Scheduling

In the deterministic part, it is the treatment of the single machine, the job shop and the open shop that have been expanded considerably. In the stochastic part, an all-new chapter focuses on planning one machine with release dates.

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Introduction

The Role of Scheduling

The processing times for the various operations are proportional to the order size, i.e. the number of bags ordered. One of the objectives of the scheduling system is to reduce the sum of these penalties.

The Scheduling Function in an Enterprise

The shop floor is not the only part of the organization that affects the scheduling process. This may concern the discussion of resources, e.g. the allocation of aircraft to gates (see Example 1.1.3), or the booking of meeting rooms or other facilities.

Fig. 1.1 Information ﬂow diagram in a manufacturing system

Outline of the Book

Actual processing times, release dates, and due dates are only made known upon completion of processing or the actual release or due date display. Appendix E contains a complexity classification of deterministic scheduling problems, while Appendix F presents a summary of stochastic scheduling problems.

Fig. 1.2 Information ﬂow diagram in a service system

Comments and References

A more recent text by Baker (1974) gives an excellent overview of many aspects of deterministic planning. A collection of papers, edited by Zweben and Fox (1994), describes a number of planning systems and their current implementations.

Deterministic Models

Preliminaries

Framework and Notation
Examples
Classes of Schedules
Complexity Hierarchy

It is the time when the job arrives in the system, i.e., the earliest time at which the job can begin its processing. Finish Date (dj) The finish date of the jobj represents the date the shipment was made or finished (ie, the date the work was promised to the customer).

Fig. 2.1 Due date related penalty functions

Exercises (Computational)

Consider the occurrence of 1||Lmax with the following processing times and due dates. a) Find the optimal sequence and calculate the value of the measure. Ouch with the following processing times and due dates. a) Find all optimal sequences and calculate the value of the objective.

Exercises (Theory)

The processing times of the four tasks on both devices are again the same as in Exercise 2.6. One of the first classification schemes for scheduling problems appeared in Conway, Maxwell, and Miller (1967).

Single Machine Models (Deterministic)

The Total Weighted Completion Time

The weighted total completion time of jobs processed before jobs j and k is not affected by the exchange. Neither is the weighted total completion time of jobs processed after jobs j and k.

Fig. 3.1 A pairwise interchange of jobs j and k

Weighted Completion Time and Chains) Whenever the machine is freed, select among the remaining chains the one with

The Maximum Lateness

The ρ factor for the remainder of the first chain is 12/6 and is determined by job 3. The first question that comes to mind is whether a preemptive version of the WSPT rule is optimal.

Minimizing Maximum Cost) Step 1

The Number of Tardy Jobs

The complement of setJ, setJc, indicates the set of tasks that have yet to be scheduled and the subsetJ of Jc indicates the set of tasks that can be scheduled immediately before setJ, that is, the set of tasks that are all inJ successors. Each node at this level has a specific task that is placed in the first position in the schedule.

Fig. 3.3 Proof of optimality of Theorem 3.2.2

Minimizing Number of Tardy Jobs) Step 1

The Total Tardiness - Dynamic Programming

In the following lemma, the sensitivity of an optimal sequence to the due dates is taken into account. Let V(S) denote the total delay during an arbitrary sequence S with respect to the first set of due dates, and let V(S) denote the total delay during sequence S with respect to the second set of due dates.

Minimizing Total Tardiness) Initial Conditions

The Total Tardiness - An Approximation Scheme

An approximation scheme A is called fully polynomial as the value of the target it achieves for example. Tj∗(S) denotes the total slowness under series S with respect to the processing times Kpj and the original due dates and rentals. Moreover, for this choice of K, the time bound O(n5Tmax(EDD)/K) becomes O(n7/4), making the approximation scheme completely polynomial.

FPTAS for Minimizing Total Tardiness) Step 1

The Total Weighted Tardiness
Discussion

Most of the problems described in this chapter can be formulated as mixed integer programs (MIP). This appendix also provides an overview of techniques that can be applied to MIPs. Consider the following preemptive version of the WSPT rule: if pj(t) denotes the remaining processing time of job j at time t, then a preemptive version of the WSPT rule sets at each instant in time the job with wj/pj ( t) ratio in the car.

Advanced Single Machine Models (Deterministic)

The Total Earliness and Tardiness

Since this problem is more difficult than the perfect delay problem, it makes sense to first analyze the special cases that are solvable. Consider the case with the property that no optimal scheduler starts processing its first job at t = 0, i.e. the due date d is a bit loose and the machine sits idle for a while before it starts processing its first job. This means that the total delay is reduced by |J2| times the shift length, while the overall convenience increases by |J1| shift times.

Minimizing Total Earliness and Tardiness with Loose Due Date)

Due to the property described in Lemma 4.1.1, the optimal schedule is often said to have an aV form. For a case where all optimal schedules start processing the first job some time after = 0, the following algorithm yields the optimal assignments of jobs to sets J1 and J2. 72 4 Advanced Single Machine Models (deterministic) whether the machine must actually remain idle before it starts processing its first job.

Minimizing Total Earliness and Tardiness with Tight Due Date)

Due to the different expiration dates, it is not necessarily optimal to process the jobs one after the other without interruption; it may be necessary to have rest periods between the processing of successive tasks. But given a predetermined order of tasks, the processing timings and idle times can be computed in polynomial time. Let us be the last orbit in cluster σr that is early, i.e. the orbit with the least earliness.

Fig. 4.1 Cost functions with common due date and diﬀerent shapes

Optimizing the Timings Given a Sequence) Step 1

Primary and Secondary Objectives

Thus, in an optimal schedule, a set of tasks with identical processing times should be arranged according to the EDD rule. The decision maker must do this for each set of tasks with identical processing times. Subject to the constraint that each job must be completed by its due date, ie. the maximum lateness with respect to the new deadlines must be zero or, equivalently, all the tasks must be completed on time.

Minimizing Total Completion Time with Deadlines) Step 1

Multiple Objectives: A Parametric Analysis

The two cases considered in the previous section are the two extremes of the exchange curve. The two extreme points of the exchange curve in Figure 4.3 therefore correspond to the problems discussed in the previous section. The algorithm that generates all Pareto-optimal solutions in the exchange curve contains two loops.

Determining Trade-Oﬀs between Total Completion Time and Maximum Lateness)

The Makespan with Sequence Dependent Setup Times

It is clear that the two extreme points of the exchange curve can be determined in polynomial time (with WSPT/EDD and EDD). The following lemma quantifies the cost of the exchangerI(j, k) applied to the sequence; the cost of this exchange is denoted by cΦI(j, k). To connect the disjoint elements (i.e., the cycles corresponding to the subtours) and construct a connected graph, additional arcs must be inserted into this undirected graph.

Fig. 4.4 Permutation mappings: (a) {0 , 1 , 2 , 3} → {2 , 3 , 1 , 0}

Finding Optimal Tour for TSP) Step 1

Job Families with Setup Times

The total cost of the resulting tour can be viewed as consisting of two components. Second, in the case where the permutation map represents an actual tour, this lower bound is shown to be greater than or equal to the total cost of the tour constructed in the algorithm. Note that qh≤nh, and that family's task batch settings at the beginning of the schedule are not included.

Minimizing the Total Weighted Completion Time) Initial Condition

94 4 Advanced Single Machine Models (Deterministic) priority job (qg, g), . ng, g) and it starts at time 0 with a bundle of works of family. Applying the WSPT rule to the two jobs of family 2 indicates that job (1,2) must appear in the schedule before job (2,2). Before describing the dynamic programming procedure for this problem, it is again necessary to establish certain properties related to optimal schedules.

Minimizing the Maximum Lateness) Initial Condition

Minimizing the Number of Tardy Jobs) Initial Condition

Batch Processing

The processing times of the jobs in a batch may not all be the same and the entire batch is not ready until the last job of the batch is completed, i.e. the completion time of the entire batch is determined by the job with the longest processing time. If the objective function γ is regular and the batch size is unlimited, then the optimal scheme is an SPT batch scheme. Let V(n+1) denote the minimum total weighted completion time of the empty set, which is zero.

Minimizing Total Weighted Completion Time – Batch Size Inﬁnite)

100 4 Advanced Single Machine Models (deterministic) completion time of an SPT group program containing jobs j,.

Minimizing Maximum Lateness – Batch Size Inﬁ- nite)

Tracing back gives the following schedule: The fact that the minimum for V(1) is reached for k= 4 means that the first three jobs are combined into one batch. This processing time pk already contributes to the range of the previous state, which can be V(j−1, u, k) or V(j−1, u−1, k), depending on whether jobj is timely or not. The previous batch ends with jobj−1 and the processing time of the new batch ispk.

Minimizing Number of Tardy Jobs – Batch Size Inﬁnite)

Minimizing Makespan – Batch Size Finite) Step 1. (Initialization)

Discussion

Letp(Bk) indicates the maximum processing time of the jobs in batch k, i.e. p(Bk) is the time it takes to process batch k. There are several articles on timing algorithms when the optimal order of the tasks is a given. An excellent overview of the literature on this subject is given in the article by Potts and Kovalyov (2000).

Fig. 4.8 Cost function with due date range

Parallel Machine Models (Deterministic)

The Makespan without Preemptions

So for the smallest counterexample, the start time of the shortest work under LPT is Cmax(LP T)−pn. The right side is an upper bound on the start time of the shortest job. It can be shown that the worst case of this arbitrary list rule satisfies the inequality.

Minimizing the Makespan of a Project)

The Makespan with Preemptions

The LFJ rule selects, each time a machine is freed, from among the available jobs the job that can be processed on the smallest number of machines, i.e. the least ﬂexible job. The least ﬂexible job that can be processed on machine 1 is job 1, since it can only be processed on. However, in the case of example 5.1.9, LFM-LFJ does not provide an optimal schedule either.

Fig. 5.2 Precedence constraints graph with critical path in Example 5.1.4

Minimizing Makespan with Preemptions) Step 1

The Total Completion Time without Preemptions
The Total Completion Time with Preemptions
Due Date Related Objectives
Online Scheduling
Discussion

Let p(j) denote the processing time of the job at position j in the sequence. Processing times should be set in such a way that the amount of products is minimized. The processing times of the three jobs on both machines are shown in the table below.

Fig. 5.6 LRPT with three jobs on two machines with preemptions allowed at integer points in time (Example 5.2.8)

Flow Shops and Flexible Flow Shops (Deterministic)

Flow Shops with Unlimited Intermediate Storage

Thus, the problem of minimizing the makespan in a machine permutation flow store is formulated as a MIP. Minimizing the makespan in a proportional permutation current store is denoted by F m | prmu, pij = pj | Cmax. In a proportional permutation flow shop with different speeds, if the first (last) machine is the bottleneck, then LPT (SPT) minimizes the makespan.

Fig. 6.1 Directed Graph for the Computation of the Makespan in F m | prmu | C max under sequence j 1 ,

Flow Shops with Limited Intermediate Storage

The two goals are equivalent, since twice the makespan is equal to the sum of 2nprocessing times plus. One popular heuristic for F m | block | Cmax is a profile fitting (PF) heuristic that works as follows: one task is selected first, preferably according to some scheme, e.g. the job with the smallest sum of processing times. After selecting the job that best matches the job for the second place, a new profile is calculated, i.e. the departure times of this second job from m machines, and the process is repeated.

Fig. 6.6 Directed graph for the computation of the makespan

Flexible Flow Shops with Unlimited Intermediate Storage

In a staged flexible flow proportional shop, the completion time of job j in the last stage occurs at the earliest cpj time units after its start time in the first stage. So under SPT the sum of the finish times is equal to the sum of the start times in the first phase plus n. Since SPT minimizes the sum of the start times in the first phase and the work must remain at least cpj unit time in .

Discussion

Consider now a flexible flow shop with the same number of machines at each stage, say. Find the optimal schedule for the proportional flow shop example F2|pij=pj|hmax with the following jobs. Apply a variant of Algorithm 3.4.4 to find an optimal schedule for the proportional flow instance storeF2|pij =pj|.

Job Shops (Deterministic)

Disjunctive Programming and Branch-and-Bound

All operations (nodes) in the same clique must be performed on the same machine. The route, i.e. the machine sequence, as well as the processing time are given in the table below. In this algorithm Ω denotes the set of all operations whose all predecessors have already been scheduled (i.e., the set of all scheduled operations) andrij.

Fig. 7.1 Directed graph for job shop with makespan as objective

Generation of all Active Schedules) Step 1. (Initial Condition)

The length of the critical path in this graph already gives a lower bound for the span at node V. This gives a better lower bound for the span at the node corresponding to the operation (1,1) that was first planned, which means that the lower bound for makespan at this node corresponding to operation (1,3) planned first also equals 28.