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					Chapter 7
 Transportation Problems (運輸問題)
 Assignment Problems (指派問題)
 Transshipment Problems (轉運問題)
 簡介
 Transportation Problems (運輸問題)
  如何以最低之運輸成本,將貨物由來源地送至目的地
 Assignment Problems (指派問題)
  如何以適當之方式,做一對一之指派
  為運輸問題之特例
 Transshipment Problems (轉運問題)
  在運輸問題中,允許貨物由來源地經由其他來源地或
  目的地轉運,最後送至目的地
 上述之線性規劃模式仍然可以單體法求解,但有特別
  之求解方式,遠較單體法來得有效率。
                               2
                                                   p.360
7.1 Transportation Problems
 A transportation problem(運輸問題) basically deals
  with the problem, which aims to find the best way
  to fulfill the demand of n demand points(需求點)
  using the capacities of m supply points (供應點).
 While trying to find the best way, generally a
  variable cost of shipping the product from one
  supply point to a demand point or a similar
  constraint should be taken into consideration.




                                                      3
                                                                   p.360
Example 1: Powerco Formulation
 Powerco has three electric power plants that
  supply the electric needs of four cities.
 The associated supply of each plant and demand of
  each city is given in the following table.

Table 1

                                           To

                                                            Supply
                City 1   City 2   City 3        City 4
    From                                                 (Million kwh)
   Plant 1        $8       $6     $10            $9           35
   Plant 2        $9     $12      $13            $7           50
   Plant 3      $14        $9     $16            $5           40
  Demand
(Million kwh)
                 45       20       30            30
                                                                         4
Example 1: Solution
 Decision Variables
  Powerco must determine how much power is sent
  from each plant to each city so
   xij = amount of electricity produced at plant i
         and sent to city j
 Constraints
     A supply constraint(供應限制)ensures that the
      total quality produced does not exceed plant
      capacity. Each plant is a supply point.
     A demand constraint (需求限制) ensures that a
      location receives its demand. Each city is a
      demand point.
     Since a negative amount of electricity can not
      be shipped all xij’s must be non negative
                                                       5
                                           To
                                                           Supply
                City 1   City 2   City 3        City 4
   From                                                  (Million kwh)

  Plant 1       8x11     6x12     10x13         9x14         ≤ 35
  Plant 2       9x21     12x22    13x23         7x24         ≤ 50
  Plant 3       14x31    9x32     16x33         5x34         ≤ 40
 Demand
                ≥ 45     ≥ 20     ≥ 30          ≥ 30
(Million kwh)

Min z = 8x11+6x12+10x13+9x14+9x21+12x22+13x23
            +7x24 +14x31+9x32+16x33+5x34
  s.t.    x11+x12+x13+x14 ≤ 35 (Supply Constraints)
          x21+x22+x23+x24 ≤ 50
          x31+x32+x33+x34 ≤ 40
          x11+x21+x31 ≥ 45         (Demand Constraints)
          x12+x22+x32 ≥ 20
          x13+x23+x33 ≥ 30
          x14+x24+x34 ≥ 30
           xij ≥ 0 (i= 1,2,3; j= 1,2,3,4)                                6
                                                 p.362
General description of
      a Transportation Problem
 A set of m supply points from which a good is
  shipped. Supply point i can supply at most si units.
 A set of n demand points to which the good is
  shipped. Demand point j must receive at least di
  units of the shipped good.
 Each unit produced at supply point i and shipped
  to demand point j incurs a variable cost of cij.
 Table 2 (p.364).




                                                         7
                                                      p.364
A Transportation Tableau

 Table 2                                     Supply

               C11       C12           C1n
                               …               S1

               C21       C22           C2n
                               …               S2


           :         :             :

            Cm1       Cm2           Cmn
                               …              Sm

 Demand    D1        D2        …   Dn



                                                         8
Let xij = number of units shipped
          from supply point i to demand point j

          i  m j n
min       ∑∑ x
            c          ij   ij
          i 1 j 1
   j n
s.t. ∑ ij ≤ si (i  1,2,...,m) (supply constraints)
      x
   j 1
   im

   ∑x        ij   ≥ dj ( j  1,2,...,n) (demand constraints)
   i 1

          xij ≥ 0 (i  1,2,...,m; j  1,2,...,n)



                                                               9
                                                p.363
Balanced Transportation Problem

      i m           j n
 If   s
      i 1
             i      d
                     j 1
                            j



 then total supply equals to total demand,
 the problem is said to be a balanced transportation
 problem (平衡運輸問題).




                                                       10
                                                                  p.364
Balanced Transportation Problem
Table 3

                    City 1    City 2    City 3    City 4    Supply
                          8         6        10         9
          Plant 1                                            35
                              10        25
                          9        12        13         7
          Plant 2                                            50
                    45                  5
                         14         9        16         5
          Plant 3                                            40
                              10                  30

          Demand     45        20           30     30        125



                                                                     11
If total supply exceeds total demand
       i=m   j=n
 If   ∑si > ∑dj
       i=1   j=1

  total supply exceeds total demand, we can balance
  the problem by adding dummy demand point (虛擬
  需求點). Since shipments to the dummy demand
  point are not real, they are assigned a cost of zero.
 Figure 2 (p.364).




                                                      12
                                                             p.364
If total supply exceeds total demand
                                                          =125

                  City 1    City 2    City 3    City 4    Supply
                        8         6        10         9
        Plant 1                                            35
                            10        25
                        9        12        13         7
        Plant 2                                            50
                  45                  5
                       14         9        16         5
        Plant 3                                            40
                            10                  30

120 =   Demand     40        20           30     30



                                                                   13
If total supply is less than total demand

 If a transportation problem has a total supply that
  is strictly less than total demand the problem has
  no feasible solution.
     No doubt that in such a case one or more of the
      demand will be left unmet.
    Generally in such situations a penalty cost is
     often associated with unmet demand and as
     one can guess the total penalty cost is desired
     to be minimum.
 Example 2 (p.365).



                                                        14
                                                  p.365
Example 2 : Handing Shortages
Two reservoirs are available to supply the water
needs of three cities. Each reservoir can supply up to
50 million gallons of water per day. Each city would
like to receive 40 million gallons per day. For each
million gallons per day of unmet demand, there is a
penalty. At city 1, the penalty is $20; at city 2, the
penalty is $22; and at city 3, the penalty is $23. The
cost of transporting 1 million gallons of water from
each reservoir to each city is shown in Table 4.
Formulate a balanced transportation problem that
can be used to minimize the sum of shortage and
transport costs.


                                                         15
Example 2 :

     Table 4
                              To
         From       City 1   City 2   City 3
      Reservoir 1    $7       $8       $10
      Reservoir 2    $9       $7        $8




                                               16
                                                         p.365
Solution :

  Daily supply = 50+50 = 100 million gallons per day
  Daily demand = 40+40+40 =120 million gallons per day


   Table 5
                                     To

       From       City 1    City 2    City 3    Supply

    Reservoir 1     $8        $6          $10    50
    Reservoir 2     $9       $12          $13    50
      Dummy        $14        $9          $16    20
     Demand         40       40           40


                                                            17
                                                    p.371
 Exercise 1 :
A company supplies goods to three customers, who
each require 30 units. The company has two
warehouses. Warehouses 1 has 40 units available, and
warehouses 2 has 30 units available. The cost of
shipping 1 unit from warehouse to customer are shown
in Table 7. There is penalty for each unmet customer
unit of demand: With customer 1, a penalty cost of
$90 incurred; with customer 2,$80; and with customer
3,$110. Formulate a balanced transportation problem
to minimize the sum of shortage and shipping costs.
       Table 7
                      Cust. 1   Cust. 2   Cust. 3
        Warehouse 1    $15       $35       $25
        Warehouse 2    $10       $50       $40         18
Solution:

               Cust. 1   Cust. 2   Cust. 3   Supply
 Warehouse 1    $15       $35       $25       40
 Warehouse 2    $10       $50       $40       30
  Shortage      $90       $80       $110      20
  Demand         30        30        30




                                                      19
Exercise 2 :

Referring to exercise 1, suppose that extra units
could be purchases and shipped to either warehouse
for a total cost of $100 per unit and that all
customer demand must be met. Formulate a
balanced transportation problem to minimize the
sum of purchasing and shipping costs.




                                                     20
Solution:

            C1     C2     C3    Dummy   Supply
  W1       $15    $35    $25      0      40
  W2       $10    $50    $40      0      30
W1 Extra   $115   $135   $125     0      20
W2 Extra   $110   $150   $140     0      20
Demand     30     30     30      30      120




                                                 21
運輸問題求解
1. 將問題建成運輸問題模式
2. 平衡運輸問題
3. 求解過程分為二階段;
   第一階段為求起始解,第二階段為求最佳解
4. 起始解之解法
     Northwest Corner Method (西北角法)
     Minimum Cost Method (最小成本法)
     Vogel’s Method (佛格法)
5. 最佳解之解法
     The Transportation Simplex Method(運輸單體法)
     Stepping Stone Method(階石法)
                                                 22
                                                     p.373
7.2 Finding Basic Feasible Solution
    for Transportation Problems
  Unlike other Linear Programming problems, a
   balanced transportation problem with m supply
   points and n demand points is easier to solve,
   although it has m + n equality constraints.
  The reason for that is, if a set of decision variables
   (xij’s) satisfy all but one constraint, the values for
   xij’s will satisfy that remaining constraint
   automatically.




                                                            23
 An ordered sequence of at least four different cells
  is called a loop if
   Any two consecutive cells lie in either the same
    row or same column
   No three consecutive cells lie in the same row or
    column
   The last cell in the sequence has a row or
    column with the first cell in the sequence




                                                         24
 The Northwest Corner Method dos not utilize
  shipping costs. It can yield an initial bfs easily but
  the total shipping cost may be very high.
 The Minimum Cost Method uses shipping costs in
  order come up with a bfs that has a lower cost.
  Often the minimum cost method will yield a costly
  bfs.
 Vogel’s Method for finding a bfs usually avoids
  extremely high shipping costs.




                                                           25
                                                  p.376
Northwest Corner Method (西北角法)

 Begin in the upper left (northwest) corner of the
  transportation tableau and set x11 as large as
  possible. x11 can clearly be no larger than the
  smaller of s1 and d1.
 Continue applying this procedure to the most
  northwest cell in the tableau that does not lie in a
  crossed-out row or column.
 Assign the last cell a value equal to its row or
  column demand, and cross out both cells row and
  column.
 Table 15 ~ Table 20.


                                                         26
                                 p.376
Northwest Corner Method

Table 15

           D1   D2   D3   D4   Supply

    S1                           5

    S2                           1

    S3                           3

Demand     2    4    2    1      9




                                        27
                                 p.377
Northwest Corner Method

Table 20

           D1   D2   D3   D4   Supply

    S1     2    3                5

    S2          1                1

    S3          0    2    1      3

 Demand    2    4    2    1      9




                                        28
                                                 p.378
The Minimum Cost Method (最小成本法)
 Find the decision variable with the smallest
  shipping cost (xij). Then assign xij its largest
  possible value, which is the minimum of si and dj
 Next, as in the Northwest Corner Method cross out
  row i and column j and reduce the supply or
  demand of the noncrossed-out row or column by
  the value of xij.
 Choose the cell with the minimum cost of shipping
  from the cells that do not lie in a crossed-out row.
 Table 21 ~ Table 26.



                                                         29
                                                      p.378
The Minimum Cost Method
 Table 21

            D1       D2       D3       D4       Supply
                 2        3        5        6
   S1                                             5

                 2        1        3        5
   S2                                            10

                 3        8        4        6
   S3                                            15

 Demand     12       8        4        6         30



                                                         30
                                                                      p.380
The Minimum Cost Method
 Table 26

                D1           D2           D3           D4       Supply
                     2            3            5            6
   S1                                                             5
            5
                     2            1            3            5
   S2                                                            10
            2            8
                     3            8            4            6
   S3                                                            15
            5                         4            6

 Demand         12           8            4            6         30



                                                                         31
                                                 p.380
The Vogel’s Method (佛格法)
 Begin with computing each row and column a
  penalty. The penalty will be equal to the difference
  between the two smallest shipping costs in the row
  or column.
 Identify the row or column with the largest penalty.
 Find the first basic variable which has the smallest
  shipping cost in that row or column.
 Assign the highest possible value to that variable,
  and cross-out the row or column as in the previous
  methods.
 Compute new penalties and use the same procedure.
 Table 28 ~ Table 32.
                                                     32
The Vogel’s Method (佛格法)
 計算每列剩餘格中最小與次小之差(penalty)
 計算每行剩餘格中最小與次小之差(penalty)
 找出Max{,}
 找出產生之列或行剩餘格中之最小成本




                             33
                                                   p.381
The Vogel’s Method
    Table 28

               D1       D2       D3       Supply
                    6        7        8
      S1                                   10

                15          80       78
      S2                                   15

    Demand     15       5        5         25




                                                      34
                                                                p.382
The Vogel’s Method
    Table 32

                    D1           D2           D3       Supply
                         6            7            8
      S1                                                10
               0             5            5
                     15              80           78
      S2                                                15
               15

    Demand          15           5            5         25




                                                                   35
                                              p.382
Exercise 1 :

              C1     C2       C3     Supply
     W1       $15   $35      $25      40
     W2       $10   $50      $40      30
   Shortage   $90   $80      $110     20
   Demand     30     30      30


  To find A bfs by the following methods .
  (1)Northwest Corner Method
  (2)Minimum Cost Method
  (3)Vogel’s Method
                                                 36
Solution: Northwest Corner Method

             C1   C2   C3   Supply
    W1       30   10   0     40
    W2            20   10    30
  Shortage                   20
   Demand    30   30   30




                                     37
Solution : Minimum Cost Method

             C1   C2   C3   Supply
    W1       0    10   30    40
    W2       30              30
  Shortage             20    20
   Demand    30   30   30




                                     38
Solution : Vogel’s Method

             C1   C2   C3   Supply
    W1                       40
    W2                       30
  Shortage                   20
   Demand    30   30   30




                                     39
                                               p.382
Exercise 2 :

             C1     C2      C3    Dummy     Supply
   W1        15     35      25       0       40
   W2        10     50      40       0       30
 W1 Extra   115     135    125       0       20
 W2 Extra   110     150    140       0       20
 Demand     30      30      30      20       120


 To find A bfs by the following methods .
 (1)Northwest Corner Method
 (2)Minimum Cost Method
 (3)Vogel’s Method
                                                     40
Solution: Northwest Corner Method

            C1   C2   C3   Dummy   Supply

   W1       30   10                 40

   W2            20   10            30

 W1 Extra             20            20

 W2 Extra                   20      20

 Demand     30   30   30    20




                                            41
Solution: Minimum Cost Method

            C1        C2        C3        Dummy     Supply
                 15        35        25         0
   W1                                                40

                 10        50        40         0
   W2                                                30

             115       135       125            0
 W1 Extra                                            20

             110       150       140            0
 W2 Extra                                            20

 Demand     30        30        30         20




                                                             42
Solution: Vogel’s Method

            C1        C2        C3        Dummy     Supply
                 15        35        25         0
   W1                                                40

                 10        50        40         0
   W2                                                30

             115       135       125            0
 W1 Extra                                            20

             110       150       140            0
 W2 Extra                                            20

 Demand     30        30        30         20




                                                             43
                                                  p.382
7.3 The Transportation Simplex Method
  Method of Multipliers (乘數法)
  Modified Distribution Method (MODI, 修正分配法)

 Step 1 Determine the enter the basis.
 Step 2 Find the loop involving the entering variable
       and some of the basic variables.
 Step 3 Counting the cells in the loop,
       label them as even cells or odd cells.
 Step 4 Find the odd cells whose variable assumes
       the smallest value. Call this value θ.
       The variable corresponding to this odd
       cell will leave the basis.

                                                        44
 To perform the pivot, decrease the value of each
  odd cell by θ and increase the value of each even
  cell by θ. The variables that are not in the loop
  remain unchanged. The pivot is now complete.
    If θ=0, the entering variable will equal 0, and
     an odd variable that has a current value of 0
     will leave the basis. In this case a degenerate
     bfs existed before and will result after the pivot.
    If more than one odd cell in the loop equals θ,
     you may arbitrarily choose one of these odd
     cells to leave the basis; again a degenerate bfs
     will result



                                                       45
 Two important points to keep in mind in the
  pivoting procedure
    Since each row has as many +20s as –20s,
     the new solution will satisfy each supply and
     demand constraint.
    By choosing the smallest odd variable (x23)
     to leave the basis, we ensured that all
     variables will remain nonnegative.




                                                     46
單體法求解步驟 (複習一下!!)

1. 建立起始可行解
2. 最佳測試, 決定entering var.
3. 決定leaving var.
4. 建立下一個可行解




                           47
                                                      p.387
Summary of the Transportation
             Simplex Method
For a min problem
Step1. Balance problem.
Step2. Find a bfs. (initial solution)
Step3. Let u1=0, cij=ui+vj for all BV. Find ui ,vj.
Step4. Let   c ij =ui+vj-cij
       If cij ≤0 for all NBV,
       then the current bfs is optimal
       else the var with the most positive c ij is a new bfs.
Step5. using the new bfs return to step 3 and 4.
For a max problem
Step4’.   cij ≥0                                          48
           vj             v1             v2             v3             v4                p.387
 ui   Si        Dj        D1             D2             D3             D4       supply
                               8              6              10             9
 u1        S1                                                                    35
                     35
                               9              12             13             7
 u2        S2                                                                    50
                     10             20             20
                               14             9              16             5
 u3        S3                                                                    40
                                                   10             30
       demand             45             20             30             30
      Z=8(35)+9(10)+12(20)+13(20)+16(10)+5(30)=1180
= ui+vj =cij                             c ij = ui+vj-cij
u u =0
   1
                      u1=0               c12 = 5
i
+ u1+v1=8             u2=1               c13 = 2                   Each unit of x32 that
v u2+v1=9             u3=4               c14 = -8                  is entered into the
j u2+v2=12            v1=8
                                         c24 = -5                  basis will decrease
- u2+v3=13            v2=11
  u3+v3=16            v3=12              c31 = -2                  cost by $6
                                                                                            49
c u3+v4=5             v4=1               c32 = 6
                                                                                           p.388
Table 35         vj =             8             11            12         1
     ui =   Si          Dj        D1            D2            D3         D4       supply
                                       8             6         10             9
      0          S1                                                                35
                             35
                                       9         12            13             7
      1          S2                                                                50
                             10            20            20
                                      14             9         16             5
      4          S3                                                                40
                                                         10         30
            demand                45            20            30         30



            x32 is the entering variable.
            Min{20,10}=10  x33 is the leaving variable.

   P.S. 閉迴路中,除起點與終點為NBV,其餘轉角點均為BV
                                                                                              50
            Z=8(35)+9(10)+12(10)+13(30)+910)+5(30)=1120
Table 36          vj =             8             11            12         7
     ui =    Si          Dj        D1            D2            D3         D4       supply
                                        8             6         10             9
      0           S1                                                                35
                              35
                                        9         12            13             7
      1           S2                                                                50
                              10            10            30
                                       14             9         16             5
     -2           S3                                                                40
                                            10                       30
             demand                45            20            30         30
             以 ui+vj =cij 計算 c ij = ui+vj-cij 選擇正最大為進入變數

             x12 is the entering variable.
             Min{35,10}=10  x22 is the leaving variable.



                                                                                            51
            Z=8(35)+6(10)+9(20)+13(30)+9(10)+5(30)=1050
Table 37          vj =             8             6             12         2
     ui =    Si          Dj        D1            D2            D3         D4       supply
                                        8             6         10             9
      0           S1                                                                35
                              25            10
                                        9            12         13             7
      1           S2                                                                50
                              20                          30
                                       14             9         16             5
      3           S3                                                                40
                                            10                       30
             demand                45            20            30         30



             x13 is the entering variable.
             Min{25,30}=25  x11 is the leaving variable.



                                                                                            52
Table 38     vj =         6             6            10         2
              Si
     ui =             D1            D2               D3     D4          supply
              Dj
                              8             6         10            9
      0       S1                                                         35
                                   10            25
                              9             12        13            7
      3       S2                                                         50
                     45                          5
                              14            9         16            5
      3       S3                                                         40
                                   10                      30
            demand    45            20               30     30

   z=6(10)+10(25)+9(45)+13(5)+9(10)+5(30)=1020




                                                                                 53
  Exercise:use the transportation simplex
           method to solve the problem.
         vj =

                  D1        D2           D3           D4       supply
ui =

                       3        11            3        10
          S1                                                    17
                                     4            13
                       1         9            2            8
          S2                                                    24
                 13                  11
                       7         4        10               5
          S3                                                    19
                           16                     3
        demand    13        16           15           16


       z=3(4)+10(13)+1(11)+2(11)+4(16)+5(3)=256
                                                                        54
Exercise:use the transportation simplex
         method to solve p.382 #1 #2




                                          55
Stepping Stone Method (階石法)
1. 對於每一個NBV xij 對應一個邊際成本 c
                             ij
2. 邊際成本c :以該NBV為entering variable, 其運輸量
        ij
           每增加1單位總運輸成本的增加量。




                                          56
               D1            D2             D3            D4        supply
                    3         11                 3             10
  S1                                                                 17
         x11                           4             13
                    1             9              2             8
  S2                                                                 24
         13                            11
                    7             4          10                5
  S3                                                                 19
                        16                           3
demand         13            16             15            16

               D1            D2             D3            D4        supply
                    3             11             3         10
  S1                                                                 17
                        x12            4             13
                    1             9              2             8
  S2                                                                 24
         13                            11
                    7             4          10                5
  S3                                                                 19
                        16                           3
demand         13            16             15            16                 57
              D1            D2             D3             D4        supply
                   3         11                 3              10
  S1                                                                 17
                                      4             13
                   1             9              2              8
  S2                                                                 24
         13                           11            x24
                   7             4          10                 5
  S3                                                                 19
                       16                           3

demand        13            16             15             16

              D1            D2             D3             D4        supply
                   3             11             3          10
  S1                                                                 17
                                      4             13
                   1             9              2              8
  S2                                                                 24
         13                           11
                   7             4          10                 5
  S3                                                                 19
                       16             x33           3
demand        13            16             15             16                 58
              D1            D2             D3            D4       supply
                   3         11                 3         10
  S1                                                               17
                                      4             13
                   1             9              2             8
  S2                                                               24
         13                           11
                   7             4          10                5
  S3                                                               19
         x31           16                           3
demand        13            16             15            16


              D1            D2             D3            D4       supply
                   3             11             3         10
  S1                                                               17
                                      4             13
                   1             9              2             8
  S2                                                               24
         13            x22            11
                   7             4          10                5
  S3                                                               19
                       16                           3
demand        13            16             15            16                59
  C11 = C11 - C13 + C23 - C21 = 1 > 0
  C12 = C12 - C14 + C34 - C23 = 2 > 0
◎ C24 = C24 - C14 + C13 - C23 = -1 < 0                      x24:entering variable
  C33 = C33 - C34 + C14 - C13 = 12 > 0
  C31 = C31 - C21 + C23 - C13 + C14 - C34 = 10 > 0
  C22 = C22 - C23 + C13 - C14 + C34 - C32 = 1 > 0

                 D1            D2            D3            D4        supply
                      3         11                3             10
      S1                                                              17
                                        4             13
                      1             9             2             8
      S2                                                              24
            13                          11            x24
                      7             4         10                5
      S3                                                              19
                          16                          3
   demand        13            16            15            16

  Min{13,11}=11  x23:leaving variable                                          60
              D1            D2          D3               D4        supply
                   3         11                  3            10
  S1                                                                17
                                      4+11           13-11
                                                                            x24:EV
                   1             9    x23        2   X24       8
  S2                                                                24      x23:LV
         13                           11-11          0+11
                   7             4           10                5
  S3                                                                19
                       16                            3
demand        13            16          15               16

              D1            D2              D3            D4       supply
                   3             11              3            10
  S1                                                                17
                                      15             2                      Z=245<256
                   1             9               2             8
  S2                                                                24
         13                                          11
                   7             4           10                5
  S3                                                                19
                       16                            3
demand        13            16             15             16                         61
              D1            D2            D3            D4       supply
                   3         11                3         10
  S1                                                              17        Next
                                     15            2
                                                                            x11:EV
                   1             9             2             8
  S2                                                              24        x14:LV
         13                                        11
                   7             4         10                5
                                                                            why?
  S3                                                              19
                       16                          3
demand        13            16            15            16

              D1            D2            D3            D4       supply
                   3         11                3         10
  S1                                                              17
         2                           15                                   Z=245
                   1             9             2             8
  S2                                                              24
         11                                        13                     degeneracy
                   7             4         10                5
  S3                                                              19
                       16                          3
demand        13            16            15            16
                                                                                  62
                                                p.393
7.5. Assignment Problems
 Assignment problems(指派問題) are a certain class
  of transportation problems for which transportation
  simplex is often very inefficient.
 In general an assignment problem is balanced
  transportation problem in which all supplies and
  demands are equal to 1.
 The assignment problem’s matrix of costs is its
  cost matrix(成本矩陣).
 All the supplies and demands for this problem are
  integers which implies that the optimal solution
  must be integers.
 Using the minimum cost method a highly
  degenerate bfs is obtained.
                                                      63
                                                 p.393
Example 4: Machine Assignment Problem
 Machineco has four jobs to be completed.
 Each machine must be assigned to complete one job.
 The time required to setup each machine for
  completing each job is shown.
     Table 43
                           Time (Hours)
                    Job1   Job2   Job3    Job4
      Machine   1    14      5      8       7
      Machine   2     2      12     6       5
      Machine   3     7      8      3       9
      Machine   4     2      4      6      10

 Machineco wants to minimize the total setup time
  needed to complete the four jobs.
                                                     64
 Solution :
Machineco must determine which machine should be
assigned to each job.
  i, j =1,2,3,4
  xij=1 (if machine i is assigned to meet the demands of job j)
  xij=0 (if machine i is not assigned to meet the demands of job j)
min Z  14x11  5x12  8x13  7x14  2x21  12x22  6x23  5x24
            7x31  8x32  3x33  9x34  2x 41  x 42  6x 43  10x 44
  s.t. x11  x12  x13  x14  1       (Machine constraint)
       x21  x22  x23  x24  1
       x31  x32  x33  x34  1
       x 41  x 42  x 43  x 44  1
       x11  x21  x31  x 41  1       (Job constraint)
       x12  x22  x32  x 42  1
       x13  x23  x33  x 43  1
       x14  x24  x34  x 44  1
                                                                         65
       xij  0 or Xij  1
指派問題之求解方法

 Simplex method (單體法)
 Transportation Simplex Method (運輸單體法)
 Hungarian Method (匈牙利法)




                                          66
Initial Solution
     — Minimum Coat Method

   Table 44   J1       J2       J3       J4        supply
               14           5        8        7
     M1                                              1

                   2    12           6        5
     M2                                              1

                   7        8        3        9
     M3                                              1

                   2        4        6        10
     M4                                              1

   demand     1        1        1        1



                                                            67
Initial Solution
     — Minimum Coat Method

   Table 44       J1           J2           J3           J4        supply
                   14               5            8            7
     M1                                                              1
                           1
                       2        12               6            5
     M2                                                              1
                                                     1
                       7            8            3            9
     M3                                                              1
                                        1
                       2            4            6            10
     M4                                                              1
              1
   demand         1            1            1            1



                                                                            68
Initial Solution
     — Minimum Coat Method

   Table 44       J1           J2           J3           J4        supply
                   14               5            8            7
     M1                                                              1
                           1            0            0
                       2        12               6            5
     M2                                                              1
                                                     1
                       7            8            3            9
     M3                                                              1
                                        1
                       2            4            6            10
     M4                                                              1
              1            0
   demand         1            1            1            1


     The current BFS is highly degenerate.
                                                                            69
Optimal Solution
     —Transportation Simplex Method

   Table 45       J1           J2           J3           J4        supply
                   14               5            8            7
     M1                                                              1
                           1                         0
                       2        12               6            5
     M2                                                              1
                                                     1
                       7            8            3            9
     M3                                                              1
                                        1
                       2            4            6            10
     M4                                                              1
              1            0            0
   demand         1            1            1            1



                                                                            70
                                               p.395
 The Hungarian Method (匈牙利法)

Step1 Find a bfs. Find the minimum element in each
      row of the m x m cost matrix. Construct a
      new matrix by subtracting from each cost the
      minimum cost in its row. For this new matrix,
      find the minimum cost in each column.
      Construct a new matrix (reduced cost matrix)
      by subtracting from each cost the minimum
      cost in its column.




                                                      71
Step2 Draw the minimum number of lines
      (horizontal and/or vertical) that are needed
      to cover all zeros in the reduced cost matrix.
      If m lines are required , an optimal solution
      is available among the covered zeros in the
      matrix. If fewer than m lines are required,
      proceed to step 3.
Step3 Find the smallest nonzero element (call its
      value k) in the reduced cost matrix that is
      uncovered by the lines drawn in step 2. Now
      subtract k from each uncovered element of
      the reduced cost matrix and add k to each
      element that is covered by two lines. Return
      to step2.

                                                       72
匈牙利法之步驟
1. 建立成本表:方陣
2. 簡化列:列-row min
3. 簡化行:行-col min
4. 最佳測試性:最少劃0線,若線條數=列數,則進行指派
5. 進一步簡化成本表
   (a)未劃線元素 - 最小未劃線元素
   (b)劃二線元素 + 最小未劃線元素
   (c)返回4.




                               73
                                                               p.396
Table 46                       row min   Table 47
14    5        8       7          5       9     0   3   2
 2    12       6       5          2       0    10   4   3
 7    8        3       9          3       4     5   0   6
 2    4        6       10         2       0     2   4   8
                                          0     0   0   2   col min

以最少之線條劃過所有的0                             未劃線元素 - 最小未劃線元素
                                         劃二線元素 + 最小未劃線元素
  Table 48                               Table 49
  9        0       3       0             10    0    3   0
  0    10          4       1             0     9    3   0

  4        5       0       4             5     5    0   4

  0        2       4       6             0     1    3   5

                                最少劃0線 若線條數=列數 則進行指派74
Exercise:p. 398 #1
 Optimal Solution -The Hungarian Method
   Table 50
              J1   J2   J3   J4   J5   Row Min
     P1       22   18   30   18   0       0
     P2       18   M    27   22   0       0
     P3       26   20   28   28   0       0
     P4       16   22   M    14   0       0
     P5       21   M    25   28   0       0
   Col Min    16   18   25   14   0

              J1   J2   J3   J4   J5
     P1        6    0    5    4   0
     P2        2   M     2    8   0
     P3       10    2    3   14   0
     P4        0    4   M     0   0
     P5        5   M     0   14   0
                                                 75
          J1    J2    J3   J4       J5
   P1      6     0     5    4       0
   P2      2    M      2    8       0    #(col)=5
   P3     10     2     3   14       0    #(line)=4
   P4      0     4    M     0       0
   P5      5    M      0   14       0

The smallest uncovered element is 2.
Subtract 2 from uncovered costs.
Add 2 to all twice covered cost.

          J1    J2    J3   J4       J5
   P1      6     0     7    4       2
   P2      0    M      2    6       0    #(col)=5
   P3      8     0     3   12       0    #(line)=5
   P4      0     4    M     0       2
   P5      3    M      0   12       0

An optimal solution is available.
                                                     76
Assignment
      J1   J2    J3   J4   J5        J1   J2   J3   J4   J5
 P1   6      0   7    4    2    P1   6    0    7    4    2
 P2   0      M   2    2    0    P2   0    M    2    2    0
 P3   8      0   3    12   0    P3   8    0    3    12   0
 P4   0      4   M    0    2    P4   0    4    M    0    2
 P5   3      M   0    12   0    P5   3    M    0    12   0

      J1   J2    J3   J4   J5        J1   J2   J3   J4   J5
 P1   6      0   7    4    2    P1   6    0    7    4    2
 P2   0      M   2    2    0    P2   0    M    2    2    0
 P3   8      0   3    12   0    P3   8    0    3    12   0
 P4   0      4   M    0    2    P4   0    4    M    0    2
 P5   3      M   0    12   0    P5   3    M    0    12   0

Person 3 is not assigned any job.Total time =18+18+14+25=75
                                                              77
Exercise:p. 398 #2
 Optimal Solution -The Hungarian Method
 Table 51
               Free   Breast   Fly   Back   Row Min
  Hall          54      54     51     53      51
  Spitz         51      57     52     52      51
  Montgomery    50      53     54     56      50
  Jastremski    56      54     55     53      53

               Free   Breast   Fly   Back
  Hall          3       3       0     2
  Spitz         0       6       1     1
  Montgomery    0       3       4     6
  Jastremski    3       1       2     0
  Column Min    0       1       0     0

                                                      78
             Free   Breast   Fly   Back
Hall          3       2       0     2     #(col)=4
Spitz         0       5       1     1     #(line)=3
Montgomery    0       2       4     6
Jastremski    3       0       2     0


             Free   Breast   Fly   Back
Hall          4       2       0     2     #(col)=4
Spitz         0       4       0     0     #(line)=4
Montgomery    0       1       3     5
Jastremski    4       0       2     0



                                                      79
 Exercise:p. 399 #3a
  Optimal Solution -The Hungarian Method
 max z=7x11 + 5x12 + 8x13 + 2x14 +... + 5x41 + 5x42 +             6x43 + 7x44
   s.t x11 + x12 + x13 + x141 (TC)
        x21 + x22 + x23 + x241 (FP)
        x31 + x32 + x33 + x341 (HF)
        x41 + x42 + x43 + x441 (ML)
        x11 + x21 + x31 + x411 (JA)
        x12 + x22 + x32 + x421 (CC)
        x13 + x23 + x33 + x431 (GP)
        x14 + x24 + x34 + x441 (JR)    xij0

Transform Max problem to Min problem, Multiplying benefits by (-1).
    Table 52
         JA    CC GP JR                    JA   CC GP JR
    TC   7     5   8    2             TC   -7   -5      -8   -2
    FP   7     8   9    4             FP   -7   -8      -9   -4
    HF   3     5   7    9             HF   -3   -5      -7   -9
    JR   5     5   6    7             JR   -5   -5      -6   -7            80
                                          JA   CC   GP   JR
      JA   CC GP        JR Row Min
                                     TC   0    2    0    6
TC    -7   -5   -8      -2      -8
                                     FP   1    0    0    5
 FP   -7   -8   -9      -4      -9
                                     HF   5    3    2    0
HF    -3   -5   -7      -9      -9
                                     JR   1    1    1    0
 JR   -5   -5   -6      -7      -7
                                          JA   CC   GP   JR
                                     TC   0    2    0    7
      JA   CC   GP      JR
TC    1    3        0       6        FP   1    0    0    6

 FP   2    1        0       5        HF   4    2    1    0
HF    6    4        2       0        JR   0    0    0    0

 JR   2    2        1       0             JA   CC   GP   JR
Col   1    1    0       0            TC   0    2    0    7
Min                                  FP   1    0    0    6
                                     HF   4    2    1    0
                                     JR   0    0    0    0    81
                                                p.400
7.6 Transshipment Problems
 A transportation problem(轉運問題) allows only
  shipments that go directly from supply points to
  demand points.
 Shipments are allowed between supply points or
  between demand points.
 Sometimes there may also be points (called
  transshipment points) through which goods can be
  transshipped on their journey from a supply point
  to a demand point.
 Fortunately, the optimal solution to a
  transshipment problem can be found by solving a
  transportation problem.

                                                      82
Step 1 If necessary, add a dummy demand point
  (with a supply of 0 and a demand equal to the
  problem’s excess supply) to balance the problem.
  Shipments to the dummy and from a point to
  itself will be zero. Let s= total available supply.
Step2 Construct a transportation tableau as follows:
  A row in the tableau will be needed for each
  supply point and transshipment point, and a
  column will be needed for each demand point and
  transshipment point.




                                                    83
 Each supply point will have a supply equal to it’s
  original supply, and each demand point will have a
  demand to its original demand.
    Let s= total available supply.
    Then each transshipment point will have
     a supply equal to (point’s original supply)+s and
     a demand equal to (point’s original demand)+s.
    This ensures that any transshipment point that is
     a net supplier will have a net outflow equal to
     point’s original supply and a net demander will
     have a net inflow equal to point’s original demand.
    Although we don’t know how much will be shipped
     through each transshipment point, we are sure
     that the total amount will not exceeds.

                                                       84
                                                                              p.400
Example 5:
•source:Memphis(150), Denver(200)
•Destination:L.A.(130), Boston(130)
•Transshipment:N.Y., Chicago (Source & Destination)

 Table 59    N.Y.       Chicago        L.A.         Boston      Dummy         supply
                    8           13            25           28             0
 Memphis                                                                       150
            130                                                 20
                  15            12            26           25             0
  Denver                                                                       200
                                                   130          70
                    0           6             16           17             0
   N.Y.                                                                        350
            220                      130
                    6           0             14           16             0
 Chicago                                                                       350
                        350
 demand       350         350          130           130             90
                                                                                       85
 Exercise:p.403 #1a
Table 60
           LA     Detroit Atlanta Houston Tampa supply
LA            0      140       100     90     225 1100
Detroit     145         0      111    110     119 2900
Atlanta     105      115         0    113      78
Houston      89      109       121      0   -
Tampa       210      117        82   -          0
demand                             2400     1500
           LA     Detroit Atlanta Houston Tampa Dummy    supply
LA            0      140       100     90    225     0    5100
Detroit     145         0      111    110    119     0    6900
Atlanta     105      115         0    113     78     0    4000
Houston      89      109       121      0   M        0    4000
Tampa       210      117        82  M          0     0    4000
           4000     4000      4000  6400    5500   100       86
 Exercise:p.403 #1b

          LA       Detroit Atlanta Houston Tampa Dummy
LA           0        140       100     90    225     0   5100
Detroit    145           0      111    110    119     0   6900
Atlanta    105        115         0    113     78     0   4000
Houston     89        109       121      0   M        0   4000
Tampa      210        117        82  M          0     0   4000
          4000       4000      4000  6400    5500   100

          LA       Detroit Atlanta Houston Tampa Dummy
LA             0     M          100     90    225     0   5100
Detroit   M              0      111    110    119     0   6900
Atlanta    105        115         0    113     78     0   4000
Houston     89        109       121      0   M        0   4000
Tampa      210        117        82  M          0     0   4000
          4000       4000      4000  6400    5500   100      87
 Exercise:p.403 #2
Table 61
           Well1   Well2   Mobile   Galv.   N.Y.    L.A.
Well 1      0       -       10       13     25       28      150
Well 2      -       0       15       12     26       25      200
Mobile      -       -        0       6      16       17
Galv.       -       -        6       0      14       16
N.Y.        -       -        -        -      0       15
L.A.        -       -        -        -     15       0
                                            140     160
       Mobile      Galv.    N.Y.     L.A.   Dummy   supply
Well 1  10          13      25        28      0      150
Well 2  15          12      26        25      0      200
Mobile   0          6       16        17      0      350
Galv.    6          0       14        16      0      350
N.Y.    M           M        0        15      0      350
L.A.    M           M       15        0       0      350
demand 350         350      490      510      50    1050
                                                                   88
Exercise:p.403 #3
Table 61
           Mobile   Galv.   NY     LA   Dummy   Supplies
Well 1      10       13      25   28      0       150
Well 2      15       12      26   25      0       200
Mobile       0       6       16   17      0
Galv.        6       0       14   16      0
NY          M        M       0    15      0
LA          M        M       15     0     0
Demands                     140   160     50
       Mobile Galv.         NY     LA   Dummy   Supplies
Well 1 10+12 13+10           M     M      0       150
Well 2 15+12 12+10           M     M      0       200
Mobile   0     6             16   17      0       350
Galv.    6     0             14   16      0       350
NY       M     M             0    15      0       350
LA       M     M             15     0     0       350
Demands 350   350           490   510     50               89
Exercise:p.403 #4

          Mobile Galv.   NY     LA   Dummy   Supplies
Well 1    10+12 13+10     M     M      0       150
Well 2    15+12 12+10     M     M      0       200
Mobile      0     6       16   17      0
Galv.       6     0       14   16      0
NY          M     M       0    15      0
LA          M     M       15     0     0
Demands                  140   160     50
       Mobile Galv.      NY     LA   Dummy   Supplies
Well 1 10+12 13+10        M     M       0      150
Well 2 15+12 12+10        M     M       0      200
Mobile   0     6          16   17       0      180
Galv.    6     0          14   16       0      150
NY       M     M          0    15       0      350
LA       M     M          15     0      0      350
Demands 180   150        140   160     50               90
Transportation Problems--總供給小於總需求

               D1   D2   D3   Supply
        S1     3    7    8     18
        S2     9    4    6      7
        S3     10   11   5     15
      Demand   21   16   13


     總供給= 40, 總需求=50

               D1   D2   D3   Supply
        S1      3    7    8     18
        S2      9    4    6      7
        S3     10   11    5     15
        S4      0    0    0     10
      Demand   21   16   13
                                       91
Transportation Problems--總供給大於總需求

               D1   D2   D3   Supply
        S1     3    7    8     21
        S2     9    4    6     16
        S3     10   11   5     13
      Demand   18   7    15


     總供給= 50, 總需求=40

               D1   D2   D3     D4     Supply
        S1      3    7    8      0      18
        S2      9    4    6      0       7
        S3     10   11    5      0      15
      Demand   21   16   13     10


                                                92
Example: 總供給大於總需求
           D1   D2   D3   D4   Supply
    S1     3    7    8    0     18
    S2     9    4    6    0      7
    S3     10   11   5    0     15
  Demand   21   16   13   10    50


若來源貨物無法送出時,分別導致每單位3、1、2的存貨成本

           D1   D2   D3   D4   Supply
    S1     3    7    8    3     18
    S2     9    4    6    1      7
    S3     10   11   5    2     15
  Demand   21   16   13   10    50


                                        93
Example: 總供給大於總需求
           D1   D2   D3   D4   Supply
    S1     3    7    8    0     18
    S2     9    4    6    0      7
    S3     10   11   5    0     15
  Demand   21   16   13   10    50


  若S1之貨物必須全部送出,以便挪出空間作為他用

           D1   D2   D3   D4   Supply
    S1     3    7    8    M     18
    S2     9    4    6    0      7
    S3     10   11   5    0     15
  Demand   21   16   13   10    50


                                        94
建立轉運供需表之步驟

1. 建立平衡之轉運供需表,ΣSi=ΣDj=T
2. 增設來源地於列;增設目的地於行。
3. 填入各路徑之單位成本。通常本站到本站之單位成本為0 。
   任兩點來回之單位成本可能相同,也可能不相。
4. 可供轉運之起點供給量增加T;可供轉運之終點供給量增加T;
   所有新增之點(無論供給量或需求量)其量均為T 。




                             95
Example :來源地與目的地不為轉運點
來源地(S1,S2)與目的地(D1,D2)之供給與需求為:
s1=75, s2=60, d1=55, d2=85 。
其運輸成本表為如下,則轉運供需表為何?

      S1   S2   D1   D2                D1   D2 supply
 S1   0    7    16   25         S1     16   25   75
 S2   7    0    12   16        S2     12   16   60
 D1   15   13   0    8          S3     0    0     5
 D2   23   17   9    0        demand   55   85   140




                                                        96
Example :來源地與目的地均為轉運點

          S1   S2   D1    D2
    S1    0    7    16    25
    S2    7    0    12    16
    D1    15   13   0     8
    D2    23   17   9     0


               S1        S2     D1     D2     supply
     S1        0         7      16     25     75+140
    S2        7         0      12     16     60+140
     D1        15        13     0      8       0+140
     D2        23        17     9      0       0+140
     S3        0         0      0      0        5
    demand 0+140        0+140 55+140 85+140    700     97
運輸問題之退化解

 若BV個數少於(m+n-1)時,即產生退化解
 可能發生於起始解階段,也可能發生於最佳解階段
 起始解階段:
    若供給點之供給量累加至某一階段與
    需求點之需求量累加至某一階段恰好相等
 最佳解階段:
    對於BV及NBV所對應之封閉迴路,若有2個以上最小
    負號轉移數量相等,則下一個BV產生退化解




                                98
運輸問題之退化解--起始解階段
                D1            D2             D3       supply 解決方法:
                     5             8              7                    加一BV,其數量為0,以ε表之
  S1                                                      35
           25            10
                     4             7              6                    最好選在轉角,使其產生封閉迴路
  S2                                                      30
                         30             ε
                     7             6          10
  S3                                                      45
                                        45
demand          25            40             45

                 vj            5             8                7
   ui                          D1            D2               D3       supply
                                    5             8       0        7
       0        S1                                                      35
                          25            10
                           0        4             7                6
   -1           S2                                                      30
                                      30              ε
                          -1        7 -5          6            10
       3        S3                                                      45
                                                      45
                                                                                         99
           demand              25            40               45
       vj          5             8                7
ui                 D1            D2               D3       supply
                        5             8       0        7
0      S1                                                   35
              25            10
               0        4             7                6
-1     S2                                                   30
                          30              ε
              -1        7 -5          6            10
3      S3                                                   45
                                          45
     demand        25            40               45


       vj          5             8                12
ui                 D1            D2               D3       supply
                        5             8   -5           7
0      S1                                                   35
              25          10
              +5        4 +5          7                6
-6     S2                                                   30
                                          30
                        7             6            10
-2     S3                                                   45
              +4            30            15
     demand        25            40               45
                                                                    100
       vj      5             8             12
ui             D1            D2            D3       supply
                    5             8   -5        7
0      S1                                            35
              25   10
              +5 4 +5             7             6
-6     S2                                            30
                                      30
              +4 7                6         10
-2     S3                                            45
                        30            15
     demand    25            40            45


       vj      5             3             7
ui             D1            D2            D3       supply
                    5 +5          8             7
0      S1                                            35
              25                      10
              +0 4 +5             7             6
-1     S2                                            30
                                      30
              -1    7             6         10
3      S3                                            45
                        40             5
     demand    25         40               45
                                                             101
       vj          5             8             12
ui                 D1            D2            D3       supply
                        5 +5          8   -5        7
0      S1                                                35
              25          10
               0        4 +5          7             6
-6     S2                                                30
                                          30
              -1        7             6         10
-2     S3                                                45
                            30            15
     demand        25            40            45


       vj          5             4             7
ui                 D1            D2            D3       supply
                        5 +4          8             7
0      S1                                                35
              20                          15
              0         4 +4          7             6
-1     S2                                                30
                                          30
                        7             6   +1 10
2      S3                                                45
               5            40
     demand        25         40               45
                                                                 102
運輸問題之退化解—最佳解階段
             D1           D2           D3       supply 解決方法:
                  5   -        3   +        7            封閉迴路中負號(-)之數量最小者,
  S1                                             11
                      5            6                     成本較小者留著,數量設為0;
         -        4   +        7            6            成本較大者則成為NVB
  S2                                             12
         5            7
         +        7            6   -    10
  S3                                              8
                                   8
demand       5            12           14        31

             D1           D2           D3       supply
                  5            8            7
  S1                                             35
                      0            11
                  4            7            6
  S2                                             30
                      12
                  7            6        10
  S3                                             45
         5                         3
demand       25           40           45                               103
運輸問題之退化解—最佳解階段
             D1           D2           D3       supply 解決方法:
                  5            8            7         封閉迴路中負號(-)之數量最小者,
  S1                                             35
                      0            11                 成本較小者留著,數量設為0;
                  4            7            6         成本較大者則成為NVB
  S2                                             30
                      12
                  7            6        10
  S3                                             45
         5                         3
demand       25           40           45




                                                                     104

				
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