Scheduling policies in the <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">G</Emphasis>/1 make-to-stock queue

In this paper, we analyse a production/inventory system modelled as an M/G/1 make-to-stock queue producing different products requiring different and general production times. We study different scheduling policies including the static first-come-first-served, preemptive and non-preemptive priority disciplines. For each static policy, we exploit the distributional Little's law to obtain the steady-state distribution of the number of customers in the system and then find the optimal inventory control policy and the cost. We additionally provide the conditions under which it is optimal to produce a product according to a make-to-order policy. We further extend the application area of a well-known dynamic scheduling heuristic, Myopic(T), for systems with non-exponential service times by permitting preemption. We compare the performance of the preemptive-Myopic(T) heuristic alongside that of the static preemptive-bμ rule against the optimal solution. The numerical study we have conducted demonstrates that the preemptive-Myopic(T) policy is superior between the two and yields costs very close to the optimal.     

Minimizing makespan in a two-machine flowshop scheduling with batching and release time

This paper studies two-machine flowshop scheduling with batching and release time, whose objective is to minimize the makespan. We formulate the scheduling problem as a mixed integer programming model and show that it is a strongly NP-hard problem. We derive a lower bound and develop dynamic programming-based heuristic algorithms to solve the scheduling problem. Computational experiments are carried out to evaluate the performance of the heuristic algorithms. The numerical results show that some of the heuristic algorithms can indeed find effective solutions for the scheduling problem.     

The Systems Approach to Hospitals

By exploiting the relationship between scheduling and sorting, this paper describes a functional heuristic algorithm for seeking a quick and approximate solution to the n-job, M-machine flowshop scheduling problem under the assumption that all jobs are processed on all machines in the same order and no passing of jobs is permitted. The proposed functional heuristic algorithm can be executed by hand for reasonably large size problems and yields solutions which are closer to optimal solutions than those obtained by Palmer's slope index algorithm.     

Dynamic fleet scheduling with uncertain demand and customer flexibility

We develop a dynamic fleet scheduling model that demonstrates how a carrier can improve fleet utilization. The fleet scheduling model presented by Lee et?al. (Eur J Oper Res 218(1):261–269, 2012) minimizes (1) a carrier’s fleet size and (2) the penalty associated with the alternative delivery times selected. The model is static since requests are collected over time and processed together. In this paper we present a stochastic, dynamic version of the fleet reduction model. As demand is revealed throughout an order horizon, decisions are made in stages by sampling anticipated demand to avoid recourse penalties in later stages. Based on computational experiments we find the following:

1. Modeling stochasticity improves the quality of solutions relative to the analogous model that does not include stochasticity. Counter-intuitively, an order lead-time distribution in which most loads are requested early can negatively impact optimal solution costs.
2. The stochastic model produces good results without requiring prohibitively large numbers of demand scenarios.
3. Consignees that place orders early in the order horizon are more often assigned their requested delivery times than those who place orders late.

     

Minimizing the weighted number of early and tardy jobs in a stochastic single machine scheduling problem

We study a static stochastic single machine scheduling problem in which jobs have random processing times with arbitrary distributions, due dates are known with certainty, and fixed individual penalties (or weights) are imposed on both early and tardy jobs. The objective is to find an optimal sequence that minimizes the expected total weighted number of early and tardy jobs. The general problem is NP-hard to solve; however, in this paper, we develop certain conditions under which the problem is solvable exactly. An efficient heuristic is also introduced to find a candidate for the optimal sequence of the general problem. Our illustrative examples and computational results demonstrate that the heuristic performs well in identifying either optimal sequences or good candidates with low errors. Furthermore, we show that special cases of the problem studied here reduce to some classical stochastic single machine scheduling problems including the problem of minimizing the expected weighted number of early jobs and the problem of minimizing the expected weighted number of tardy jobs which are both solvable by the proposed exact or heuristic methods.     

Simultaneously scheduling multiple turns for steel color-coating production

This paper investigates a large-scale scheduling problem in the iron and steel industry, called color-coating production scheduling (CCPS). The problem is to generate multiple production turns for the galvanized coils that dynamically arrive from upstream lines within a given scheduling horizon, and at the same time determine the sequence of these turns so that the productivity and product quality are maximized while the production cost and the number of generated turns are minimized. We formulate this problem as a mixed integer nonlinear program and propose a tabu search heuristic to obtain satisfactory solutions. Results on real production instances show that the presented model and heuristic are more effective and efficient with comparison to manual scheduling. A practical scheduling system for CCPS combining the model and heuristic has been developed and successfully implemented in a major iron and steel enterprise in China.     

建筑废料现场分拣协调调度优化研究

工程项目施工现场废料分拣的效率对施工进度具有重要的影响。通过分析现场废料分拣实施过程,建立了带有限中间缓冲的混合流水车间调度模型。提出了收集阶段作业排序的动态自适应算法和后续设备分配问题的考虑缓冲区的设备分配规则,在此基础上设计了废料分拣模型的启发式算法,平衡各分区工作量,进一步搜索最优解,并推导了问题的一个低界。实验结果表明,所提出算法能很好地对施工现场废料分拣问题进行求解,具有良好的收敛性和较高的时间效率。     

Resource capacity allocation to stochastic dynamic competitors: knapsack problem for perishable items and index-knapsack heuristic

In this paper we propose an approach for solving problems of optimal resource capacity allocation to a collection of stochastic dynamic competitors. In particular, we introduce the knapsack problem for perishable items, which concerns the optimal dynamic allocation of a limited knapsack to a collection of perishable or non-perishable items. We formulate the problem in the framework of Markov decision processes, we relax and decompose it, and we design a novel index-knapsack heuristic which generalizes the index rule and it is optimal in some specific instances. Such a heuristic bridges the gap between static/deterministic optimization and dynamic/stochastic optimization by stressing the connection between the classic knapsack problem and dynamic resource allocation. The performance of the proposed heuristic is evaluated in a systematic computational study, showing an exceptional near-optimality and a significant superiority over the index rule and over the benchmark earlier-deadline-first policy. Finally we extend our results to several related revenue management problems.     

Single machine scheduling to minimize total weighted earliness subject to minimal number of tardy jobs

Motivated by just-in-time manufacturing, we consider a single machine scheduling problem with dual criteria, i.e., the minimization of the total weighted earliness subject to minimum number of tardy jobs. We discuss several dominance properties of optimal solutions. We then develop a heuristic algorithm with time complexity O(n³) and a branch and bound algorithm to solve the problem. The computational experiments show that the heuristic algorithm is effective in terms of solution quality in many instances while the branch and bound algorithm is efficient for medium-size problems.     

Economic lot sizes and vehicle scheduling

In this paper a relationship between the vehicle scheduling problem and the dynamic lot size problem is considered. For the latter problem we assume that order quantities for different products can be determined separately. Demand is known over our n-period production planning horizon. For a certain product our task is to decide for each period if it should be produced or not. If it is produced, what is its economic lot size? Our aim here is to minimize the combined set-up and inventory holding costs. The optimal solution of this problem is given by the well-known Wagner-Whitin dynamic lot size algorithm. Also many heuristics for solving this problem have been presented. In this article we point out the analogy of the dynamic lot size problem to a certain vehicle scheduling problem. For solving vehicle scheduling problems the heuristic algorithm developed by Clark and Wright in very often used. Applying this algorithm to the equivalent vehicle scheduling problem we obtain by analogy a simple heuristic algorithm for the dynamic lot size problem. Numerical results indicate that computation time is reduced by about 50% compared to the Wagner-Whitin algorithm. The average cost appears to be approximately 0.8% higher than optimum.     

On-line scheduling to minimize max flow time: an optimal preemptive algorithm

We investigate the maximum flow time minimization problem of on-line scheduling jobs on m identical parallel machines. When preemption is allowed, we derive an optimal algorithm with competitive ratio 2-1/m. When preemption is not allowed and m=2, we show that the First In First Out heuristic achieves the best possible competitive ratio.     

An Optimization Model for Stochastic Project Networks with Cash Flows

Project networks – or PERT networks – can be characterized by random completion times of activities and positive or negative cash flows throughout the project. In these cases the decision maker’s problem consists of determining a feasible activities schedule, to maximize the project financial value, where the financial value is measured by the net present value (npv) of cash flows.The analysis of these networks is a difficult computational task for the following reason. First, suppose that a schedule is fixed using a heuristic rule. Then the expected npv is calculated. But, due to stochastic job completion times, this problem belongs to the ♯-P complete difficulty class, e.g. problems that involve finding all the Hamiltonian cycles in a network. The problem is such that evaluating one project alone is not sufficient, but the optimal one has to be selected. This involves a further increase in computational time.This paper proposes a stochastic optimization model to determine a heuristic scheduling rule, that provides an approximate solution to finding the optimal project npv. A feature of this approach is that the scheduling rule is completely deterministic and defined when the project begins. Therefore an upper bound of the expected npv, that is an optimistic estimate, can be calculated through linear programming and a lower bound, that is a pessimistic estimate, can be calculated using simulation before the project begins.     

A single-machine scheduling problem with maintenance activities to minimize makespan

We study a single-machine scheduling problem with periodic maintenance activity under two maintenance stratagems. Although the scheduling problem with single or periodic maintenance and nonresumable jobs has been well studied, most of past studies considered only one maintenance stratagem. This research deals with a single-machine scheduling problem where the machine should be stopped for maintenance after a fixed periodic interval (T) or after a fixed number of jobs (K) have been processed. The objective is to minimize the makespan for the addressed problem. A two-stage binary integer programming (BIP) model is provided for driving the optimal solution up to 350-job instances. For the large-sized problems, two efficient heuristics are provided for the different combinations of T and K. Computational results show that the proposed algorithm Best-Fit-Butterfly (BBF) performs well because the total average percentage error is below 1%. Once the constraint of the maximum number of jobs (K) processed in the machine’s available time interval (T) is equal or larger than half of jobs, the heuristic Best-Fit-Decreasing (DBF) is strongly recommended.     

Dynamic lot-sizing in sequential online retail auctions

Retailers often conduct non-overlapping sequential online auctions as a revenue generation and inventory clearing tool. We build a stochastic dynamic programming model for the seller’s lot-size decision problem in these auctions. The model incorporates a random number of participating bidders in each auction, allows for any bid distribution, and is not restricted to any specific price-determination mechanism. Using stochastic monotonicity/stochastic concavity and supermodularity arguments, we present a complete structural characterization of optimal lot-sizing policies under a second order condition on the single-auction expected revenue function. We show that a monotone staircase with unit jumps policy is optimal and provide a simple inequality to determine the locations of these staircase jumps. Our analytical examples demonstrate that the second order condition is met in common online auction mechanisms. We also present numerical experiments and sensitivity analyses using real online auction data.     

Production scheduling with supply and delivery considerations to minimize the makespan

In this paper we study a scheduling model that simultaneously considers production scheduling, material supply, and product delivery. One vehicle with limited loading capacity transports unprocessed jobs from the supplier’s warehouse to the factory in a fixed travelling time. Another capacitated vehicle travels between the factory and the customer to deliver finished jobs to the customer. The objective is to minimize the arrival time of the last delivered job to the customer. We show that the problem is NP-hard in the strong sense, and propose an O(n) time heuristic with a tight performance bound of 2. We identify some polynomially solvable cases of the problem, and develop heuristics with better performance bounds for some special cases of the problem. Computational results show that all the heuristics are effective in producing optimal or near-optimal solutions quickly.     

The study of a dynamic dial-a-ride problem under time-dependent and stochastic environments

This paper studies a dynamic dial-a-ride problem bearing complex constraints on a time-dependent network. A flexible scheduling scheme is proposed to dynamically cope with different stochastic events, such as the travelling time fluctuation, new requests, absences of customers, vehicle breakdowns, cancellations of requests, traffic jams and so on. A fast heuristic is proposed to re-optimize the schedule when a new event occurs. This heuristic consists of a properly organized local search strategy and uses a secondary objective function to drive the search out of local optima. Intensive computational simulations were carried out to evaluate the performance of this scheduling scheme and the influence of different stochastic factors. The simulation results of different scenarios with different percentage of dynamic requests reveal that this scheduling scheme can generate high quality schedules and is capable of coping with various stochastic events.     

Two-machine flowshop scheduling problems involving a batching machine with transportation or deterioration consideration

This paper considers two scheduling problems for a two-machine flowshop where a single machine is followed by a batching machine. The first problem is that there is a transporter to carry the jobs between machines. The second problem is that there are deteriorating jobs to be processed on the single machine. For the first problem with minimizing the makespan, we formulate it as a mixed integer programming model and then prove that it is strongly NP-hard. A heuristic algorithm is proposed for solving this problem and its worst case performance is analyzed. The computational experiments are carried out and the numerical results show that the heuristic algorithm is effective. For the second problem, we derive the optimal algorithms with polynomial time for minimizing the makespan, the total completion time and the maximum lateness, respectively.     

Minimizing makespan with multiple-orders-per-job in a two-machine flowshop

We investigate a new scheduling problem, multiple-orders-per-job (MOJ), in the context of a two-machine flowshop. Lower bounds for the makespan performance measure are provided for combinations of lot-processing and item-processing machines. An optimization model is presented that addresses both job formation and job sequencing. We define a heuristic to minimize the makespan for the MOJ problem for two-machine item-processing flowshops. The heuristic obtains solutions within 2% of a tight lower bound and runs in O(HF) time, where H is the number of orders and F is the restricted number of jobs.     

Markovian analysis for performance evaluation and scheduling in m machine stochastic flow-shop with buffers of any capacity

This paper deals with performance evaluation and scheduling in m machine static stochastic permutation flow-shop with buffers of any capacity (unlimited, limited or null). The processing time of a given job for a given machine is assumed to be exponentially distributed with a known rate. We propose a theorem which provides recursive scheme based on Markov chains and Chapman–Kolmogorov equations to compute the expected completion time of the last job for any sequence of jobs. This scheme is combined with metaheuristics based on simulated annealing for the scheduling problem. Computational results are given.     

Approximation results in parallel machnies stochastic scheduling

We consider scheduling a batch of jobs with stochastic processing times on parallel machines. We derive various new formulae for the expected flowtime and weighted flowtime under general scheduling rules. Smith's Rule, which orders job starts by decreasing ratio of weight to expected processing time provides a natural heuristic for this problem. We obtain a bound on the worst case difference between the expected weighted flow time under Smith's Rule and under an optimal policy. For a wide class of processing time distributions, this bound is of oderO(1) and does not increase with the number of jobs.This research was supported in part by NSF Grant ECS-8712798.