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Preface
Coverage of the Text
For the reader interested in a more extensive link to the research literature than our text covers, we offer a set of online research notes. Research Notes represent unique material that expands the book's coverage and builds an intellectual bridge to the research literature on sequencing and planning.
Historical Background
The second appendix contains background derivations related to the "critical ratio rule", which occurs frequently in safe planning models. Finally, for readers who want to read research articles directly from the source, we occasionally need to discuss topics that are not essential to the text but that frequently arise in the literature.
New in the Second Edition
This book updates the coverage of ESS and adds coverage of secure scheduling as well as traditional stochastic scheduling. Because the new material comes from active researchers, the book surpasses competing texts in terms of its topicality.
Acknowledgments
Introduction
Introduction to Sequencing and Scheduling
When we specify the tasks and resources, we effectively define the boundary of the scheduling problem. Many of the early developments in the field of scheduling were motivated by problems that arose in manufacturing.
Scheduling Theory
The choice of a suitable method depends mainly on the nature of the model and the choice of objective function. On the other hand, if the function is O(2n), then the algorithm is non-polynomial (exponential in this case).
Philosophy and Coverage of the Book
Then we treat safe planning problems as extensions of the deterministic models, in the spirit of building from the specific to the general. Again, the use of algorithms is not an end in itself, but rather a way to reinforce the logic of the analysis.
Bibliography
In Chapter 8, we relax several of the elementary assumptions and analyze the problem structures that arise. The understanding of models, techniques and insights that we develop in the previous chapters is to a large extent integrated into the study of the job shop.
Single-machine Sequencing
Introduction
After discussing some introductory points in Section 2.2, we review basic sequencing results in Section 2.3 for problems that do not contain due dates and in Section 2.4 for problems that do include due dates. In the next chapter, we examine several general methodologies that can be applied to single-machine problems.
Preliminaries
Measures of schedule performance are usually functions of the set of completion times in a schedule. C Can only increase if at least one of the completion times in the schedule increases.
Problems Without Due Dates: Elementary Results
- Flowtime and Inventory
- Minimizing Total Flowtime
- Minimizing Total Weighted Flowtime
The intimate connection between these two goals can be illustrated in the basic single machine model. The "low inventory" objective can be interpreted as reducing the average number of jobs in the system.
Problems with Due Dates: Elementary Results
- Lateness Criteria
- Minimizing the Number of Tardy Jobs
- Minimizing Total Tardiness
In the example, the jobs are already indexed with EDD as required in step 1 of the algorithm. The proof follows directly from the adjacency analysis of pair exchange with the same interpretation as in the proof of Theorem 2.6.
Flexibility in the Basic Model
- Due Dates as Decisions
- Job Selection Decisions
To measure the stringency of a set of deadlines, we use the sum of the deadlines or. In the basic sequencing model, the workload is given, and deadlines are given, and the scheduling task is to find the best sequence.
Summary
In the identical parameter case, the problem reduces to finding two sets – a time set and its complement. Moreover, the general form of the problem involves finding three sets—a timely set of accepted jobs, a late set of accepted jobs, and the set of rejected jobs.
Exercises
Now suppose you want to use a due date setting rule to assign due dates to different orders. For a given sequence of tests, the element is subjected to each test in sequence until the tests pass the element; if the item is rejected by any test, no further tests are performed.
Optimization Methods for the Single-machine Problem
Introduction
Letj denote the last job in the sequence, and let G(S) denote the highest cost among non-iandj jobs. If this job is removed from consideration, we repeat the process for the remaining (n-1) jobs and continue until all jobs are in sequence.
Adjacent Pairwise Interchange Methods
In contrast, the decision rule resulting from an adjacent pairwise exchange in the F-problem involves only a comparison of the processing times of the jobs being exchanged. For such rules as SWPT, EDD and MST, the optimal sequence is characterized by a transitive pairwise ordering of the tasks.
A Dynamic Programming Approach
However, in the case of the criterion T, we can only conclude that the optimal ordering rule (whatever it may be) is not transitive. This fast-access lookup for the value of G(J−j) lies at the heart of the calculations.
Dominance Properties
If one of the conditions applies to the pair of jobsiandj, then the sizes of BjandAi increase. The labeling scheme consists of a mechanism for assigning labels to jobs; then the label of a particular subset is simply the sum of the labels of the jobs contained in the subset.
A Branch-and-bound Approach
The bounding procedure calculates a lower bound for the solution of each subproblem generated during the branching process. In the next version of the algorithm, the active list will be ordered by lower bound, smallest first.
Integer Programming
- Minimizing the Weighted Number of Tardy Jobs
- Minimizing Total Tardiness
The objective function in Eq. 3.9) is the sum of the delay values, which need not be integer values. The IP model for the T problem has more variables and constraints than the model for the Uw problem in the previous section.
Summary
The dynamic programming approach (Section 3.3) is a very flexible implicit enumeration strategy that can be directly applied to many single-machine sorting problems. A complication with both dynamic programming and branch-and-bound is that there are no general solvers.
Heuristic Methods for the Single-machine Problem
Introduction
The ability to solve 30-job single-machine problems does not imply that we can optimally solve 30 jobs in more complex problems. In multimachine models, single machine submodels may need to be solved repeatedly, perhaps as many as 2n times.
Dispatching and Construction Procedures
Next, keeping the relative order of the first two jobs fixed, find the best placement. These test problems, reproduced in Table 4.8 at the end of the chapter, are known to be relatively difficult to optimize.
Random Sampling
Three different sample sizes were tested, and the results are shown in Table 4.2 and compared with the random dispatch and the greedy algorithm of Table 4.1. 4 Heuristic methods for the single machine problem 78 . random dispatch procedure is equivalent to random sampling with a sample size of N= 1.). These probabilities are "biased" in the sense that they favor the first position on the list over the second, the second over the third, and so on.
Neighborhood Search Techniques
An improvement occurs in the first neighborhood and the sequence with T= 12) becomes the new seed because it is the first improvement in the neighborhood. Again, an improvement is found in finding a new neighborhood, and the new seed is a sequence with T= 10.
Tabu Search
In each phase, the procedure selects the best solution from the neighborhood residents that are not on the tabu list. While the nearby search contains a built-in termination device – the discovery of a local optimal – a termination rule must be imposed in the tabu search.
Simulated Annealing
After taking samples from the environment of the seed a certain number of times, we lower the temperature and continue the search. It is clear that the performance of simulated annealing is sensitive to the planned computational effort, as measured here by the number of phases in the temperature scheme.
Genetic Algorithms
In sequencing problems, the simplest mechanism for generating offspring follows one parent for the first few jobs and takes the remaining jobs in the same order as in the other parent. Therefore, a complementary offspring can be constructed in which the last few positions of one parent are copied and the first positions appear in the same order as in the other parent.
The Evolutionary Solver
When the ASP opens, it overlays its task pane on the right side of the spreadsheet, as shown in Figure 4.3. Alternatively, with cell C8 selected, select the folder icon for Target in the task pane and click the green cross in the header of the task pane.
Summary
In addition, the neighborhood search approach is flexible and can be adapted to a number of different problem structures. The neighbor search procedure also provides a framework for more sophisticated search algorithms—such as tabu search, simulated annealing, and GAs—that overcome the local optimality trap.
Earliness and Tardiness Costs
Introduction
However, a useful organizational principle is to think in terms of two main models: one with a common due date and one with different due dates. The primary role of timeliness and tardiness cost functions is to guide solutions toward the goal of meeting all deadlines accurately.
Minimizing Deviations from a Common Due Date
- Four Basic Results
- Due Dates as Decisions
Finally, the schedule start time is the difference between the due date and the total processing time at B. Treating the due date as a decision in the E/T problem is equivalent to solving the unconstrained version of the basic problem by Algorithm 5.1∗ .
The Restricted Version
In other words, the optimal total cost in an unlimited case of the basis problem is constant as the expiration date varies. Basically, we can think of the relationship between optimal total cost and due date as shown in the graph in Figure 5-1.
Asymmetric Earliness and Tardiness Costs
In the restricted version of the problem, Theorems 5.1 and 5.2 still hold, and the V-shaped arrangements represent the dominant set. In addition, we can generalize the condition that characterizes the start delay of the schedule toe > nβ/(α+β).
Quadratic Costs
The most important observations are (i) that the average error is usually less than 1%, and (ii) that the heuristic finds an optimal solution about one-third of the time. Therefore, the unconstrained version of the quadratic E/T problem is equivalent to minimizing the variance of completion times.
Job-dependent Costs
Distinct Due Dates
Added to the block, job 3 finishes at time 9, and the three-job block has a cost of 8. If the block starts at time zero, its total cost drops to 7, so we start the schedule at time zero.
Summary
Minimizing the average deviation of task completion times from a common due date. Naval Research Logistics Quarterly28: 643–651. Single machine scheduling to minimize absolute deviation of completion times from a common due date. Naval Research Logistics Quarterly.
Sequencing for Stochastic Scheduling
Introduction
In contrast, the SPT sequence is not well defined because the processing times are not known in advance. For example, the stochastic counterpart of the F-problem is the stochastic scheduling problem in which the objective function is the expected total flow time, E[F].
Basic Stochastic Counterpart Models
In Table 6.4 we can recognize the expected delay value of 11.1, and we can see the expected value of the other 6 listed performance measures. In our second example, processing times are independent, and we can illustrate the simulation interpretation of the sampling-based approach.
The Deterministic Counterpart
And related to this, when is the optimal sequence for the deterministic counterpart also optimal for the stochastic problem. In these problems, the deterministic counterpart may not provide an optimal value of the objective function.
Minimizing the Maximum Cost
Proposition 6.1 The sequences that minimize the maximum expected cost and the expected maximum cost are not necessarily identical. Let ZL and ZU denote the maximum expected cost and the expected maximum cost of S1.
The Jensen Gap
Stochastic Dominance and Association
But by definition, two associated random variables have a non-negative covariance, and the variance of a sum with positive covariance is higher than the sum of the variances. This relation in turn implies that the variance of performance measures based on processing times, which are associated random variables, is also higher than the variance of independent processing times.
Using Analytic Solver Platform
The deterministic model is modified slightly, as shown in Figure 6.4, and contains two identical rows. ASP displays a probability distribution function (pdf ), as shown in Figure 6.5, and in the Parameters window on the right, we can specify the mean and standard deviation by referring to cells C6 and C7, respectively.
Non-probabilistic Approaches: Fuzzy and Robust Scheduling
However, in this example, the minimal regret is actually produced by a sequence that is not one of the four identified so far. But minimum regret optimization remains challenging (and perhaps this is why thermobust planning is commonly interpreted as being driven by minimum regret rather than minimum cost).
Summary
In addition, the importance of the worst case may be overemphasized, even if it is extremely unlikely. In addition, the parameters of the cost function gj(Tj) =δ(Tj)(aj+bjTj) are given in the following table.
Safe Scheduling
Introduction
As an example of the economic approach, we study the stochastic version of the E/T problem in Section 7.4, again treating due dates as decisions. In Section 7.7, we discuss the level-of-service approach to the stochastic counterpart of the U problem and introduce the economic approach in Section 7.8.
Meeting Service Level Targets
- Sample-based Analysis
- The Normal Model
Due dates corresponding to the other service level targets are shown in the table, leading to D= 62.89. Therefore, keeping jobia in front of jobk minimizes the sum of the two, for anyyk.
Trading Off Tightness and Tardiness
- An Objective Function for the Trade-off
- The Normal Model
- A Branch-and-bound Solution
In the stochastic version of this problem, the goal is to find the expected value of the function in Eq. Suppose we have a partial sequenceπ and we want to calculate a lower bound on the value of the objective function that can be obtained by completing the sequence.
The Stochastic E/T Problem
G(d), or expected total E/T cost, becomes our objective function in the stochastic case, and we can express it as. The optimal choice of due dates is again determined by a critical fraction rule, as indicated in the following result.
