Transportation Problem

Question

I want to create a transportation model. I have identified importing and exporting countries with their estimated demand and supply. The cost of transportation per one uni is already calculated for each ex- and importer. Some exporters transport the hydrogen by ship and others by pipeline.

The problem I have is that I need to introduce losses when it is transported by ship. About 20% is lost and won't reach the importer. For pipeline transport there is no losses. I am having trouble to implement this into the objective function or the constraints. Is there a way to integrate this into the model?

Answer 1

You can add an index to the $x$ variables to indicate whether transport is via pipeline or by ship: $x_{ij}^k$ denotes the quantity shipped on link $(i,j)$ via transport mode $k\in\{\mbox{pipeline, ship}\}$.

The problem becomes: $$ \min \sum_k\sum_i\sum_jc_{ij}^kx_{ij}^k $$ subject to

• supply constraints: $$ \sum_k\sum_jx_{ij}^k \le a_i \quad \forall i \tag{1} $$
• demand constraints: $$ \sum_k\sum_ix_{ij}^k \ge b_j \quad \forall j \tag{2} $$

To take into account $20\%$ loss when transported by ship, you can consider that a dummy copy of node $j$ absorbs the $20\%$ loss. Constraints $(2)$ should be modified as follows:

$$ \sum_k\sum_ix_{ij}^k \ge b_j + 0.2\sum_ix_{ij}^{\mbox{ship}}\quad \forall j \tag{3} $$

Alternatively, as proposed by @Sune, you could simply say that only $80\%$ of the shipped quantity reaches the destination:

$$ \sum_ix_{ij}^{\mbox{pipeline}} + 0.8\sum_ix_{ij}^{\mbox{ship}} \ge b_j \quad \forall j \tag{4} $$

Both constraints are of course equivalent, as $$ \sum_k\sum_ix_{ij}^k = \sum_ix_{ij}^{\mbox{pipeline}} +0.8\sum_ix_{ij}^{\mbox{ship}} +0.2\sum_ix_{ij}^{\mbox{ship}} \tag{5} $$

Answer 2

You can try as below as: $\sum_{i}(x_{i,j}P_i + 0.8x_{i,j}(1-P_i)) \ge b_j \quad \forall j$
where known parameter $P_i = \begin{cases} 1, \text{if transport by pipe from i} \\ 0, \text{by ship} \end{cases}$

Edit: correction made as suggested below.