Are my linear program equations correct?

Question

Here's the problem: "An electronics company has a contract to deliver 21,475 radios within the next four weeks. The client is willing to pay 20 dollars for each radio delivered by the end of the first week, 18 dollars for those delivered by the end of the second week, 16 dollars by the end of the third week, and 14 dollars by the end of the fourth week. Since each worker can assemble only 50 radios per week, the company cannot meet the order with its present labor force of 40; hence it must hire and train temporary help. Any of the experienced workers can be taken off the assembly line to instruct a class of three trainees; after one week of instruction, each of the trainees can either proceed to the assembly line or instruct additional new classes.

At present, the company has no other contracts; hence some workers may become idle once the delivery is completed. All of them, whether permanent or temporary, must be kept on the payroll 'til the end of the fourth week. The weekly wages of a worker, whether assembling, instructing, or being idle, are 200 dollars; the weekly wages of a trainee are 100 dollars. The production costs, excluding the worker's wages, are 5 dollars per radio. The company's aim is to maximize the total net profit. Formulate this as an LP problem (not necessarily in the standard form)."

I feel that this is a pretty complex linear program (at least as far as my abilities go). I feel like I'm going ape with the amount of variables I'm using, to the point of complicating the LP to the point it can't be solved with Excel Solver (that's the homework assignment, is to formulate the LP and then use a computer to solve it). But this is what I got:

Variables (with i ranging from 1 to 4)

Wi = radios produced in week i

Pi = producers of radios in week i

Ti = trainers in week i

Yi = trainees in week i

Maximize: 20W1 + 18W2 + 16W3 + 14W4 - 5(W1 + W2 + W3 + W4) - 200(Pi + Ti) - 100(Yi) Subject to: W1+W2+W3+W4 = 21475

P1 + T1 + (0)Y1 <= 40

P2 + T2 - Y2 <= 40

P3 + T3 - Y2 - Y3 <= 40

P4 - Y2 - Y3 - Y4 <= 40

Wi = 40(Pi + Yi)

Yi = 3Ti-1

(all variables greater than or equal to 0)

I feel -5(W1 + W2 + W3 + W4) could be left out, as it is just a constant. Is there a better way of doing this than I have shown here? This feels too sloppy to me. Any help would be greatly, GREATLY appreciated.

Answer 1

I think following needs consideration in objective fn.

1. objective fn should also include pay for weeks for which trainees are employed therefore it should be 100(Y1)*4+100(Y2)*3+100(Y3)*2 and no Y4 since they will be of no use for four weeks.
2. Pay for permanent employees 200*(Pi + Ti) * 4 is also a constant in objective fn.
3. Leving out Wi variable it can be replaced with W1=P1*50, W2=(P2+T1)*50,W3=(P3+T1+T2)*50 and W4=(40+T1+T2+T3)*50 (no training in last week)

For constrains

1. constrain for Yi can be taken as Yi <= 3Ti, Ti+Pi=40
2. I think following constrains are not required
3. P1 + T1 + (0)Y1 <= 40
4. P2 + T2 - Y2 <= 40
5. P3 + T3 - Y2 - Y3 <= 40
6. P4- Y2 - Y3 - Y4 <= 40 and Wi = 40(Pi + Yi)