Nonrenewable Resource Constraints Research Articles

A simplified swarm optimization algorithm to minimize makespan on non-identical parallel machines with unequal job release times under non-renewable resource constraints

A simplified swarm optimization algorithm to minimize makespan on non-identical parallel machines with unequal job release times under non-renewable resource constraints

New complexity and approximability results for minimizing the total weighted completion time on a single machine subject to non-renewable resource constraints

We consider single machine scheduling problems with additional non-renewable resource constraints. Examples for non-renewable resources include raw materials, energy, or money. Usually they have an initial stock and replenishments arrive over time at a-priori known time points and quantities. The jobs have some requirements from the resources and a job can only be started if the available quantity from each of the required resources exceeds the requirements of the job. Upon starting a job, it consumes its requirements which decreases the available quantities of the respective non-renewable resources. There is a broad background for this class of problems. Most of the literature concentrate on the makespan, and the maximum lateness objectives. This paper focuses on the total weighted completion time objective for which the list of the approximation algorithms is very short. We extend that list by considering new special cases and obtain new complexity results and approximation algorithms.

Integrated Planning and Scheduling of Multiple Manufacturing Projects Under Resource Constraints Using Raccoon Family Optimization Algorithm

Today's dynamic environment and increasing demand for highly customized products have significantly increased the number of companies operating in the project environment. Project planning and scheduling are one of the major problems faced by managers due to resource constraints. Enterprises have to execute several projects simultaneously while sharing limited resources (i.e., human resources, equipment, and budget) among the projects to effectively meet the deadlines. Therefore, this work investigates the integrated planning and scheduling problem of multiple projects with different release dates and execution modes while considering the renewable and non-renewable resource constraints. Moreover, the raccoon family optimization (RFO) algorithm is proposed to maximize the net profit while considering the early completion bonus, penalty cost, and resource costs. In the proposed RFO algorithm, greedy search and modified genetic operators are introduced to enhance the performance and efficiency. Effectiveness of the proposed RFO algorithm is compared with the genetic algorithm (GA), raccoon optimization algorithm (ROA), and artificial bee colonial (ABC) algorithm on test cases as well as an industrial case study. The results indicate that the proposed RFO algorithm outperforms the other compared algorithms, both in terms of effectiveness and efficiency.

Automatic Clustering and Multi-Directional Scheduling Scheme for Multi-Mode Scheduling Projects

In this study, a cluster-based heuristic algorithm to solve multi-mode resource constrained project scheduling problem (MMRCPSP) is presented. The MMRCPSPs are usually challenging to schedule because each activity has multiple execution modes with renewable and non-renewable resources that are categorized as an NP-hard problem. In this paper, we propose and apply three ideas to solve an MMRCPSP. Firstly, an MMRCPSP with two objectives namely, makespan and executive cost is converted into a single mode resource-constrained project scheduling problem (RCPSP) with three objectives by relaxing the non-renewable resource constraints and then defining a corresponding penalty value. In the second phase, in the light of precedence and renewable resource constraints, the new RCPSP are scheduled by calling the multi-directional scheduling schemes (multi-dss). In the end, on the basis of three objectives’ value (namely makespan, cost, and penalty value), we defined a density-based fitness function for an evolutionary approach regarding the clustering methods on the new solution space. Our numerical results showed that the multi-dss with higher dimension regarding the automatic cluster-based fitness functions affected the performance of the evolutionary approach. Furthermore, the evolutionary approach based on multi-directional scheduling schemes prevents to trap into the local optimums by increasing the solutions’ diversity.

Minimizing total weighted completion time on a single machine subject to non-renewable resource constraints

In this paper, we describe new complexity results and approximation algorithms for single-machine scheduling problems with non-renewable resource constraints and the total weighted completion time objective. This problem is hardly studied in the literature. Beyond some complexity results, only a fully polynomial-time approximation scheme (FPTAS) is known for a special case. In this paper, we discuss some polynomially solvable special cases and also show that under very strong assumptions, such as the processing time, the resource consumption and the weight is the same for each job; minimizing the total weighted completion time is still NP-hard. In addition, we also propose a 2-approximation algorithm for this variant and a polynomial-time approximation scheme (PTAS) for the case when the processing time equals the weight for each job, while the resource consumptions are arbitrary.

A PTAS for a resource scheduling problem with arbitrary number of parallel machines

In this paper we study a parallel machine scheduling problem with non-renewable resource constraints. That is, besides the jobs and machines, there is a common non-renewable resource consumed by the jobs, which has an initial stock and some additional supplies over time. Unlike in most previous results, the number of machines is part of the input. We describe a polynomial time approximation scheme for minimizing the makespan.

Multi-objective algorithms to multi-mode resource-constrained projects under mode change disruption

The present paper studies the Multi-mode Resource-Constrained Project Scheduling Problem (MRCPSP) wherein tasks have different execution modes, each implying a duration and different resource requirements. During execution, the project may encounter some problems that may cause schedule deterioration or even unfeasibility. Its utmost goals are firstly the proposition of a set of new disruption measures that can be used in new objective functions, and then the resolution of a bi-objective problem minimizing the project makespan and a disruption measure. The latter is solved by an Evolutionary algorithm and a new reactive multi-objective heuristic. Considering that the project is in progress, the paper focuses mainly on problem feasibility, Pareto Front performance, and algorithm CPU time. Given that the feasibility problem of static MRCPSP is NP-Complete (Kolisch & Drexl, 1997), we transform the bi-objective MRCPSP problem to a multi-objective problem by including a penalty to cope with the potential violation of non-renewable resource constraints, and enlarge the search space. We show that schedule adjustment is necessary for the problem solving process. A diverse set of non-time-consuming operators is also included in the algorithm.

A Honey Bee Swarm Optimization Algorithm for Minimizing the Total Costs of Resources in MRCPSP

In this paper, we solve an extended type of multimode project scheduling problem considering renewable and non-renewable resource constraints and finish-to-start precedence relations among project activities. We suppose that renewable resources are hired and are not accessible in the whole duration of the project. Therefore, there is an assigned ready date as well as a due date for each renewable resource type so that no resource can be rented before its ready date. But, the resources are allowed to be used after their due dates by incurring penalty costs. The aim is to minimize the whole charges of both renewable and nonrenewable resource usage. This problem is introduced by Khalilzadeh et al. 7 as multimode resource-constrained project scheduling problem minimizing the net present value of entire weighted resource tardiness penalty costs (MRCPSP-DCTWRTPC) in which both renewable and nonrenewable resource restrictions rely on activity mode. The problem we proposed in this paper is the extended form of the problem introduced by Khalilzadeh et al. 7 with considering the discounted cash flows of the resource costs. We call this problem multi-mode resource-constrained project scheduling problem, minimization of discounted cash flows of total weighted resource tardiness penalty cost (MRCPSP-DCTWRTPC). For this problem, we propose a novel meta-heuristic algorithm based on a Honey Bee Swarm Optimization (HBSO) approach together with a prioritization rule for project tasks and numerous improvement and local search procedures. Computational results prove the effectiveness and efficiency of the presented method. Keywords: Discounted Cash Flow, Honey Bee Swarm Optimization, Multi Modes, Project Scheduling, Resource Cost, Tardiness Penalty Cost

Approximation schemes for single machine scheduling with non-renewable resource constraints

In this paper we discuss exact and approximation algorithms for scheduling a single machine with additional non-renewable resource constraints. Given the initial stock levels of some non-renewable resources (e.g., raw materials, fuel, money), and time points along with replenishment quantities, a set of resource consuming jobs has to be scheduled on the machine such that there are enough resources for starting each job, and the makespan is minimized. We show that the problem admits a pseudo-polynomial time algorithm when the number of replenishments is not part of the input, and also present an FPTAS when there is only a single resource, and it is replenished only once. We also describe a PTAS for the problem with a constant number of replenishments.

Ant colony optimization-based multi-mode scheduling under renewable and nonrenewable resource constraints

Abstract An ant colony optimization (ACO)-based methodology for solving the multi-mode resource-constrained project scheduling problem (MRCPSP) considering both renewable and nonrenewable resources is presented. With regard to the MRCPSP solution consisting of activity sequencing and mode selection, two levels of pheromones are proposed to guide search in the ACO algorithm. Correspondingly, two types of heuristic information and probabilities as well as related calculation algorithms are introduced. Nonrenewable resource-constraint and elitist-rank strategy are taken into account in updating the pheromones. The flowchart of the proposed ACO algorithm is described, where a serial schedule generation scheme is incorporated to transform an ACO solution into a feasible schedule. The parameter-selection and the resultant performance of the proposed ACO methodology are investigated through a series of computational experiments. It is expected to provide an effective alternative methodology for solving the MRCPSP by utilizing the ACO theory.

A Genetic Algorithm for Solving Multi-Mode Critical Chain Project Scheduling Problem

This paper studies the application of multi-mode critical chain project scheduling problem based on genetic algorithm. A model of multi-mode critical chain project scheduling problem is addressed by combining the characteristics of multi-mode project scheduling problem and the critical chain method. Comprehensively considering the renewable resource constraints, nonrenewable resource constraints, activities with multi-mode and the uncertainty of activity duration, a bi-level genetic algorithm to solve multi-mode critical chain project scheduling problem is proposed. We use the shared benchmark from PSPLIB to test this new genetic algorithm and the results show that the algorithm in this paper has obvious superiority over most currently existing approach.

Multi-mode resource-constrained project scheduling using RCPSP and SAT solvers

This paper reports on a new solution approach for the well-known multi-mode resource-constrained project scheduling problem (MRCPSP). This problem type aims at the selection of a single activity mode from a set of available modes in order to construct a precedence and a (renewable and non-renewable) resource feasible project schedule with a minimal makespan. The problem type is known to be NP-hard and has been solved using various exact as well as (meta-)heuristic procedures. The new algorithm splits the problem type into a mode assignment and a single mode project scheduling step. The mode assignment step is solved by a satisfiability (SAT) problem solver and returns a feasible mode selection to the project scheduling step. The project scheduling step is solved using an efficient meta-heuristic procedure from literature to solve the resource-constrained project scheduling problem (RCPSP). However, unlike many traditional meta-heuristic methods in literature to solve the MRCPSP, the new approach executes these two steps in one run, relying on a single priority list. Straightforward adaptations to the pure SAT solver by using pseudo boolean non-renewable resource constraints has led to a high quality solution approach in a reasonable computational time. Computational results show that the procedure can report similar or sometimes even better solutions than found by other procedures in literature, although it often requires a higher CPU time.

A hybrid rank-based evolutionary algorithm applied to multi-mode resource-constrained project scheduling problem

We consider the multi-mode resource-constrained project scheduling problem (MRCPSP), where a task has different execution modes characterized by different resource requirements. Due to the nonrenewable resources and the multiple modes, this problem is NP-hard; therefore, we implement an evolutionary algorithm looking for a feasible solution minimizing the makespan. In this paper, we propose and investigate two new ideas. On the one hand, we transform the problem of single objective MRCPSP to bi-objective one to cope with the potential violation of nonrenewable resource constraints. Relaxing the latter constraints allows to visit a larger solution set and thus to simplify the evolutionary operators. On the other hand, we build the fitness function not on a priori grid of the bi-objective space, but on an adaptive one relying on clustering techniques. This proposed idea aims at more relevant fitness values. We show that a clustering-based fitness function can be an appealing feature in multi-objective evolutionary algorithms since it may promote diversity and avoid premature convergence of the algorithms. Clustering heuristics require certainly computation time, but they are still competitive with respect to classical niche formation multi-objective genetic algorithm.

Project Scheduling Under Partially Renewable Resource Constraints

We consider a generalization of the classical resource constrained project scheduling problem. We introduce so-called partially renewable resources by assuming for each resource a capacity on subsets of periods. The concept of partially renewable resources is a fundamental tool in order to make, e.g., timetabling and shift scheduling aspects, amenable to project scheduling. In addition, partially renewable resources serve to model complicated labor regulations. Furthermore, they cover traditional renewable and nonrenewable resource constraints as special cases. We consider makespan minimization as objective. For the exact solution of the problem we employ a basic enumeration scheme. In order to speed up convergence, we formulate bounds which take into account future resource consumption of partially renewable resources. Moreover, we generalize the serial scheduling scheme in order to get fast approximation methods. A rigorous assessment of the procedures is provided by solving ProGen instances generated under a full factorial test design. Besides the well-known problem parameters we employ additionally three parameters which control the generation of partially renewable resources.

A genetic algorithm approach to a general category project scheduling problem

A genetic algorithm (GA) approach is proposed for the general resource-constrained project scheduling model, in which activities may be executed in more than one operating mode, and renewable as well as nonrenewable resource constraints exist. Each activity's operation mode has a different duration and requires different amounts of renewable and nonrenewable resources. The objective is the minimization of the project duration or makespan. The problem under consideration is known to be one of the most difficult scheduling problems, and it is hard to find a feasible solution for such a problem, let alone the optimal one. The GA approach described in this paper incorporates problem-specific scheduling knowledge by an indirect chromosome encoding that consists of selected activity operating modes and an ordered set of scheduling rules. The scheduling rules in the chromosome are used in an iterative scheduling algorithm that constructs the schedule resulting from the chromosome. The proposed GA is denoted a hybrid GA (HGA) approach, since it is integrated with traditional scheduling tools and expertise specifically developed for the general resource-constrained project scheduling problem. The results demonstrate that HGA approach produces near-optimal solutions within a reasonable amount of computation time.

On scheduling project activities with variable expenditure rates

Commercial projects are undertaken with the expectation of maximizing financial returns. There are a variety of cash flow models in the literature in which payments, whose amounts depend on the type of contract quoted between the contractor and the client, are tied to the project events or to the completion times of activities. These models are of deterministic nature with respect to cash inflows, that is, the amount and timing of the payments, once determined, are fixed. However, there are situations specifically in the housing industry where the contractor is the owner of the project. In this case, the contractor starts with an initial capital to cover the activity expenditures and then, capital is augmented by the sale of flats which take place randomly over the progress of the project. In this risky environment, the contractor has to decide on the rate of expenditure at each decision time in order to maintain a positive cash balance. Hence, activities are represented by multiple performing modes with different activity durations and the same total cost. A heuristic to construct and re-construct schedules during the progress of the project is proposed here with the aim of maximizing the project Net Present Value while completing the project on time. The heuristic incorporates dynamic mode selection criteria which change adaptively according to the current status of the project. Computational experiments with the heuristic demonstrates that it provides satisfactory results regarding the feasibility of the schedules with respect to the project due date and the nonrenewable resource constraints.

A framework for an interactive project scheduling system under limited resources

In this paper, we propose a framework for an interactive project scheduling system under limited resources. The framework includes a modelling module ( model) and a scheduling module ( scheduler). The modelling module model allows the Decision Maker (DM) to develop his/her own model with features such as alternative operating modes for activities; renewable, nonrenewable and/or doubly-constrained resource constraints; general cash flow patterns, related to the realization of activities or events; and progress payments distributed over the project span. The performance criteria include the maximization of Net Present Value (NPV), and either the minimization of maximum tardiness (when a project due date exists) or the minimization of the project duration (when there is no project due date). The scheduler is developed on a constraint-based scheduling algorithm, which is called Local Constraint Based Analysis ( lcba) and which has previously been tested and shown to produce near-optimal results with respect to the criterion of minimizing project duration. The decisions taken in the scheduler consist of determining the start times of activities and the specific operating modes in which they are to be realized. The decisions are taken by activating relevant essential conditions in lcba and in cases where resource conflicts are not resolved, the DM reaches a final decision by testing the alternatives proposed by lcba through a what-if routine. The scheduler represents a realistic scheduling system which is useful not only in the planning phase of a project but can also be employed during the progress of a project for updating the project plan, if necessary. An important feature is that the project plan can be updated by performing the least modification of future commitments. It is possible to freeze the activities already scheduled in the near future while admitting the changes in the activity/network information.

A survey on the resource-constrained project scheduling problem

In this paper, research on the resource-constrained project scheduling problem is classified according to specified objectives and constraints. Each classified area is extensively surveyed, and special emphasis is given to trends in recent research. Specific papers involving nonrenewable resource constraints and time/cost-based objectives are discussed in detail because they present models that are close representations of real-world problems. The difficulty of solving such complex models by optimization techniques is noted. For the purposes of this survey, a set of 78 optimally solved test problems from the literature and a second set of 110 benchmark problems have been subjected to analysis with some well-known dispatching rules and a scheduling algorithm that consists of a decision-making process utilizing the problem constraints as a base of selection. The computational results are reported and discussed in the text. Constructive scheduling algorithms that are directly based on the problem constraints...

A local constraint based analysis approach to project scheduling under general resource constraints

In the doubly-constrained project scheduling problem where nonrenewable resources are constrained both on a period and a project basis, the combinatorial complexity arising from interrelated precedence, renewable and nonrenewable resource constraints necessitates a solution approach which takes all such considerations into account including the objectives. In this context an activity is permitted to operate in one of its modes, each of which represents the trade off between different choices of resource requirements (types) and operating durations. The proposed heuristic approach to handle this problem is local constraint based analysis (LCBA) where the selection of activities and their respective modes is made locally at every decision point. LCBA imposes precedence relationships on activities which guarantee the feasibility of the current makespan constrained by the available resource levels at that decision point as long as a complete sequence of schedulable activities is found. The procedure is also adapted to various model extensions such as flexible resource requirement levels. The procedure can also be utilized in a dynamic environment where resource absenteeism, activity duration changes and readjustment of the project network configuration are accounted for. Numerical results are obtained for LCBA and three well-reputed dispatching rules to show that considerably better solutions are obtained by the LCBA approach in encouragingly short computation times.

Job Shop Scheduling under a Non-Renewable Resource Constraint

In this paper we consider the job shop scheduling problem under a discrete non-renewable resource constraint. We assume that jobs have arbitrary processing times and resource requirements and there is a unit supply of the resource at each time period. We develop an approximation algorithm for this problem and empirically test its effectiveness in finding the minimum makespan schedules. Most of the research done in the area of scheduling deals with the allocation of a single scarce resource over time to perform a collection of tasks. In this study, the scheduling environment is extended to include an additional non-renewable resource. The term non-renewable implies that the resource is actually being consumed by the jobs competing for it. Financial constraints are typical examples of such constraints and for this reason non-renewable resource constraints are often called financial constraints1. Materials shared by different products can also be regarded as a non-renewable resource in manufacturing environments. Considering these constraints explicitly is likely to yield more realistic schedules. A limited amount of research has been done in the area of non-renewable resource constrained scheduling. There are some polynomially bounded solution algorithms for precedence constrained scheduling problems2 and for pre-emptive scheduling of independent jobs on parallel machines1. In the case of arbitrary resource requirements and availabilities, Slowinski' states that Carlier has shown that even the non-preemptive single machine scheduling problem is NP-complete if job processing times are different from unity. In Toker et al.3, it is shown that when the amount of resource available at each time period is constant, the single machine non-renewable resource constrained problem is equivalent to a resource- free, two-machine flowshop problem. Hence it is solvable in polynomial time. They also extended their results to the m-machine case and showed how to transform different types of resource constrained problems into equivalent, unconstrained problems. The scheduling environment we consider is a job shop. A single non-renewable resource becomes available over time in equal quantities (say unity) and there is a set of jobs to be processed. Each operation requires an arbitrary amount of the non-renewable resource which must be available at the start of that operation and which is consumed during its processing. We used makespan as the performance measure. Optimization algorithms for the general job shop scheduling problem are restricted to implicit enumeration techniques, such as branch-and-bound based procedures4-7. On the other hand, approximation algorithms usually use a dispatching rule to give priorities to operations to be scheduled8. The non-renewable resource constrained job shop scheduling problem is NP-complete as, when all resource requirements are zero, it reduces to the unconstrained job shop problem which is NP-complete9. This complexity result serves as a formal justification to use approximation algorithms for the constrained problem. We first describe the approximation algorithm developed and then introduce two lower bounds for the n-job, m-machine resource-constrained job shop scheduling problem. Next, we discuss the computational results. Finally, we present our conclusions.