CHAPTER 8: POST OPTIMALITY ANALYSIS

What is Post Optimality Analysis? Post Optimality Analysis

– Sensitivity Analysis– What-If Analysis

what happens to the optimal solution if: – THE INPUTS CHANGE: – THE CONDITIONS OF THE BUSINESS OBJECTIVE

CHANGE

– THE FACTORY PARAMETERS CHANGE

Example– Let X = the number of Tables made per week,

Let Y = the number of Chairs made per week,

– Maximise Profit = 4X + 3Y Objective Function

Subject to

4X+1Y 90 Wood

2X+1Y 50 Machine-Time

1X +1Y 40 Polishing-Time

X, Y 0

Shadow Costs, Binding & Non-Binding Constraints Wood constraint

– 4X + 1Y 90 – Optimal Solution(X,Y)=(10,30), – the wood used 4*10+1*30=70 kilograms– 20 kilograms unused---slack value – Let RHS 90 90+1 or 90-1– The optimal solution does not change.– The addition or removal of one kilogram of Wood

makes no difference to the optimal production plan. The Wood constraint is said to be a Non-Binding constraint.

Definition of a Non-Binding Constraint – If the availability of an additional unit of a reso

urce has no effect on the production plan, then that constraint is said to be non-binding.

Definition Shadow Cost – The Shadow Cost of a resource is the additional

profit generated by an additional unit of that resource.

– Example: WOOD is a Non-Binding Constraint and definitionally non-binding constraints have a Shadow Cost = 0

Machine-Time constraint– 2X + 1Y 50 – Let RHS 50 51 or 49– The optimal solution change.– (X,Y) =(10,30) (X,Y)=(11,29)– Change=(1,-1)

Definition Binding Constraint – If the availability of an additional unit of resource alters

the optimal production plan, thereby increasing the profit, then that constraint is said to be a 'Binding Constraint'.

Shadow Cost of Machine-Time – The new value of the objective

function :4*11+3*29=131

– The shadow cost of the resource Machine-Time: £ 1

Shadow Cost: A way of pricing the value of the resource

The consequences of linear systems – if increasing 2 hours or decreasing 5 hours of

Machine-Time became available, the new production plan would be:

• (10, 30) +2*(1,-1) = (10, 30) + (2, -2) = (12, 28)

• (10, 30) +(-5)*(1,-1) = (10, 30) + (-5, 5) = (5, 35)

– The new profit would be :• Old profit + 2*Shadow Cost

• £130 + 2*£1 = £132

• Old profit + (-5)*Shadow Cost

• £130 + (-5)*£1 = £125

Polishing-Time constraint– Polishing-Time is a binding constraint – The shadow price for Polishing-Time is £2

SummaryRESOURCE BINDING/NON

BINDINGSLACK SHADOW COST

WOOD NON-BINDING 20Kgms of WOOD £0

MACHINE TIME BINDING 0 Hours £1

POLISHING TIME BINDING 0 Hours £2

Resource Ranging

How much is it possible to increase or decrease the units of a resource whilst the shadow price is effective?

Ranging for the WOOD resourceLower Limit: 70

– Existing Resource: 90 Upper Limit No upper: limit.

Ranging for the MACHINE-TIME resource

Lower Limit 40 Existing Resource 50 Upper Limit 56.67

The ranging for the resource POLISHING-TIME Lower Limit 30

Existing Resource 40Upper Limit 50

Summary

RESOURCE LOWER LIMIT EXISTING RESOURCE UPPER LIMIT

WOOD 70 90 No upper limit.

MACHINE-TIME 40 50 56.67

POLISHING-TIME 30 40 50

Changes in the Objective Function Coefficient Introductory details

– Profit = aX + bY – Y=Profit/b-a/b X

• Example: Profit = 4X + 3Y • Y = Profit/3 -4/3 X

– The slope of the Machine-Time constraint • Y = 50 - 2X

– The slope of Polishing-Time constraint• Y=40-1X

– The slope of objective function must lie between the two constraint slopes

Y = 40-1X

Y = Profit/b -a/b X (Y=130/3-a/3 X)

Y = 50 - 2X

– The coefficient of Y can vary : 2 to 4

Slopes    

Machine-Time Objective Polishing-Time

-2 -a/3 -1

– The coefficient of X can vary : 3 to 6

Slopes    

Machine-Time Objective Polishing-Time

-2 -4/b -1

The Factory Parameter Changes If the factory were to invest in different

plant, new technology that had the effect of making more effective use of the input resources, how would this modify the solution to the linear programming?

The practical way to see the effect this has on the linear programming is to resolve the linear programming with the new constraint