An electronics company has a contract to deliver 20,000 radios within the next four weeks. The client is willing to pay $20 for each radio delivered by the end of the first week, $18 for those delivered by the end of the second week, $16 by the end of the third week, and $14 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 till the end of the fourth week. The weekly wages of a worker, whether assembling, instructing, or being idle, are $200; the weekly wages of a trainee are $100. The production costs, excluding the workers wages, are $5 per radio.
For example, the company could adopt the following plan.
- First Week: 10 assemblers, 30 instructors, 90 trainees
- Workers' wages: $8,000
- Trainees' wages: $9,000
- Profit from 500 radios: $7,500
- Net loss: $9,500
- Second Week: 120 assemblers, 10 instructors, 30 trainees
- Workers' wages: $26,000
- Trainees' wages: $3,000
- Profit from 6,000 radios: $78,000
- Net profit: $49,000
- Third Week: 160 assemblers
- Workers' wages: $32,000
- Profit from 8,000 radios: $88,000
- Net profit: $56,000
- Fourth Week: 110 assemblers, 50 idle
- Workers' wages: $32,000
- Profit from 5,500 radios: $49,500
- Net profit: $17,500
This plan, leading to a total net profit of $113,000, is one of many possible plans. The company's aim is to maximize the total net profit.
1. Formulate this problem as an LP (not necessarily in the standard form). Your model sho