chAPTER 8 POST OPTIMALITY ANALYSIS
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
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 XY≥0
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
10 0∞元 WOOD CONSTRAINT MACHINE TIME CONSTRAINT ASE户G POLLSHING TIME CONSTRAINT No, Tables
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 rhs90→90+1or90-1 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
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 resource 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
Definition of a Non-Binding Constraint – If the availability of an additional unit of a resource 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 rhs50→51or49 The optimal solution change (X,Y)=(10,30)(XY)=(11,29) Change=(l, -1) WOOD CONSTRAINT 09 MACHINE TIME SONSTRAINT POLISHING CONSTRAINT 00 15 25 30 No. Tables
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 t 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.i 1 Shadow Cost: A way of pricing the value of the resource
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)*E1=25
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 ◆ Summary RESOURCE BINDING NON SLACK SHADOW COST BINDING WOOD NON-BINDING of WOoD MACHINE TIME BINDING POLISHING TIME BINDING 0 Hours
Polishing-Time constraint – Polishing-Time is a binding constraint – The shadow price for Polishing-Time is £2 Summary RESOURCE BINDING/NON BINDING SLACK SHADOW COST WOOD NON-BINDING 20Kgms of WOOD £0 MACHINE TIME BINDING 0 Hours £1 POLISHING TIME BINDING 0 Hours £2