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```									Transportation, Assignment,
and Transshipment

Slide 1
Chapter 7
Transportation, 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

Slide 2
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.

Slide 3
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

Slide 4
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.

Slide 5
Transportation Problem

   Network Representation

1   d1
c11
s1   1         c12
c13
2   d2
c21    c22

s2   2
c23
3   d3

SOURCES              DESTINATIONS

Slide 6
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
ij

s.t.   Sxij < si for each origin i
j

Sxij = dj for each destination j
i

xij > 0 for all i and j

Slide 7
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 > Lij
• Maximum route capacity from i to j:
xij < Lij
• Unacceptable routes:
delete the variable

Slide 8
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

Slide 9
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)

Slide 10
Transportation Problem

   Network Representation of BBC-1

Northwood
1        25
24
Plant
50      1               30

40               Westwood    45
2
30
Plant             40
30      2          42
Eastwood
10
3

SOURCES                 DESTINATIONS
Slide 11
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 < 50 (Plant 1 capacity)
x21 + x22 + x23 < 30 (Plant 2 capacity)
x11 + x21 = 25 (Northwood demand)
x12 + x22 = 45 (Westwood demand)
x13 + x23 = 10 (Eastwood demand)
all xij > 0 (Non-negativity)

Slide 12
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

Slide 13
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.

Slide 14
Assignment Problem

   Network Representation

c11
1                        1
c12
c13

c21
c22
2                        2
c23

c31
c32
3   c33                  3

WORKERS                  JOBS
Slide 15
Assignment Problem

   Linear Programming Formulation

Min SScijxij
ij

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.

Slide 16
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)?

Slide 17
Example: Assignment

   Network Representation
50
West.         36
A

16

28
30
Fed.                      B
18

35    32
20
Gol.                      C

25     25
14
Univ.
Slide 18
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)

Slide 19
Example: Assignment

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

Slide 20
Example: Assignment

   LP Formulation
• Constraints
x11 + x12 + x13 < 1       (no more than one
x21 + x22 + x23 < 1        project assigned
x31 + x32 + x33 < 1           to any one
x41 + x42 + x43 < 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   > 0 (non-negativity)

Slide 21
Example: Assignment

   Optimal Assignment

Subcontractor Project Distance
Westside         C       16
Federated       A        28
Universal        B       25
Goliath         (unassigned)
Total Distance = 69 miles

Slide 22
Variations of Assignment Problem

   Total number of agents not equal to total number of
   Maximization objective function
   Unacceptable assignments

Slide 23
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.

Slide 24
Transshipment Problem

   Network Representation

c36
3
c13
c37
s1   1           c14                                 6   d1
c15                     c46
4    c47
c23
c24            c56              7   d2
s2    2
c25
5         c57

SOURCES                    INTERMEDIATE         DESTINATIONS
NODES

Slide 25
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 < 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 > 0       for all i and j

Slide 26
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.

Slide 27
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

Slide 28
Example: Transshipping

   Network Representation
ZROX
Zrox       50
5

Arnold      5                   1
ARNOLD                Thomas
75      1                     3           5
8                   8
Hewes
HEWES
6         60

3
7
Supershelf            Washburn           4
WASH
75       2                    4
BURN
4                   4
Rockwright
7        40

Slide 29
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

Slide 30
Example: Transshipping

   Constraints Defined
Amount out of Arnold:          x13 + x14 < 75
Amount out of Supershelf:      x23 + x24 < 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 > 0, for all i and j.

Slide 31
Variations of Transshipment Problem

   Total supply not equal to total demand
   Maximization objective function
   Route capacities or route minimums
   Unacceptable routes

Slide 32

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