Linear Programming and Limiting Factors

Question 1

Linear Programming

What is it used for(3)

What can we not use?

Answer 1

• A linear optimisation method used to find best-case outcomes based on objective linear relationships.
• We use linear programming to solve limiting factor problems where:
  • There is more than one constraint; and
  • Products rank differently for those constraints.

In these cases we cannot use the contribution per limiting factor approach, which depends on only one constraint.

Question 2

Two or More Limiting factors

What would we assume? (3)

Answer 2

We would assume that:

• Firms tend to maximise contribution or minimise cost
• The inputs change in a linear function with production
• Only two products are produced

Question 3

Shadow price

What is shadow price?
What do they arise from?
For each constraint what does the shadow price indicate?

Answer 3

• Shadow price is the additional contribution margin or profit that could be achieved if one additional unit of a constrained resource were made available, while holding other factors constant.
• Shadow prices arise from linear programming solutions, where businesses aim to optimize an objective function (e.g., maximize profit or minimize cost) subject to constraints.
• For each constraint, the shadow price indicates how much the objective function would improve per additional unit of the constrained resource.

Question 4

Shadow price (example)

Answer 4

Suppose a company manufactures two products, A and B, and the production is constrained by machine hours. If the shadow price of machine time is £50, it implies that each additional hour of machine availability would increase the total profit by £50. This information can guide decisions on whether to invest in additional machine capacity or revise production schedules.

Question 5

Zero Shadow Price

What is this?

Answer 5

If a constraint has a shadow price of zero, it means the constraint is not binding, and increasing the resource will not improve the outcome. For example, if machine time is not fully utilized, additional machine time will not increase profit.

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