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Resource investment problem with discounted cash flows (RIPDCFs) is a class of project scheduling problem. In RIPDCF, the availability levels of the resources are considered decision variables, and the goal is to find a schedule such that the net present value of the project cash flows optimizes. In this paper, we consider a new RIPDCF in which tardiness of project is permitted with defined penalty. We mathematically formulated the problem and developed a heuristic method to solve it. The results of the performance analysis of the proposed method show an effective solution approach to the problem.

Project scheduling is a major objective of most models, and methods propose to aid planning and management of projects. Project scheduling problems are combination of precedence constraints, resource constraints, and some side constraints in which the goal is to find a schedule such that an objective function like project duration, project total costs, or net present value (NPV) optimizes. For a comprehensive survey of project scheduling problems refer to [

Resource investment problem (RIP) is a class of project scheduling problem. RIP is known as the problem of minimizing renewable resource costs subject to a project due date. In RIP, we are concerned about completing a project consisting of a set of activities, such that a given deadline is met in time and a set of resources needed for the execution of the activities over the project is utilized. Since costs incur to provide resources, the goal is to find a schedule and resource requirement levels such that total costs of the resource utilizations minimizes.

Möhring [

Many of the recent researches in project scheduling focus on maximizing the NPV of the project using the sum of positive and negative discounted cash flows throughout the life cycle of the project. Russell [

In addition, Najafi and Niaki [

In [

In this paper we consider an RIPDCF in which tardiness is permitted with delay penalty and call it RIPDCFT. The rest of the paper is organized as follows. In Section

A project is given with a set of

According to assumptions and notations introduced in Section

starting time of activity

occurrence time for payment

required level of resource

providing time of resource

expulsion time of resource

a binary variable where it is one if activity

In addition,

The objective function (

In this section, based on the priority rules of the RIPDCF, we propose a heuristic method to solve the problem. To do this, first we state some definitions that are required in the procedure.

Negative cash flow of an activity. It includes discounted cash flow of the resource usage cost and fixed cost at the activity starting time. It can be stated as

Positive cash flow of an activity. If the precedent activity set of payment occurrence contains only one activity, then we set positive cash flow of the activity to be equal to the discounted cash flow of that payment at the activity starting time. In this case, we define the positive cash flow of the activity as

Cash flow of an activity. Cash flow of an activity equals to the sum of the negative and the positive cash flows of an activity. In other words, we have

The amount of nonusage resource at a period. With (

In order to develop the solution procedure, we use the structure of the objective function given in (

Now we are ready to describe the executive steps of the proposed algorithm as follows:

Let problem

The

Eliminate constraint (

Add all resources in a set, named resource candidate list.

From the list of resource candidates, select the resource with the highest discounted cost of nonusage

If the temporary objective function value is more than the active objective function value add the selected constraint to the active problem. Then, consider the acquired problem, related scheduling, and the temporary objective function value as an active problem and go to step four. Otherwise, do not add the selected resource constraint to the active problem.

Eliminate the selected resource from the resource candidate list and go to step seven.

If the resource candidate list is empty, stop. The active schedule is the solution of the proposed algorithm. Otherwise, go to step four.

In this section, we present the performance of the proposed procedure introduced in the previous sections. For the purpose of this section we needed a set of solved problems. Since the RIPDCFT is a newly defined problem, no standard test problems could be found to examine the performance of the proposed procedure. Therefore, we are forced to use the RIPDCF test problems, suggested by Najafi and Niaki [

Table

Computational results.

No. of Activities | No. of Instances | A | Avg. of dev% | Avg. of | Avg. of |

LINGO CPU time (Sec.) | the proposed method CPU time (Sec.) | ||||

10 | 15 | 10 | −1.9% | 388 | 1 |

20 | 15 | 7 | −2.3% | 1253 | 2 |

30 | 15 | 4 | −2.8% | 2407 | 4 |

The results of experiments showed that (a) there are many instances that the solver software is unable to solve in 3600 seconds, but there is a solution by the proposed method, (b) for problems in which LINGO was able to find a solution, there is no significant difference between the solutions obtained by LINGO and the ones obtained by the proposed method and (c) while actually there is no difference between the solutions obtained by LINGO and the proposed method, the amount of CPU time for the proposed method is much less than that of those obtained by LINGO.

In this paper, we introduced a new resource investment problem with discounted cash flows in which tardiness is permitted with penalty. We mathematically formulated the problem. In order to solve the problem we came up with a heuristic approach and through some generated test problems, we showed that it works relatively well.

The extension of this research would be to investigate an RIP/max problem in which the goal is to maximize the NPV of the project and tardiness is permitted with penalty, too. One of the other potential interests would be to develop some metaheuristics methods, such as genetic algorithm, simulated annealing, neural networks, ant colony algorithm, and so forth, to solve the problem.

The authors thank the anonymous referees for their useful suggestions that improved the presentation of the paper.