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It is stated in the text titled: Linear Programming, by: James Hadley; as introduction to the Transportation problem, on page #7-8, with the last line, as stated below, causing confusion; as it states the nonzero coefficients of $x_{ij}$ to be $1,$ though that shows the number of units transported from origin $i$ to the destination $j.$

Hence, the nonzero coefficients can be any positive value permissible, and should be the goal to find; given the values for $c_{ij}$ :

A typical transportation problem can be described as follows:

$\cdots$
Assuming that it is possible to ship from any one warehouse to any one retail outlet, we are interested in determining the minimum-cost routing from the warehouses to the retail outlets.
We shall suppose that there are $m$ origins and $n$ outlets. Take $x_{ij}$ to be the number of units, shipped from origin $i$ to the destination $j.$
$\cdots$
Here, we have a total of $mn$ terms for $x_{ij}.$
Let $a_i$ be the number of units of the product available at origin $i,$ and $b_j$ be the number of units required at destination $j.$
We cannot ship more goods from any one origin than are available at that origin. Hence, summing over all destinations, we have
$\sum_{j=1}^nx_{ij} = x_{i1}+ x_{i2}+ \cdots+x_{in}\leq a_i, i = 1,2, \cdots, m$
There are $m$ such constraints, one for each origin. We must supply each destination with the number of units desired; thus
$\sum_{j=1}^mx_{ij} = x_{i1}+ x_{i2}+ \cdots+x_{in}= b_j, j = 1,2, \cdots, n$
The total amount received at any destination is the sum over the amounts received from each origin. The needs of the outlets can be satisfied iff
$\sum_{i=1}^ma_{ij} \geq \sum_{j=1}^nb_j$
$\vdots$
We now summarize the transportation problem as follows:
Find $x_{ij} \geq 0$ which minimize $z = \sum_{j=1}^n \sum_{i=1}^m c_{ij}x_{ij},$ subject to
$\sum_{j=1}^mx_{ij} \leq a_i,\ \ i = 1,2,\cdots, m,$
$\sum_{j=1}^mx_{ij} = b_j,\ \ j = 1,2,\cdots, n.$
This is a linear programming problem in $mn$ variables, with $m+n$ constraints.
It will be observed that the constraints in the transportation problem have a particularly simple form.
All the nonzero coefficients of the $x_{ij}$ are ones.
$\vdots$

jiten
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1 Answers1

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They are referring to the coefficient.

$x_{ij}$ is the number of units from $i$ to $j$ but in the constraint, the coefficient is binary.

$$\color{blue}1 \cdot _{1}+\color{blue}1\cdot _{2}+\ldots +\color{blue}1\cdot _{} \le _$$

The coefficients are binary because we just need to sum up the units going from/to a particular location.

Siong Thye Goh
  • 153,832
  • Seems the author means that the value of $x_{ij}$ can be either zero, or positive; depending on if there is any unit transported from origin $i$ to outlet $j,$ or not. So, if it is nonzero, then need to have for each of the $m$ such constraints (one for each origin) of the form: $\sum_{j=1}^nx_{ij} = x_{i1}+ x_{i2}+ \cdots+x_{in}\leq a_i, i = 1,2, \cdots, m;$ an individual term denoted as: $1.x_{ij}.$ Else, that individual term would anyway be $0.x_{ij}.$ – jiten Dec 07 '24 at 06:37
  • Please see my Edit #2, in my post here: https://math.stackexchange.com/q/5015985/424260 – jiten Dec 26 '24 at 15:06