2.4 Exercise 1:

Suppose we have a set of 3 products P = {1, 2, 3} and a set of 4 factory locations F = {1, 2, 3, 4}, where these products are manufactured. The products are transported from one factory to another at unit cost ci,j with i,j F. Furthermore, each product has a price vp with p P. Suppose also, that we transport xp,i,j unities of product p from factory i to factory j with i,j F and p P. The data are as follows (Since x is 3-dimensional,we use the notation xp=1 to “slice” (or extract) the matrix xi,j where p = 1):

                       (              )          (                )
                         -   1  5   2               -   48  37  45
     (       )         | 2  -   4   2 |          | 36   -   43  46|
v =   6  2  3      c = |(              |)   xp=1 = |(                |)
                         1   2  -   3              44   36  -   33
                         1   3  1   -              36   39  34  -

       (                )           (               )
       | -   46   35  39|           | -   32  32  40|
xp=2 = | 32   -   47  35|    xp=3 = | 30  -   41  30|
       ( 45  49   -   39)           ( 45  43  -   45)
         47  46   30  -               44  41  34  -

Using these data, we now can calculate various data using indexed notation (please verify as an exercise):

(1) The total transportation costs T is calculated as follows:

        ∑
T  =          ci,j ⋅ xp,i,j = 3180
     p∈P, i,j∈F

(2) The total value transported is as follows:

       ∑
V =          vp ⋅ xp,i,j = 5213
     p∈P, i,j∈F

(3) The total transportation costs Cp for each product p P is:

      (                    )    (     )
∧.            ∑                   1061
        Cp =      ci,j ⋅ xp,i,j  = ( 1096)
   p∈P         i,j∈F                1023

(4) The value Wi,j transported between all factories (i,j) F × F is:

                                 (                    )
 .      (                   )       -   476  388   468
∧               ∑                || 370   -   475   436||
   i,j∈F  Wi,j =     vp ⋅ xp,i,j  = ( 489  443   -    411)
                 p∈P
                                   442  449  366    -

(5) The maximum quantity Mi transported from a factory i F is:

∧.    (                  )    (               )
        Mi =   max   xp,i,j   =  48  47  49  47
  i∈F        p∈P, j∈F

(6) For each product p P the sum of the maximum quantity Si transported from a factory i F is:

                                  (    )
 .    (          (         ) )      130
∧      S  =  max   ∑   x       =  || 125||
  i∈F    i   p∈P        p,i,j       ( 133)
                   j∈F              123

   2.4.1 Indexed index sets
   2.4.2 Ordering