15.053
February 5, 2002
 The Geometry of Linear Programs
– the geometry of LPs illustrated on GTC
Handouts: Lecture Notes
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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: $.60 per football per month
 With your partner: write an LP to describe the
problem
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On the formulation
 Choose decision variables.
– Let xj= the number of footballs produced in
month j (in 1000s)
– Let yj= the number of footballs held in
inventory from year j to year j + 1. (in 1000s)
– y0= 5
 Then write the constraints and the
objective.
Pigskin Spreadsheet
3
Data for the GTC Problem
Wrenches Pliers
Available
Steel
1.5
1.0
15,000 tons
Molding
Machine
1.0
1.0
12,000 hrs
Assembly
Machine
.4
.5
5,000 hrs
Demand Limit
8,000
10,000
Contribution
($ per item)
$.40
$.30
Want to determine the number of wrenches and pliers
to produce given the available raw materials, machine
hours 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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Graphing the Feasible Region
Graph the Constraint:
P + 1.5W ≤15
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Graphing the Feasible Region
Graph the Constraint:
W + P ≤12
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Graphing the Feasible Region
Graph the Constraint:
.4W + .5P ≤5
What happened to the
constraint :
W + P ≤12?
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Graphing the Feasible Region
Graph the Constraint:
W ≤8, and P≤10
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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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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?
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How do we find an optimal solution?
Maximize z = 400W + 300P
Can you see what the
optimal solution will be?
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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=?
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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
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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.
Maximize z = 400W + 300P
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: 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
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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
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.
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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 nonbinding 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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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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Finding the New Optimum Solution
Maximize
z = 400W + 300P
Binding
Constraints:
1.5W + P = 15 + ∆
.4W + .5P = 5
W = 50/7 +(10/7)∆
Solution:
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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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
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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
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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
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