Linear Programming Tasks

Linear programming task without solution. See if you can solve them. Discuss your solution in seminaries of course Modeling of Decision Processes.

Task 1

Two students, Ann and Charles, work X and Y hours per week. Their combined work hours may not exceed 40 hours per week. Considering the rules for temporary workers Ann may work maximally 8 hours longer then Charles, but Charles may work only 6 hour longer then Ann. Additional constrain is 18 <= 2y + x. Try to find optimal solution for:

  1. Ann 15$ and Charles 17$ per hour
  2. Ann 17$ and Charles 15$ per hour
  3. Ann and Charles 16$ per hour

Find maximal combined income.

Task 2

Company produces two products (X and Y) and uses two machines to produce them (A and B). To produce every unit of product X it is necessary to machine A for 50 minutes and B for 30 minutes. Every unit of product Y requires 24 minutes of machine A and 33 minutes of machine B.

At beginning of the week there is 30 units of product X and 90 units of product Y in storage. Available time on machine A is estimated to be 40 hours and 35 hours for machine B.

Demand for product X is estimated to be 75 units and 95 units of Y for this week. Company strategy for production is to maximallize combined number of products X and Y in storage at the end of the week.

How much of each product should company produce.

Task 3

Company produces two products (X and Y). For production company uses two resources – machines for automated production and workers for finishing of the products. Product X requires 13 minutes of machine and 20 minutes of worker’s time, while product Y requires 19 minutes of machine and 29 minutes of worker’s time.

In next week the company may use machines for 40 hours, but only 35 hours of worker’s time. Costs of using machines are 10$ and 2$ for workers per hour. Company does not pay the workers the time, when they do not produce.

The price of X is 20$ and Y 30$. Company has contract to produce 10 products X. Company tries to maximize its profit – what should it produce?

Task 4

Florist Azalea won the contract to prepare wedding flower arrangement. The flower will be of two kinds and their number must not exceed 30. To produce flower 1 20 minutes are required and there is profit 56 Kč (price charged 145 Kč) . Production of flower 2 requires 15 minutes with profit 43 Kč (price charged 130 Kč).

The total costs of flower arrangement must not exceed 4100 Kč and preparation of the flowers must not take longer than 9 hours. What combination of flowers should florist choose to maximalize its profit?

Florist has enough resources for both types of flowers a can produce both of them at same quality level.

Task 5

Tea producer has available 3 kg of dried mint and 1.5 kg of dried St. John’s leafs. Producer can produce two types of teas – mint tea or combination of mint and St. John’s leafs in mixture 3:2. Herbs are filled into sack 10g each. When producing it is necessary to calculate with evaporation 5% for mint and 8% for St. John’s leaves.

Producer is capable to sell maximally 100 sacks of mint with profit 2 Kč. The profit of selling the mixture is 3 Kč. How much sacks of the tea of each type should the producer produce if he should maximize its profit?

Producer has available 3 000 g of mint and 1 500 g of St. John’s leafs.