Question

Note that in this part you are only asked to model the problem. Do NOT try to solve and find the optimal solution. In models if there are two many parameters or decisions variables, like in M4, you may use parametric modeling structure, like E ci Xi <Di for every j where Di demand-j cij cost at i-j Then give an example for j=1 this constraint is 12 X1 + 8 X2 <200. Where c11=12, c12=8 and Di= 200 with decision variables X1 and X2. A company manufactures four variants of the same product and in the final part of the manufacturing process there are assembly, polish and pack operations. For each variant the speed of each operation is shown below (in units produced per hour) and the profit per unit sold. Product Variant Assembly Polishing Packing Profit ($ per 1 2 3 4 30 15 20 9 20 30 20 15 30 20 30 12 unit) 1.50 2.00 3.45 4.20 Given the current state of the labour force the company estimate that, each year, they have 100000 minutes of assembly time, 50000 minutes of polishing time and 60000 minutes of packing time available. a) Formulate an LP model to calculate, how many of each variant should the company make per year to maximize the associated profit. b) Suppose now that the company is free to decide how much time to devote to each of the three operations (assembly, polishing and packing) within the total allowable time of 210000 (= 100000+ 50000+60000) minutes. Update the model to incorporate this new condition.

Answer 1

Before doing anything, first, convert the speed into cycle time in minutes by using the following expression:

Cycle time = 60 / Speed in units per hr.

Product	Assembly	Polishing	Packing
1	60/30 = 2	3	2
2	4	2	3
3	3	3	2
4	20/3	4	5

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(a)

Let X_j = number of units of Product-j to be produced for j=1,2,3,4

Max Z = 1.5 X₁ + 2.0 X₂ + 3.45 X₃ + 4.2 X₄

Subject to,

2 X₁ + 4 X₂ + 3 X₃ + (20/3) X₄ <= 100000 (Assembly minutes)

3 X₁ + 2 X₂ + 3 X₃ + 4 X₄ <= 50000 (Polishing minutes)

2 X₁ + 3 X₂ + 2 X₃ + 5 X₄ <= 60000 (Packing minutes)

X₁, X₂, X₃, X₄ >= 0

(b)

In this case, the three constraints will be replaced just by the following single constraint:

2 X₁ + 4 X₂ + 3 X₃ + (20/3) X₄ + 3 X₁ + 2 X₂ + 3 X₃ + 4 X₄ + 2 X₁ + 3 X₂ + 2 X₃ + 5 X₄ <= 210000
or,
7 X₁ + 9 X₂ + 8 X₃ + (47/3) X₄ <= 210000