Academic journal article Journal of Management Information and Decision Sciences

Scheduling of Projects under Penalty and Reward Arrangements: A Mixed Integer Programming Model and Its Variants

Academic journal article Journal of Management Information and Decision Sciences

Scheduling of Projects under Penalty and Reward Arrangements: A Mixed Integer Programming Model and Its Variants

Article excerpt


In this paper, we develop a mixed integer linear programming (MILP) model for scheduling projects under penalty and reward arrangements. Two variants of the model (model variants I and II) are also developed. The main MILP model and its variants are applicable for scheduling for-profit and not-for-profit projects. The developments of each of the model variants are preceded by the descriptions of the conditions/assumptions under which they are applicable. Numerical examples given to test and compare the models show that they work very well. The results of the examples show that both the main model and its variants produce the same optimal values of project/activity durations and event times, bonus/penalty, and total project costs. The model variants are much easier to solve than the main MILP model. They have much less number of variables and constraints than the main model. The number of iterations before obtaining optimal solution to the main model is four to six times as large as the number of iterations before obtaining an optimal solution to each of its two variants. A remarkable and interesting finding is that the number of zero-one variables that are required for the applications of model variants I and II are 3 and 4 respectively, irrespective of project or model size. The number of the main MILP's zero-one variables is many times larger than these and the number increases rapidly with increase in project or model size. However, an advantage of the main model is that it is applicable under many conditions.

Keywords: computational efficiency, for-profit and not-for-profit organizations, project costs, project management, pro-rating variables, time-cost trade-off.

(ProQuest: ... denotes formulae omitted.)


The first published research in the applications of formal optimization techniques in project planning and scheduling is by Kelly & Walker (1959). Kelly & Walker (1959) developed what is now popularly known as time-cost-trade-off model (TCTM) with time constraints. Their pioneering work elicited a lot of research interests among many scholars and this has led to the developments and publications of different types of project scheduling models. The basic techniques of most of these models are rooted in mathematical programming. These models/problems are of different types and categories. Some of these include:

i. Client-contractor interaction problems (Szmerekovskey, 2005; Ulusory & Cabelli, 2000; Westney, 1 992).

ii. Discount cash flows/maximization of net present value of projects ( Etgar & Shtub, 1999; Ulusory & Cabelli, 2000; Vanhoucke et al, 2003).

iii. Resource-constrained project-scheduling problems (a few examples can be seen in Ballestin & Leus, 2009; Hurink et al., 2009; Ranjbar, 2008; Sabzehparvar & Seyed-Hosseini, 2008; Vails, Ballestin, & Quintamilla, 2004, 2005),

iv. Time-cost trade-off problems (Brucker et al., 199; Castro-Lacouture et al., 2009; Jolayemi & Oluleye, 1994; Jolayemi & Pennington, 2007; Ke & Liu, 2010; Kolish & Hartmann, 2006), and

v. Time-dependent cash flows ( Dayanand & Padman 2001 ; Herroelen et al., 1997; Jolai, 2008; Vanhoucke et al., 2001, 2003).

Some important elements of project planning and scheduling, many of which are subproblems or components of the major problems listed above, have also caught the attention of some authors. Among these important elements/sub-problems are:

* Fixed price contract arrangements (Gilbreath, 1992; Herroelen et al., 1997; Westney, 1992)

* Penalty and/or reward arrangement (Jolayemi, 2002; Jolayemi & Olorunniwo, 2003; Jolayemi & Pennington, 2007).

* Progress payments (Herroelen at al., 1997; Szmerekovskey, 2005), and

* Target contract arrangements (Gilbreath, 1992; Herroelen et al., 1997).

The importance of each of these key elements/sub-problems in the successful planning and execution of projects requires that much more attention be paid to the developments of more models/techniques for their efficient and effective planning and management. …

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