Question

A computer manufacturer produces three types of computers: laptop, desktop, and tablet The manufacturer has designed their supply chain in such a way so that the final assembly of their computers 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. We assume that there are three parts: CPU, RAM, and Hard Drive, each labels, respectively, as part 0, part 1, and part 2. The manufacturer 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. Use this information to construct a linear program and to answer the questions below.

Now suppose the manufacturer would like to keep four times the amount of RAM than CPUs in inventory after selling (for every one CPU in inventory, there should be 4 units of RAM in stock). Which constraint(s) would be added to accomplish this?

(a) b1 − (a1,0x0 + a1,1x1 + a1,2x2 + a1,3x3 + a1,4x4) ≥ 4(b3 − (a3,0x0 + a3,1x1 + a3,2x2 + a3,3x3 + a3,4x4))

(b) b1 − (a1,0x0 + a1,1x1 + a1,2x2 + a1,3x3) ≥ 4 b3 − (a3,0x0 + a3,1x1 + a3,2x2 + a3,3x3) ≥ 4

(c) b0 − (a0,0x0 + a0,1x1 + a0,2x2) ≥ 4(b1 − (a1,0x0 + a1,1x1 + a1,2x2))

(d) b0 − (a0,0x0 + a0,1x1 + a0,2x2) ≥ 4 b1 − (a1,0x0 + a1,1x1 + a1,2x2) ≥ 4

Answer 1

Given, The manufacturer produces three types of computers: laptop, desktop and tablet.

Now assume,

x0, x1 and x2 represent the quantity of each type of computers to be produced. p0, p1 and p2 represent the respective profit per unit for each type of computer.

There are three three parts, part 0, part 1 and part 2. h0, h1 and h2 represent the respective per-unit fee for each part.

and b0, b1, b2 represent the avaialble quantity of each part.

Let,

aij represent the quantity of part i used in production of computer type j, where i and j ={0,1,2}

for example a_0,0 represent quantity of CPU (part 0) used in making one laptop (computer type 0), a_0,1 represent quantity of CPU (part 0) used in making one desktop (computer type 1), and so on ...

In the question, it asks for RAM (part 1), but the answer option is given for CPU (part 0), this might be a typo error. Rest of the expression is correct. Expression "b0 − (a0,0x0 + a0,1x1 + a0,2x2))" represent the left-over inventory after selling the computers, the manufacturer would like this to be greater than 4 times the number of tablets sold. So, right hand side of the constraint is 4*x2 . Resulting expression is: b0 − (a0,0x0 + a0,1x1 + a0,2x2) ≥ 4x2

The left hand side of this expression shows the leftover quantity of CPU. For RAM, the constraint would be "b1 − (a1,0x0 + a1,1x1 + a1,2x2) ≥ 4x2

Option a is incorrect because its left hand side represents the raw number of CPUs used, Not the leftover quantity. Options b and c include variables x3 and x4, which are non-existent in the model. So, options a, b, c are incorrect.