Resource allocation in dynamic PERT networks with finite capacity

This article models the resource allocation problem in dynamic PERT networks with finite capacity of concurrent projects (COnstant Number of Projects In Process (CONPIP)), where activity durations are independent random variables with exponential distributions, and the new projects are generated according to a Poisson process. The system is represented as a queuing network with finite concurrent projects, where each activity of a project is performed at a devoted service station with one server located in a node of the network. For modeling dynamic PERT networks with CONPIP, we first convert the network of queues into a stochastic network. Then, by constructing a proper finite-state continuous-time Markov model, a system of differential equations is created to solve and find the completion time distribution for any particular project. Finally, we propose a multi-objective model with three conflict objectives to optimally control the resources allocated to the servers, and apply the goal attainment method to solve a discrete-time approximation of the original multi-objective problem.     

Multi-objective time–cost trade-off in dynamic PERT networks using an interactive approach

We develop a multi-objective model for the time–cost trade-off problem in a dynamic PERT network using an interactive approach. The activity durations are exponentially distributed random variables and the new projects are generated according to a renewal process and share the same facilities. Thus, these projects cannot be analyzed independently. This dynamic PERT network is represented as a network of queues, where the service times represent the durations of the corresponding activities and the arrival stream to each node follows a renewal process. At the first stage, we transform the dynamic PERT network into a proper stochastic network and then compute the project completion time distribution by constructing a continuous-time Markov chain. At the second stage, the time–cost trade-off problem is formulated as a multi-objective optimal control problem that involves four conflicting objective functions. Then, the STEM method is used to solve a discrete-time approximation of the original problem. Finally, the proposed methodology is extended to the generalized Erlang activity durations.     

Discrete-continuous project scheduling with discounted cash inflows and various payment models—a review of recent results

In this work discrete-continuous project scheduling problems with discounted cash flows are considered. These problems are characterized by the fact that activities of a project simultaneously require discrete and continuous resources for their execution. A class of these problems is considered, where the number of discrete resources is arbitrary, and there is one continuous, renewable resource, whose total amount available at a time is limited. Activities are non-preemptable, and the processing rate of an activity is a continuous, increasing function of the amount of the continuous resource allotted to the activity at a time. A positive cash flow (cash inflow) is associated with each activity, and the objective is to maximize the net present value (NPV). The discrete-continuous resource-constrained project scheduling problem with discounted cash flows (DCRCPSPDCF) is defined. Four payment models are considered: lump-sum payment at the completion of the project, payments at activity completion times, payments at equal time intervals, and progress payments. Some properties of optimal schedules are proved for two important classes of processing rate functions: all functions not greater than a linear function (including linear and convex functions), and concave processing rate functions.     

Two-parallel machines scheduling with rate-modifying activities to minimize total completion time

This paper considers the two-parallel machines scheduling problem with rate-modifying activities. In this model, each machine has a rate-modifying activity that can change the processing rate of machine under consideration. Hence the actual processing times of jobs vary depending on whether the job is scheduled before or after the rate-modifying activity. We need to make a decision on when to schedule the rate-modifying activities and the sequence of jobs to minimize some objective function. We provide polynomial and pseudo-polynomial time algorithms to solve the total completion time minimization problem and total weighted completion time minimization problem under agreeable ratio condition.     

Single machine scheduling with exponential sum-of-logarithm-processing-times based learning effect

In this paper we consider the single machine scheduling problems with exponential sum-of-logarithm-processing-times based learning effect. By the exponential sum-of-logarithm-processing-times based learning effect, we mean that the processing time of a job is defined by an exponent function of the sum of the logarithm of the processing times of the jobs already processed. We consider the following objective functions: the makespan, the total completion time, the sum of the quadratic job completion times, the total weighted completion time and the maximum lateness. We show that the makespan minimization problem, the total completion time minimization problem and the sum of the quadratic job completion times minimization problem can be solved by the smallest (normal) processing time first (SPT) rule, respectively. We also show that the total weighted completion time minimization problem and the maximum lateness minimization problem can be solved in polynomial time under certain conditions.     

带有退化效应和序列相关运输时间的排序问题

考虑带有退化效应和序列相关运输时间的单机排序问题. 工件的加工时间是其开工时间的简单线性增加函数. 当机器单个加工工件时, 极小化最大完工时间、(加权)总完工时间和总延迟问题被证明是多项式可解的, EDD序对于极小化最大延迟问题不是最优排序, 另外, 就交货期和退化率一致情形给出了一最优算法. 当机器可分批加工工件时, 分别就极小化最大完工时间和加权总完工时间问题提出了多项式时间最优算法.     

Single machine scheduling with a general exponential learning effect

In this paper we consider the scheduling problem with a general exponential learning effect and past-sequence-dependent (p-s-d) setup times. By the general exponential learning effect, we mean that the processing time of a job is defined by an exponent function of the total weighted normal processing time of the already processed jobs and its position in a sequence, where the weight is a position-dependent weight. The setup times are proportional to the length of the already processed jobs. We consider the following objective functions: the makespan, the total completion time, the sum of the δ ? 0th power of completion times, the total weighted completion time and the maximum lateness. We show that the makespan minimization problem, the total completion time minimization problem and the sum of the quadratic job completion times minimization problem can be solved by the smallest (normal) processing time first (SPT) rule, respectively. We also show that the total weighted completion time minimization problem and the maximum lateness minimization problem can be solved in polynomial time under certain conditions.     

Single machine scheduling with exponential learning functions

In this paper we consider the single machine scheduling problem with exponential learning functions. By the exponential learning functions, we mean that the actual job processing time is a function of the total normal processing times of the jobs already processed. We prove that the shortest processing time (SPT) rule is optimal for the total lateness minimization problem. For the following three objective functions, the total weighted completion time, the discounted total weighted completion time, the maximum lateness, we present heuristic algorithms according to the corresponding problems without exponential learning functions. We also analyse the worst-case bound of our heuristic algorithms. It also shows that the problems of minimizing the total tardiness and discounted total weighted completion time are polynomially solvable under some agreeable conditions on the problem parameters.     

Branch and bound based solution algorithms for the budget constrained discrete time/cost trade-off problem

The time/cost trade-off models in project management aim to reduce the project completion time by putting extra resources on activity durations. The budget problem in discrete time/cost trade-off scheduling selects a time/cost mode for each activity so as to minimize the project completion time without exceeding the available budget. There may be alternative modes that solve the budget problem optimally and each solution may have a different total cost value. In this study we consider the budget problem and aim to find the minimum cost solution among the minimum project completion time solutions. We analyse the structure of the problem together with its linear programming relaxation and derive some mechanisms for reducing the problem size. We solve the reduced problem by branch and bound based optimization and heuristic algorithms. We find that our branch and bound algorithm finds optimal solutions for medium-sized problem instances in reasonable times and the heuristic algorithms produce high quality solutions very quickly.     

Project Scheduling under Time Dependent Costs – A Branch and Bound Algorithm

In a given project network, execution of each activity in normal duration requires utilization of certain resources. If faster execution of an activity is desired then additional resources at extra cost would be required. Given a project network, the cost structure for each activity and a planning horizon, the project compression problem is concerned with the determination of optimal schedule (duration) of performing each activity while satisfying given restrictions and minimizing the total cost of project execution. This paper considers the project compression problem with time dependent cost structure for each activity. The planning horizon is divided into several regular time intervals over which the cost structure of an activity may vary. But the cost structure of the activities remains the same (constant) within a time interval. Key events of the project attract penalty for finishing earlier or later than the corresponding target times. The objective is to find an optimal project schedule minimizing the total project cost. We present a mathematical model for this problem, develop some heuristics and an exact branch and bound algorithm. Using simulated problems we provide an insight into the computational performances of heuristics and the branch and bound algorithm.     

A new path-based cutting plane approach for the discrete time-cost tradeoff problem

Determining discrete time-cost tradeoffs in project networks allows for the control of the processing time of an activity via the amount of non-renewable resources allocated to it. Larger resource allocations with associated higher costs reduce activities’ durations. Given a set of execution modes (time-cost pairs) for each activity, the discrete time-cost tradeoff problem (DTCTP) involves selecting a mode for each activity so that either: (i) the project completion time is minimized, given a budget, or (ii) the total project cost is minimized, given a deadline, or (iii) the complete and efficient project cost curve is constructed over all feasible project durations. The DTCTP is a problem with great applicability prospects but at the same time a strongly N P{\mathcal N}\,P-hard optimization problem; solving it exactly has been a real challenge. Known optimal solution methodologies are limited to networks with no more than 50 activities and only lower bounds can be computed for larger, realistically sized, project instances. In this paper, we study a path-based approach to the DTCTP, in which a new path-based formulation in activity-on-node project networks is presented. This formulation is subsequently solved using an exact cutting plane algorithm enhanced with speed-up techniques. Extensive computational results reported for almost 5,000 benchmark test problems demonstrate the effectiveness of the proposed algorithm in solving to optimality for the first time some of the hardest and largest instances in the literature. The promising results suggest that the algorithms may be embedded into project management software and, hence, become a useful tool for practitioners in the future.     

Optimal production inventory policy for defective items with fuzzy time period

In this paper, an optimal production inventory model with fuzzy time period and fuzzy inventory costs for defective items is formulated and solved under fuzzy space constraint. Here, the rate of production is assumed to be a function of time and considered as a control variable. Also the demand is linearly stock dependent. The defective rate is taken as random, the inventory holding cost and production cost are imprecise. The fuzzy parameters are converted to crisp ones using credibility measure theory. The different items have the different imprecise time periods and the minimization of cost for each item leads to a multi-objective optimization problem. The model is under the single management house and desired inventory level and product cost for each item are prescribed. The multi-objective problem is reduced to a single objective problem using Global Criteria Method (GCM) and solved with the help of Fuzzy Riemann Integral (FRI) method, Kuhn–Tucker condition and Generalised Reduced Gradient (GRG) technique. In optimum results including production functions and corresponding optimum costs for the different models are obtained and then are presented in tabular forms.     

A multi-objective lead time control problem in multi-stage assembly systems using genetic algorithms

In this paper, we develop a multi-objective model to optimally control the lead time of a multi-stage assembly system, using genetic algorithms. The multi-stage assembly system is modelled as an open queueing network. It is assumed that the product order arrives according to a Poisson process. In each service station, there is either one or infinite number of servers (machines) with exponentially distributed processing time, in which the service rate (capacity) is controllable. The optimal service control is decided at the beginning of the time horizon. The transport times between the service stations are independent random variables with generalized Erlang distributions. The problem is formulated as a multi-objective optimal control problem that involves four conflicting objective functions. The objective functions are the total operating costs of the system per period (to be minimized), the average lead time (min), the variance of the lead time (min) and the probability that the manufacturing lead time does not exceed a certain threshold (max). Finally, we apply a genetic algorithm with double strings using continuous relaxation based on reference solution updating (GADSCRRSU) to solve this multi-objective problem, using goal attainment formulation. The results are also compared against the results of a discrete-time approximation technique to show the efficiency of the proposed genetic algorithm approach.     

人力资源约束下的项目群调度问题建模与求解

基于人员胜任力是影响工作绩效的关键因素,将资源受限项目调度问题中的可更新资源通过一系列科学合理的方法或者手段转变为存在胜任力差异的人力资源,由此构建起一个强调胜任力差异的人力资源约束项目调度问题模型,此模型最突出的优势在于选取了能够客观合理评估人员胜任力的指标,提供了严谨科学的关系式,将复杂的多项目总工期与总成本的双目标最小化问题转换为综合指标单目标最大化问题,建立数学优化模型,采用遗传算法求解。通过算例研究证实,相较于传统多模式模型,基于人员胜任力水平差异的模型明显更胜一筹,其优势集中表现为最优工期更短、最优成本更低。考虑了胜任力差异的数学优化模型更符合研发项目群管理实践,同时遗传算法在求解方面不仅效率高,并且更容易获得客观准确的结果。     

Some single-machine scheduling with sum-of-processing-time-based and job-position-based processing times

The single-machine scheduling problems with position and sum-of-processing-time based processing times are considered. The actual processing time of a job is defined by function of its scheduled position and total normal processing time of jobs in front of it in the sequence. We provide optimal solutions in polynomial time for some special cases of the makespan minimization and the total completion time minimization. We also show that an optimal schedule to be a V-shaped schedule in terms of the normal processing times of jobs for the total completion time minimization problem and the makespan minimization problem.     

Scheduling problems in transportation networks of line topology

In this paper we consider online scheduling problems for linear topology under various objective functions: minimizing the maximum completion time, minimizing the largest delay, and minimizing the sum of completion times. We give optimal solutions for uni-directional version of the problem for each of the objectives and show that for the two-directional versions of each problem, no online algorithm can deterministically achieve the optimal solution for any of the considered objective functions. We also propose 2-approximation on-line algorithms for the MinMakespan and the MinSum minimization objectives. We also prove that no online algorithm can deterministically achieve the optimal solution for any of the considered objective functions for the weighted case of uni-directional scenarios.     

Solving project scheduling problem with the philosophy of fuzzy random programming

Project scheduling problem is to determine the schedule of allocating resources to achieve the trade-off between the project cost and the completion time. In real projects, the trade-off between the project cost and the completion time, and the uncertainty of the environment are both considerable aspects for managers. Due to the complex external environment, this paper considers project scheduling problem with coexisted uncertainty of randomness and fuzziness, in which the philosophy of fuzzy random programming is introduced. Based on different ranking criteria of fuzzy random variables, three types of fuzzy random models are built. Besides, a searching approach by integrating fuzzy random simulations and genetic algorithm is designed for searching the optimal schedules. The goal of the paper is to provide a new method for solving project scheduling problem in hybrid uncertain environments.     

Optimization models and a GA-based algorithm for stochastic time-cost trade-off problem

In real-life projects, both the trade-off between the project cost and the project completion time, and the uncertainty of the environment are considerable aspects for decision-makers. However, the research on the time-cost trade-off problem seldom concerns stochastic environments. Besides, optimizing the expected value of the objective is the exclusive decision-making criterion in the existing models for the stochastic time-cost trade-off problem. In this paper, two newly developed alternative stochastic time-cost trade-off models are proposed, in which the philosophies of chance-constrained programming and dependent-chance programming are adopted for decision-making. In addition, a hybrid intelligent algorithm integrating stochastic simulations and genetic algorithm is designed to search the quasi-optimal schedules under different decision-making criteria. The goal of the paper is to reveal how to obtain the optimal balance of the project completion time and the project cost in stochastic environments.     

Single-machine scheduling problems with an actual time-dependent deterioration

In this paper, we consider the single machine scheduling problems with an actual time-dependent deterioration effect. By the actual time-dependent deterioration effect, we mean that the processing time of a job is defined by increasing function of total actual processing time of jobs in front of it in the sequence. We show that even with the introduction of an actual time-dependent deterioration to job processing times, makespan minimization problem, total completion time minimization problem, the total lateness, and the sum of the quadratic job completion times minimization problem remain polynomially solvable, respectively. We also show that the total weighted completion time minimization problem, the discounted total weighted completion time minimization problem, the maximum lateness minimization problem, and the total tardiness minimization problem can be solved in polynomial time under certain conditions.