Chapter 4 - Planning With Limitng Factors

Question 0

A linear programming problem involving two variables

Answer 0

Linear programming is used to maximise contribution and/ or minimise costs.

The steps involved in linear programming are:
Step 1: Define the variables

Step 2: Define and formulate the objective (maximise contribution)

Step 3: Formulate the constraints

Step 4: Draw a graph identifying the feasible region

Step 5: Solve for the optimal production plan

Step 6: Answer the question

Question 1

Planning with one limiting factor

Answer 1

If there is one limiting factor, then the problem is best solved using key factor analysis.

Step 1: Identify the scarce resource.

Step 2: Calculate the contribution per unit for each product.

Step3: Calculate the contribution per unit of the scarce resource for each product.

Step 4: Rank the products in order of the contribution per unit of the scarce resource.

Step 5: Allocate resources using this ranking and answer the question.

Question 2

Slack

Answer 2

Slack:

1. The amount by which a resource is under-utilised. It will occur when the optimum point does not fall on a given resource line.
2. Is important because unused resources can be put to another use.

Question 3

Shadow (Dual) prices

Answer 3

Shadow (dual) price:

1. Found by calculating the increase in value (usually extra contribution) which would be created by having one additional unit of a limiting resource at its original cost.
2. It represents the maximum premium that the firm should be willing to pay for one extra unit of each constraint.
3. Non-critical constraints will have zero shadow prices as slack exists already.

Question 4

Calculating shadow prices

Answer 4

Step 1: Take the equations of the straight lines that intersect at the optimal point. Add one unit to the constraint concerned while leaving the other unchanged.

Step 2: Use simultaneous equations

Step 3: Calculate the revised optimal contribution. The increase is the shadow price for the constraint under consideration

Question 5

Implications of shadow prices

Answer 5

1. Management can use shadow prices as a measure of the maximum premium they would be willing to pay for one more unit of the scarce resource.
2. However, e.g. If the shadow price of labour is $20 per hour, it may be possible to negotiate a lower price than this.
3. If more of the critical constraint is obtained, then the constraint line will move outwards and alter the shape of the feasible region. Then after a certain point, there will be little point in buying more of the scarce resource since any non- critical constraints will be critical.

Question 6

Limiting factor analysis assumptions

Answer 6

Assumptions
1. There is a single quantifiable objective, to maximise contribution.
In reality, there may be multiple objectives such as maximising return whilst minimising risk.

2. Each unit always uses the same quantity of the scarce resource per unit. In reality, learning effects may be enjoyed.
3. The contribution per unit is constant. In reality, the selling price may be lowered to sell more and discounts for buying in bulk.
4. Products are independent. In reality, customers may expect to buy both products together and the products may be manufactured jointly together.
5. The scenario is short term which allows us to ignore fixed costs.