1 The Geometry of Linear Programs –the geometry of LPs illustrated on GTC Handouts: Lecture Notes February 5, 2002

2 But first, the pigskin problem (from Practical Management Science) Pigskin company makes footballs All data below is for 1000s of footballs Forecast demand for next 6 months –10, 15, 30, 35, 25 and 10 Current inventory of footballs: 5 Max Production capacity: 30 per month Max Storage capacity: 10 per month Production Cost per football for next 6 months: –$12:50, $12.55, $12.70, $12.80, $12.85, $12.95 Holding cost: $.60 per football per month With your partner: write an LP to describe the problem

3 On the formulation Choose decision variables. –Let x j = the number of footballs produced in month j (in 1000s) –Let y j = the number of footballs held in inventory from year j to year j + 1. (in 1000s) –y 0 = 5 Then write the constraints and the objective. Pigskin Spreadsheet

4 Data for the GTC Problem Wrenches PliersAvailable Steel ,000 tons Molding Machine ,000 hrs Assembly Machine.4.55,000 hrs Demand Limit8,000 10,000 Contribution ($ per item) $.40$.30 Want to determine the number of wrenches and pliers to produce given the available raw materials, machine hours and demand.

5 Formulating the GTC Problem P = number of pliers manufactured (in thousands) W = number of wrenches manufactured (in thousands) Maximize Profit = Steel: Molding: Assembly: Plier Demand: Wrench Demand: Non-negativity: 300 P W P W ≤15 P + W ≤ P W ≤5 P ≤10 W ≤8 P,W ≥0

6 Graphing the Feasible Region We will construct and shade the feasible region one or two constraints at a time.

7 Graphing the Feasible Region Graph the Constraint: P + 1.5W ≤15

8 Graphing the Feasible Region W + P ≤12 Graph the Constraint:

9 Graphing the Feasible Region.4W +.5P ≤5 Graph the Constraint: What happened to the constraint : W + P ≤12?

10 Graphing the Feasible Region W ≤8, and P≤10 Graph the Constraint:

11 How do we find an optimal solution? Maximize z = 400W + 300P Is there a feasible solution with z = 1200?

12 How do we find an optimal solution? Maximize z = 400W + 300P Is there a feasible solution with z = 1200? Is there a feasible solution with z = 3600?

13 How do we find an optimal solution? Maximize z = 400W + 300P Can you see what the optimal solution will be?

14 How do we find an optimal solution? Maximize z = 400W + 300P What characterizes the optimal solution? What is the optimal solution vector? W = ? P = ? What is its solution value? z = ?

15 Optimal Solution S Binding constraints Maximize z = 400W + 300P 1.5W + P ≤15.4W +.5P ≤5 plus other constraints A constraint is said to be binding if it holds with equality at the optimum solution. Other constraints are non-binding

16 How do we find an optimal solution? Optimal solutions occur at extreme points. In two dimensions, this is the intersection of 2 lines. Maximize z = 400W + 300P 1.5W + P ≤15.4W +.5P ≤5 Solution:.7W = 5, W = 50/7 P = /7 = 30/7 z = 29,000/7 = 4,142 6/7

17 Finding an optimal solution in two dimensions: Summary The optimal solution (if one exists) occurs at a “corner point” of the feasible region. In two dimensions with all inequality constraints, a corner point is a solution at which two (or more) constraints are binding. There is always an optimal solution that is a corner point solution (if a feasible solution exists). More than one solution may be optimal in some situations

18 Preview of the Simplex Algorithm In n dimensions, one cannot evaluate the solution value of every extreme point efficiently. (There are too many.) The simplex method finds the best solution by a neighborhood search technique. Two feasible corner points are said to be “adjacent” if they have one binding constraint in common.

19 Preview of the Simplex Method Maximize z = 400W + 300P Start at any feasible extreme point. Move to an adjacent extreme point with better objective value. Continue until no adjacent extreme point has a better objective value.

20 Preview of Sensitivity Analysis Suppose the plier demand is decreased to 10 -∆. What is the impact on the optimal solution value? Theshadow price of a constraint is the unit increase in the optimal objective value per unit increase in the RHS of the constraint. Changing the RHS of a non- binding constraint by a small amount has no impact. The shadow price of the constraint is 0.

21 Preview of Sensitivity Analysis Suppose slightly more steel is available? 1.5W + P ≤15 +∆ What is the impact on the optimal solution value?

22 Shifting a Constraint Steel is increased to 15 + ∆. What happens to the optimal solution? What happens to the optimal solution value?

23 Shifting a Constraint Steel is increased to 15 + ∆. What happens to the optimal solution? What happens to the optimal solution value?

24 Finding the New Optimum Solution Maximize z = 400W + 300P Binding Constraints: Solution: 1.5W + P = 15 + ∆.4W +.5P = 5 W = 50/7 +(10/7)∆ P= 30/7 -8/7∆ z = 29,000/7 +(1,600/7)∆ Conclusion: If the amount of steel increases by ∆units (for sufficiently small∆) then the optimal objective value increases by(1,600/7)∆. The shadow price of a constraint is the unit increase in the optimal objective value per unit increase in the RHS of the constraint. Thus the shadow price of steel is 1,600/7 = 228 4/7.

25 Some Questions on Shadow Prices Suppose the amount of steel was decreased by ∆units. What is the impact on the optimum objective value? How large can the increase in steel availability be so that the shadow price remains as 228 4/7? Suppose that steel becomes available at $1200 per ton. Should you purchase the steel? Suppose that you could purchase 1 ton of steel for $450. Should you purchase the steel? (Assume here that this is the correct market value for steel.)

26 Shifting a Constraint Steel is increased to 15 + ∆. What happens to the optimal solution? Recall that W <= 8. The structure of the optimum solution changes when ∆=.6, and W is increased to 8

27 Shifting a Cost Coefficient The objective is: Maximize z = 400W + 300P What happens to the optimal solution if 300P is replaced by (300+  )P How large can  be for your answer to stay correct?.4W +.5P = 5

28 Summary: 2D Geometry helps guide the intuition The Geometry of the Feasible Region –Graphing the constraints Finding an optimal solution –Graphical method –Searching all the extreme points –Simplex Method Sensitivity Analysis –Changing the RHS –Changing the Cost Coefficients