Logistics scheduling with batching and transportation

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Abstract

This paper studies a general two-stage scheduling problem, in which jobs of different importance are processed by one first-stage processor and then, in the second stage, the completed jobs need to be batch delivered to various pre-specified destinations in one of a number of available transportation modes. Our objective is to minimize the sum of weighted job delivery time and total transportation cost. Since this problem involves not only the traditional performance measurement, such as weighted completion time, but also transportation arrangement and cost, key factors in logistics management, we thus call this problem logistics scheduling with batching and transportation (LSBT) problem.

We draw an overall picture of the problem complexity for various cases of problem parameters accompanied by polynomial algorithms for solvable cases. On the other hand, we provide for the most general case an approximation algorithm of performance guarantee.

Introduction

Most of classical scheduling literature considers job processing without taking into account the delivery issue (see, for example, Blazewicz et al. [2], Brucker [3] and Pinedo [14]) or with the assumption that job delivery can be done instantaneously (see, for example, Herrmann and Lee [9], Chen [4], Cheng et al. [6] and Hall and Potts [8]).

Lee and Chen [10] consider transportation issue in machine scheduling problems, in which there are either a certain number of transporters to transfer jobs from one-machine to another (Type-I), or transfer finished jobs to the customers or warehouses (Type-II). Both transportation time and transporter capacity were considered. Hall et al. [7] consider extensively scheduling problems in an environment in which the finished jobs can be delivered only on fixed dates. Matsuo [12] considers a similar problem in the single machine environment. Li et al. [11] consider a scheduling problem similar to Type-II in [10] yet with two customer locations and hence involving routing issue. None of the above four papers considers the transportation cost. Chen and Vairaktarakis [5] consider a two-stage scheduling problem in which the first stage is manufacturing facility and the second is the delivery to customers, i.e., type-II problem in [10]. Their objective function is a combination of customer service level and total distribution cost, where customer service level is measured as a function of the job delivery times to customers. Wang and Lee [16] also consider a two-stage scheduling problem in which the second stage involves the transportation mode selection. There are two different transportation modes available and the one with shorter delivery time will incur a higher delivery cost.

In this paper, we consider a more general two-stage scheduling problem where in the second stage of transportation there are multiple (more than two) transportation modes to select and multiple destinations. We also incorporate job weights into our model to take job priority issue into account. More specifically, we consider the following problem of logistics scheduling: There are n independent jobs of respective weights w1,,wn, which need to be processed by a single processor for respective uninterrupted p1,p2,,pn time units. The processor can handle at most one job at a time. Once completed on the processor, each job j needs to be delivered to a pre-specified destination sj,sj{1,,k}. Up to B jobs of the same destination may be batched together for simultaneous delivery, which will incur a (batch) transportation time ti(s) and a (batch) transportation cost ci(s), where s{1,,k} and i{1,,m} denote, respectively, the pre-specified destination of the batch of jobs and the transportation mode for delivering the batch. Note that the batch transportation time and cost are independent of the batch size. Delivery of a batch takes place immediately after all jobs of the batch are completed on the processor, independent of other batches.

Without loss of generality, we assume that speedier delivery incurs higher cost, i.e., we assume that t1(s)>t2(s)>>tm(s) and c1(s)<c2(s)<<cm(s) for any destination s=1,,k. The delivery time Dj of job j is Bj+tmj(sj), the completion time Bj on the processor of the last job of the batch job j belongs to plus transportation time tmj(sj) of the batch of job j, where mj is the transportation mode in which the batch of job j is delivered. The problem is to construct a schedule S based on a three-fold decision: (a) the sequence these jobs are processed, (b) how completed jobs are batched, and (c) which transportation mode should be used for delivery of each batch. The objective is to minimize the sum ϕ(S) of weighted job delivery time and total transportation cost in schedule S, i.e.,ϕ(S)=j=1nwjDj+all batchesicm¯i(s¯i),where i is the batch index, m¯i and s¯i are the transportation mode and destination of batch i, respectively. Since this problem involves not only the traditional performance measurement, such as weighted completion time, but also transportation arrangement and cost, key factors in logistics management, we thus call this problem logistics scheduling with batching and transportation (LSBT) problem.

In this paper, under a practical assumption that both the number k of delivery destinations and the number m of available transportation modes are fixed constants, we provide, on the one hand, an overall picture of the problem complexity for various cases of problem parameters, which is accompanied by polynomial algorithms for solvable cases. The picture is complete except for one minor case, which we leave as an open problem. On the other hand, we provide for the most general case an approximation algorithm of performance guarantee. Note that complexity issues are also considered by Soukhal et al. [15] for LSBT problem of flow shop (with equal job weights) and by Albers and Brucker [1] with batch setup times.

The rest of the paper is organized as follows. After establishing a preliminary result in Section 2, we deal with various special cases in Section 3 and general cases in Section 4. Two polynomial approximation algorithms will be provided in Section 5 for the general problem together with analysis of performance guarantee.

Section snippets

Preliminaries

We establish an observation in this section, which will be used in subsequent sections.

Lemma 1

Suppose jobs j1 and j2 are of the same destination. If pj1>pj2 and wj1wj2, then job j1 cannot be processed in a separate batch earlier than the one job j2 belongs to in any optimal schedule.

Proof

Suppose to the contrary that jobs j1 and j2 are processed in two separate batches and job j1 is in an earlier batch. Let C1 < C2 be the respective completion times of the two batches and W1, W2 be their respective sums of

Special cases

In this section we solve to optimality three categories of special cases: (a) the weights are agreeable, i.e., pj < pk implies wj  wk; (b) the number of distinct processing times or distinct weights of jobs of each destination is bounded by a constant, i.e., either {pj(s),j=1,,ns}N or {wj(s),j=1,,ns}N for any s = 1,  , k, where the superscript (s) on the processing time and weight indicates the destination of the corresponding job, ns is the number of jobs of destination s, and N is a

General case

In this section, we do not assume any special condition for the job processing times and weights. We will see our scheduling problem LSBT becomes extremely hard.

Theorem 6

For any fixed B  3, the LSBT problem with batch size bounded by B is strongly NP-hard, even if there is only one transportation mode (m = 1) and one delivery destination (k = 1).

Proof

Let us first assume B = 3. We will transform the strongly NP-complete 3-Partition to the decision version of our scheduling problem. Given any instance of 3-Partition,

Approximation

Given the high difficulty of our problem, we establish some approximation results in this section. Recall that ϕ(S) denotes the objective value of schedule S, i.e., the sum of weighted delivery time and total transportation cost of schedule S. The first result is derived easily from Theorem 4.

Theorem 7

The general LSBT problem can be approximated within a factor of B in O(n log n) time.

Proof

According to Theorem 4, we find in O(n log n) time schedule S1 that is optimal among all schedules without batching. Let S

Acknowledgement

This research is supported in part by Hong Kong RGC Earmark Grant: HKUST 6010/02E.

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