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Transportation, assignment, and Trasshipment

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Title: Transportation, assignment, and Trasshipment


1
Transportation, Assignment, and
Transshipment
Professor Ahmadi

2
Chapter 7Transportation, Assignment, and
Transshipment Problems
  • The Transportation Problem The Network Model and
    a Linear Programming Formulation
  • The Assignment Problem The Network Model and a
    Linear Programming Formulation
  • The Transshipment Problem The Network Model and
    a Linear Programming Formulation

3
Transportation, Assignment, and Transshipment
Problems
  • A network model is one which can be represented
    by a set of nodes, a set of arcs, and functions
    (e.g. costs, supplies, demands, etc.) associated
    with the arcs and/or nodes.
  • Transportation, assignment, and transshipment
    problems are all examples of network problems.

4
Transportation, Assignment, and Transshipment
Problems
  • Each of the three models of this chapter
    (transportation, assignment, and transshipment
    models) can be formulated as linear programs.
  • For each of the three models, if the right-hand
    side of the linear programming formulations are
    all integers, the optimal solution will be in
    terms of integer values for the decision
    variables.
  • These three models can also be solved using a
    standard computer spreadsheet package.

5
Transportation Problem
  • The transportation problem seeks to minimize the
    total shipping costs of transporting goods from m
    origins (each with a supply si) to n destinations
    (each with a demand dj), when the unit shipping
    cost from an origin, i, to a destination, j, is
    cij.
  • The network representation for a transportation
    problem with two sources and three destinations
    is given on the next slide.

6
Transportation Problem
  • Network Representation

1
d1
c11
1
c12
s1
c13
2
d2
c21
c22
2
s2
c23
3
d3
SOURCES
DESTINATIONS
7
Transportation Problem
  • LP Formulation
  • The linear programming formulation in terms of
    the amounts shipped from the origins to the
    destinations, xij, can be written as
  • Min SScijxij
  • i j
  • s.t. Sxij lt si for
    each origin i
  • j
  • Sxij dj for
    each destination j
  • i
  • xij gt 0 for all i
    and j

8
Transportation Problem
  • LP Formulation Special Cases
  • The following special-case modifications to the
    linear programming formulation can be made
  • Minimum shipping guarantees from i to j
  • xij gt Lij
  • Maximum route capacity from i to j
  • xij lt Lij
  • Unacceptable routes
  • delete the variable

9
Example BBC-1
  • Building Brick Company (BBC) has orders for 80
    tons of bricks at three suburban locations as
    follows Northwood -- 25 tons, Westwood -- 45
    tons, and Eastwood -- 10 tons. BBC has two
    plants. Plant 1 produces 50 and plant 2 produces
    30 tons per week.
  • How should end of week shipments be made to fill
    the above orders given the following delivery
    cost per ton
  • Northwood Westwood
    Eastwood
  • Plant 1 24
    30 40
  • Plant 2 30 40
    42

10
Example BBC-1
  • LP Formulation
  • Decision Variables Defined
  • xij amount shipped from plant i to
    suburb j
  • where i 1 (Plant 1) and 2 (Plant
    2)
  • j 1 (Northwood), 2 (Westwood),
  • and 3 (Eastwood)

11
Transportation Problem
  • Network Representation of BBC-1

Northwood 1
25
24
Plant 1
50
30
Westwood 2
45
40
30
Plant 2
40
30
42
Eastwood 3
10
DESTINATIONS
SOURCES
12
Example BBC-1
  • LP Formulation
  • Objective Function
  • Minimize total shipping cost per week
  • Min 24x11 30x12 40x13 30x21
    40x22 42x23
  • Constraints
  • s.t. x11 x12 x13 lt 50 (Plant 1
    capacity)
  • x21 x22 x23 lt 30 (Plant 2 capacity)
  • x11 x21 25 (Northwood demand)
  • x12 x22 45 (Westwood demand)
  • x13 x23 10 (Eastwood demand)
  • all xij gt 0 (Non-negativity)

13
Example BBC-1
  • Optimal Solution
  • From To
    Amount Cost
  • Plant 1 Northwood 5 120
  • Plant 1 Westwood 45
    1,350
  • Plant 2 Northwood 20
    600
  • Plant 2 Eastwood 10
    420
  • Total Cost 2,490

14
Assignment Problem
  • An assignment problem seeks to minimize the total
    cost assignment of m workers to m jobs, given
    that the cost of worker i performing job j is
    cij.
  • It assumes all workers are assigned and each job
    is performed.
  • An assignment problem is a special case of a
    transportation problem in which all supplies and
    all demands are equal to 1 hence assignment
    problems may be solved as linear programs.
  • The network representation of an assignment
    problem with three workers and three jobs is
    shown on the next slide.

15
Assignment Problem
  • Network Representation

c11
1
1
c12
c13
c21
c22
2
2
c23
c31
c32
3
3
c33
WORKERS
JOBS
16
Assignment Problem
  • Linear Programming Formulation
  • Min SScijxij
  • i j
  • s.t. Sxij 1
    for each worker i
  • j
  • Sxij 1
    for each job j
  • i
  • xij 0
    or 1 for all i and j.
  • Note A modification to the right-hand side of
    the first constraint set can be made if a worker
    is permitted to work more than one job.

17
Example Assignment
  • A contractor pays his 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.

  • Project
  • A B C
  • Westside 50 36 16
  • Subcontractors Federated 28 30
    18
  • Goliath 35 32 20
  • Universal 25 25
    14
  • How should the contractors be assigned to
    minimize total distance (and total cost)?

18
Example Assignment
  • Network Representation

50
West.
A
36
16
28
B
Fed.
30
18
32
35
C
Gol.
20
25
25
Univ.
14
19
Example Assignment
  • LP Formulation
  • Decision Variables Defined
  • xij 1 if subcontractor i is assigned to
    project j
  • 0 otherwise
  • where i 1 (Westside), 2 (Federated),
  • 3 (Goliath), and 4 (Universal)
  • j 1 (A), 2 (B), and 3 (C)

20
Example Assignment
  • LP Formulation
  • Objective Function
  • Minimize total distance
  • Min 50x11 36x12 16x13 28x21 30x22
    18x23
  • 35x31 32x32 20x33 25x41 25x42
    14x43

21
Example Assignment
  • LP Formulation
  • Constraints
  • x11 x12 x13 lt 1 (no more than one
  • x21 x22 x23 lt 1 project assigned
  • x31 x32 x33 lt 1 to any one
  • x41 x42 x43 lt 1 subcontractor)
  • x11 x21 x31 x41 1 (each project
    must
  • x12 x22 x32 x42 1 be assigned to
    just
  • x13 x23 x33 x43 1 one
    subcontractor)
  • all xij gt 0 (non-negativity)

22
Example Assignment
  • Optimal Assignment
  • Subcontractor Project
    Distance
  • Westside C 16
  • Federated A
    28
  • Universal B
    25
  • Goliath
    (unassigned)
  • Total Distance 69 miles

23
Variations of Assignment Problem
  • Total number of agents not equal to total number
    of tasks
  • Maximization objective function
  • Unacceptable assignments

24
Transshipment Problem
  • Transshipment problems are transportation
    problems in which a shipment may move through
    intermediate nodes (transshipment nodes)before
    reaching a particular destination node.
  • Transshipment problems can be converted to larger
    transportation problems and solved by a special
    transportation program.
  • Transshipment problems can also be solved as
    linear programs.
  • The network representation for a transshipment
    problem with two sources, three intermediate
    nodes, and two destinations is shown on the next
    slide.

25
Transshipment Problem
  • Network Representation

c36
3
c13
c37
6
1
s1
d1
c14
c46
c15
4
c47
c23
c24
7
c56
2
d2
s2
c25
5
c57
INTERMEDIATE NODES
SOURCES
DESTINATIONS
26
Transshipment Problem
  • Linear Programming Formulation
  • xij represents the shipment from node i to
    node j
  • Min S cijxij
  • all arcs
  • s.t. S xij - S xij lt si
    for each origin node i
  • arcs out arcs in
  • S xij - S xij 0
    for each intermediate
  • arcs out arcs in
    node
  • S xij - S xij
    -dj for each destination arcs out
    arcs in node j (Note the order)
  • xij gt 0
    for all i and j

27
Example Transshipping
  • Thomas Industries and Washburn Corporation
    supply three firms (Zrox, Hewes, Rockwright) with
    customized shelving for its offices. They 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, and 40 for Rockwright.
    Both Arnold and Supershelf can supply at most 75
    units to its customers.
  • Additional data is shown on the next slide.

28
Example Transshipping
  • Because of long standing contracts based on
    past orders, unit costs from the manufacturers to
    the suppliers are
  • Thomas
    Washburn
  • Arnold 5
    8
  • Supershelf 7
    4
  • The cost to install the shelving at the various
    locations are
  • Zrox Hewes
    Rockwright
  • Thomas 1 5
    8
  • Washburn 3 4
    4

29
Example Transshipping
  • Network Representation

Zrox 5
ZROX
50
1
Arnold 1
Thomas 3
5
ARNOLD
75
5
8
8
Hewes 6
60
HEWES
3
7
Supershelf 2
Washburn 4
4
WASH BURN
75
4
4
Rockwright 7
40
30
Example Transshipping
  • LP Formulation
  • Decision Variables Defined
  • xij amount shipped from manufacturer i to
    supplier j
  • xjk amount shipped from supplier j to
    customer k
  • where i 1 (Arnold), 2
    (Supershelf)
  • j 3 (Thomas), 4 (Washburn)
  • k 5 (Zrox), 6 (Hewes), 7
    (Rockwright)
  • Objective Function Defined
  • Minimize Overall Shipping Costs
  • Min 5x13 8x14 7x23 4x24 1x35 5x36
    8x37
  • 3x45 4x46 4x47

31
Example Transshipping
  • Constraints Defined
  • Amount out of Arnold x13 x14 lt
    75
  • Amount out of Supershelf x23 x24 lt 75
  • Amount through Thomas x13 x23 - x35 -
    x36 - x37 0
  • Amount through Washburn x14 x24 - x45 -
    x46 - x47 0
  • Amount into Zrox x35 x45
    50
  • Amount into Hewes x36 x46
    60
  • Amount into Rockwright x37 x47
    40
  • Non-negativity of variables xij gt 0, for
    all i and j.

32
Variations of Transshipment Problem
  • Total supply not equal to total demand
  • Maximization objective function
  • Route capacities or route minimums
  • Unacceptable routes
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