### 16.1.3 Transportation problem

The objective of a transportation problem is to minimize the cost of distributing a product from m sources to n destinations. It is determined by three parameters:

- s=(s1,s2,…,sm), the supply vector, where sk∈ℤ+ is the maximum number of units that can be delivered from the kth source for k=1,2,…,m.
- d=(d1,d2,…,dn), the demand vector d=(d1,d2,…,dn) , where dk∈ℤ+ is the minimum number of units required by the kth destination for k=1,2,…,n.
- C=[cij]m× n, the cost matrix, where cij∈ℝ+ is the cost of transporting one unit of product from the ith source to the jth destination, for i=1,2,…,m and j=1,2,…,n.

An optimal solution is represented by the matrix X∗=(xij∗), where xij∗ is number of units that must be transported from the ith source to the jth destination for i=1,2,…,m and j=1,2,…,n.

The tpsolve command solves a transportation problem.

- tpsolve takes three arguments:
  - s, the supply vector.
  - d, the demand vector.
  - C, the cost matrix.
- tpsolve(s,d,C) returns a sequence c,X∗, where X∗ is the optimal solution and c=∑i=1m∑j=1ncij xij∗ is the total (minimal) cost of transportation.
- If the total supply and total demand are equal, i.e. if ∑i=1m si=∑j=1n dj holds, the transportation problem is said to be closed or balanced, otherwise it is said to be unbalanced. The excess supply/demand is covered by adding a dummy demand/supply point with zero cost of “transportation” from/to that point. The tpsolve command handles such cases automatically.
- Sometimes it is desirable to forbid transportation on certain routes. That is achieved by setting a very high cost, represented by a symbol (e.g. M), to these routes. All instances of the symbol in the cost matrix are automatically replaced by a practically infinite number (precisely (m n+1)/epsilon, which forces the algorithm to avoid the associated routes.

#### Examples

s:=[12,17,11]:; d:=[10,10,10,10]:; C:=[[50,75,30,45],[65,80,40,60],[40,70,50,55]]:; tpsolve(s,d,C)

2020,

⎡
⎢
⎢
⎣

0 | 0 | 2 | 10
0 | 9 | 8 | 0
10 | 1 | 0 | 0

⎤
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⎦

s:=[7,10,8,8,9,6]:; d:=[9,6,12,8,10]:; C:=[[36,40,32,43,29],[28,27,29,40,38],[34,35,41,29,31], [41,42,35,27,36],[25,28,40,34,38],[31,30,43,38,40]]:; tpsolve(s,d,C)

1275,

⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣

0 | 0 | 2 | 0 | 5
0 | 0 | 10 | 0 | 0
0 | 0 | 0 | 0 | 5
0 | 0 | 0 | 8 | 0
9 | 0 | 0 | 0 | 0
0 | 6 | 0 | 0 | 0

⎤
⎥
⎥
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⎦

s:=[95,70,165,165]:; d:=[195,150,30,45,75]:; C:=[[15,M,45,M,0],[12,40,M,M,0],[0,15,25,25,0],[M,0,M,12,0]]:; tpsolve(s,d,C)

2820,

⎡
⎢
⎢
⎢
⎣

20 | 0 | 0 | 0 | 75
70 | 0 | 0 | 0 | 0
105 | 0 | 30 | 30 | 0
0 | 150 | 0 | 15 | 0

⎤
⎥
⎥
⎥
⎦