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1 Introduction
What constraints can a firm face?

1 Introduction
 limited demand
 limited skilled labour and other production resources
 limited finance (‘capital rationing’).


2 Planning with one limiting factor
What are the five steps using key factor analysis if there is one limiting factor?

2 Planning with one limiting factor
Step 1: identify the bottleneck constraint
Step 2: calculate the contribution per unit for each product
Step 3: calculate the contribution per unit of the bottleneck resource for each product
Step 4: rank the products in order of the contribution per unit of the bottleneck resource
Step 5: allocate resources using this ranking and answer the question


3 Several limiting factors – linear programming Steps involved for linear programming 
3 Several limiting factors – linear programming 

3 Several limiting factors – linear programming
Limiting factor analysis assumptions

1) There is a single quantifiable objective – e.g. maximise contribution. In reality there may be multiple objectives such as maximising return while simultaneously minimising risk. 2) Each product always uses the same quantity of the scarce resource per unit. In reality this may not be the case. For example, learning effects may be enjoyed 3) The contribution per unit is constant. In reality this may not be the case  the selling price may have to be lowered to sell more  there may be economies of scale, for example a discount for buying in bulk 4) Products are independent – in reality  customers may expect to buy both products together  the products may be manufactured jointly together 5) The scenario is short term. This allows us to ignore fixed costs 

4 Shadow prices and slack
1) What is slack?
2) Why is it important?

4 Shadow prices and slack
1) Slack is the amount by which a resource is underutilised. It will occur when the optimum point does not fall on a given resource line.
2) Slack is important because unused resources can be put to another use, e.g. hired out to another manufacturer.


4 Shadow prices and slack
1) What are shadow (duel) prices? 2) How is the shadow cost found? 3) What about noncritical constraints? 
4 Shadow prices and slack
1) It represents the maximum premium that the firm should be willing to pay for one extra unit of each constraint
2) The shadow price of a resource can be found by calculating the increase in value (usually extra contribution) which would be created by having available one additional unit of a limiting resource at its original cost.
3) Noncritical constraints will have zero shadow prices as slack exists already


4 Shadow prices and slack
How to calculate shadow costs 
4 Shadow prices and slack
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 critical constraint unchanged.
Step 2: Use simultaneous equations to derive a new optimal solution
Step 3: Calculate the revised optimal contribution. The increase is the shadow price for the constraint under consideration.


4 Shadow prices and slack
What are the implications of shadow prices? 
4 Shadow prices and slack
 Management can use shadow prices as a measure of the maximum premium that they would be willing to pay for one more unit of the scarce resource
 However, the shadow price should be considered carefully. For example, the shadow price of labour may be calculated as $20 per hour. However, it may be possible to negotiate a lower shadow price than this.
 In addition, if more of the critical constraint is obtained, the constraint line will move outwards altering the shape of the feasible region. After a certain point there will be little point in buying more of the scarce resource since any noncritical constraints will become critical
