Module 2

Shadow Price (Dual Price):

(Dual Price): The change is the objective function value per one unit increase in the RHS of the constraint. Is only good within the "allowable range"Allowable range for RHS: Range of right hand side (RHS) changes while not "changing the solution" - The term shadow price is an economic term. It indicates the change in the optimal value of the objective when the right side of some constraint changes by one unit.

Example: High Point Farms40 •Sensitivity Analysis Report Questions

(a) What happens if 10 more workers are hired? • Optimal solution, x(20,20,0), changes but it is still a combination of wheat and corn. Profit increases by 25*10 (a) What happens if 40 more workers are hired? •Answer: Optimal solution, x(20,20,0), changes. Workers can be increased at most by 20 without changing the optimal solution. We need to rerun Excel to answer this question c) What happens if HP Farm gets 5 more acers? Should they get any more? •No, they should not get any! Shadow price is 0. There is no value of getting any more acres! (d) What if HP Farm could buy 20 tons of fertilizer for $1000 (total)? Would they want to buy them? • This means that HP farm need to pay 1000/20=$50 per ton. • They gain $62.5 for each ton. Since 62.5 >50 they should get this offer. (Note that at most they can buy 40) • You can alternative work on this with the totals to get the same decision. • 62.5 *20 - 1000 = 1250 - 1000 = 250. This means they would earn $250 so they should get this offer.

feasible solution space

(feasible space, feasible region) is the set of all feasible solutions

by 5

+ 5

Steps of Constraint RHS Changes

1. Check if the change is within the allowable increase/decrease. • Shadow price changes. • If not, change spreadsheet + re‐run solver. 2. If the change is allowable. • Shadow price does not change • Profit will change by (Change in RHS) x (Shadow Price)

to 5$

5 - current calculate yourself

redundant constraint

A constraint that does not affect the feasible region and thus cannot affect the optimal solution is called a

Slack variable

A variable added to the left‐hand side of a less‐than‐or‐equal‐to constraint to convert the constraint into an equality. The value of this variable can usually be interpreted as the amount of unused resource.

Surplus variable

A variable subtracted from the left‐hand side of a greater‐than‐or‐equal to constraint to convert the constraint into an equality. The value of this variable can usually be interpreted as the amount over and above some required minimum level. Any excess quantity corresponding to a > constraint is referred to as__

slack associated with the constraint.

Any unused or idle capacity for a constraint is referred to as the

Step 2

Find the optimal which is intercepted in step 1 and plug it into the min/max equation to get the value moving the contour closer to the optimization point is better

Reduced cost

Impact on the objective function when a currently unused variable is raised to 1. ( How much more profitable would variables have to be to be included in the solution?) At the optimal solution, either the value of a variable is zero, or its reduced cost is 0. - The reduced cost for any decision variable with value 0 in the optimal solution indicates how much better that coefficient must be before that variable enters at a positive level. The reduced cost for any decision variable at its upper bound in the optimal solution indicates how much worse its coefficient must be before it will decrease from its upper bound. The reduced cost for any variable between 0 and its upper bound in the optimal solution is irrelevant.

Contour line

Line on which all solutions have the same objective value

Graphical solutions have two steps

Step 1 Draw feasible region - Defined by constraints - Only Check constraints Step 2 Use objective function to find the best solution in this region

Alternative optimal solutions

The case in which more than one solution provides the optimal value for the objective function.

Infeasibility

The situation in which no solution to the linear programming problem satisfies all the constraints.

Unbounded

The situation in which the value of the solution may be made infinitely large in a maximization linear programming problem or infinitely small in a minimization problem without violating any of the constraints.

Sensitivity Analysis Examples

Things to look out for: • OFC vs RHS change • Maximization Problem vs. Minimization Problem•(<= vs >= constraints) • Increase vs. decrease • Also, changes in absolute values vs. you need to calculate changes • Negative shadow prices (!!!)

Step 1 of Graphing

To be able to graph constraints Treat X1 and X2 as if it was X and Y and then take turns solving for each by making each 0 and solving. (should be 2 equations for X1 and X2) 2X1+5X2 < 60 X1+X2 < 20 To find the intercept solve systems of equations 2X1+5X2=60 X1+X2=20 You can do substitution or elimination (leave as a fraction)

Steps of Objective Function Coefficient (OFC) Changes

Two Steps 1) Check to see if the change we're considering is within the allowable increase or allowable decrease Is -(Allowable Decrease) < change < Allowable Increase • If it's not, the change is too large to analyze using the sensitivity report. We would have to change the actual objective coefficient in our original spreadsheet and run the solver again to find out what would happen. 2) If it is within the allowable range, Then: The current solution will still be the optimal solution. The value of the objective function will change by the amount: (Change in Coefficient) * (Value of decision variable)

Inside or outside lines

X1 and X2 < # (under) X1 and X2 > # (over)

Sensitivity Analysis Report Example for Kooky Candy

Yellow area is what we look at to be changed.

graphical solution

You can use _______ to find extreme points and select the one that gives the best objective value.

Example: High Point Farms Sensitivity Analysis Report

[Reduced Cost] What does ‐550 mean?• Objective function value decreases by 550 if we plant 1 unit of Cotton Implicit cost to produce one acre: 1 acre: cost =0 6 workers: 8: tons of fertilizer Total: 650 Profit‐implicit cost = 100‐650 =‐550 (reduced cost) [Allowable Increase] What does 250 mean? • While profit of Wheat is less than 450, we plant Wheat and Corn [Allowable Decrease] What does 50 mean? 1E+30? • While profit of Wheat is greater than 150, we plant Wheat and Corn • Cotton is currently not profitable. Decreased profit?

Example: High Point Farms Sensitivity Analysis Report Cont.

[Shadow Price] What does 0 mean? 25? • Objective function value remains same if we have 1 more acre [Allowable Increase] What does 1E+30 mean? • We have currently unused land. More land? [Allowable Decrease] What does 5 mean? 53.3? • While Acre is greater than 40, we plant Wheat and Corn • While Fertilizer is greater than 66.7, ...

Active/binding constraints

are constraints satisfied at equality at the optimal solution.

infeasible result when on excel means...

constraints are too conflicting

Binding / Active constraints

constraints satisfied at equality at the optimal solution

optimal solution

is a feasible solution that has an objective function value at least as good as any other feasible solution The last point on the contour line will touch before it leaves the feasible area

feasible solution

is a solution that satisfies all constraints

infeasible solution

is a solution that violates one or more constraints.

The optimum LP solution

is always associated with a corner point of the solution space (extreme points).

constraint issue

max/min is incorrect

Non‐binding constraints have

slack or surplus.

Resource Availability and Shadow Prices

• If a resource constraint is binding in the optimal solution, the company is willing to pay up to some amount, the shadow price, to obtain more of the resource. This is because the objective improves by having more of the resource. • However, there is typically a decreasing marginal effect: As the company owns more and more of the resource, the shadow price tends to decrease. This is usually because other constraints become binding, which causes extra units of this resource to be less useful (or not useful at all).

Sensitivity Analysis

• Performed after you find the optimal solution • It gives you a better understanding of the solution•Helps you answer "what‐if" questions • Even with long‐term strategy question • Helps you understand the impact of any uncertainty in your data. • Is the optimal solution sensitive to changes in input parameters? • Sensitivity analysis determines the effect on the optimal solution of changes in parameter values of the objective function and constraint equations.

Characteristics of Objective Function Coefficient (OFC) Changes

• There is no effect on the feasible region. • The slope of the profit line changes • If the slope changes enough, a different corner point will become optimal. - there is a range for each OFC where the current optimal corner point remains optimal. ----This is called "allowed range", which is the range between allowable increase and allowable decrease.

The Effect of Constraints on the Objective

•If a constraint is added or an existing constraint becomes more constraining (for example, less of some resource is available), the objective can only get worse; it can never improve. •The easiest way to understand this is to think of the feasible region. When a constraint is added or an existing constraint becomes more constraining, the feasible region shrinks, so some solutions that were feasible before, maybe even the optimal solution, are no longer feasible. •The opposite is true if a constraint is deleted or an existing constraint becomes less constraining. In this case, the objective can only improve; it can never get worse. Again, the idea is that when a constraint is deleted or an existing constraint becomes less constraining, the feasible region expands. In this case, all solutions that were feasible before are still feasible, and there are some additional feasible solutions available

What type of changes do you expect when you change one of the constraints right-hand side (RHS)?

•The constraint line shifts, which could change the feasible region •Slope of constraint line does not change •Corner point locations can change •The optimal solution can change