Linear Programming

Network Flow Problems Transportation Assignment Transshipment Production and Inventory

Linear Programming

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

Linear Programming

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

Linear Programming

Define Variables - BBC

Let:

xij = # of units shipped from Plant i to Destination j

Linear Programming

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

Linear Programming

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

Linear Programming

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

Linear Programming

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

Linear Programming

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

Linear Programming

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

Linear Programming

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

Linear Programming

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

Linear Programming

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

Linear Programming

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

Linear Programming

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

Linear Programming

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

Linear Programming

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

Linear Programming

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

Linear Programming

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

Linear Programming

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

Linear Programming

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

Linear Programming

Network Flow Problems Transportation Assignment Transshipment Production and Inventory

Linear Programming

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

Linear Programming

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

Linear Programming

Define Variables - ABC

Let:

xij = 1 if contractors i is assigned to Project j and equals zero if not assigned

Linear Programming

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

Linear Programming

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

Linear Programming

Network Flow Problems Transportation Assignment Transshipment Production and Inventory

Linear Programming

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.

Linear Programming

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

Linear Programming

Define Variables - Transshipment

Let:

xij = # of units shipped from node i to node j

Linear Programming

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

Linear Programming

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

Linear Programming

Network Flow Problems Transportation Assignment Transshipment Production and Inventory

Linear Programming

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

Linear Programming

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

Linear Programming

Define Variables - Inventory

Let:

xij = # of units flowing from node i to node j

Linear Programming

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