Transportation III Transportation Management Lecture 17 ESD.260 Fall 2003 Caplice.
Lecture 15 Transportation Algorithm - Image Formation and...
Transcript of Lecture 15 Transportation Algorithm - Image Formation and...
Lecture 15
Transportation Algorithm
October 14, 2009
Lecture 15
Outline
• Recap the last lecture
• Selection of the initial basic feasible solution
� Northwest-corner method
• Computing reduced costs of nonbasic variables
� Thorugh the use of shadow prices
• Basis change
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Lecture 15
Sun-Ray Transportation ModelGrain from Silos to Mills - Balanced problem
Mill 1 Mill 2 Mill 3 Mill 4 Supply
Silo 1 10 2 20 11 15
Silo 2 12 7 9 20 25
Silo 3 4 14 16 18 10
Demand 5 15 15 15
Northwest-corner method
Mill 1 Mill 2 Mill 3 Mill 4
Silo1 10 x11 = 5 2 x12 = 10 20 11
Silo 2 12 7 x22 = 5 9 x23 = 15 20 x24 = 5
Silo 3 4 14 16 18 x34 = 10
Working with the simplex method would require 12 variables, of which 6
are basic variables. We resort to a more compact representation:
- the use of the preceding table.
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Simplex Method
Having the initial table (with initial basic feasible solution), we perform the
typical simplex iteration
Step 1 Reduced Cost Computation
Compute the reduced costs of the nonbasic variables
Step 2 Optimality Check
Looking at the reduced cost values, we check the optimality
• In the transportation minimization problem: optimality requires non-
negative reduced cost
Step 3 Basis Change
If not optimal, we perform change of basis and update the table
• In the transportation minimization problem: solution is not optimal
when reduced cost is positive for some nonbasic variable
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Reduced Cost Computation
We do not have Basis Inverse, so we have to rely on the dual problem and
the fact that the reduced costs of the basic variables are zero
We use shadow prices - hence, we need to look at the dual of the
transportation problem
minimizem∑
i=1
n∑j=1
cijxij
subject ton∑
j=1
xij = bi for i = 1, . . . , m ←−(ui)
m∑i=1
xij = dj for j = 1, . . . , n ←−(vj)
xij ≥ 0 for all i, j
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The dual of the general transportation problem
maximizem∑
i=1
biui +n∑
j=1
djvj
subject to ui + vj ≤ cij for all (i, j)− pairs (xij)
• Reduced cost c̄ij of variable xij is given by
c̄ij = ui + vj − cij
where cij is the original cost of the variable xij as given in the objective
of the transportatation problem
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Step 1: Computing the reduced costs of nonbasic
variables
1. Determine the shadow prices of the problem corresponding to the current
basic feasible solution, using the fact that the reduced costs of the basic
variables are 0, i.e.
c̄ij = ui + vj − cij = 0 for all basic variables xij
• We have m + n unknowns ui, vj, and m + n− 1 equations
• One degree of freedom: set u1 = 0 (or other than u1 variable)
2: Using the shadow prices determined in item 1, compute the reduced
costs of the nonbasic variables
c̄ij = ui + vj − cij for all nonbasic variables xij
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Back to Sun-Ray Case: Reduced cost
Mill 1 Mill 2 Mill 3 Mill 4
Silo1 10 x11 = 5 2 x12 = 10 20 11
Silo 2 12 7 x22 = 5 9 x23 = 15 20 x24 = 5
Silo 3 4 14 16 18 x34 = 10
1. Determining the shadow prices from c̄ij = 0 for currently basic variables
(with u1 = 0)
x11 u1 + v1 = 10 & u1 = 0 −→ v1 = 10
x12 u1 + v2 = 2 u1 = 0 −→ v2 = 2
x22 u2 + v2 = 7 v2 = 2 −→ u2 = 5
x23 u2 + v3 = 9 u2 = 5 −→ v3 = 4
x24 u2 + v4 = 20 u2 = 5 −→ v4 = 15
x34 u3 + v4 = 18 v4 = 15 −→ u3 = 3
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2. Using the shadow prices, compute the reduced costs c̄ij for nonbasic
variables
c̄ij = ui + vj − cij
Mill 1 Mill 2 Mill 3 Mill 4
Silo1 10 x11 = 5 2 x12 = 10 20 11
Silo 2 12 7 x22 = 5 9 x23 = 15 20 x24 = 5
Silo 3 4 14 16 18 x34 = 10
x13 c̄13 = u1 + v3 − c13 = 0 + 4− 20 = −16
x14 c̄14 = u1 + v4 − c14 = 0 + 15− 11 = 4x21 c̄21 = u2 + v1 − c21 = 5 + 10− 12 = 3x31 c̄31 = u3 + v1 − c31 = 3 + 10− 4 = 9x32 c̄32 = u3 + v2 − c32 = 3 + 2− 14 = −9
x33 c̄33 = u3 + v3 − c33 = 3 + 4− 16 = −9
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Step 2: Optimality Check
x13 c̄13 = −16
x14 c̄14 = 4x21 c̄21 = 3x31 c̄31 = 9x32 c̄32 = −9
x33 c̄33 = −9
We can choose any of x14, x21, and x31.
We may choose the one with the largest reduced cost - x31.
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Simplex Method
Having the initial table (with initial basic feasible solution), we perform the
typical simplex iteration
Step 1 Reduced Cost Computation (DONE)
Compute the reduced costs of the nonbasic variables
Step 2 Optimality Check (DONE)
Looking at the reduced cost values, we check the optimality
• In the transportation minimization problem: optimality requires non-
negative reduced cost
Step 3 Basis Change (We are HERE, x31 enters the basis)
If not optimal, we perform change of basis and update the table
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Step 3: SunRay Basis Change
Mill 1 Mill 2 Mill 3 Mill 4
Silo1 10 x11 = 5 2 x12 = 10 20 11
Silo 2 12 7 x22 = 5 9 x23 = 15 20 x24 = 5
Silo 3 4 14 16 18 x34 = 10
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Figure 1: Graph representation of the current basic faesible solution
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Figure 2: x31 entering the current basis. Flow push along a cycleformed by arcs of the old solution and the new arc (3,1) of thevariable entering the basis
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Figure 3: Table representation of the cycle and the flow push.
The maximum flow that can be send along the cycle cannot exceed the
amount of the flow on the backward traversed arcs (corresponds to removal
of the existing flow on these arcs):
θ − 5 ≥ 0, θ − 5 ≥ 0, θ − 10 ≥ 0 =⇒ θ = 5
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Figure 4: The resulting table after sending 5 units of flow along thecycle. There are two variables that can leave the basis: those whoseflow dropped to 0. Thus, either x11 or x22 may leave the basis.Suppose we choose x11 to leave.
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Figure 5: The resulting basic feasible solution after x11 left the basis.
Now we have completed a simplex iteration.
We go to the next iteration:
We repeat steps 1,2,3 for this basic solution.
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