Lecture 15 Minimum Cost Network Flows

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Lecture 15 Minimum Cost Network Flows

• Alg 10B On Page 527 Network Simplex
• Example Requirements At Nodes 1,2,3 5,0,-5

Node 1 has a supply of 5, Node 3 has a demand of 5
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Graphical Representation
5
0
0,1,4
1
2
flow,cost, bound
4,3,4
0,1,2
1,10,10
requirement
3
-5
Step 0. Feasible Flow 0, 0, 4, 1 with obj value
22 Bold arcs are basic.
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Basis Spanning Tree
1
2
3
Nonbasic arcs are 2 (2,3) and 3 (1,3) Flow on
nonbasics must be either 0 or bound!
4
Current Solution
Flow 0
5
1
2
Flow 4
Flow 0
Flow 1
3
-5
5
Step 1. Add nonbasic (2,3) to the basis tree
Cost for this direction 11-10 -8 implies an
improving direction.
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The Improving Simplex Cycle
0,1,4
1
2
flow, cost, bound
0,1,2
1,10,10
3
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Step 3. Step Size

• Flow is increasing in (1,2) max increase is
  (4-0) 4
• Flow is increasing in (2,3) max increase is
  (2-0) 2
• Flow is decreasing in (1,3) max decrease 1
• Step size ? min 4, 2, 1 1

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Step 4. Advance
Current obj value 111214 an improvement.
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Step 5 New Basis
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New Basis Spanning Tree
Flow 1
5
1
2
Where is the rest of the flow needed to satisfy
the demand of 5?
Flow 1
3
-5
11
Nonbasic arcs have flow of 0 or bound
Flow 1
5
1
2
Satisfies flow conservation at every node
Flow 1
Flow 4
3
nonbasic
-5
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Step 1- Iteration 2 decrease flow in arc 3
Cost 11-3 -1, hence an improving direction
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3. Step Size
Flow 1
1
2
Flow 1
Flow 4
3
For (1,2) max increase 4-1 3 For (2,3) max
increase 2-1 1 For (1,3) max decrease 4
? 1
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Step 4 Advance
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Step 5 New Basis
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Step 1 Simplex Direction
Flow 2
5
1
2
Where is the rest of the flow that is needed to
satisfy flow conservation at nodes 2 and 3?
Flow 3
3
-5
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Check Feasibility
Flow 2
Nonbasic arc with flow at bound
1
2
Flow 2
Flow 3
Flow conservation is satisfied.
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Obj value cost 22913, an improvement.
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Two Possible Cycles
Flow 2
1
Flow 2
2
1
2
Flow 0
Flow 3
Flow 2
Flow 3
3
3
Cycle 1-3-1 Cost 10-37 not improving
Cycle 3-2-1-3 Cost -1-131 not improving
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Optimal Solution Obtained
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Modeling Fixed Charges Page 558
cost
0, if x 0 Cost fcx, otherwise
c
1
f
x
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Network Problem With Fixed Costs

• Consider a network model with fixed costs on some
  of the arcs

Cost functions g1(x1) 0, if x10 and
105x1, otherwise g2(x2) 0, if x20 and
20x2, otherwise g3(x3) 3x3
0
5
-5
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The Cost Functions

g1(x1)
g2(x2)
5
1
1
10
1
20
x1
x2
g3(x3)
3
x3
1
0
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Model Of Fixed Charge Network Problem

• var x1 gt 0 var x2 gt 0 var x3 gt 0, lt 3
• var y1 binary var y2 binary
• subject to Node1 x1 x3 5
• subject to Node2 -x1 x2 0
• subject to Node3 -x2 -x3 -5
• subject to FCx1 x1 lt 100y1
• subject to FCx2 x2 lt 100y2
• minimize FixedCosts
• 10y1 5x1 20y2 x2 3x3

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Optimal Solution
0
Cost x1 10(2)(5) 20 x2 20(2)(1) 22
x3 (3)(3) 9 Total 51
x12
2
5
1
x22
3
x33
-5
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Chapter 11 Discrete Optimization Models

• 11.4 page 561 A knapsack model.
• An Integer Linear Program (ILP) with one
  constraint.
• max cx
• s. t. ax lt b
• xj binary for all j This problem can be
  very hard.
• How many possible solutions for a problem with
  100 variables?
• 2100 is about
  100010 1030

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Example of a knapsack problem

• var x1 binary var x2 binary var x3 binary var
  x4 binary
• var x5 binary var x6 binary var x7 binary var
  x8 binary
• var x9 binary var x10 binary
• maximize benefit
• 8x12x24x31x44x57x62x71x85x95x10
  
• subject to knapsack
• 9x14x22x37x43x59x69x77x83x99x10
  
• lt 47
• What is 210? 1024

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Solution Obtained using AMPL and CPLEX

• Optimal Objective 36
• Include items 1,2,3,4,5,6,9,10 in the knapsack.
• Omit 7 and 8 only.
• What is the bang/buck for items 7 and 8?
• 7 2/9 8 1/7
• What about others 1 8/9 2 2/4 3 4/2 4 1/7

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Mutually Exclusive Constraints

• Select at most one from the set M
• sum j in M xj lt 1
• x1 x4 x27 x54 lt 1
• all binary var
• Select exactly one from the set M
• sum j in M xj 1
• Select at least one from the set M
• sum j in M xj gt 1

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Dependancies

• Example 1 Select 5 only if 3 is selected
• x5 lt x3
• Example 2 NASA Problem - If you do any of the
  missions 4 thru 7, then you must do mission 3.
• x4 lt x3
• x5 lt x3
• x6 lt x3
• x7 lt x3

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Dependancies Example 3

• You only have to do mission 3 if you do all
  missions 4 thru 7.

x4x5x6x7 lt 3 x3
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Dependancies Example 4

• You must do mission 3 if and only if you do all
  missions 4 thru 7.

x4x5x6x7 lt 3 x3 x3 lt x4 x3 lt x5 x3 lt
x6 x3 lt x7
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The End