FIT3158 Note - W3 Sensitivity analysis

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Aug 10, 2022 08:11 AM

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What sorta questions are we answering from the sensitivity report?

Amounts by which objective function coefficients can change without changing the optimal solution.

The impact on the optimal objective function value of changes in constrained resources.

The impact on the optimal objective function value of forced changes in decision variables.

The impact changes in constraint coefficients will have on the optimal solution.

Excel notation (Answer report) — optimal solution, final value of decision variables, resource usage

Section 1 and 2 together tell us the optimal values for the decision variables and the optimal value of the objective function (e.g. profit or return).

Section 3 gives info. on the constraints:

Resource constraints:

Which are binding i.e. which resources are all used up

Which are non-binding i.e. some of the resource is not used

Non-negativity constraints

Which decision variables have values greater than their lower bound of zero

Excel notation (Sensitivity report) — 1) Adjustable cells – final value, reduced cost, allowable increase/decrease; 2) Constraints – final value, shadow price, allowable increase/decrease;

Variable section

Further explanation of Reduced Costs

Def.:

Products whose marginal profits are less than the marginal value of the goods required for their production will not be produced in an optimal solution (unless a constraint forces the product to be produced)

Reduced costs tell us by how much we would have to increase the profitability of an item before it would be included in the optimal solution (for a maximisation problem). And vice-versa for a minimisation problem.

Constraint section

Why is the shadow price always 0 when constraint = Non-binding ?

(because adding more of resource that has not been used up is no help!

Degenerate

The 100% Rule determines if the current solution remains optimal, when more than one objective function coefficient changes.

Problem: But what if I wanna examine the if the current solution remains optimal WHEN there is more than one objective function coefficient changes?

Solution: Then we need The 100% rule.

Two situations could arise when applying this rule:

Case 1. All variables whose objective function coefficients change have nonzero reduced costs. (全部RC 唔係0)

Case 2. At least one variable whose objective function coefficient changes has a reduced cost of zero.

How to perform:

Excel notation (Limits report) — how the objective function varies as each variable ranges between its limits;

Summarises the value of the objective function as each variable cell ranges through its limits (and the others remain constant).

To be studied further:

Comments About Changes in Constraint RHS Values (p.25 - p. 27)

Further explanation of Reduced Costs

10 M + 15O + 10T ≤ 60