Discrete Dynamics in Nature and Society

Volume 2015, Article ID 586087, 10 pages

http://dx.doi.org/10.1155/2015/586087

## Robust Proactive Project Scheduling Model for the Stochastic Discrete Time/Cost Trade-Off Problem

^{1}School of Management, Shanghai University, 99 Shangda Road, Shanghai 200444, China^{2}School of Economics and Management, Beihang University, 37 Xueyuan Road, Beijing 100191, China

Received 21 August 2014; Accepted 4 November 2014

Academic Editor: Hua Gong

Copyright © 2015 Hongbo Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We study the project budget version of the stochastic discrete time/cost trade-off problem (SDTCTP-B) from the viewpoint of the robustness in the scheduling. Given the project budget and a set of activity execution modes, each with uncertain activity time and cost, the objective of the SDTCTP-B is to minimize the expected project makespan by determining each activity’s mode and starting time. By modeling the activity time and cost using interval numbers, we propose a proactive project scheduling model for the SDTCTP-B based on robust optimization theory. Our model can generate robust baseline schedules that enable a freely adjustable level of robustness. We convert our model into its robust counterpart using a form of the mixed-integer programming model. Extensive experiments are performed on a large number of randomly generated networks to validate our model. Moreover, simulation is used to investigate the trade-off between the advantages and the disadvantages of our robust proactive project scheduling model.

#### 1. Introduction

In project management, the project duration can usually be shortened by allocating more resources to critical activities. The number of resources tends to be discrete, such as the numbers of workers and machines. These resources are usually treated as nonrenewable resources and measured by capital (or cost), resulting in the discrete time/cost trade-off problem (DTCTP) [1]; Harvey and Patterson [2] and Hindelang and Muth [3] first proposed the DTCTP, which is a special case of the multimode resource-constrained project scheduling problem [4].

In the deterministic DTCTP, each activity has multiple execution modes that are characterized by specific time and cost combinations. In terms of the types of the objective function, the DTCTP can be divided into three versions: the deadline problem (DTCTP-D), the budget problem (DTCTP-B), and the time/cost trade-off curve problem (DTCTP-C). In the DTCTP-D, given a set of modes and a project deadline, the objective is to minimize the total project cost by specifying for each activity an execution mode. In the DTCTP-B, a project budget is given and the objective is to determine the modes that minimize the project makespan. In the DTCTP-C, the goal is to determine the Pareto curve that minimizes the project cost and makespan simultaneously.

Once the mode of each activity is determined, we can determine the baseline schedule by calculating the earliest start time of each activity in accordance with the critical path method (CPM).

However, during project execution, due to considerable uncertainties, the optimal baseline schedule which is obtained based on a deterministic environment and complete information may deviate from our expectations or even become unfeasible. Possible sources of uncertainties may be a shortage of machineries, a delayed delivery of materials, the absence of workers, fluctuations in the exchange rates, and so forth [5, 6]. As a consequence, issues such as schedule delays and/or budget overruns may occur and project time and cost objectives will be threatened. Therefore, there is a pressing need for new procedures for the DTCTP under uncertainties to obtain project schedules which are insensitive to disruption.

Recent studies have paid more attention to the stochastic DTCTP (SDTCTP), which accounts for uncertainties by treating the time and cost of activities as stochastic variables, with the objective of optimizing the expected project performance. As early researchers in the field, Gutjahr et al. [7] presented a stochastic branch-and-bound procedure to solve the DTCTP deadline problem based on the assumption that the times of each activity are mutually independent random variables. Laslo [8] used the fractal method to construct time/cost curves for a single activity of stochastic duration. Cohen et al. [9] implemented a robust optimization to solve the time/cost trade-off problem. Ke et al. [10] used chance-constrained programming and dependent-chance programming to model the stochastic DTCTP; the authors designed an intelligent algorithm to search the quasi-optimal schedules while balancing the project duration and cost. Klerides and Hadjiconstantinou [11] proposed a path-based two-stage stochastic integer programming approach to decide how and when to execute each activity to minimize the project duration or cost using realized activity durations. Ma et al. [12] studied the stochastic time-cost-quality trade-off problem where the activity durations are uncertain and developed a hybrid genetic algorithm.

However, the above-mentioned research papers primarily focused on optimizing the system performance in an average sense and these prior approaches cannot guarantee the performance of the baseline schedule during a single project execution. Therefore, determining a robust baseline schedule under uncertainty is increasingly attracting the attention of scholars. To achieve a robust baseline schedule, the use of robust optimization is a natural choice. Robust optimization can determine a solution with certain robustness by optimizing the worst-case performance of the system. Although robust optimization has been used to solve some classic project scheduling problems [13, 14], little effort has been applied to the study of the SDTCTP. To the best of our knowledge, Hazır et al. [6] is the only research article that combines the SDTCTP and the robust optimization approach. Hazır et al. propose three robust optimization models, in which cost uncertainty is modeled via intervals for the SDTCTP deadline problem. The aim of their model is to minimize the effect of unexpected events on project performance. The limitations of their models are that the activity cost is still assumed to be deterministic and that only the parameters in the objective function are subject to uncertainty (i.e., the parameters in the constraints are deterministic).

To the best of our knowledge, addressing both time- and cost-uncertainty and applying robust optimization in solving the SDTCTP-B have not been taken into account in both the project scheduling and the robust optimization literature. The contributions of this paper are as follows.(1)We proposed a proactive scheduling model for the SDTCTP budget problem (SDTCTP-B) based on robust optimization theory. Our model uses interval numbers to model the uncertain time and cost of the activities that can follow any type of probability distribution. The objective of our model is to generate a stable baseline schedule that can account for some of the uncertainties during project execution to ensure, to the extent possible, that each activity begins at their respective planned start time.(2)We conducted a detailed experimental analysis for our proposed model. We used experimental design to randomly generate a large number of instances to validate our model. In addition, robust optimization improves the schedule stability at the cost of prolonging the project duration. Therefore, we used simulation to investigate the trade-off between the advantages and the disadvantages of robust optimization. Specifically, we analyzed the impact of the number of activities, the network order strength, and the number of modes on the schedule stability by using discrete systems simulation.

This paper is organized as follows. Section 2 provides a description of the SDTCTP. In Section 3, we use interval numbers to model the uncertain parameters and present a proactive model to solve the SDTCTP-B based on robust optimization. Considering the nonlinear characteristics of the proposed model, we converted our model into its robust counterpart, which has the form of a mixed integer linear programming model, and used the branch-and-cut algorithm to solve the model. In Section 4, we present the experimental results. Finally, Section 5 concludes the paper.

#### 2. Problem Description

The stochastic discrete time/cost trade-off problem can be described as follows. A project network is represented in the activity-on-node format, where the set of nodes denotes the activities and the set of directed arcs represents the finish-start, zero-lag precedence relations . The nodes are topologically numbered from the single-start node 1 to the single-terminal node , , where nodes 1 and represent two dummy activities. The duration and cost of activity are random variables, denoted as and , respectively. For activity , represents the set of its modes. Each activity has modes, which are characterized by a duration-cost pair , . The duration of an activity is a discrete, nonincreasing function of the amount of the single nonrenewable resource committed to it; that is, if , then, for the expected duration and cost, we have and . We assume that the dummy activities 1 and have only one execution mode of zero duration/cost.

Given the project budget , the objective of the SDTCTP-B is to minimize the expected project makespan by assigning a mode to each activity and determining the start time of each activity. If we substitute the random duration and cost of the activities by their most likely values and , respectively, for the above problem, the SDTCTP will become the deterministic DTCTP. For DTCTP-B, we have the following integer programming model: where and are decision variables. are 0-1 variables that determine whether a mode of an activity is selected. The objective function (1) minimizes the start time of the dummy end node , which is equivalent to minimizing the project makespan. Equation (2) ensures that for each activity only one execution mode is selected. Equation (3) defines the precedence relationship constraints for the activities. Equation (4) ensures that the total project cost does not exceed the budget . Equation (5) ensures that the start time of each activity is nonnegative. The budget problem of DTCTP is strongly NP-hard [1, 15, 16].

Faced with the uncertainty in the duration and cost of each activity, the baseline schedule generated by the above deterministic model is not expected to be executed as determined, thereby resulting in failure to achieve the desired project objective. When the uncertainty is considered, we notice that the uncertain parameters mainly affect constraints (3) and (4). Therefore, in the following section, we model the uncertain parameters as interval numbers and develop a proactive scheduling model for the SDTCTP-B based on robust optimization, thereby obtaining a robust baseline schedule.

#### 3. The Proactive Scheduling Model for SDTCTP-B

##### 3.1. Modeling Uncertain Parameters as Interval Numbers

In practice, it is usually easier for decision-makers to estimate the range and the most likely value of the duration and cost of activities rather than their probability distribution. Therefore, we use interval numbers to model the uncertain duration and cost of the activity. For mode , let and be the most likely value of the activity duration and cost , respectively; and take a value in the interval and , respectively (i.e., , ). We define the maximum deviation of the activity duration and cost as and , respectively, which represent the maximum difference between the planned activity duration and cost and the actual activity duration and cost that can be tolerated by the decision-makers.

##### 3.2. The Proactive Scheduling Model

For each activity , we introduce a parameter , which is not necessarily an integer, that takes on a value in the interval . is used to adjust the robustness of our model (i.e., the level of conservatism of the solutions). For each activity , we assume that modes take on the values at their upper bounds of , , the value for one mode can deviate and , and the remaining modes are set to their most likely values of .

Our proactive scheduling model for SDTCTP-B based on robust optimization [17, 18] is as follows:

The above model introduces the uncertain parameters into the deterministic DTCTP model and is able to generate robust solutions. is a subset of , and the modes belonging to take on the worst case values. The cardinality of is determined by . In addition, the objective function (7) is a deterministic function rather than an expectation function. Therefore, the time-consuming expectation calculation is avoided.

Our model has two primary advantages. The first advantage is that the robustness level of the obtained schedule can be freely adjusted. The greater the value of the parameter , the higher the level of robustness is. If , the model will become the deterministic DTCTP model. The second advantage is that, although the proposed model is nonlinear, it can be easily reformulated as an equivalent linear mixed-integer programming (MIP) model according to the robust optimization theory [17, 18]. To obtain the equivalent MIP model, we first let .

means that we need to determine a subset that includes elements, such that is maximized. Therefore, we introduce decision variables , , . Given a vector , equals the objective function value of the following linear programming:

The constraints of problem (10) ensue that the resulting optimal objective function value is equivalent to . It is clear that the linear programming problem (10) has an optimal solution with of decision variables taking on the value of 1 and the remaining taking on the value 0.

The dual of problem (10) iswhere and are dual variables. According to strong duality, because problem (10) has an optimal solution, then problem (11) also has an optimal solution, and their optimal values are the same. In addition, equals the objective function value of problem (11).

Similarly, the “max” part of (9) can be converted to the following linear programming model:

Then we can obtain the equivalent mixed-integer linear optimization model of Model 1 by substituting (11) and (12) into it:

##### 3.3. Example

We use an example project network in Figure 1 to illustrate our model. Activities 0 and 5 are dummy activities. Activities 1 and 2 have only one mode. Activities 3 and 4 have two modes. We assume that the project budget is .