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Suppose we have a transportation problem similar to pickup and delivery problems. So, we have a set of drivers and a set of passengers. each passenger has predefined origins and destinations. I'd like to know if we could compute the distance between a passenger's destinations and the drop-off nodes without having access to the solver's output.

Every riders could be picked-up and drooped-off by multiple drivers. And there's no particular order on nodes of the graph which is denoted by $N$.

We have a binary variables of $ x_{i,j}^{d,r}$ if driver $d$ drives $r$ on edge $(i,j)$ and we already know for each $r$ their start's $s_r$ and their destination's $d_r$ points ($s_r \in N$ and $d_r in N$). I'm wondering without knowing the path and values of variables from the solver how to minimize the objective as a distance between the last drop-off point and the destinations of each rider. Is this possible to compute this value implicitly and have it as an objective function or do we simply need more information?

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Define binary $e_{i}^r,s_{i}^r$ as drop-off & pickup nodes for rider $r$.
Then assuming a rider $r$ is picked/dropped by same driver
$\sum_d(\sum_j x_{j,i}^{d,r} -\sum_jx_{i,j}^{d,r}) = e_{i}^r - s_{i}^r$

$ s_{i}^r+ e_{i}^r \le 1 \quad \forall i \ \forall r$: A node $i$ can either be origin/destination.

If you don't want to use $ s_i$ then (as used in another response by Dr. Rob)
$ 2e_{i}^r - 1 \le \sum_d(\sum_j x_{j,i}^{d,r} -\sum_jx_{i,j}^{d,r}) \le e_{i}^r$

Then distance is:
if $ d_r$ may either be a node or 2-D location coordinates, then every node $i$ will also have location coordinates. So predefine distance matrix to compute
$ \sum_r \sum_i dist_{d_r,i}e_{i}^r $

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  • $\begingroup$ @linkho yes they are binary variables. If drivers are changing, so summing over driver $d$ makes sense. This is what I have done to make it work because if you imagine a node $i$ with its edges over $j$, along that arc it should be 1 driver. Summing over drivers $d$, basically summing means 'any' driver, should make all incoming edges $\le 1$ & similarly outgoing edges $\le 1$. Then destination is always the node with only incoming edge, no outgoing. Similarly for start its only outgoing. So the relation will be either $ 1-0$ or $ 0-1$ or $ 1-1$: route, $ 0-0$: no travel. $\endgroup$ Mar 19, 2023 at 21:26
  • $\begingroup$ @linkho right, my bad, no summing over $d$ in 2nd constraint. $\endgroup$ Mar 19, 2023 at 21:28
  • $\begingroup$ @linkho, I assumed drop-off node $i$ is part of the arc that solver needs to figure out. Pickup node will be presented by $ s_i$. Final destination for rider $r$ is given by predefined parameter $d_r$. Is $ d_r$ one of the nodes $ i$? Or its a location coordinate? Either way need to preset a distance matrix between all $d_r$ and nodes. Then distance becomes $\sum_i dist[d_r, i]*e_i$ where $ dist[d_r,i]$ is predefined distance between $ d_r$ & every node $i$. I have updated the answer $\endgroup$ Mar 19, 2023 at 22:00
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    $\begingroup$ Yes, still for a rider $r$, if dropped off by driver $d1$, then picked up by driver $d2$, at node say $i$ still there's going to be incoming & outgoing. That's why summing over drivers. Only final drop-off node $i$ will have only 1 incoming, even when summed over all drivers and summing over all other nodes $j$ connected to that node $i$. $\endgroup$ Mar 20, 2023 at 16:26
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    $\begingroup$ So the node $i$ with only 1 incoming, no outgoing (1-0) will turn $e$ to 1. $\endgroup$ Mar 20, 2023 at 16:32

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