Middle-Mile Consolidation Network Design

This is the base model MMC of [Greening23] with only the truckload mode. Lead-time requirements are handled by the route sets, which contain only lead-time-feasible routes. Variables: \(x_r\in\{0,1\}\) selects route \(r\), and \(f_l\in\mathbb{Z}_{\geq 0}\) is the number of truckloads dispatched per week on leg \(l\in\mathcal{L}\), where \(\mathcal{L}\) holds the vendor-to-FC, FC-to-FC, and FC-to-LMD legs. The leg volume \(v_l\) is substituted out and the truckload lower bound is 0, so the model is

In these instances every candidate route uses at least one leg of \(\mathcal{L}\), so there is no direct vendor-to-LMD option. The single-leg FC-to-LMD routes of FC-origin commodities have \(C_r=0\). An instance has \(|\mathcal{R}|\) binaries, \(|\mathcal{L}|\) general integers, and \(|\mathcal{K}|+|\mathcal{L}|\) constraints.

This is the middle-mile consolidation with waiting times model MMCW of [Greening23]. Vendor-to-FC and FC-to-FC legs use TL only. FC-to-LMD legs may use TL or one of the three LTL buckets (mode set \(\mathcal{M}_l\)). The dispatch frequency of lane \((l,m)\) is chosen from \(\mathcal{F}_{lm}=\{1,\dots,14\}\) for TL and \(\{1,\dots,5\}\) for LTL; in the Very hard-BC and Very hard-BC4 instances, a few high-volume legs (mostly FC-to-FC) allow more TL dispatches, up to 40 per week. Lead times include the waiting delay between dispatches and must be met in expectation (conservatism level \(\rho_r\)). Direct vendor-to-LMD routes are pre-processed: their cost is included in \(C_r\) and they use no leg of \(\mathcal{L}\). Every other physical path usually appears as two route variables, one shipping its final FC-to-LMD leg by TL and one by LTL (a few paths have only the TL variant). \(\bar{\mathcal{R}}\) denotes the non-direct routes, \(l_r\) the final FC-to-LMD leg of \(r\), and \(\mathcal{M}_r\) the modes allowed on \(l_r\) for route \(r\) (\(\{TL\}\) with \(\omega\in\{1,\dots,14\}\), or \(\{LTL1,LTL2,LTL3\}\) with \(\omega\in\{1,\dots,4\}\)). Each non-direct route has a lead-time parameter \(\tau_r=\hat W_r/\rho_r\), where \(\hat W_r\) is the allowable waiting delay (in days) and \(|r|\) is the number of legs of \(r\). Variables: \(x_r\in\{0,1\}\) selects route \(r\), \(z_{lm\omega}\in\{0,1\}\) means lane \((l,m)\) dispatches \(\omega\) loads per week, and \(v_{lm}\geq 0\) is the volume on lane \((l,m)\).

Constraint (4) selects one route per commodity. (5) computes the volume on each leg and splits it among its modes. (6) and (7) set the dispatch frequency from the load-size bounds (\(Q^{\max}=12000,2000,2700,4000\) and \(Q^{\min}=0.1,0.1,2000,2700\) lbs for TL, LTL1, LTL2, LTL3). (8) lets each leg use at most one mode and one frequency. (9) is the linearized lead-time chance constraint \(\rho_r\sum_{l\in r}1/f_l\leq\hat W_r\) divided by \(\rho_r\): the waiting delay on a leg with \(\omega\) weekly dispatches is uniform on \([0,1/\omega]\); in the instances its coefficient is capped at \(1/14\) week, so frequencies above 14 per week do not shorten the counted delay further. (10) ties a route variant to the mode used on its final leg. With \(|\mathcal{L}|\) legs, \(\sum_l|\mathcal{M}_l|\) lanes, and \(|\bar{\mathcal{R}}|\) non-direct routes, an instance has \(|\mathcal{R}|+\sum_{l,m}|\mathcal{F}_{lm}|\) binaries, \(\sum_l|\mathcal{M}_l|\) continuous variables, and \(|\mathcal{K}|+2|\mathcal{L}|+2\sum_l|\mathcal{M}_l|+2|\bar{\mathcal{R}}|\) constraints.