unit Profits for Each Model Model Compact Economy Mid-Size SUV Per-Unit Profit $5000 $6000 $8000 $10000 Quantity of Each Part Required to Produce a Single Unit of Each Model Compact Economy Mid-S 4 Part Framing lires Paint ize 4 4 4 Steering Wheel Window:s 4 12 6. Suppose now that the total number of SU/Vs and Compact cars sold must be greater than 10. Which constraintls) should we add to reflect this? x,と10 2 10 と10 7. Now suppose that the manufacturee would like to sell double the amount ol SLIVs than compact cars. Which constraints) should we add to reflect this? + 22 8. Now suppose the manufacturer would like to keep four tmes the aount of tirest steering wheels in inventory after selling for every one steering wheed in inventory, th should be 4 tires in stock). Which constrainats) would be added to accomplish this? b:-(aph 14,.xi + a12Xy + )と3(, .. (an4 + aix: + a12x2 + as4)) 9. Which of the following functioes is a linear function? (c) f(x0.xi)-apo + a.xi + 1 10. Which of the following functioes is a non-linear function?

Answer 1

Linear program is following:

Let X0, X1, X2, X3 be the number of Compact, Economy, Midsize and SUV models to be manufactured resp.

and P0, P1, P2, P3, P4 be the number of parts leftover of respective index number

Max 5000X0+6000X1+8000X2+10000X3-2P0-1.5P1-3P2-2.5P3-4.25P4

s.t.

2X0+1X1+4X2+6X3+P0 = 50000

4X0+4X1+5X2+5X3+P1 = 100000

4X0+5X1+6X2+1X3+P2 = 45000

1X0+1X1+1X2+1X3+P3 = 60000

4X0+3X1+6X2+12X3+P4 = 45000

X0,X1,X2,X3,P0,P1,P2,P3,P4 >= 0

Q6. (b) X0+X3 >= 10

where, X0 refers to number of Compact cars and X3 refers to SUVs

Q7. (d) X3 >= 2X0

Q8. (a) b1-(a_1,0X0+a_1,1X1+a_1,2X2+a_1,3X3) >= 3(b3-(a_3,0X0+A_3,1X1+A_3,2X2+a_3,3X3))

b1 is the current inventory of tires and b3 is the current inventory of steering wheels .

(a_1,0X0+a_1,1X1+a_1,2X2+a_1,3X3) represents the number of tires used. so remaining number of tires in ending inventory is b1-(a_1,0X0+a_1,1X1+a_1,2X2+a_1,3X3)

(a_3,0X0+A_3,1X1+A_3,2X2+a_3,3X3) represents the number of steering wheels used. so remaining number of steering wheels in ending inventory is b3-(a_3,0X0+A_3,1X1+A_3,2X2+a_3,3X3)

(there is an typo error in all the answer options, the question says tires should be 4 times, but in the answer options, 3 is written)

Q9. (c) f(x0,x1) = a0x0+a1a1+1

Q10. (d) f(x0,x1) = a0x2/x1 -a1x0+1