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									Transportation, Assignment,
    and Transshipment




        Professor Ahmadi




                              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
    standard computer spreadsheet package.




                                                        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
    tasks
   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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