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Transportation Problem

    By : Alvin G. Niere
    Misamis University
Aim of Transportation Model


To find out optimum transportation
 schedule keeping in mind cost of
 transportation to be minimized.
What is a Transportation Problem?
• The transportation problem is a special type of
  LPP where the objective is to minimize the cost of
  distributing a product from a number of sources
  or origins to a number of destinations.

• Because of its special structure the usual simplex
  method is not suitable for solving transportation
  problems. These problems require special
  method of solution.
The Transportation Problem
• The problem of finding the minimum-cost
  distribution of a given commodity
  from a group of supply centers (sources) i=1,…,m
  to a group of receiving centers (destinations)
  j=1,…,n
• Each source has a certain supply (si)
• Each destination has a certain demand (dj)
• The cost of shipping from a source to a
  destination is directly proportional to the number
  of units shipped
Simple Network Representation
              Sources               Destinations


  Supply s1     1                              Demand d1
                                        1

  Supply s2     2
                                        2      Demand d2
              …




                                      …
                             xij

                                        n      Demand dn
  Supply sm     m

                        Costs cij


                                                           Transportation-5
Application of Transportation Problem

 Minimize shipping costs

 Determine low cost location

 Find minimum cost production schedule

 Military distribution system
Two Types of Transportation Problem


• Balanced Transportation Problem
  where the total supply equals total demand

• Unbalanced Transportation Problem
  where the total supply is not equal to the
  total demand
Phases of Solution of Transportation
               Problem

• Phase I- obtains the initial basic feasible
  solution

• Phase II-obtains the optimal basic solution
Initial Basic Feasible Solution

North West Corner Rule (NWCR)

Row Minima Method

Column Minima Method

Least Cost Method

Vogle Approximation Method (VAM)
Optimum Basic Solution

Stepping Stone Method



Modified Distribution Method a.k.a. MODI Method
Optimum Basic Solution:
      Stepping-Stone Method
1. Select any unused square to evaluate
2. Beginning at this square, trace a closed path
   back to the original square via squares that
   are currently being used
3. Beginning with a plus (+) sign at the unused
   corner, place alternate minus and plus signs at
   each corner of the path just traced
Stepping-Stone Method
4. Calculate an improvement index by first
   adding the unit-cost figures found in each
   square containing a plus sign and subtracting
   the unit costs in each square containing a
   minus sign
5. Repeat steps 1 though 4 until you have
   calculated an improvement index for all
   unused squares. If all indices are ≥ 0, you have
   reached an optimal solution.
Problem Illustration

        TO           A.          B.           C.           FACTORY
FROM            ALBUQUERQUE   BOSTON       CLEVELAND       CAPACITY

D. DES MOINES             5            4            3
                                                                100

E. EVANSVILLE             8            4               3
                                                                300

F. FORT                   9            7               5
LAUDERDALE                                                      300

WAREHOUSE
DEMAND              300         200           200               700
Initial Feasible Solution using
              Northwest Corner Rule
        TO           A.                B.              C.           FACTORY
FROM            ALBUQUERQUE         BOSTON          CLEVELAND       CAPACITY

D. DES MOINES                 5                4               3
                      100                                                 100

E. EVANSVILLE                8                 4               3
                     200              100                                 300

F. FORT                      9                 7                5
LAUDERDALE                             100             200                300

WAREHOUSE
DEMAND                300              200               200              700

IFS= DA + EA +EB + FB + FC = 100(5) + 200(8) + 100(4) + 100(7) + 200(5)
= 500 + 1600 + 400 + 700 + 1000 = 4200
Optimizing Solution using
                          Stepping-Stone Method
                         To       (A)               (B)          (C)        Factory
   From                       Albuquerque         Boston      Cleveland     capacity

                                         $5              $4           $3
   (D) Des Moines              100
                                     -                                          100         Des Moines-
                                              +
                                         $8              $4           $3
                                                                                              Boston index
   (E) Evansville              200            100                               300
                                     +        -                                             = $4 - $5 + $8 - $4
                                         $9              $7           $5
   (F) Fort Lauderdale                        100             200               300         = +$3
   Warehouse
   requirement                   300               200          200             700




                                                                99                     $5           1             $4
                                                                           100
                                                                            -                             +


                                                                            +                              -
                                                               201                     $8           99            $4
Figure C.5                                                                 200                           100
Stepping-Stone Method
                           To       (A)             (B)             (C)       Factory
     From                       Albuquerque       Boston         Cleveland    capacity
                                           $5           $4 Start         $3
     (D) Des Moines              100                                           100
                                       -                   +
                                           $8           $4               $3
     (E) Evansville              200             100                           300
                                       +        -
                                           $9           $7               $5
     (F) Fort Lauderdale                         100             200           300
                                                +            -
      Warehouse
      requirement                  300            200              200         700

                                Des Moines-Cleveland index
Figure C.6                      = $3 - $5 + $8 - $4 + $7 - $5 = +$4
Stepping-Stone Method
                      To       (A)            (B)          (C)         Factory
From                       Albuquerque      Boston      Cleveland      capacity
                                     $5           $4              $3
(D) Des Moines              100                                         100

                                     $8           $4              $3
(E) Evansville              200            100                          300

                                     $9            $7           $5
(F) Fort Lauderdale                        100          200             300
                              Evansville-Cleveland index
Warehouse                     = $3 - $4 + $7 - $5 = +$1
requirement                   300             200         200           700
                              (Closed path = EC - EB + FB - FC)
                              Fort Lauderdale-Albuquerque index
                              = $9 - $7 + $4 - $8 = -$1
                              (Closed path = FA - FB + EB - EA)
Stepping-Stone Method
1. If an improvement is possible, choose the
   route (unused square) with the largest
   negative improvement index
2. On the closed path for that route, select the
   smallest number found in the squares
   containing minus signs
3. Add this number to all squares on the closed
   path with plus signs and subtract it from all
   squares with a minus sign
Stepping-Stone Method
                             To       (A)           (B)         (C)      Factory
       From                       Albuquerque     Boston     Cleveland   capacity
                                           $5           $4          $3
       (D) Des Moines              100                                    100

                                           $8           $4          $3
       (E) Evansville              200           100                      300
                                      -         +
                                           $9           $7          $5
       (F) Fort Lauderdale                       100         200          300
                                      +         -
       Warehouse
       requirement                  300             200         200       700
                                    1.    Add 100 units on route FA
                                    2.    Subtract 100 from routes FB
                                    3.    Add 100 to route EB
                                    4.    Subtract 100 from route EA
Figure C.7
Stepping-Stone Method
                       To       (A)          (B)         (C)        Factory
 From                       Albuquerque    Boston     Cleveland     capacity
                                     $5          $4           $3
 (D) Des Moines              100                                      100

                                     $8          $4           $3
 (E) Evansville              100          200                         300

                                     $9          $7           $5
 (F) Fort Lauderdale         100                      200             300

 Warehouse
 requirement                   300         200          200           700

Total Cost        = $5(100) + $8(100) + $4(200) + $9(100) + $5(200)
                  = $4,000
                                                                   Figure C.8
Special Issues in Modeling

 Demand not equal to supply
    Called an unbalanced problem
    Common situation in the real world
    Resolved by introducing dummy sources
     or dummy destinations as necessary with
     cost coefficients of zero
Total Cost
            Special50($8) + 200($4)in50($3) + 150($5) + 150(0)
             = 250($5) +
                         Issues + Modeling
                   = $3,350
                        To        (A)          (B)         (C)                  Factory
                                                                     Dummy      capacity
  From                        Albuquerque    Boston     Cleveland

                                       $5          $4           $3          0
  (D) Des Moines               250                                               250

                                       $8          $4           $3          0
  (E) Evansville                50          200          50                      300

                                       $9          $7           $5          0
  (F) Fort Lauderdale                                   150          150         300

  Warehouse
  requirement                    300         200          200         150        850


                                                                     New
  Figure C.9                                                         Des Moines
                                                                     capacity
Special Issues in Modeling

 Degeneracy
    To use the stepping-stone
     methodology, the number of occupied
     squares in any solution must be equal to
     the number of rows in the table plus the
     number of columns minus 1
    If a solution does not satisfy this rule it is
     called degenerate
Special Issues in Modeling
                      To    Customer      Customer      Customer     Warehouse
      From                      1             2             3         supply
                                    $8            $2            $6
      Warehouse 1           100                                        100

                                    $10           $9            $9
      Warehouse 2             0           100           20             120

                                    $7            $10           $7
      Warehouse 3                                       80               80

      Customer
      demand                  100           100           100          300

                    Initial solution is degenerate
                    Place a zero quantity in an unused square and
Figure C.10         proceed computing improvement indices

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Transportation Problem

  • 1. Transportation Problem By : Alvin G. Niere Misamis University
  • 2. Aim of Transportation Model To find out optimum transportation schedule keeping in mind cost of transportation to be minimized.
  • 3. What is a Transportation Problem? • The transportation problem is a special type of LPP where the objective is to minimize the cost of distributing a product from a number of sources or origins to a number of destinations. • Because of its special structure the usual simplex method is not suitable for solving transportation problems. These problems require special method of solution.
  • 4. The Transportation Problem • The problem of finding the minimum-cost distribution of a given commodity from a group of supply centers (sources) i=1,…,m to a group of receiving centers (destinations) j=1,…,n • Each source has a certain supply (si) • Each destination has a certain demand (dj) • The cost of shipping from a source to a destination is directly proportional to the number of units shipped
  • 5. Simple Network Representation Sources Destinations Supply s1 1 Demand d1 1 Supply s2 2 2 Demand d2 … … xij n Demand dn Supply sm m Costs cij Transportation-5
  • 6. Application of Transportation Problem  Minimize shipping costs  Determine low cost location  Find minimum cost production schedule  Military distribution system
  • 7. Two Types of Transportation Problem • Balanced Transportation Problem where the total supply equals total demand • Unbalanced Transportation Problem where the total supply is not equal to the total demand
  • 8. Phases of Solution of Transportation Problem • Phase I- obtains the initial basic feasible solution • Phase II-obtains the optimal basic solution
  • 9. Initial Basic Feasible Solution North West Corner Rule (NWCR) Row Minima Method Column Minima Method Least Cost Method Vogle Approximation Method (VAM)
  • 10. Optimum Basic Solution Stepping Stone Method Modified Distribution Method a.k.a. MODI Method
  • 11. Optimum Basic Solution: Stepping-Stone Method 1. Select any unused square to evaluate 2. Beginning at this square, trace a closed path back to the original square via squares that are currently being used 3. Beginning with a plus (+) sign at the unused corner, place alternate minus and plus signs at each corner of the path just traced
  • 12. Stepping-Stone Method 4. Calculate an improvement index by first adding the unit-cost figures found in each square containing a plus sign and subtracting the unit costs in each square containing a minus sign 5. Repeat steps 1 though 4 until you have calculated an improvement index for all unused squares. If all indices are ≥ 0, you have reached an optimal solution.
  • 13. Problem Illustration TO A. B. C. FACTORY FROM ALBUQUERQUE BOSTON CLEVELAND CAPACITY D. DES MOINES 5 4 3 100 E. EVANSVILLE 8 4 3 300 F. FORT 9 7 5 LAUDERDALE 300 WAREHOUSE DEMAND 300 200 200 700
  • 14. Initial Feasible Solution using Northwest Corner Rule TO A. B. C. FACTORY FROM ALBUQUERQUE BOSTON CLEVELAND CAPACITY D. DES MOINES 5 4 3 100 100 E. EVANSVILLE 8 4 3 200 100 300 F. FORT 9 7 5 LAUDERDALE 100 200 300 WAREHOUSE DEMAND 300 200 200 700 IFS= DA + EA +EB + FB + FC = 100(5) + 200(8) + 100(4) + 100(7) + 200(5) = 500 + 1600 + 400 + 700 + 1000 = 4200
  • 15. Optimizing Solution using Stepping-Stone Method To (A) (B) (C) Factory From Albuquerque Boston Cleveland capacity $5 $4 $3 (D) Des Moines 100 - 100 Des Moines- + $8 $4 $3 Boston index (E) Evansville 200 100 300 + - = $4 - $5 + $8 - $4 $9 $7 $5 (F) Fort Lauderdale 100 200 300 = +$3 Warehouse requirement 300 200 200 700 99 $5 1 $4 100 - + + - 201 $8 99 $4 Figure C.5 200 100
  • 16. Stepping-Stone Method To (A) (B) (C) Factory From Albuquerque Boston Cleveland capacity $5 $4 Start $3 (D) Des Moines 100 100 - + $8 $4 $3 (E) Evansville 200 100 300 + - $9 $7 $5 (F) Fort Lauderdale 100 200 300 + - Warehouse requirement 300 200 200 700 Des Moines-Cleveland index Figure C.6 = $3 - $5 + $8 - $4 + $7 - $5 = +$4
  • 17. Stepping-Stone Method To (A) (B) (C) Factory From Albuquerque Boston Cleveland capacity $5 $4 $3 (D) Des Moines 100 100 $8 $4 $3 (E) Evansville 200 100 300 $9 $7 $5 (F) Fort Lauderdale 100 200 300 Evansville-Cleveland index Warehouse = $3 - $4 + $7 - $5 = +$1 requirement 300 200 200 700 (Closed path = EC - EB + FB - FC) Fort Lauderdale-Albuquerque index = $9 - $7 + $4 - $8 = -$1 (Closed path = FA - FB + EB - EA)
  • 18. Stepping-Stone Method 1. If an improvement is possible, choose the route (unused square) with the largest negative improvement index 2. On the closed path for that route, select the smallest number found in the squares containing minus signs 3. Add this number to all squares on the closed path with plus signs and subtract it from all squares with a minus sign
  • 19. Stepping-Stone Method To (A) (B) (C) Factory From Albuquerque Boston Cleveland capacity $5 $4 $3 (D) Des Moines 100 100 $8 $4 $3 (E) Evansville 200 100 300 - + $9 $7 $5 (F) Fort Lauderdale 100 200 300 + - Warehouse requirement 300 200 200 700 1. Add 100 units on route FA 2. Subtract 100 from routes FB 3. Add 100 to route EB 4. Subtract 100 from route EA Figure C.7
  • 20. Stepping-Stone Method To (A) (B) (C) Factory From Albuquerque Boston Cleveland capacity $5 $4 $3 (D) Des Moines 100 100 $8 $4 $3 (E) Evansville 100 200 300 $9 $7 $5 (F) Fort Lauderdale 100 200 300 Warehouse requirement 300 200 200 700 Total Cost = $5(100) + $8(100) + $4(200) + $9(100) + $5(200) = $4,000 Figure C.8
  • 21. Special Issues in Modeling  Demand not equal to supply  Called an unbalanced problem  Common situation in the real world  Resolved by introducing dummy sources or dummy destinations as necessary with cost coefficients of zero
  • 22. Total Cost Special50($8) + 200($4)in50($3) + 150($5) + 150(0) = 250($5) + Issues + Modeling = $3,350 To (A) (B) (C) Factory Dummy capacity From Albuquerque Boston Cleveland $5 $4 $3 0 (D) Des Moines 250 250 $8 $4 $3 0 (E) Evansville 50 200 50 300 $9 $7 $5 0 (F) Fort Lauderdale 150 150 300 Warehouse requirement 300 200 200 150 850 New Figure C.9 Des Moines capacity
  • 23. Special Issues in Modeling  Degeneracy  To use the stepping-stone methodology, the number of occupied squares in any solution must be equal to the number of rows in the table plus the number of columns minus 1  If a solution does not satisfy this rule it is called degenerate
  • 24. Special Issues in Modeling To Customer Customer Customer Warehouse From 1 2 3 supply $8 $2 $6 Warehouse 1 100 100 $10 $9 $9 Warehouse 2 0 100 20 120 $7 $10 $7 Warehouse 3 80 80 Customer demand 100 100 100 300 Initial solution is degenerate Place a zero quantity in an unused square and Figure C.10 proceed computing improvement indices