One-dimensional vehicle scheduling with a front-end depot and non-crossing constraints

Abstract

This paper considers the scheduling problem of multiple vehicles executing a given set of jobs in parallel along a shared pathway. The job set consists of storage and retrieval tasks, transporting goods between a front-end depot and given storage locations on the line. Non-crossing constraints need to be applied to the vehicle movements. This problem setting is relevant when a train is loaded with containers by multiple straddle carriers on the landside of a container terminal. Other potential applications exist in multi-shuttle automated storage and retrieval systems and multi-stage production systems where items are transported by parallel hoists. We formalize the problem, analyze its computational complexity, and develop exact and heuristic solution procedures.

Appendix

1.1 Mixed-integer linear program for VS1D with two vehicles

Given the notation summarized in Table 5, the VS1D with two vehicles can be formulated as a mixed-integer linear program as follows:

$$\begin{aligned} \hbox {Minimize}\; Z(X,Y)=\sum _{k=1}^{l^{\max }} y_k \end{aligned}$$

(1)

subject to

$$\begin{aligned} y_k \ge \sum _{j \in J} x_{jkv} \cdot p_j \qquad&\forall \, k=1,\ldots ,l^{\max }; \, v \in V \end{aligned}$$

(2)

$$\begin{aligned} j \cdot x_{jk1} \le \sum _{j^{\prime } \in J} x_{j^{\prime }k2} \cdot j^{\prime } \qquad&\forall \, k=1,\ldots ,l^{\max }; \, j \in J \end{aligned}$$

(3)

$$\begin{aligned} \sum _{j \in J} x_{jk2} \le 1 \qquad&\forall \, k=1,\ldots ,l^{\max } \end{aligned}$$

(4)

$$\begin{aligned} \sum _{k=1}^{l^{\max }} \sum _{v \in V} x_{jkv} = 1 \qquad&\forall \, j \in J \end{aligned}$$

(5)

$$\begin{aligned} x_{jkv} \in \{0,1\} \qquad&\forall \, j \in J; \, k=1,\ldots ,l^{\max }; \, v \in V \end{aligned}$$

(6)

Using objective function (1), the makespan, i.e., the sum of the cycles’ durations, is minimized. Constraint (2) calculates the cycle duration to be either the maximum sum of processing times assigned to both vehicles, or zero if no job is assigned. Note that $l^{\max }$ is an upper bound on the number of cycles required, which can, for instance, be initialized as $l^{\max }=|J|$. The non-crossing constraint (3) means that the jobs performed by the inner vehicle are always closer to the depot (i.e., have a lower index number) than the job executed by the outer vehicle in each cycle. Constraints (4) and (5) ensure that a single job, at most, can be processed by the outer vehicle during a single cycle and that each job is processed, respectively. Finally, (6) defines the binary variables.

Table 5 Notation for VS1D with two vehicles

1.2 Lower bound on the makespan for VS1D

The lower bound on the objective value of the VS1D problem is based on two considerations:

• First, even if it were possible that all straddle carriers always work in parallel without obstruction, the optimal makespan can obviously never be less than the total workload divided by the number of vehicles; i.e.,

  $$\begin{aligned} {\hbox {LB}}_1 = \left\lceil \frac{\sum \nolimits _{j \in J}{p_j}}{m} \right\rceil . \end{aligned}$$

• Second, if there are $n$ jobs and $m$ vehicles, then at least one vehicle must perform at least $\underline{n} = \left\lceil \frac{n}{m} \right\rceil$ jobs. Let $\underline{J} \subseteq J$ be the set containing the $\underline{n}$ jobs with the lowest processing times. The workload of at least one crane cannot be lower than

  $$\begin{aligned} {\hbox {LB}}_2 = \sum _{j \in \underline{J}}{p_j}. \end{aligned}$$

We can thus derive the lower bound

$$\begin{aligned} {\hbox {LB}} = \max \{{\hbox {LB}}_1; {\hbox {LB}}_2\}. \end{aligned}$$