In case you're using AP instances, you have to use the Euclidean distance measure for the given coordinates to compute the nodetonode distance. This would be your initial distance matrix. In case you're using CAB instances, you already have the distance matrix. To compute the transportation costs for each origindestination pair given a specific hub pair (note that the hub pair could of consists of just on hub node), you have to sum three costs: consolidation (origin to first hub), transportation (first hub to second hub), and distribution (second hub to destination). Each of these terms are computed as the corresponding distance multiplied by its respective economies of scale coefficient (namely \(\chi\), \(\alpha\), and \(\sigma\)). Depending on the formulation you're using, the best way to compute the cost would differ. For example, in case your using the EK 3index formulation, you have to consider distance matrix as the cost matrix and then multiply the necessary cost parameters in the model by their respective economies of scale coefficients. In case you're using the Campbell's 4index formulation, you should avoid defining a 4index parameter for the cost (as represented in the model) and just compute the cost for each allocation decision variable using the expanded formula (\((\chi.c_{ik}+\alpha.c_{km}+\sigma.c_{mj}).X_{ijkm}\)). answered 25 Apr '13, 07:51 Ehsan ♦ thanks for your answer. It was really efficient. firstly, what do you mean by EK 3index and 4index.would you please explain more? my model is expansion of campbell hub location problem. I solved my model using random generated cost matrix,but now I have to compute the exact cost.
(26 Apr '13, 15:47)
nadia
In the Campbell formulation (I attributed it to O'Kelly by mistake), the allocation variables are denoted by 4 indices (\(X_{ijkm}\)). Hence, it is sometimes referred to as the "4index formulation. In 1996, Ernst and Krishnamoorthy proposed a new formulation with variables of at most 3 indices (see the paper here). The 4index formulation results in tighter LPR bounds, but it requires huge amount of memory. On the other hand, the 3index formulation has a weaker LPR bound but requires much less amount of memory.
(26 Apr '13, 16:04)
Ehsan ♦
thank you. I think I got the point. do you know how I can get access to CAB data set? I looked for it every where in the internet, but I did not find any thing.
(27 Apr '13, 08:00)
nadia
I have found it finally. thanks :D
(28 Apr '13, 05:10)
nadia
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