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MT 235 3
Network Flow Problems - Transportation
Building Brick Company (BBC) manufactures bricks. One of BBC’s main concerns is transportation costs which are a very significant percentage of total costs. BBC has orders for 80 tons of bricks at three suburban locations as follows: Northwood – 25 tons Westwood – 45 tons Eastwood – 10 tons
BBC has two plants, each of which can produce 50 tons per week. BBC would like to minimize transportation costs. How should end of week
shipments be made to fill the above orders given the following delivery cost per ton?
$/ton Northwood Westwood Eastwood
Plant 1 24 30 40
Plan 2 30 40 42
MT 235 4
Network Representation - BBC
1Northwood
2Westwood
3Eastwood
1Plant 1
2Plant 2
50
50
25
45
10
Plants(Origin Nodes)
DestinationsTransportationCost per Unit
Distribution Routes - arcs DemandSupply
$24
$30
$40
$30
$40
$42
MT 235 6
General Form - BBC
Min
24x11+30x12+40x13+30x21+40x22+42x23
s.t.
x11 +x12 +x13 <= 50
x21 +x22+ x23 <= 50
x11 + x21 = 25
x12 + x22 = 45
x13 + x23 = 10
xij >= 0 for i = 1, 2 and j = 1, 2, 3
Plant 1 Supply
Plant 2 Supply
North Demand
West Demand
East Demand
MT 235 11
Network Flow Problems Transportation Problem Variations
Total supply not equal to total demand Total supply greater than or equal to total demand Total supply less than or equal to total demand
Maximization/ minimization Change from max to min or vice versa
Route capacities or route minimums Unacceptable routes
MT 235 12
Network Flow Problems - Transportation
Building Brick Company (BBC) manufactures bricks. One of BBC’s main concerns is transportation costs which are a very significant percentage of total costs. BBC has orders for 80 tons of bricks at three suburban locations as follows: Northwood – 25 tons Westwood – 45 tons Eastwood – 10 tons
BBC has two plants, each of which can produce 50 tons per week. BBC would like to minimize transportation costs. How should end of week
shipments be made to fill the above orders given the following delivery cost per ton?
Suppose demand at Eastwood grows to 50 tons.$/ton Northwood Westwood Eastwood
Plant 1 24 30 40
Plan 2 30 40 42
MT 235 13
Network Representation - BBC
1Northwood
2Westwood
3Eastwood
1Plant 1
2Plant 2
50
50
25
45
10
Plants(Origin Nodes)
DestinationsTransportationCost per Unit
Distribution Routes - arcs DemandSupply
$24
$30
$40
$30
$40
$42
50
MT 235 14
General Form - BBC
Min
24x11+30x12+40x13+30x21+40x22+42x23
s.t.
x11 +x12 +x13 <= 50
x21 +x22+ x23 <= 50
x11 + x21 = 25
x12 + x22 = 45
x13 + x23 = 10
xij >= 0 for i = 1, 2 and j = 1, 2, 3
Plant 1 Supply
Plant 2 Supply
North Demand
West Demand
East Demand
Min
24x11+30x12+40x13+30x21+40x22+42x23
s.t.
x11 +x12 +x13 = 50
x21 +x22+ x23 = 50
x11 + x21 <= 25
x12 + x22 <= 45
x13 + x23 <= 50
xij >= 0 for i = 1, 2 and j = 1, 2, 3
MT 235 19
Network Flow Problems Transportation Problem Variations
Total supply not equal to total demand Total supply greater than or equal to total demand Total supply less than or equal to total demand
Maximization/ minimization Change from max to min or vice versa
Route capacities or route minimums Unacceptable routes
MT 235 20
Network Flow Problems - Transportation
Building Brick Company (BBC) manufactures bricks. BBC has orders for 80 tons of bricks at three suburban locations as follows: Northwood – 25 tons Westwood – 45 tons Eastwood – 10 tons
BBC has two plants, each of which can produce 50 tons per week. BBC would like to maximize profit. How should end of week shipments be
made to fill the above orders given the following profit per ton?
$/ton Northwood Westwood Eastwood
Plant 1 24 30 40
Plan 2 30 40 42
MT 235 21
Network Representation - BBC
1Northwood
2Westwood
3Eastwood
1Plant 1
2Plant 2
50
50
25
45
10
Plants(Origin Nodes)
DestinationsTransportationCost per Unit
Distribution Routes - arcs DemandSupply
$24
$30
$40
$30
$40
$42
Profitper Unit
MT 235 22
General Form - BBC
Min
24x11+30x12+40x13+30x21+40x22+42x23
s.t.
x11 +x12 +x13 <= 50
x21 +x22+ x23 <= 50
x11 + x21 = 25
x12 + x22 = 45
x13 + x23 = 10
xij >= 0 for i = 1, 2 and j = 1, 2, 3
Plant 1 Supply
Plant 2 Supply
North Demand
West Demand
East Demand
Max
MT 235 27
Network Flow Problems Transportation Problem Variations
Total supply not equal to total demand Total supply greater than or equal to total demand Total supply less than or equal to total demand
Maximization/ minimization Change from max to min or vice versa
Route capacities or route minimums Unacceptable routes
MT 235 28
Network Flow Problems - Transportation
Building Brick Company (BBC) manufactures bricks. One of BBC’s main concerns is transportation costs which are a very significant percentage of total costs. BBC has orders for 80 tons of bricks at three suburban locations as follows: Northwood – 25 tons Westwood – 45 tons Eastwood – 10 tons
BBC has two plants, each of which can produce 50 tons per week. BBC would like to minimize transportation costs. How should end of week
shipments be made to fill the above orders given the following delivery cost per ton?
BBC has just been instructed to deliver at most 5 tons of bricks to Eastwood from Plant 2.
$/ton Northwood Westwood Eastwood
Plant 1 24 30 40
Plan 2 30 40 42
MT 235 29
Network Representation - BBC
1Northwood
2Westwood
3Eastwood
1Plant 1
2Plant 2
50
50
25
45
10
Plants(Origin Nodes)
DestinationsTransportationCost per Unit
Distribution Routes - arcs DemandSupply
$24
$30
$40
$30
$40
$42
At most 5 tons Delivered from Plant 2
MT 235 30
General Form - BBCMin24x11+30x12+40x13+30x21+40x22+42x23
s.t. x11 +x12 +x13 <= 50
x21 +x22+ x23 <= 50 x11 + x21 = 30 x12 + x22 = 45 x13 + x23 = 10
x23 <= 5
xij >= 0 for i = 1, 2 and j = 1, 2, 3
Plant 1 Supply
Plant 2 Supply
North Demand
West Demand
East Demand
Route Max
MT 235 35
Network Flow Problems Transportation Problem Variations
Total supply not equal to total demand Total supply greater than or equal to total demand Total supply less than or equal to total demand
Maximization/ minimization Change from max to min or vice versa
Route capacities or route minimums Unacceptable routes
MT 235 36
Network Flow Problems - Transportation
Building Brick Company (BBC) manufactures bricks. One of BBC’s main concerns is transportation costs which are a very significant percentage of total costs. BBC has orders for 80 tons of bricks at three suburban locations as follows: Northwood – 25 tons Westwood – 45 tons Eastwood – 10 tons
BBC has two plants, each of which can produce 50 tons per week. BBC would like to minimize transportation costs. How should end of week
shipments be made to fill the above orders given the following delivery cost per ton?
BBC has just learned the route from Plant 2 to Eastwood is no longer acceptable.
$/ton Northwood Westwood Eastwood
Plant 1 24 30 40
Plan 2 30 40 42
MT 235 37
Network Representation - BBC
1Northwood
2Westwood
3Eastwood
1Plant 1
2Plant 2
50
50
25
45
10
Plants(Origin Nodes)
DestinationsTransportationCost per Unit
Distribution Routes - arcs DemandSupply
$24
$30
$40
$30
$40
$42
Route no longeracceptable
MT 235 38
General Form - BBC
Min
24x11+30x12+40x13+30x21+40x22+42x23
s.t.
x11 +x12 +x13 <= 50
x21 +x22+ x23 <= 50
x11 + x21 = 30
x12 + x22 = 45
x13 + x23 = 10
xij >= 0 for i = 1, 2 and j = 1, 2, 3
Plant 1 Supply
Plant 2 Supply
North Demand
West Demand
East Demandx13 = 10
24x11+30x12+40x13+30x21+40x22
x21 +x22 <= 50
MT 235 44
Network Flow Problems - Assignment
ABC Inc. General Contractor pays their subcontractors a fixed fee plus mileage for work performed. On a given day the contractor is faced with three electrical jobs associated with various projects. Given below are the distances between the subcontractors and the projects. How should the contractors be assigned to minimize total distance (and total cost)?
Project
Subcontractors A B C
Westside 50 36 16
Federated 28 30 18
Goliath 35 32 20
Universal 25 25 14
MT 235 45
Network Representation - ABC
1A
2B
3C
1West
2Fed
1
1
1
1
1
Contractors(Origin Nodes)
Electrical Jobs(Destination Nodes)
TransportationDistance
Possible Assignments - arcsDemandSupply
50
36
1628
3018
3Goliath
4Univ.
1
1
3532
2025
25
14
MT 235 46
Define Variables - ABC
Let:
xij = 1 if contractor i is assigned to Project j and equals zero if not assigned
MT 235 47
General Form - ABC
Min50x11+36x12+16x13+28x21+30x22+18x23+35x31+32x32+20x33+25x41+25x42+14x43
s.t. x11 +x12 +x13 <=1
x21 +x22 +x23 <=1
x31 +x32 +x33 <=1
x41 +x42 +x43 <=1
x11 +x21 +x31 +x41 =1 x12 +x22 +x32 +x42 =1 x13 +x23 +x33 +x43 =1
xij >= 0 for i = 1, 2, 3, 4 and j = 1, 2, 3
MT 235 48
Network Flow Problems Assignment Problem Variations
Total number of agents (supply) not equal to total number of tasks (demand)
Total supply greater than or equal to total demand Total supply less than or equal to total demand
Maximization/ minimization Change from max to min or vice versa
Unacceptable assignments
MT 235 54
Network Flow Problems - Transshipment Thomas Industries and Washburn Corporation supply three firms (Zrox, Hewes, Rockwright) with
customized shelving for its offices. Thomas and Washburn both order shelving from the same two manufacturers, Arnold Manufacturers and Supershelf, Inc.
Currently weekly demands by the users are: 50 for Zrox, 60 for Hewes, 40 for Rockwright.
Both Arnold and Supershelf can supply at most 75 units to its customers. Because of long standing contracts based on past orders, unit shipping costs from the manufacturers
to the suppliers are:
Thomas Washburn
Arnold 5 8
Supershelf 7 4
The costs (per unit) to ship the shelving from the suppliers to the final destinations are:
Zrox Hewes Rockwright
Thomas 1 5 8
Washburn 3 4 4
Formulate a linear programming model which will minimize total shipping costs for all parties.
MT 235 55
Network Representation - Transshipment
5Zrox
6Hewes
7Rockwright
3Thomas
4Washburn
75
75
50
60
40
Warehouses(Transshipment Nodes)
Retail Outlets(Destinations Nodes)
TransportationCost per Unit
Distribution Routes - arcs DemandSupply
$1
$5
$8
$3$4
$4
TransportationCost per Unit
1Arnold
2Super S.
$5
$8
$7
$4
Plants(Origin Nodes)
Flow In150
Flow Out150
Resembles Transportation Problem
MT 235 57
General Form - Transshipment
Min5x13+8x14+7x23+4x24+1x35+5x36+8x37+3x45+4x46+4x47
s.t. x13 +x14 <= 75
x23 +x24 <= 75
x35 +x36 +x37 = x13 +x23
x45 +x46 +x47 = x14 +x24
+x35 +x45 = 50 +x36 +x46 = 60 +x37 +x47 = 40
xij >= 0 for all i and j
Flow In150
Flow Out150
MT 235 58
Network Flow Problems Transshipment Problem Variations
Total supply not equal to total demand Total supply greater than or equal to total demand Total supply less than or equal to total demand
Maximization/ minimization Change from max to min or vice versa
Route capacities or route minimums Unacceptable routes
MT 235 64
Network Flow Problems – Production & Inventory A producer of building bricks has firm orders for the next four weeks.
Because of the changing cost of fuel oil which is used to fire the brick kilns, the cost of producing bricks varies week to week and the maximum capacity varies each week due to varying demand for other products. They can carry inventory from week to week at the cost of $0.03 per brick (for handling and storage). There are no finished bricks on hand in Week 1 and no finished inventory is required in Week 4. The goal is to meet demand at minimum total cost. Assume delivery requirements are for the end of the week, and assume carrying
cost is for the end-of-the-week inventory.
(Units in thousands) Week 1 Week 2 Week 3 Week 4
Delivery Requirements 58 36 52 70
Production Capacity 60 62 64 66
Unit Production Cost ($/unit) $28 $27 $26 $29
MT 235 65
Network Representation – Production and Inventory
1Week 1
62
Production Nodes Demand NodesProduction Costs
Production - arcs
DemandProductionCapacity
2Week 2
3Week 3
4Week 4
64
66
605
Week 1
6Week 2
7Week 3
8Week 4
36
52
70
58$28
$27
$26
$29
$0.03
$0.03
$0.03
InventoryCosts
MT 235 67
General Form - Production and Inventory
Min28x15+27x26+26x37+29x48+.03x56+.03x67+.03x78
s.t.
x15 <= 60
x26 <= 62
x37 <= 64
x48 <= 66
x15 = 58+x56
x26 +x56 = 36+x67
x37 +x67 = 52+x78
x48 +x78 = 70
xij >= 0 for all i and j