Transportation

1
Chapter 5

• Transportation
• Assignment
• Transshipment
• Inventory

2
Network Flow Problems

• Transportation
• Assignment
• Transshipment
• Production and Inventory

3
Network Flow Problems

• Different authors hold different conventions.
• For example, I suggest the convention of
  denominating supplies as negative values (in this
  case, all constraints are greater-than-or-equal).
• Conversely, supplies can be denominated as
  positive values in which case their constraints
  become less-than-or-equals.
• Some indicate demands as equal constraints.
  While this is conceptually correct, the logical
  equivalent for a cost minimization problem is
  greater-than-or-equal. Do you see why?
• The bottom line is that there are often multiple
  ways to correctly model a business situation. It
  is up to you to understand the underlying logic
  so that valid interpretations of the results can
  be made.

4
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

5
Network Flow Problems - Transportation

• Building Brick Company (BBC) manufactures bricks.
  One of BBCs 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?

6
Network Representation - BBC
Destinations
Transportation Cost per Unit
Plants (Origin Nodes)
1 Northwood
1 Plant 1
24
25
50
30
40
2 Westwood
45
30
2 Plant 2
40
50
42
3 Eastwood
10
Distribution Routes - arcs
Demand
Supply
7
Define Variables - BBC

• Let
• xij of units shipped from Plant i to
  Destination j

8
General Form - BBC

• Min
• 24x11 30x12 40x13 30x21 40x22 42x23
• s.t.
• 1x11 1x12 1x13 0x21 0x22
  0x23 lt 50
• 0x11 0x12 0x13 1x21 1x22
  1x23 lt 50
• 1x11 0x12 0x13 1x21 0x22
  0x23 gt 25
• 0x11 1x12 0x13 0x21 1x22
  0x23 gt 45
• 0x11 0x12 1x13 0x21 0x22
  1 x23 gt10
• xij gt 0 for i 1, 2 and j 1, 2, 3

Plant 1 Supply
Plant 2 Supply
North Demand
West Demand
East Demand
9
Network Flow Problems

• Transportation
• Assignment
• Transshipment
• Production and Inventory

10
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

11
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)?

12
Network Representation - ABC
Transportation Distance
Contractors (Origin Nodes)
Electrical Jobs (Destination Nodes)
1 West
1 A
50
1
36
1
16
2 Fed
28
1
30
2 B
18
1
35
3 Goliath
32
1
20
3 C
25
25
1
4 Univ.
14
1
Possible Assignments - arcs
Demand
Supply
13
Define Variables - ABC

• Let
• xij 1 if contractor i is assigned to Project j
  and equals zero if not assigned

14
General Form - ABC

• Min
• 50x1136x1216x1328x2130x2218x2335x3132x3220
  x3325x4125x4214x43
• s.t.
• 1x11 1x12 1x13 0x21 0x22 0x23
  0x31 0x32 0x33 0 x41 0x42 0x43 lt1
• 0x11 0x12 0x13 1x21 1x22 1x23
  0x31 0x32 0x33 0 x41 0x42 0x43 lt1
• 0x11 0x12 0x13 0x21 0x22 0x23
  1x31 1x32 1x33 0 x41 0x42 0x43 lt1
• 0x11 0x12 0x13 0x21 0x22 0x23
  0x31 0x32 0x33 1x41 1x42 1x43 lt1
• 1x11 0x12 0x13 1x21 0x22 0x23
  1x31 0x32 0x33 1x41 0x42 0x43 gt1
• 0x11 1x12 0x13 0x21 1x22 0x23
  0x31 1x32 0x33 0 x41 1x42 0x43 gt1
• 0x11 0x12 1x13 0x21 0x22 1x23
  0x31 0x32 1x33 0 x41 0x42 1x43 gt1
• xij gt 0 for i 1, 2, 3, 4 and j 1, 2, 3

15
Network Flow Problems

• Transportation
• Assignment
• Transshipment
• Production and Inventory

16
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

17
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

• The costs (per unit) to ship the shelving
  from the suppliers to the final destinations are

• Formulate a linear programming model which
  will minimize total shipping costs for all
  parties.

18
Network Representation - Transshipment
Transportation Cost per Unit
Transportation Cost per Unit
Retail Outlets (Destinations Nodes)
Warehouses (Transshipment Nodes)
Plants (Origin Nodes)
5 Zrox
3 Thomas
1 Arnold
1
50
5
75
5
8
8
6 Hewes
Flow In 150
Flow Out 150
60
3
7
4 Washburn
4
2 Super S.
4
75
4
7 Rockwright
40
Distribution Routes - arcs
Demand
Supply
19
Define Variables - Transshipment

• Let
• xij of units shipped from node i to node j

20
General Form TransshipmentShowing supplies as
negative values

• Min
• 5x138x147x234x241x355x368x373x454x464x47
• s.t.
• 1x13 1x14 0x23 0x24 0x35 0x36 0x37
  0x45 0x46 0x47 gt -75
• 0x13 0x14 1x23 1x24 0x35 0x36 0x37
  0x45 0x46 0x47 gt -75
• -1x13 0x14 - 1x23 0x24 1x35 1x36
  1x37 0x45 0x46 0x47 0
• 0x13 - 1x14 0x23 - 1x24 0x35 0x36
  0x37 1x45 1x46 1x47 0
• 0x13 0x14 0x23 0x24 1x35 0x36 0x37
  1x45 0x46 0x47 50
• 0x13 0x14 0x23 0x24 0x35 1x36
  0x37 0x45 1x46 0x47 60
• 0x13 0x14 0x23 0x24 0x35 0x36
  1x37 0x45 0x46 1x47 40
• xij gt 0 for all i and j

Flow In 150
Flow Out 150
21
General Form TransshipmentShowing supplies as
positive values

• Min
• 5x13 8x14 7x23 4x24 1x35 5x36 8x37
  3x45 4x46 4x47
• s.t.
• 1x13 1x14 0x23 0x24 0x35 0x36 0x37
  0x45 0x46 0x47 lt 75
• 0x13 0x14 1x23 1x24 0x35 0x36 0x37
  0x45 0x46 0x47 lt 75
• -1x13 0x14 - 1x23 0x24 1x35 1x36
  1x37 0x45 0x46 0x47 0
• 0x13 - 1x14 0x23 - 1x24 0x35 0x36
  0x37 1x45 1x46 1x47 0
• 0x13 0x14 0x23 0x24 1x35 0x36 0x37
  1x45 0x46 0x47 50
• 0x13 0x14 0x23 0x24 0x35 1x36
  0x37 0x45 1x46 0x47 60
• 0x13 0x14 0x23 0x24 0x35 0x36
  1x37 0x45 0x46 1x47 40
• xij gt 0 for all i and j

Flow In 150
Flow Out 150
22
Network Flow Problems

• Transportation
• Assignment
• Transshipment
• Production and Inventory

23
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.

24
Network Representation Production and Inventory
Production Nodes
Demand Nodes
Production Costs
1 Week 1
5 Week 1
28
60
58
Inventory Costs
0.03
2 Week 2
6 Week 2
27
62
36
0.03
3 Week 3
7 Week 3
26
64
52
0.03
4 Week 4
8 Week 4
29
66
70
Production - arcs
Production Capacity
Demand
25
Define Variables - Inventory

• Let
• xij of units flowing from node i to node j

26
General Form - Production and Inventory

• Min
• 28x1527x2626x3729x48.03x56.03x67.03x78
• s.t.
• x15
  
  lt 60
• x26
  
  lt 62
• x37
  
  lt 64
• x48
  lt
  66
• x15
  
  58x56
• x26
  x56
  36x67
• x37
  x67
  52x78
• x48
  x78 70
• xij gt 0 for all i and j