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There is an allocation problem, while we have to assign logistics routes to multiple candidate carriers.

For simplicity, let's assume there are only two routes, $A$ and $B$, with two candidate carriers, $P$ and $Q$. The total number of orders for route $A$ is $N_a$ and for route $B$ is $N_b$. Carrier $P$ provides unit prices $P_a$ and $P_b$ for routes $A$ and $B$ respectively, while carrier $Q$ provides unit prices $Q_a$ and $Q_b$ for routes $A$ and $B$ respectively. Both carrier $P$ and $Q$ also provide stepped pricing types based on the number of orders they undertake (i.e., order quantity step), with two steps assumed for simplicity.

For carrier $P$, the discount coefficient for the total freight cost is $c_1$ when the number of orders is between $[0, M_1)$, and $c_2$ when the number of orders is $[M_1, inf)$. The problem is how to allocate the orders for these two routes to the two carriers, $P$ and $Q$, in such a way as to minimize the total cost of the freight.


For example, let's say the total order volumes for routes $A$ and $B$ are $N_a = 100$ and $N_b = 200$, respectively, with unit prices of $P_a = 2$ and $P_b = 3$, and unit prices of $Q_a = 3$ and $Q_b = 2$. The step details are $M_1 = 120$, $c_1 = 0.8$, and $c_2 = 0.6$.

Assuming that carrier $P$ is assigned 60 orders for route $A$ and 80 orders for route $B$, while carrier $Q$ is assigned the remaining orders, i.e., 40 orders for $A$ and 120 orders for $B$, the total cost (without discount) for carrier $P$ is calculated as 60 * 2 + 80 * 3 = 360. The total shipment volume is 60 + 80 = 140, which falls in the second step, with a discount factor of 0.6. Therefore, the total freight charge payable to carrier P is 360 * 0.6.

Similar calculations can be made for carrier $Q$.


To formulate this problem, let $I$ be the set of routes and $J$ be the set of carriers. The decision variable $x_{ij}$ represents the quantity of orders assigned from route $i$ to carrier $j$.

The first constraints is $\sum\limits_{j}{x_{ij}} = N_i$.

Then we introduce a binary variable $\mu_{kj}$ to indicates whether the total number of orders assigned to carrier $j$ follows into the $k$th interval, thus we have $\sum\limits_{k}{\mu_{kj}} = 1,\forall j$.

Then the continuous variable $\pi_{kj}$ to indicate the actual number of orders in interval $k$, thus we have $M_{k-1,j}\mu_{kj} \leq \pi_{kj} \leq M_{kj} \mu_{kj}$

The second constraint is $\sum\limits_{i}{x_{ij}} = \sum\limits_{k}{\pi_{kj}},\forall j$, which represent the number of orders undertake by each carrier $j$.

The total freights of carrier $j$ is $\sum\limits_{i}{p_i x_{ij}}$ multiplied by $c_{kj}$, which indicated by $\mu_{kj}$, will introduce the nonlinearity.

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your $x_{i,j}$ is continuous and $u_{k,j}$ is binary in the expression $ \sum_j p_jx_{i,j}\sum_k c_{j,k}u_{j,k}$ or $ \sum_j \sum_k p_jc_{j,k}x_{i,j}u_{k,j} $. So basically if you wish to linearize $xu$ you can
Introduce a continuous variable (of same domain as $x$), say $z$ and couple of constraint sets

$ 0 \le z_{i,j} \le N_ju_{jk}$: where $N_j$ is total order possible for carrier $j$ or any other bigger constant
$ x_{i,j} + N_j(u_{j,k}-1) \le z_{i,j} \le x_{i,j} + N_j(1-u_{j,k})$

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  • $\begingroup$ thanks, I carefully considered and realized that this would require the introduction of a large number of continuous variables, say $z_{ijk}$. The right handside of the last constraint could simplify to $z_{ijk} \leq x_{ij}$ $\endgroup$
    – Ying
    Sep 24, 2023 at 0:45
  • $\begingroup$ Since for each carrier $j$, only one $\pi_{kj} > 0$, others will be 0. Can we use $\pi$ variables (multiplied by $p_j$ and $c_k$) to represent the total freight? $\endgroup$
    – Ying
    Sep 24, 2023 at 3:29
  • $\begingroup$ Yes $\pi$ summed over $k$ is equivalent to $x$ summed over routes $i$. It will work. $\endgroup$ Sep 24, 2023 at 13:56
  • $\begingroup$ Thank you. I wonder $\pi$ variables represent the freight will eliminate the nonlinearity? $\endgroup$
    – Ying
    Sep 25, 2023 at 1:38

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