Formulating and Solving Multiple Scarce Resource Problems

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Linear programming

When there are two or more resources in short supply, linear programming is required to find the solution.

Linear programming is used to:

1. maximise contribution and/or
2. minimise costs

Linear programming assumptions

  1. a single quantifiable objective

  2. each product always uses the same quantity of the scarce resource per unit.

  3. the contribution (or cost) per unit is constant for each product, regardless of the level of activity. Therefore, the objective function is a straight line.

  4. products are independent

  5. the focus is short-term

  6. all costs either vary with a single volume-related cost driver or they are fixed for the period under consideration.

Illustration

A company manufactures two types of boxes, Box A and Box B.

Contribution / box A = $20 Contribution / box B = $45

Two materials, X and Y are used in the manufacturing of each box. 
Each material is in short supply.

Material X = 3,000 kg available         Material Y = 2,700 kg available

Each box A uses 20 kg of Material X and 15 kg of Material Y
Each box B uses 30 kg of Material X and 50 kg of Material Y

The objective of this company is to maximize profits.

  • Variables

    Let A be the number of boxes of A produced and sold
    Let B be the number of boxes of B produced and sold

  • Objective function

    Max C = 20A + 45B

  • Constraints

    Mat X = 20A + 30B ≤ 3,000 
    Mat Y = 15A + 50B ≤ 2,700

    Non-negativity = A,B ≥ 0

  • Material X

    20A + 30B = 3,000
    When A is 0, B = 100
    When B is 0, A = 150

  • Material Y

    15A + 50B = 2700
    When A = 0, B = 54
    When B = 0, A = 180

NotesQuizPaper examCBE