The Geometry of Linear Programs the geometry of LPs illustrated on GTC Handouts: Lecture Notes

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The Geometry of Linear Programs– the geometry of LPs illustrated on GTC

Handouts: Lecture Notes

15.053 February 5, 2002

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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 proble

m

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On the formulation

Choose decision variables. – Let xj= the number of footballs produced in m

onth j (in 1000s)

– Let yj= the number of footballs held in inventory from year j to year j + 1. (in 1000s)

– y0= 5

Then write the constraints and the objective.

Pigskin Spreadsheet

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Data for the GTC Problem

Wrenches Pliers Available

Steel 1.5 1.0 15,000 tons

Molding Machine

1.0 1.0 12,000 hrs

Assembly Machine

.4 .5 5,000 hrs

Demand Limit 8,000 10,000

Contribution

($ per item)

$.40 $.30

Want to determine the number of wrenches and pliersto produce given the available raw materials, machinehours and demand.

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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 + 400 W

P + 1.5 W ≤15

P + W ≤12

0.5 P + 0.4 W ≤5

P ≤10

W ≤8

P,W ≥0

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Graphing the Feasible Region

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

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Graph the Constraint:

P + 1.5W ≤15

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W + P ≤12Graph the Constraint:

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.4W + .5P ≤5Graph the Constraint:

What happened to the constraint :W + P ≤12?

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W ≤8, and P≤10Graph the Constraint:

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How do we find an optimal solution?

Maximize z = 400W + 300P

Is there a feasible solution with z = 1200?

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Can you see what the optimal solution will be?

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What characterizes the optimal solution?

What is the optimal solution vector? W = ? P = ?

What is its solution value? z = ?

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Optimal Solution SBinding constraints


1.5W + P ≤15

.4W + .5P ≤5

plus other constraints

A constraint is said to be bindingif it holds with equality at the optimum solution.

Other constraints are non-binding

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How do we find an optimal solution?Optimal solutions occur at extreme points. In two dimensions, this is the intersection of 2 lines.


1.5W + P ≤15

.4W + .5P ≤5

Solution:

.7W = 5, W = 50/7

P = 15 -75/7 = 30/7

z = 29,000/7 = 4,142 6/7

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Finding an optimal solution in two dimensions: SummaryThe optimal solution (if one exists) occurs at

a “corner point” of the feasible region.In two dimensions with all inequality constrai

nts, 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

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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.

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Preview of the Simplex Method


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.

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Preview of Sensitivity AnalysisSuppose 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.

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Preview of Sensitivity Analysis

Suppose slightly more steel is available? 1.5W + P ≤15 +∆

What is the impact on the optimal solution value?

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Shifting a Constraint

Steel is increased to 15 + ∆.

What happens to the optimal solution?

What happens to the optimal solution value?

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What happens to the optimal solution value?

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Finding the New Optimum SolutionMaximize 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.

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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.)

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Recall that W <= 8.

The structure of the optimum solution changes when ∆= .6, and W is increased to 8

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Shifting a Cost Coefficient

The objective is:


What happens to the optimal solution if 300P is replaced by (300+δ)P

How large can δbe for your answer to stay correct?

.4W + .5P = 5

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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

The Geometry of Linear Programs the geometry of LPs illustrated on GTC Handouts: Lecture Notes

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