An integer programming formulation for the project scheduling problem with irregular time–cost tradeoffs Joseph G. Szmerekovsky a,n , Prahalad Venkateshan b a North Dakota State University, College of Business, NDSU Department 2420, PO Box 6050, Fargo, North Dakota 58108-6050, USA b Indian Institute of Management, Vastrapur, Ahmedabad 380015, India article info Available online 24 August 2011 Keywords: Project scheduling Integer programming formulations Time/cost trade-offs Irregular costs abstract Four integer programming formulations are studied for the irregular costs project scheduling problem with time/cost trade-offs (PSIC). Three formulations using standard assignment type variables are tested against a more novel integer programming formulation. Empirical tests show that in many instances the new formulation performs best and can solve problems with up to 90 activities in a reasonable amount of time. This is explained by a reduced number of binary variables, a tighter linear programming (LP) relaxation, and the sparsity and embedded network structure of the constraint matrix of the new formulation. & 2011 Elsevier Ltd. All rights reserved. 1. Introduction This research considers integer programming formulations for the irregular costs project scheduling problem with time/cost trade-offs (PSIC). The interest in this problem is motivated by the need to reconcile two areas of the project management literature: time/cost trade-off problems and problems with financial objec- tive functions. The variety of time/cost trade-off problems that have been studied in the literature are reviewed by Brucker et al. [1], De et al. [2] and Icmeli et al. [3]. Recent research on time/cost trade-off problems has considered PERT networks [4,5], uncer- tainty in budgets [6], discrepancies in quality between activity modes [7], and time-switch constraints [8,9]. Surprisingly, despite the vast diversity of models that have been studied none have yet considered explicitly the impact of the contractor’s chosen sche- dule on the client’s benefits. Specifically, in those versions of the time/cost trade-off problem known as deadline problems, the objective is to minimize the cost of completing the project subject to a due date. When only cost is considered, the contractor will clearly prefer the largest due date possible so that activities can be processed slowly and cheaply without consideration to poten- tial financial rewards for earlier project completion. Hence, the due date exists primarily as a proxy for the client’s interest in the contractor’s choice of schedule. In practical settings the dynamic between contractor and client is much more sophisticated and can include time-dependent bonuses and penalties, payment retention, the selection of milestone activities and periodic payment schemes. That is, the contractor’s and client’s interests are both more fully represented by the payment schedule rather than a simple project due date. The contractor will select a processing mode and completion time for each activity, called the activity schedule, so as to maximize some financial objective. Given the contractor’s behavior, the client will strive to negotiate a payment plan that maximizes his own financial objective. A variety of possible financial objectives exist in the literature including net-present value (NPV), cost, and cash availability (CA). The NPV objective for project scheduling originally appeared in Russell [10] and has since received significant attention. Reviews can be found in Icmeli et al. [3], Herroelen et al. [11] and Brucker et al. [1]. The problem of how the contractor should schedule linear time-dependent cash flows has been studied in Etgar and Shtub [12] and Vanhoucke et al. [13]. Erenguc et al. [14] use generalized Bender’s decomposition to solve the problem without resource constraints and heuristics are provided for the resource constrained problem by Sunde and Lichtenberg [15] and Icmeli and Erenguc [16]. In addition Ulusoy et al. [17] provide a genetic algorithm for four different payment models used with a multi- mode resource constrained project scheduling problem. As mentioned, most literature focuses primarily on the con- tractor’s NPV. Works that consider the NPV of the client include Bey et al. [18], Dayanand and Padman [19], Ulusoy and Cebelli [20], Dayanand and Padman [21] and Szmerekovsky [22]. The client’s interests are studied in Dayanand and Padman [19] and Dayanand and Padman [21] which consider the payment sche- duling problem in which the client has complete control, simul- taneously scheduling activities and determining the amount and timing of payments. Both the activity schedule and the payment Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/caor Computers & Operations Research 0305-0548/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.cor.2011.08.011 n Corresponding author. E-mail addresses: Joseph.Szmerekovsky@ndsu.edu (J.G. Szmerekovsky), prahalad@iimahd.ernet.in (P. Venkateshan). Computers & Operations Research 39 (2012) 1402–1410