Sensitivity of the Right Hand Side Coefficients Linear Programming Sensitivity of the Right Hand Side Coefficients

Sensitivity of RHS Coefficients RHS coefficients usually give some maximum limit for a resource or some minimum requirement that must be met. Changes to the RHS can happen when extra units of the resource become available or when some of the original resource becomes unavailable. Or the minimum requirement is loosened (made less) or strengthened (made greater). Extra units may be available for a price. The question becomes how much would an extra unit add to the value of the objective function, that is, “what is the most we would be willing to pay for extra units of the resource?”

Finding the Optimal Point - Review X2 1000 900 800 700 600 500 400 300 200 100 X1 Max 8X1 + 5X2 s.t. OPTIMAL POINT (320,360) = 4360 2X1 + 1X2 ≤ 1000 (Plastic) 3X1 + 4X2 ≤ 2400 (Time) 1X1 + 1X2 ≤ 700 (Limit) 1X1 - 1X2 ≤ 350 (Mix) X1, X2 ≥ 0

Optimal Point With One Extra Unit of Plastic 1000 900 800 700 600 500 400 300 200 100 X1 Shadow Price (for Plastic) 4363.40 – 4360 = (new profit) - (old profit) $3.40 Max 8X1 + 5X2 s.t. New OPTIMAL POINT (320.8,359.4) = 4363.40 1001 2X1 + 1X2 ≤ 1000 (Plastic) 3X1 + 4X2 ≤ 2400 (Time) 1X1 + 1X2 ≤ 700 (Limit) 1X1 - 1X2 ≤ 350 (Mix) X1, X2 ≥ 0 Still determined by Plastic and Time constraints

Shadow Prices The shadow price for a constraint is the amount the objective function value will change given: 1 additional unit on the RHS of the constraint No other changes This shadow price is valid as long as the same constraints (including the one whose RHS is changing) determine the optimal point. In this case plastic and production time It can be shown that if the RHS for plastic were 1002 the profit would increase another $3.40 to $4366.80. It can also be shown that if the RHS for plastic were 999 the profit would decrease by $3.40 to $4356.60.

Allowable Increase and Allowable Decrease of a RHS Value The shadow prices remain valid as long as the same constraints (called the binding constraints) determine the optimal point. When the RHS of the constraint is increased or decreased to the point that another constraint replaces one of the binding constraints to determine the optimal point a new shadow price becomes valid for the constraint. The amount the RHS can increase or decrease before another constraint becomes one of the binding constraints is what Excel calls the Allowable Increase and the Allowable Decrease respectively.

Increasing the Right Handside for Plastic X2 1000 900 800 700 600 500 400 300 200 100 X1 Max 8X1 + 5X2 s.t. 1030 1060 1100 2X1 + 1X2 ≤ 1000 (Plastic) 3X1 + 4X2 ≤ 2400 (Time) 1X1 + 1X2 ≤ 700 (Limit) 1X1 - 1X2 ≤ 350 (Mix) X1, X2 ≥ 0 Plastic and Time constraints still determine the optimal solution to this point.

Further Increasing the Right Hand Side for Plastic X2 1000 900 800 700 600 500 400 300 200 100 X1 Max 8X1 + 5X2 s.t. 1030 1060 1100 1102 2X1 + 1X2 ≤ 1000 (Plastic) 3X1 + 4X2 ≤ 2400 (Time) 1X1 + 1X2 ≤ 700 (Limit) 1X1 - 1X2 ≤ 350 (Mix) X1, X2 ≥ 0 Further increases to the RHS side of plastic have now made the plastic and the Limit constraints as the ones that determine the optimal point. The shadow prices will now CHANGE

Decreasing the RHS for Plastic X2 1000 900 800 700 600 500 400 300 200 100 X1 Max 8X1 + 5X2 s.t. 600 700 850 1000 2X1 + 1X2 ≤ (Plastic) 3X1 + 4X2 ≤ 2400 (Time) 1X1 + 1X2 ≤ 700 (Limit) Redundant 1X1 - 1X2 ≤ 350 (Mix) X1, X2 ≥ 0 Optimal solution determined by Plastic and Time Constraints and by X2 axis!

Further Decreasing the RHS for Plastic X2 1000 900 800 700 600 500 400 300 200 100 X1 Max 8X1 + 5X2 s.t. 590 600 700 850 1000 2X1 + 1X2 ≤ (Plastic) 3X1 + 4X2 ≤ 2400 (Time) 1X1 + 1X2 ≤ 700 (Limit) Redundant 1X1 - 1X2 ≤ 350 (Mix) X1, X2 ≥ 0 Optimal Point now determined by the plastic constraint and the X2-axis The shadow prices will now CHANGE

Comparison With Excel Here is the printout out of the sensitivity analysis dealing with the objective RHS coefficients for the original Galaxy Industries problem. Shadow Price for each constraint Range of Feasibility for RHS1 1000 – 400  1000 + 100 600  1100 Range of Feasibility for RHS2 2400 - 650  2400 + 100 1750  2500 Range of Feasibility for RHS3 700 – 20  700 + ∞ 680  ∞ Range of Feasibility for RHS4 350 – 390  350 + ∞ - 40  ∞ Range of Feasibility is the range of values that an RHS coefficient can assume without changing the shadow prices as long as no other changes are made.

Exact Meaning of Shadow Prices A shadow price always means the amount the objective function will change given a one unit increase in the RHS value of a constraint. But does this mean that this is the value (the most you would be willing to pay) for an extra unit? The answer depends on how the objective function coefficients were calculated. If the objective function coefficients did not take the value of the resource into consideration, these are sunk costs. Shadow price = the value of an extra unit of the resource. If the objective function coefficients did take the value of the resource into consideration, these are included costs. Shadow price = a premium above the current price of the item that one would be willing to pay for an extra unit.

Plastic is an included cost EXAMPLE Suppose the $8 objective function coefficient for dozens of Space Rays and the $5 objective function coefficient for dozens of Zappers were calculated as follows: DOZ. DOZ. SPACE RAYS ZAPPERS Selling Price $24 $26 Costs Plastic ($3/lb) $ 6 (2 lbs.) $ 3 (1 lb.) Other Variable Costs $10 $18 =========== ========== Total Profit Per Dozen $ 8 $ 5 Plastic is an included cost Production time is a sunk cost The $3.40 shadow price for plastic means we would be willing to pay up to $3.40 more than the current price of $3 per pound (that is up to $6.40/ lb.) for extra plastic. It is not included in the objective function coefficient calculation. The $0.40 shadow price is the value of an extra minute of production time.

Complementary Slackness Complementary slackness also holds for RHS values. This property for RHS values states: Again, it can happen, that both are 0. Complementary Slackness For RHS Coefficients For each constraint, either the slack (difference between RHS – LHS) is 0 or its shadow price will be 0. Plastic: Shadow Price ≠ 0; Slack = 1000-1000 = 0 Time: Shadow Price ≠ 0; Slack = 2400-2400 = 0 Prod. Limit: Slack = 700-680 ≠ 0; Shadow Price = 0 Prod. Mix: Slack = 350-(-40) ≠ 0; Shadow Price = 0

Review Shadow price Range of Feasibility Complementary Slackness Found by subtracting the original objective function value from the objective function value with one more unit of the resource on the RHS Meaning Included Cost Sunk Cost Range of Feasibility Range of RHS value in which shadow price does not change The same constraints determine the optimal solution in the range of feasibility Complementary Slackness Either the slack is 0 or the shadow price is 0