Integer programming models for the re-entrant shop scheduling problems

A new theoretical formulation for re-entrant flow shop scheduling problem is derived to plan an optimum production schedule by mixed integer programming. Flow time, which is an index of the evaluation of the planned schedule, is minimized to decrease amount of work-in-process. Cycle time, which is another important index, is determined by the total load time on a bottleneck machine to manufacture the products at the highest productivity. Both objective function and constraint conditions are described in linear expressions and a general optimization tool is adopted to solve the formulated scheduling problems. Another objective function is also introduced in order to shorten the time for the optimization. Computational scheduling results obtained by the optimization are evaluated and discussed in the present paper.

Optimizing blocking flow shop scheduling problem with total completion time criterion

Bi-objective unrelated parallel machine joint scheduling of jobs and preventive maintenance with a dynamic speed-scaling technique

A genetic algorithm for the simultaneous delivery and pickup problems with time window

The sugarcane transport system is very complex and uses a daily schedule, consisting of a set of locomotives runs, to satisfy the requirements of the mill and harvesters. The total cost of sugarcane transport operations is very high; over 35% of the total cost of sugarcane production in Australia is incurred in cane transport. Producing efficient schedules for sugarcane transport can reduce the cost and limit the negative effects that this system can have on the raw sugar production system. In this paper, the sugarcane rail operations are formulated as a blocking job shop scheduling problem. A mixed integer programming approach is used to formulate the shop job scheduling problem. Mixed integer programming and constraint programming search techniques are integrated for solving the problem. A case study is solved to test the approach.

This paper proposes a modified shifting bottleneck heuristic (MSBH) for the reentrant job shop scheduling problem (RJSSP) with makespan minimization objective. Recently, the reentrant job shop has come into prominence as a new type of manufacturing shop. The principle characteristic of a reentrant job shop is that a job may visit certain machines more than once during the process flow, whereas in the classic job shop, each job visits a machine only once. The shifting bottleneck heuristic (SBH) is one of the most successful heuristic approaches for the classical job shop scheduling problem, which decomposes the problem into a number of single-machine subproblems. This paper adapts the SBH for the RJSSP and proposes a new sequencing heuristic for the single-machine maximum lateness subproblem considering the reentrant jobs in order to handle large-size RJSSPs. It also uses a subproblem criticality measure that further shortens the implementation time. The proposed MSBH is tested by using instances up to 20 machines and 100 jobs, and it is illustrated that good quality solutions can be obtained in reasonable computational times. A real-life application of the MSBH is also given as a case study to evaluate its performance.

This article examines the multi-objective scheduling of a reentrant hybrid flow shop. This type of shop is composed of several stages made of several identical parallel machines. When a task has to be processed on a stage, it is assigned to the machine with the smallest workload. This problem shows a reentrant structure: each task must be processed several times at each stage. In this paper, this problem is solved by minimizing two objectives: the makespan (maximum completion time of the jobs) and the total tardiness of the tasks. A new method is improved with different local searches: Adjacent and Non Adjacent Pairwise Interchange, Extract and Backward-Shifted Reinsertion, and Extract and Forward-Shifted Reinsertion. Every local search is tuned with statistical method (design of experiment) and the best one is worked out. This method is compared with the best one in several instances. The results involve three different measures.

A Large Scale Printer (LSP) is a Cyber Physical System (CPS) printing thousands of sheets per day with high quality. The print requests arrive at run-time requiring online scheduling. We capture the LSP scheduling problem as online scheduling of re-entrant flowshops with sequence dependent setup times and relative due dates with makespan minimization as the scheduling criterion. Exhaustive approaches like Mixed Integer Programming can be used, but they are compute intensive and not suited for online use. We present a novel heuristic for scheduling of LSPs that on average requires 0.3 seconds per sheet to find schedules for industrial test cases. We compare the schedules to lower bounds, to schedules generated by the current scheduler and schedules generated by a modified version of the classical NEH (MNEH) heuristic [1], [2]. On average, the proposed heuristic generates schedules that are 40% shorter than the current scheduler, have an average difference of 25% compared to the estimated lower bounds and generates schedules with less than 67% of the makespan of schedules generated by the MNEH heuristic.

In this article, we present a new approach to the solution of optimization problems with discrete control variables modeled by binary integer programs (BIP). The solution of BIP is computationally demanding when the number of the BIP variables increase. Two instances that increase the number BIP variables in practical applications are the reduction of the discretization sampling time and the increase of the optimization time period. The proposed approach transforms a single BIP optimization into a linear program (LP) and N feasibility BIP's, with less number of variables. The reduction of the number of variables increases the algorithm speed in providing a solution. The approach permits to solve optimization problems with longer time intervals and with a higher number of control variables, while being computationally tractable. A case study on the electricity cost minimization in a pumping station shows the applicability of the method.

This article presents an overview and a flavour of almost two decades of MARKAL model developments and selected applications. The MARKAL family of models has been contributing to energy/environmental planning since the early J980’s. Under the auspices of the International Energy Agency’s (IEA) Energy Technology Systems Analysis Programme (ETSAP) the model started as a linear programming (LP) application focused strictly on the integrated assessment of energy systems. It was followed by a non-linear programming (NLP) formulation which combines the ‘bottom-up’ technology model with a ‘top-down’ simplified macro-economic model. In recent years, the family was enlarged by members to model material flows, to employ stochastic programming (SP) to address future uncertainties, to model endogenous technology learning using mixed integer programming (MIP) techniques, and to model multiple regions (NLP/LP).

This paper presents a makespan minimization of -jobs -machines re-entrant flow shop scheduling problem (RFSP) under fuzzy uncertainties using Genetic Algorithm. The RFSP objective is formulated as a mathematical programme constrained by number of jobs and resources availability with traditional scheduling policies of First Come First Serve (FCFS) and the First Buffer First Serve (FBFS). Jobs processing times were specified by fuzzy numbers and modelled using triangular membership function representations. The modified centroid defuzzification technique was used at different alpha-cuts to obtain fuzzy processing times (FPT) of jobs to explore the importance of uncertainty. The traditional GA schemes and operators were used together with roulette wheel algorithm without elitism in the selection process based on job fuzzy completion times. A test problem of five jobs with specified Job Processing and Transit Times between service centres, Job Start Times and Job Due times was posed. Results obtained using the deterministic and fuzzy processing times were compared for the two different scheduling policies, FCFS and FBFS. The deterministic optimal makespan for FBFS schedule was 61.2% in excess of the FCFS policy schedule.  The results also show that schedules with fuzzy uncertainty processing times provides shorter makespans than those for deterministic processing times and those under FCFS performing better than those under FBFS policy for early jobs while on the long run the FBFS policy performs better. The results underscore the need to take account of comprehensive fuzzy uncertainties in job processing times as a trade-off between time and costs influenced by production makespan. http://dx.doi.org/10.4314/njt.v36i3.21

Some complexity results in cyclic scheduling

Recently, wireless power transmission technology (WPT) is used in wireless rechargeable unmanned aerial vehicle (UAV) networks (WRUNs) to replenish energy for UAVs. In WRUN, if UAVs cannot replenish energy in time, the task will fail or the task completion time will be greatly prolonged. So, when and how long should the wireless static chargers (WSCs) be scheduled to replenish energy for UAVs while the energy utilization rate of WSCs can be optimized is very critical. However, the existing methods only optimize the charging time periods, that is the time durations during which WSCs release energy, ignoring the impact of the adjustable source power of WSCs. In fact, inappropriate source power will lead to the low energy utilization rate of WSCs. This paper takes both the charging time periods and the adjustable source power of WSCs into account and formulates the power-aware charging time scheduling (PACTS) problem as an optimization problem. Namely, how to decide the optimal charging time periods and the appropriate source power in each charging time period simultaneously to improve the energy utilization rate of WSCs. To solve PACTS, we first discretize the flight paths of UAVs spatiotemporally and then present a novel method to reformulate PACTS as a binary integer programming (BIP) problem that can be solved by existing algorithms. Finally, experiments are conducted to evaluate the performance of our scheme.

Today's enterprises face the challenge of managing large, complex information technology (IT) systems consisting of software applications, servers, network routers, and other type of resources. Change management, and especially change scheduling, is known to be one of the most challenging problems in managing IT operations. Change scheduling must comply with strict availability Service Level Agreements (SLAs) and tackle complex IT environments. Various challenges arise from complex IT component dependencies, unpredictable durations of work processes, and large volume of concurrent changes and affected IT components. In this chapter, we address the problem of optimal IT change scheduling. We illustrate how the optimization problem can be formulated and solved using a mixed integer programming technique so that IT change management practitioners can efficiently schedule large change volumes and determine schedules that are robust to uncertainty in durations and costs with lower risks of SLA violations.

For a binary integer program (IP) \(\max c^\mathsf {T}x, Ax \le b, x \in \{0,1\}^n\), where \(A \in \mathbb {R}^{m \times n}\) and \(c \in \mathbb {R}^n\) have independent Gaussian entries and the right-hand side \(b \in \mathbb {R}^m\) satisfies that its negative coordinates have \(\ell _2\) norm at most n/10, we prove that the gap between the value of the linear programming relaxation and the IP is upper bounded by \(\text {poly}(m)(\log n)^2 / n\) with probability at least \(1-1/n^7-2^{-\text {poly}(m)}\). Our results give a Gaussian analogue of the classical integrality gap result of Dyer and Frieze (Math. of O.R., 1989) in the case of random packing IPs. In constrast to the packing case, our integrality gap depends only polynomially on m instead of exponentially. By recent breakthrough work of Dey, Dubey and Molinaro (SODA, 2021), the bound on the integrality gap immediately implies that branch and bound requires \(n^{\text {poly}(m)}\) time on random Gaussian IPs with good probability, which is polynomial when the number of constraints m is fixed.