Incentive contract design for projects: The owner׳s perspective

Omega ∎ (∎∎∎∎) ∎∎∎–∎∎∎ Contents lists available at ScienceDirect Omega journal homepage: www.elsevier.com/locate/omega Incentive contract design for projects: The owner's perspective$ L.P. Kerkhove a, M. Vanhoucke a,b,c,n a b c Faculty of Economics and Business Administration, Ghent University, Ghent, Belgium Technology and Operations Management Area, Vlerick Business School, Ghent, Belgium UCL School of Management, University College London, London, UK art ic l e i nf o a b s t r a c t Article history: Received 14 November 2014 Accepted 3 September 2015 Due to the adoption of more and more complex incentive contract structures for projects, designing the best contract for a specific situation has become an increasingly daunting task for project owners. Through the combination of findings from contracting literature with knowledge from the domain of project management, a quantitative model for the contract design problem is constructed. The contribution of this research is twofold. First of all, a comprehensive and quantitative methodology to analyse incentive contract design is introduced, based on an extensive review of the existing literature. Secondly, based on this methodology, computational experiments are carried out, which result in a set of managerial guidelines for incentive contract design. Our analysis shows that substantial improvements can often be attained by using contracts which include incentives for cost, duration as well as scope simultaneously. Moreover, nonlinear and piecewise linear formulae to calculate the incentive amounts are shown to improve both the performance and robustness across different projects. & 2015 Elsevier Ltd. All rights reserved. Keywords: Incentives Contracting Project Management Decision Making Strategy 1. Introduction The project owner and the contractor executing the project are two separated economic actors, each with their own set of potentially conflicting objectives [53,80]. Hence, when the owner expedites work to a contractor, a relationship must be established. The nature of this relationship can be plotted on a spectrum between an explicitly negotiated contract and an alliance in which both parties are formally unified into a single economic actor for the duration of the project. For projects where complexity is limited, an explicit contract which specifies the deliverables can suffice [82]. For complex projects on the other hand, it may be more favourable to unify both actors in an alliance structure, effectively forming a single economic entity [84]. Although valuable arguments can be made in favour of such alliance structures [61], the implementation of such a structure is often highly complex [10]. Hence, the inclusion of incentive clauses, which form the middle ground in the relational spectrum between explicit contracts and alliances, can provide a more workable alternative [9]. Performance related pay in general [25,27,90], and the design of incentivised agreements for projects in particular (see Table 1) have been widely studied in academic literature over the last decades. Notwithstanding these recent advances, little guidance is available for project owners on how to identify the best contract for a specific project environment. The aim of this paper is to provide a quantitative framework for incentive contract design in projects, which can be used by the project owner to select the most adequate contract for any given project environment. This quantitative framework consists of three components: a trade-off model describing the nature of the project, an evaluation model describing the valuation of the different outcomes of the project for both the owner and contractor, and a contract model which is capable of representing the majority of (incentivised) contractual agreements used in practice. Using these models, computational experiments have been carried out to investigate the impact of different project environments on the performance of contract types. These experiments take an economical rather than psychological perspective on the problem, and therefore assume that the contractor is a risk neutral profit-maximising actor. This risk-neutrality can be assumed since we are considering economic actors rather than individuals [32]. The desirability of different types of contracts is judged by taking into account both the expected profit of the owner, as well as the degree to which the motivations of the owner and contractor are aligned. 2. Literature review ☆ This manuscript was processed by Associate Editor Jozefowska. Corresponding author at: Faculty of Economics and Business Administration, Ghent University, Ghent, Belgium. n Literature relevant to incentive contract design for projects can be divided into two main categories: literature dealing with the http://dx.doi.org/10.1016/j.omega.2015.09.002 0305-0483/& 2015 Elsevier Ltd. All rights reserved. Please cite this article as: Kerkhove LP, Vanhoucke M. Incentive contract design for projects: The owner's perspective. Omega (2015), http://dx.doi.org/10.1016/j.omega.2015.09.002i 2 L.P. Kerkhove, M. Vanhoucke / Omega ∎ (∎∎∎∎) ∎∎∎–∎∎∎ trade-offs in project management and literature concerned with the design and implications of incentive contracts. Project management literature dictates that the properties of a project can be described along three dimensions: the costs associated with the project, the duration of the project and the scope of a project (also known as the iron triangle [47]). These three dimensions are viewed as an interrelated trade-off mechanism. Ceteris paribus, decreasing the cost of a project will be accompanied by an increase in duration and/or a decrease in scope. Similar statements are also true for the duration and scope dimensions. Within the context of this paper these three dimensions are viewed as the outcomes of the project, as perceived by the project owner. The cost reflects the financial payment the owner has to make to the contractor to compensate for the work performed by the latter, as well as the resources used in the project (insofar as this amount is variable). The duration represents the time needed by the contractor to complete the project. The scope of the project Table 1 Overview of literature on incentive contracting and associated trade-offs. Author(s) Meinhart and Delionback [50] McCall [49] Cukierman and Shiffer [24] Hiller and Tollison [35] Weitzman [87] Stukhart [75] Herten and Peeters [34] McAfee and McMillan [48] Ryan et al. [64] William and Ashley [88] Abu-Hijleh and Ibbs [1] Veld and Peeters [83] Rosenfeld and Geltner [63] Chapman and Ward [18] Herbsman [33] Jaraiedi et al. [37] Ward and Chapman [86] Jaafari [36] Al-Subhi Al-Harbi [3] Arditi and Yasamis [5] Berends [7] Paquin et al. [55] Perry and Barnes [56] Boukendour and Bah [8] El-Rayes [28] Dayanand and Padman [25] Bower et al. [9] Broome and Perry [11] Bubshait [12] Shr and Chen [71] Shr and Chen [69] Shr et al. [70] Turner [80] Bayiz and Corbett [6] El-Rayes and Kandil [29] Kandil and El-Rayes [38] Pollack-johnson and Liberatore [57] Tang et al. [76] Tareghian and Taheri [78] Afshar et al. [2] Lee and Thomas [42] Rosandich [59] Sillars [72] Chapman and Ward [19] Rose [62] Stenbeck [74] Tang et al. [77] Ghodsi et al. [31] Ramón and Cristóbal [58] Anvuur and Kumaraswamy [4] Chan et al. [14] Love et al. [44] Mihm [52] Rose and Manley [60] Shahsavari Pour et al. [68] Zhang and Xing [89] [22] Chan et al. [15] Chan et al. [17] Dimensionsa C D S E I I – I I I I I I I I I I I – I I I I I I – I I – – I I I T – T I – T I T I T T – I T T I T I T T I I I I I T T T I I I – I – – I I – – I I I – – I I – I – I – – – – I I I T I I I I – I T I T I T T I I I T I – I T T I – I – I T T I T – I – – – – I I – – I I I – – – I – T – T – T – – – – I T I – – – – – T – T I T T T I – – I I I T T I – I – I T T – T – – – – – – – – – – – T – – T – – – – – T – – – – T – – – – – – – – T T T – – – – T – T – – – – – – – – – – – – – – – – Contract natureb Cost contract typesc A þ Bd Validation5 L L L L L L P L L – P P L L L L P P L L P – L P L – P P L L, N L L L L – – – – – – L L L L P L L – – L L P L L – – L L L FPI/TCC, CPIF FFP, FPI/TCC, CPFF – FPI/TCC, CPFF, CPPF FFP, CPIF, CPFF FFP, GMP, FPI/TCC, CPFF, CPPF FFP, FPI/TCC, CPIF, CPFF, CPPF FFP, CPIF, CPFF, CPPF FFP, FPI/TCC, CPFF FFP, CPFF, CPPF FFP, GMP, CPIF, CPFF, CPPF FFP, FPI/TCC, CPIF, CPFF, CPPF FFP, GMP, FPI/TCC, CPIF, CPFF, CPPF FFP, CPFF, CPPF – FPI/TCC, CPIF FFP, FPI/TCC, CPIF, CPFF FFP, FPI/TCC FFP, FPI/TCC, CPIF, CPFF, CPPF FFP, FPI/TCC, CPIF, CPFF FFP, FPI/TCC, CPIF, CPFF – FPI/TCC GMP – – FFP, CPIF FPI/TCC, CPIF, CPFF FPI/TCC, CPIF, CPFF, CPPF – – – FFP, GMP, FPI/TCC, CPIF, CPFF, CPPF FFP – – – – – – – FPI/TCC – FPI/TCC, CPFF FFP, CPIF – – – – GMP GMP, FPI/TCC – FPI/TCC CPIF – – – FFP, GMP, FPI/TCC GMP, FPI/TCC – – – – – – – – – – – – – – x x – – – x – – – – x – – – – x x x – – x – – – – – x – x – – – – – – – – – – – – – – – – – – – CS TE – CS, Sur CS – Sur CS Sur TE – Sur – – CS, Sim – Sur CS TE TE – TE TE CS CS Sur CS CS CS – TE TE – TE Sur TE TE CS TE, Sim CS CS CS CS CS, Sur TE CS CS Sur Sur – CS TE CS CS Sur Sur Please cite this article as: Kerkhove LP, Vanhoucke M. Incentive contract design for projects: The owner's perspective. Omega (2015), http://dx.doi.org/10.1016/j.omega.2015.09.002i 3 L.P. Kerkhove, M. Vanhoucke / Omega ∎ (∎∎∎∎) ∎∎∎–∎∎∎ Table 1 (continued ) Author(s) Rose and Manley [61] Chan et al. [13] Choi and Kwak [21] Keren and Cohen [40] Mackley [45] Meng and Gallagher [51] Zhang et al. [91] Lippman et al. [43] Tavana et al. [79] Cohen and Iluz [23] Chen et al. [20] Dimensionsa C D S E I I T T T I T I T T T I – I T I I T – T T I I – – T – I T – T T – – – T – – – – T – – – Contract natureb Cost contract typesc A þ Bd Validation5 L L L – P L – L – – P – FFP, GMP, FPI/TCC – – – FFP, FPI/TCC, CPIF, CPFF – FFP, CPIF, CPFF, CPPF – – – – – – – – – – – – – – CS Sur CS TE CS CS, Sur TE – – Sur – a Indicates if the authors cover incentive contracting (I) or the trade-offs (T) associated with the cost (C), duration (D), scope (S) and effort (E) dimensions. Incentive contracts can be linear (L), piecewise linear (P) or nonlinear (N). c Indicates which of the traditional cost contract types are treated, these include: firm fixed price (FFP), guaranteed maximum price (GMP), fixed price incentive/target cost contract (TCC), cost plus incentive fee (CPIF) and cost plus fixed fee (CPFF) contracts. b Fig. 1. Visualisation of the modelling approach and the interrelations between models. is a very general term which encompasses the amount of work performed as well as the quality level of the performance. When a project is expedited to a contractor, it is the latter who controls the precise way in which the project is carried out. Depending on the way in which the contractor decides to execute the project, (s)he effectively selects one of the possible trade-off points on the cost-duration-scope spectrum. Due to the fact that the contractor is introduced as a second party, a fourth dimension has to be introduced to extend the traditional cost-duration-scope trade-off: contractor effort. The reason for this is that not all actions of the contractor can be directly controlled or perceived by the owner. This concept has already appeared widely in literature (see Section 2.1), albeit sometimes under other aliases. Contractor effort can be defined as an investment in the project by the contractor, which enhances the outcome of one or more of the three traditional dimensions, but is not directly perceived by the owner. Effectively, an increased effort from the contractor increases the total utility which can be derived from the project by the owner. Naturally, an increase in contractor effort comes at a cost to the contractor, which (s)he will wish to compensate by earning a larger incentive fee (see Section 2.1 for more information on contractor effort). The second body of literature studies the design and implementation of incentive contracts, as well as their (perceived) effectiveness. A project owner can have several reasons for including incentive clauses, such as encouraging more innovative solutions [16], efficiently allocating risk, [19], improving communication between both parties [53], or creating a more cooperative mindset [61]. This research focusses on the two most prominently used and quantifiable measures: the owner's expected profit, and the degree to which the motives of the owner and contractor are aligned. The owner's profit is viewed as an objective measure for the performance of the project [82,61,16,9], assuming that all relevant elements have been quantified. A second quantifiable motivation for the implementation of contracts is the degree to which the contract succeeds in transferring risks and motivations to the contractor [9]. The primary motivation for this is the removal of the risk associated with agency-conflicts [65,32]. Moreover, the degree to which this is done also heavily influences other beneficial effects such as the cooperative atmosphere between the two parties [16,9]. Hence, the implementation of an incentive agreement can be viewed as being related to economic portfolio theory, maximising the return (i.e. owner profit) for a given level of risk (i.e. objective alignment) [19,46]. Practically, the owner's (expected) profit can be determined by assuming that the contractor is a profit maximising economic actor [82]. Under this assumption, the manner in which the project is executed will guarantee maximal attainable profits for the contractor. Hence, the expected profit for the owner is equal to the profit associated with this scenario. Measuring the alignment of motivations is done by comparing the relative outcomes for both parties across the possible project outcomes. The most relevant works in both domains are listed in Table 1, where the “Dimensions” columns indicate whether the paper deals with incentive contracts (I) for a specific dimension, or the tradeoff problem (T) related to a specific dimension. The trade-off between the cost, duration, scope, and effort levels which is controlled by the contractor should not be confused with the balancing of the different dimensions of the incentive contract, which are linked to the former three dimensions of the contractor's trade-off. Please cite this article as: Kerkhove LP, Vanhoucke M. Incentive contract design for projects: The owner's perspective. Omega (2015), http://dx.doi.org/10.1016/j.omega.2015.09.002i 4 L.P. Kerkhove, M. Vanhoucke / Omega ∎ (∎∎∎∎) ∎∎∎–∎∎∎ The difference between these concepts is shown by Fig. 1, which indicates that the trade-off is a representation of the nature of the project as perceived and controlled by the contractor. The same illustration also shows that the contract uses the outcomes of the project (i.e. the trade-offs) as an input, but the structure of the contract itself is a separate decision controlled by the owner rather than the contractor. To prevent any confusion regarding these concepts, the remainder of this paper will only use “trade-off” when referring to the trade-off decision of the contractor. 2.1. Trade-off literature The most traditional trade-off problem in project management is the one between the cost and duration of a project. The earliest authors modelled this problem using a simple linear relationship [39], but recent authors generally agree that the most realistic model for this trade-off is a convex curve [71,21]. This trade-off can be extended by considering the scope of the project, thus forming the well-known concept of the iron triangle [47]. Again the nature of the trade-offs between scope and cost or time are generally assumed to be of convex nature [29]. The scope concept is used in a broad sense, encompassing not only the amount of work performed, but also the quality of this work and any other area of performance which is valued by the owner [61,9]. Hence, it can also be considered to encompass concepts such as occupational health and safety, environmental impact and coordination which have received increasing amounts of attention in recent years [51,77,30]. Nevertheless since it is possible to incentivise these components separately, a disaggregation of the scope concept could be interesting for future research. As stated above, modelling the owner-x-contractor relationship requires a fourth dimension to be added to the trade-off: contractor effort. Though the nomenclature differs between authors, several examples of this dimension can be found in literature, as shown by the “E”-Dimension column in Table 1. The simplest example of contractor effort is managerial attention from the contractor [1,5], which can improve project performance, at a cost to the contractor. Another example is the proactive reduction of uncertainty through the development of contingency plans and by taking out insurance policies [18]. For contracts where the personnel costs are not included in the project cost dimension, but assumed to be included in the basic fee of the contractor, the allocation of additional staff can also be considered to be contractor effort [28,29]. Another very specific example is presented by [42], who considered a road refurbishment project where the contractor hired an on-site towing service to prevent delays due to uncleared accidents. Bayiz and Corbett [6] also use a similar dimension which they define as an unobservable element to the project owner, which comes at a certain cost to the contractor. Moreover, they also assume the trade-off relationship to be convex, something which also returns in the axioms used to construct the trade-off model in Section 5.1. 2.2. Incentives literature Incentive contracts for projects can be categorised along two dimensions. The first dimension being the trade-offs which are covered by the specific contract. An incentive contract can be linked to any of the three trade-off dimensions which are perceived by the owner: cost (C), duration (D) and scope (S). Naturally, an incentive contract can include multiple dimensions. Moreover, an important decision in incentive contract design can be finding the right balance between these different dimensions [1]. The second dimension is the nature of the equations used to calculate the incentive amount earned, based on the value of the relevant outcome dimension. These equations can be linear (L), piecewise linear (P), or nonlinear (N). The Dimensions and Contract Nature columns of Table 1 show how the existing literature on incentive contracts can be categorised along these two dimensions. 2.2.1. Literature on cost incentive contracts Cost incentives are the most researched incentive category, as can easily be seen from Table 1. Moreover, the contracts which are discussed can frequently be categorised as one of six basic contract types [87]: firm fixed price (FFP), guaranteed maximum price (GMP), fixed price incentive (FPI) (or target cost (TCC)), cost plus incentive fee (CPIF), cost plus fixed fee (CPFF) and cost plus percentage fee (CPPF) contracts. This set of cost contract types forms a risk-transfer spectrum, where the FFP contract represents the situation where all the risk is carried by the contractor, and the CPPF is the situation where all the risk is carried by the owner [3]. Variations on these archetypes are also often used to manage cost performance in production environments [85]. These contract types all use a single sharing ratio, sometimes extended with a safeguard preventing excessive disincentives to be allocated to contractors [80]. Hence, these types will be defined as linear (L) contracts. In practice, piecewise linear (P) schemes are frequently used to improve upon the basic cost incentive types presented above, several practical examples of such incentive schemes are given by Broome and Perry [11]. 2.2.2. Literature on duration incentive contracts Since no open accounting standards are needed to agree upon the outcomes of the time dimension, the duration incentive is arguably the easiest incentive to implement [1]. A very fruitful area of research related to time incentive contracting is the research done on the A þ B contracting method employed by the US government in highway contracting (see column “Aþ B” in Table 1). This is a bidding method where the contractors have to submit a bid for both the expected cost and the expected duration of a project, whereafter the road user cost (i.e. the economic damage of closing the road for a day) is used to evaluate the total cost of all offers. The implementation of such contracts is then often accompanied by a per diem time incentive. Moreover, this type of bidding allows the owner to get a better perspective on the nature of the trade-offs from the perspective of the contractor [22], thus aiding the owner in estimating the possible reaction of a contractor to a given incentive contract more accurately. Piecewise linear contracts for the duration dimension are discussed less frequently than piecewise linear contracts for the cost trade-off dimensions. Nevertheless piecewise linear contracts for time are used [44] and even nonlinear contracts using a quadratic formula have been discussed in literature [71]. 2.2.3. Literature on scope incentive contracts Scope incentives are the least commonly used of all incentive contracts, however in cases where they are used, they are often considered the most influential incentive dimension of the contract [77]. Table 1 shows that scope incentives are rarely used when cost and time incentives are not already in place. This is most likely due to the added complexity of the valuation of the performance along this dimension. Nevertheless, several authors have proposed methods to facilitate the evaluation of scope performance. Most of these methods use key performance indicators [62] or balanced scorecard techniques [77]. Along the same lines, Pollack-johnson and Liberatore [57] use the analytic hierarchy method to link these techniques to the work breakdown structure of a typical project environment. Although they are the least commonly used incentive category, the adoption of scope incentives is not novel, Herten and Peeters [34] discuss an interesting example of a scope-based incentive Please cite this article as: Kerkhove LP, Vanhoucke M. Incentive contract design for projects: The owner's perspective. Omega (2015), http://dx.doi.org/10.1016/j.omega.2015.09.002i 5 L.P. Kerkhove, M. Vanhoucke / Omega ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 4. Contract model Any incentive contract can be split up into its three components: cost, duration and scope. Hence, the model presented here also consists of three distinct components which can be used either jointly or separately for cases where one or more dimensions are irrelevant to the owner. For each dimension, the owner can opt to implement a linear, piecewise linear or nonlinear contract. For the duration dimension in particular, a lump sum incentive attached to a specific deadline can also be included. 4.1. Cost incentive model Fig. 2. Piecewise linear and nonlinear clauses. contract from 1908 when the Wright brothers were awarded an incentive based on the airspeed attained by an airplane. 3. Modelling the problem The contract design problem faced by the contractor is quantified using a combination of three models: the contract model, trade-off model, and the evaluation model (see Fig. 1). The underlying assumptions for each of these models are based on existing literature. The contract model represents the structure of the incentive contract and is based on contracting literature. This model was designed to be capable of representing the majority of incentive contracts used in practice, thus allowing for a comprehensive search of the solution space. The trade-off model model describes the nature of the project on which the incentive contract is to be used. The foundations for this second model originate from project management literature, from which six basic axioms have been derived. These six axioms fully describe the relationship between the four discernible tradeoff dimensions of the contractor: direct cost, duration, scope and contractor effort. Because the contractor is assumed to be a riskneutral profit-maximiser it suffices to define the trade-offs deterministically. Nevertheless, relaxing this assumption and working with a stochastic definition of the project is an interesting topic for future research. The goal of the evaluation model is simply the valuation of a project and contract combination. The model itself uses financial valuations, rather than utility theory, since the latter is often much harder to apply in practical situations [11]. This model quantifies the evaluations of both parties, as well as their alignment in a way which allows for an objective evaluation of the adequacy of a contract. In any project these three building blocks can be defined: there is always a contract, there are always different ways of executing a project,1 and both the contractor and owner always associate certain costs and profits with different outcomes. For this research paper we have designed the models in order to cover as much practical cases as possible, but naturally it may be necessary to extend these models for highly specific cases. The details of these three models are further explained in Sections 4, 5 and 6 respectively. For an overview of the introduced notation, the reader is referred to Appendix A. 1 Should this not be the case, it would be irrelevant to implement any kind of incentive contract. Since the outcome of this dimension is already expressed as a monetary amount, setting cost targets and evaluating the performance is relatively straightforward. The basic principle of a cost incentive can be illustrated using a simple linear incentive contract [56]. In such a contract the cost incentive (IC) paid to the contractor is determined by multiplying the difference between the cost target (Ct) and the direct cost incurred (C) by a sharing ratio (sC A ½0; 1Š), as given by Eq. (1). When the cost incurred is lower than the target cost, the difference is positive and a positive incentive (IC) is awarded, the inverse also being true: I C ¼ sC ðC t CÞ ð1Þ All of the traditional linear cost contract types (see Section 2.2.1) can be expressed as variations of this equation. This is done by varying the sharing ratio sC, and in some cases capping the risk exposure of either owner or contractor for extreme outcome scenarios (e.g. when the cost exceeds a certain threshold the disincentive no longer increases and the owner carries the remainder of the downside risk, avoiding potential bankruptcy of the contractor). An implementation of such a piecewise linear contract is visualised in Fig. 2, where the horizontal axis represents the possible performance outcomes for the contractor in terms of direct costs (C), and the vertical axis represents the incentive earned by the contractor (IC). The piecewise linear contract in Fig. 2 defines three regions, numbered r ¼ f0; 1; 2g. Each region r is defined by its lower bound BrC. For each of these regions a sharing ratio is defined as srC. The contract also contains a target for the direct cost (Ct), which can be situated in any of the available regions. Eq. (2) shows a mathematical expression for the 3-piecewise linear contract shown in Fig. 2, analogous expressions can of course be created for contracts with more dimensions and/or targets lying in other regions: 8 C s
ðC t > > < 1 t I C ¼ sC1
ðC > > : sC
ðC t 1 BC2 Þ þsC2
ðBC2 CÞ if BC1 r C r BC2 CÞ BC1 Þ þsC0 if C 4 BC2
ðBC1 CÞ ð2Þ if C o BC1 An alternative to the piecewise linear approach is the quadratic contract form as presented in [71]. Other nonlinear forms can of course also be used, but since their use is infrequent, only the quadratic form is included in this paper. This quadratic form requires the owner to specify upper (UBC) and lower (LBC) bounds, which outline the region over which he wishes to spread his (dis) incentive. Note that these express the distance from the target cost, rather than an absolute cost. The magnitude of the incentive itself is defined using two parameters: I Cmax A ½0; þ 1½ signifies the maximal possible incentive and I Cmin A Š 1; 0Š which specifies the biggest disincentive amount. The target cost (Ct) has to be specified as well. For this contract type, the cost incentive amount can Please cite this article as: Kerkhove LP, Vanhoucke M. Incentive contract design for projects: The owner's perspective. Omega (2015), http://dx.doi.org/10.1016/j.omega.2015.09.002i 6 L.P. Kerkhove, M. Vanhoucke / Omega ∎ (∎∎∎∎) ∎∎∎–∎∎∎ be calculated as follows: 0 8 !2 1 > > Ct C > C C @ A > min I max ; I max
> > > LBC < 0 IC ¼ !2 1 > > C Ct > C C > @ A > max I min ; I min
> > : UBC if C rC t ð3Þ if C 4C t 4.3. Scope incentive model This type of incentive contract is visualised by the dotted line in Fig. 2. An owner wishing to use such a contract is left to determine the direct cost target (Ct), the maximal incentive and disincentive amounts (I Cmax , I Cmin ), and the range wherein the incentive is to be distributed (LBC, UBC). 4.2. Duration incentive model Duration incentives are also frequently used in project management. Since the finishing time of a project is easy to observe in an objective manner, it is also one of the easiest to implement. However, it does require the owner to attach a monetary valuation to the timeliness of project completion. An example of such a valuation is the road user cost used by the US government when evaluating different bids for road refurbishment [29,72]. Once this valuation has been determined, the owner has to decide on the fraction of this valuation to be awarded to the contractor as an incentive. Naturally the value transferred to the contractor should never exceed the gain of the owner. Similar to the cost incentive contracts of Section 4.1, this can be done using a piecewise linear contract (Eq. (2)), or a nonlinear contract (Eq. (3)). Eq. (4) shows how the piecewise linear cost contract can be adjusted to a piecewise linear duration contract with similar assumptions. The difference between these two contracts is that rather than using a sharing ratio sCr A ½0; 1Š, a valuation parameter vrD is used. This parameter defines the monetary amount the contractor earns per unit of time saved in a specific region, if the region is below the target duration specified in the contract. If the region is situated above the contract's target duration, the parameter signifies the disincentive amount. The nonlinear contract for the duration dimension is identical to the nonlinear contract for the cost dimension, and will not be repeated here: 8 D D DÞ if D 4 BD vD
ðDt BD > 2 Þ þ v2
ðB2 2 > < 1 t D DÞ if BD ð4Þ I D ¼ vD 1 rD rB2 1
ðD > > : vD
ðDt BD Þ þ vD
ðBD DÞ if D o BD 1 1 0 1 such lump-sum incentive amounts can be combined with other duration-related incentive provisions such as piecewise linear or nonlinear incentive contracts, since the presence of a deadline does not mean that additional time savings or delays have no value to the project owner. 1 The concept of time in a project-environment is inherently related to specific project deadlines. Oftentimes, projects have to be completed prior to a certain date to ensure their value to the project owner (e.g. facilities which have to be constructed for Olympic games). Since delivering a project prior to an exact date can be so important to the owner, incentive contracts are of course adjusted to represent this. Ensuring that the contractor's valuation of this date is aligned with the owner's disproportionately large valuation (compared to the dates prior and after this deadline) can be done using a single lump sum incentive amount. When the contractor delivers the project prior to the agreed upon date, the agreed incentive amount is wholly awarded. In case the contractor fails to deliver on time, he does not receive anything. This very simple principle can be expressed as follows: ( D I ls if D r Dtls ID ¼ ð5Þ 0 if D 4 Dtls where Dlst is the target date associated with the lump sum payment and IlsD is the amount of the lump-sum payment. Naturally, The implementation of a scope incentive requires the adoption of a measurement method. Because scope envelops a large number of concepts, its measurement often relies on subjective estimates [26]. However, such subjective estimates are of little use as a basis for awarding incentive amounts, since the estimates of the owner and contractor will most likely be skewed in opposite directions. Stukhart [75] has proposed the use of an external party to provide a more objective measurement in such cases, but even so there is potential for disputes between both parties. Moreover, the cost of hiring a third party also decreases the project profitability. The use of key performance indicators has been advocated by Rose [62] as a method of making the performance measurement more objective. Whereas this method does indeed succeed in removing the subjectivity of scope performance, it does not include a formal method for aggregating the scope performance over the complete project. A solution for this issue is a top-down approach which defines a hierarchical structure for the complete project, defining the relative importance of various components until a degree of detail where objective evaluations are possible is reached [73,57,44]. By using such techniques, the scope performance of the complete project can be measured on a ratio scale. Hence, a value of 0 will always represent no work performed and comparative calculations can be made using different values of the scope (i.e. a scope of 2 is twice as valuable to the owner as a scope of 1). By taking this approach, the analysis of the scope dimension can happen in a way which is largely analogous to the preceding two dimensions. Assuming that the contract specifies a minimally acceptable scope as well as a scope target, similar equations as those constructed for the cost incentive in Section 4.1 can be constructed. The piecewise linear incentive contract can be adjusted for application on the scope dimension as follows: 8 S v
ðBS2 St Þ þ vS2
ðS > > < 1 t S I ¼ vS1
ðS S Þ > > : vS
ðBS St Þ þ vS
ðS 1 1 0 BS2 Þ if S 4 BS2 if BS1 r S rBS2 BS1 Þ if ð6Þ S o BS1 The key difference between the piecewise linear equation for the duration dimension (Eq. (4)) is that the statements between brackets are inverted. This is logical since a larger scope value is more valuable than a smaller value. Again the valuation is done using the vrS parameters. These parameters represent the monetary valuation of a unit of scope within a certain region r: 0 8 !2 1 > > S St > S S @ A > min I max ; I max
> > > UBS < 0 IS ¼ !2 1 > > St S > S S > @ A > max I min ; I min
> > LBS : if S 4St ð7Þ if S rS t The adjustment of the nonlinear contract for the scope dimensions is shown by Eq. (7). The sole difference between this equation and the expression for the cost dimension (Eq. (3)) is that the positive and negative senses are inverted. Please cite this article as: Kerkhove LP, Vanhoucke M. Incentive contract design for projects: The owner's perspective. Omega (2015), http://dx.doi.org/10.1016/j.omega.2015.09.002i 7 L.P. Kerkhove, M. Vanhoucke / Omega ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 5. Trade-off dynamics model Contractors subjected to an incentive contract face a fourdimensional trade-off decision. These dimensions are: direct cost, duration, scope and contractor effort. The first three of these dimensions are directly observed by the project owner and can potentially be used as a basis for an incentive provision. The fourth dimension represents the possibility for the contractor to enhance the project outcome using his own means. 5.1. Axioms A number of axioms which are an extension of those proposed by Ghodsi et al. [31] are used as a basis for the generation of realistic problem environments for the contractor. These axioms Convex relation Direct Cost A1 A3 Duration Multiplicator A2 Scope A4 A5 A6 Contractor Effort Fig. 3. Schematic overview of the components of the contractor's trade-off and the associated axioms. Axiom 1 are assumed to be valid on the level of the (aggregated) project as well as on the level of individual activities. For sake of simplicity, the relationship between the different dimensions is considered on a two-by-two basis. A conceptual representation of these axioms is given by Fig. 3, and a visual representation of their implications is given by Fig. 4. The axioms can be summarised as follows: 1 The direct cost is a non-increasing convex function of the duration of the project. (The contractor will first use the cheapest methods for time reduction.) Or in mathematical terms: ∂C=∂D r 0 and ∂2 C=∂D2 Z 0. 2 The direct cost is an non-decreasing convex function of the scope of the project. (Increasing the scope becomes progressively more expensive as the low hanging fruit is picked first.) Or in mathematical terms: ∂C=∂S Z0 and ∂2 C=∂S2 Z 0. 3 The direct cost is a non-increasing convex function of the contractor effort of the project. In mathematical notation this becomes ∂C=∂E r0 and ∂2 C=∂E2 Z0. This effort can be viewed as a way to influence the other relationships, i.e. making a decrease in duration or an increase in scope less expensive in terms of direct costs. Note that the cost in this case signifies the cost to the owner, not the contractor. The contractor's costs naturally increase when the effort is increased. 4 Shortening the duration of a high-scope project causes a cost increase which is greater than or equal to the cost increase in a low-scope project. Or in mathematical terms: 8 S1 4 S2 : ∂CðS1 Þ= ∂D r ∂CðS2 Þ=∂D r 0. Similarly, increasing scope will be at least as Axiom 4 (1/2) DC S+ C C Linear Approx Convex Axiom 4 (2/2) D D S Axiom 2 Axiom 5 (1/2) Axiom 5 (2/2) DC C C E- S D E Axiom 3 Axiom 6 (1/2) Axiom 6 (2/2) E C S+ C C E- S E Fig. 4. The axioms visualised. Please cite this article as: Kerkhove LP, Vanhoucke M. Incentive contract design for projects: The owner's perspective. Omega (2015), http://dx.doi.org/10.1016/j.omega.2015.09.002i 8 L.P. Kerkhove, M. Vanhoucke / Omega ∎ (∎∎∎∎) ∎∎∎–∎∎∎ costly for a shorter duration project: 8 D1 o D2 : ∂CðD1 Þ=∂S Z ∂CðD2 Þ=∂S Z 0. 5 Decreasing the duration of a project will be less than or equally costly when the effort invested in the project is greater. This can be noted as 8 E1 o E2 : ∂CðE1 Þ=∂D r∂CðE2 Þ=∂D r 0. Analogously, a smaller project duration will – ceteris paribus – result in a steeper slope of the relation between invested effort and project cost: 8 D1 oD2 : ∂CðD1 Þ=∂E r ∂CðD2 Þ=∂E r 0. 6 Increasing the scope of a project will be relatively cheap when the effort invested in the project is greater, and vice versa. In mathematical terms this can be expressed as: 8 S1 4 S2 : ∂CðS1 Þ=∂E r ∂CðS2 Þ=∂E r 0. Analogously, a change in effort level will have a greater impact on the project cost when the effort invested is smaller, or in mathematical notation: 8 E1 o E2 : ∂CðE1 Þ=∂S Z ∂CðE2 Þ=∂S Z 0. Note that the effect of the multiplicators presented in axioms 4–6 is represented visually in Fig. 4 by the change in slopes of the upwardly translated functions, and not the upward translation itself, which is a consequence of the first three axioms. In order to quantify these relationships the direct cost is assumed to be the dependent variable, and the three other dimensions are considered to be independent variables. This is merely a design choice which is made to improve the comprehensibility of the model. This design choice is visualised by image in Fig. 4, where the first three axioms represent direct relationships to the costs, and the latter three axioms are shown as multiplicators. An important note being that although the cost is the dependent variable in this model, it is still possible for the contractor to make trade-offs between the independent variables by making choices which inversely impact the cost of the project. 5.2. Mathematical representation The relationships expressed above will now be quantified. The complete model of the relationships is built in three stages: In the first stage, the basic relationships are approximated using linear relationships (axioms 1–3). Secondly, the interaction effects are modelled, using the multiplicators which represent axioms 4–6. Thirdly, the simplified linear relationships are transformed to more realistic convex relationships. All the trade-offs in this model are considered to be discrete, meaning that the functions they create are piecewise linear functions which connect a discrete number of points. These discrete points are indexed with subscript i, j and k for duration, scope and effort respectively, assuming that: i; j; k ¼ 0; 1; …; n. These discrete points can be seen on the left panel of Fig. 5, counting from i¼0 for the rightmost point up to i¼ n for the leftmost point of the trade-off. The following analysis assumes that there is an equal number of points (n þ 1) for each of the independent dimensions. This is not a prerequisite for the analysis to be valid but is simply a choice to simplify the notation which is introduced. Rather than modelling the direct cost (C), duration (D), scope (S) and effort (E) directly, the model defines Δ-variables which represent the distance of these values from their lowest cost option, which is indexed 0 (see the left panel of Fig. 5): ΔDi ¼ D0 Di ð8Þ ΔS j ¼ S i S 0 ð9Þ ΔE k ¼ E 0 E k ð10Þ Fig. 5 illustrates this notation. The left panel shows the traditional relationship between the duration (D) on the horizontal axis and the cost (C) on the vertical axis. For each of the possible discrete duration steps, an associated distance can be defined to the lowest cost option, as illustrated by ΔD4 . Associated with this ΔDi -value, a cost value can be defined as ΔC D4 , which is in effect the cost increase due to the decreased duration of the project. The right panel of Fig. 5 shows that by using these delta values on the axes, an increasing convex relationship can be defined. By defining all the variables in this manner the formulae are simplified, since every dimension now has an increasing convex relationship to the cost dimension. Moreover, each of these relations starts at the origin (e.g. ðΔD0 ; ΔC 0 Þ ¼ ð0; 0Þ). This also simplifies the definition of the convexity magnitude (see Section 5.2.3). 5.2.1. Linear relationships The first expression is a basic linear relationship between the cost (as a dependent) and the three other dimensions. This relationship is in effect the sum of the impacts of the duration, scope and effort levels respectively. This is expressed by Eq. (11), where ΔC Di represents the impact of selecting duration mode i on the costs, and ΔC Sj and ΔC Ek are similar metrics for scope and effort respectively. This is also illustrated on the right panel of Fig. 5, where the vertical axis shows ΔC D i graphically. In order to define this relationship, the lowest possible direct cost has to be given as a parameter ðC min ¼ C j ΔDi ¼ ΔSj ¼ ΔEk ¼ 0Þ. Also for each of the independent dimensions, the slope of the curve has to be defined: SD, SS and SE. This slope represents both the slope of the linear approximation as well as the average slope of the convex curve, since the start and end points of the linear approximation and the convex curve are made to coincide. Due to the Δ transformation of the independent variables, these slopes can all be represented as positive numbers, hence the absolute values in Fig. 5. Illustration of the mathematical description, using the relationship between duration and cost as an example. Please cite this article as: Kerkhove LP, Vanhoucke M. Incentive contract design for projects: The owner's perspective. Omega (2015), http://dx.doi.org/10.1016/j.omega.2015.09.002i 9 L.P. Kerkhove, M. Vanhoucke / Omega ∎ (∎∎∎∎) ∎∎∎–∎∎∎ Eq. (12): C ijk ¼ C min þ ΔC ijk ¼ C min þ Δ CD i þ Δ C Sj þ C ijk ¼ C min þ jSD jΔDi þ jSS jΔSj þ jSE jΔEk Δ C Ek ð11Þ ð12Þ Eq. (12) is in effect a simplified linear representation of the first three axioms. Note that the independent dimensions are indexed separately, since these dimensions can naturally be at different positions (i.e. he contractor can select the positions of these dimensions separately, the effect of which will be reflected in the total cost of the project). 5.2.2. Interaction effects Axioms 4–6 are then added to the linear expression through the definition of the slope multiplicators. These are defined in the following form: mab is the maximal impact of dimension b on the slope of the relationship between the direct cost and dimension a, expressed as a fraction added to the original slope of the curve. For example mSD ¼ 0:2 indicates that the slope of the duration curve will be 20% steeper when ΔS is at its maximal value.    ΔS j ΔE k C ijk ¼ C min þ jSD jΔDi mSD þ1 mED þ1 ΔS n ΔE n    ΔD i ΔE k þ1 mES þ1 þ jSS jΔSj mD S ΔD n ΔE n    ΔS j ΔDi þ1 mSE þ1 ð13Þ þ jSE jΔEk mD E ΔDn ΔS n The effect of adding these multiplicators is shown in Fig. 4, where the upwardly translated curves have a steeper slope than the original curves. 5.2.3. Convexity The third and final step is the conversion of these linear relationships into more realistic convex functions. To express the degree of convexity of a relation quantitatively, the convexity magnitude (CM) metric is introduced in this paper. The logic behind this metric can be explained by looking at the right panel of Fig. 5 where a triangle is formed by the simplified linear relationship and the projection towards the horizontal axis. Two zones are defined within this triangle such that zone A is the surface between the simplified linear relationship and the convex curve and zone B is the area below the convex curve. Hence, the total surface of the triangular surface is equal to A þ B. The CM is now defined as the fraction of the triangle consisting of zone A. Assuming an uniform step size (ΔΔDi ¼ ΔDn =n; 8 i ¼ 1; …; n), for the duration dimension this metric can be calculated as follows (see Appendix B for a full derivation): CM D ¼ B¼ A ¼1 AþB n ΔDn X n i¼1 " Δ 2
B ð14Þ ΔDn
ΔC Dn CD i 1þ ΔC Di ΔC Di 2 1 # ð15Þ In the formula above ΔDn /ΔC D can of course be replaced by ΔSn /ΔC S or ΔEn /ΔC E to model the degree of convexity of the S E scope (CM ) and effort (CM ) respectively. Logically the value of this metric lies in the interval ½0; 1½, where a value of 0 represents a perfectly linear relation and higher values represent different gradations of convexity. The actual upper bound on the convexity depends on the number of discrete points n: CMD max ¼ ðn 1Þ=n. Although this upper bound is attainable, trade-off curves approaching this bound will often have non-increasing regions and very steep angles. 5.2.4. Specific project cases The trade-off points for a specific scenario can either be defined explicitly when full information on all feasible execution methods is available, or fitted using partial information in combination with the aforementioned axioms. Hence, although it is assumed that the majority of projects will comply with the aforementioned assumptions, this is not a prerequisite for the application of the model. 6. Contract evaluation model The preceding two models enable a quantitative representation of contract and project trade-off structures respectively. The contract evaluation model presented in this section focusses on how appropriate the use of a certain contract is for a given project. From the perspective of the owner, the optimal contract has three properties. First of all, it maximises the gain the owner can expect from the project. Secondly, it guarantees that the evaluation of the contractor is well aligned with that of the owner. Thirdly, the contract should guarantee an adequate return for the contractor. An adequate contractor return has two facets, on the one hand the absolute incentive amount has to be substantial enough to be relevant to the contractor, on the other hand, the range between the highest and lowest incentive amount has to be large enough such that the contractor is motivated to locate the optimum payoff. The model presented in this section uses a single index l ¼ 1; …; m which represents the m different possible outcome scenarios, in no particular order. Such an outcome scenario is simply an attainable point of the trade-off model (i.e. a combination of cost, duration, scope and effort), for which the associated incentive amount has been calculated. When combined with a trade-off model as presented in Section 5 the value of m is equal to ðn þ 1Þ3 . 6.1. Maximisation of expected owner gain First and foremost, the goal of the project owner is to maximise his expected return. The contract evaluation model presented here calculates this amount by assuming that the contractor is a riskneutral profit-maximiser [32]. Given that the choice of a trade-off point is controlled by the contractor, the expected profit of the project owner is the profit for the trade-off point where the contractor's profits are maximised. Hence, the first step in calculating the expected net owner gain (E½NOGŠ) is the maximisation of the contractor's profit function. When evaluating the discrete set of scenarios from the perspective of the contractor, two elements have to be considered for every possible project outcome scenario l. First of all, the cost of effort invested by the contractor (Elcost) and secondly, the total incentive amount earned (Iltot). This total incentive amount is simply the sum of the cost incentive (IlC), duration incentive (IlD) and the scope incentive (IlS) amounts (which are defined by the contract model, as presented in Section 4). Using these two elements, the net contractor gain (NCGl) can be calculated for every scenario: NCGl ¼ I tot l S Ecost ¼ I Cl þI D l l þ Il Ecost l ð16Þ However, it must not be neglected that there is an opportunity cost associated with investing effort in a project, since the same effort could also earn a certain return elsewhere. Hence, the net contractor gain (NCGl) has to be corrected for this opportunity cost using the return on investment (ROIE) this effort could earn in other projects or activities. The adjusted NCG can be calculated as NCGadj ¼ I tot l l Ecost ð1 þ ROI E Þ l ð17Þ Please cite this article as: Kerkhove LP, Vanhoucke M. Incentive contract design for projects: The owner's perspective. Omega (2015), http://dx.doi.org/10.1016/j.omega.2015.09.002i 10 L.P. Kerkhove, M. Vanhoucke / Omega ∎ (∎∎∎∎) ∎∎∎–∎∎∎ This adjustment decreases the valuation of the scenarios with the opportunity cost of the effort investment. By doing this, the different NCGladj values can be directly compared. In the case where there is no ulterior investment possibility for the contractor's effort, ROIE can be set to zero. The owner's net gain from a specific scenario (NOGl) is equal to the value derived from the direct cost (Cl), duration (Dl) and scope (Sl) dimensions diminished by the total incentive amount awarded to the contractor (Iltot): NOGl ¼ OVðC l ; Dl ; Sl Þ I tot l ð18Þ where OV is a function expressing the financial value of a certain outcome of the project for different values of direct cost, duration and scope. This function can be further detailed as a simple sum of the valuation of each of these three dimensions: C D S OVðC l ; Dl ; Sl Þ ¼ OV ðC l Þ þ OV ðDl Þ þ OV ðSl Þ ð19Þ C As for the cost component, the owner's valuation (OV ) is inversely proportional to the direct cost outcome: OV C ðC l Þ ¼ Cl ð20Þ The time valuation is slightly more complex, containing both the possibility of a linear value per time unit such as a road user cost, and the possibility of a lump sum value representing an important deadline” ( μD
ð Dl Þ if Dl Z DL D ð21Þ OV ðDl Þ ¼ μD
ð Dl Þ þ μls if Dl o DL In Eq. (21), μD represents the monetary valuation of an unit of time by the owner (e.g. 1000 euro/day). This is similar to the road user cost which is often defined in A þ B contracting methods for roadwork manufacturing [72]. Analogously, the parameter μls is used to represent the monetary value to the owner of satisfying a certain deadline DL. Note that the parameter DL is different from the Dtls parameter introduced in Section 4.2, the former signifies a deadline relevant to the owner, whereas the latter is a deadline clausule included in an incentive contract to which the contractor is subjected. These two dates are not necessarily the same, since the owner can choose to insert additional buffer in the contract by setting the contractor's deadline prior to his own (i.e. DL ZDtls for rationally designed contracts). A similar construction is created for the valuation of scope, as seen in Eq. (22). Based on the premise that scope can be measured on a ratio scale (see Section 4.3), it suffices to introduce μS as the monetary value of an unit of scope to the owner: OV S ðSl Þ ¼ μS
Sl ð22Þ Using the formulae above, the payoff functions for both the owner and the contractor have been fully defined as NOGl and NCGladj respectively. As stated above, the optimal contract design has three evaluation criteria: a maximal return for the owner, an accurate alignment of the evaluation of the contractor and the owner, and an adequate payoff for the contractor. The first of these criteria can be expressed as follows:   h i adj ¼ max NCG ð23Þ E½NOGŠ ¼ max NOGl j NCGadj l l Eq. (23) effectively states that the expected gain of the owner is equal to the gain he receives when the contractor selects the scenario which maximises the contractor's own return. In the case where there are multiple scenarios in which the contractor's profits are maximised, the maximal net owner gain is used. The reasoning behind this being that given the opportunity to do so, the contractor will chose to maximise the owner's profits if this does not affect his own return. The top two panels of Fig. 6 illustrate the expected owner profit (E½NOGŠ) maximisation objective. On the horizontal axis all the potential outcome scenarios (i.e. the tradeoff points which can be chosen by the contractor) are listed in no particular order. The vertical axis represents the profit for the owner and contractor for the respective scenarios. As was formerly stated, the contractor is a profit maximiser and will therefore always select the scenario ¼ maxl ðNCGadj Þ). Hence, the which maximises his profits (NCGadj l l owner profit (NOGl) associated with this scenario is defined as the expected net owner gain (E½NOGŠ). This expected outcome is indicated by the rectangle and the letter E in Fig. 6. Assuming the owner is confronted with a choice between the contract represented by the upper left panel and the contract represented by the upper right panel, the owner will prefer the right contract, since this is likely to result in an outcome which yields a much greater net owner gain (NOG). The situation in the top left pane is in fact equivalent to rewarding a certain strategy, while hoping a different strategy will be followed [41]. Fig. 6. Illustration of the optimisation objectives. Please cite this article as: Kerkhove LP, Vanhoucke M. Incentive contract design for projects: The owner's perspective. Omega (2015), http://dx.doi.org/10.1016/j.omega.2015.09.002i 11 L.P. Kerkhove, M. Vanhoucke / Omega ∎ (∎∎∎∎) ∎∎∎–∎∎∎ Note that the example presented in Fig. 6 is simplified in that a change in the contract structure is likely to change the owner's payoff function as well as the function of the contractor. This is one of the reasons for the complex nature of incentive contract design. 6.2. Evaluation alignment The two bottom panels of Fig. 6 present an intuitive example of the importance of evaluation alignment. Again the horizontal axis lists all possible outcome scenarios in no particular order, and the vertical axis represents the contractor (NCGladj) and owner (NOGl) gains for the respective scenarios. When comparing the two bottom panels, it is clear that both contracts have an identical expected net owner gain E½NOGŠ. Hence, if this was the only evaluation criterion used, the owner would be indifferent between the contract resulting in the bottom left and the contract resulting in the bottom right scenario. However, when looking at the most attractive regions for the contractor, excluding the contractor's optimal point, a case can be made for the contract represented in the bottom right. Namely, it is clear that for the bottom left contract the attractive region coincides with the least attractive points for the contractor. Hence, should external influence cause the optimal point to be unattainable, it is likely that the outcome will gravitate towards such an unattractive point. The opposite is true for the contract on the bottom right, where even if the expected point should not be obtained for whichever reason, the other points still result in relatively attractive outcomes for the owner. The theoretical example in the bottom panels of Fig. 6 can be linked to the concept of balanced contract design in project management practice [1]. For example, assume the only timerelated incentive of a contract is a lump-sum provision awarded in case a certain deadline is met, but the contract also includes a substantial cost incentive to encourage cost savings. Should it at some point during the execution of this project become apparent to the contractor that it will be impossible to deliver the project before said deadline, it is highly likely that the contractor will attempt to maximise his cost incentive and completely neglect the time dimension. This would of course have a dramatic effect on the value of the project to the contract owner, who of course attaches a substantial value to the timely delivery of the project. The concept of payoff alignment is a way of formalising this issue. In order to quantify the alignment of the evaluation of the owner and contractor, both values are expressed in relative terms, meaning that they are rescaled in the range ½0; 1Š. This results in the following relative expressions for contractor and owner respectively: NCGadj l RNCGadj ¼ l NCGadj max RNOGl ¼ NCGadj min NCGadj min ð24Þ NOGl NOGmin NOGmax NOGmin ð25Þ Using these relative expressions, a perfect evaluation alignment of both parties can be defined as the situation where the following ¼ RNOGl 8 l ¼ 1; …; m. Naturally this optimal is true: RNCGadj l situation can never be attained in practice. Hence the need to develop metrics to measure the degree of deviation from this optimal situation. The following two metrics are used in order to express the degree of alignment (i.e. the deviation from this optimal situation): MD ¼ m h 1X RNCGadj l ml¼1 RNOGl i ð26Þ MAD ¼ m 1X RNCGadj l ml¼1 RNOGl ð27Þ Eq. (26) measures the mean deviation (MD) of the relative gains of both parties. The value of this metric lies within the range ½ 1; 1Š, negative values indicating that there are more scenarios where the owner gains are larger than those of the contractor and positive values indicating that there are more scenarios where the inverse is true. Hence, the value of this metric should be as close to zero as possible. The major flaw in Eq. (26) is of course that opposite deviations cancel each other out. Hence, the mean absolute deviation (MAD) is presented in Eq. (27). The MAD always has a value in the range ½0; 1Š, and gives an accurate reading of the overall deviations, regardless of their sign. By using the MAD to measure the average alignment in combination with the MD to measure possible skewness, the overall alignment of the evaluations can be quantified. 6.3. Constraints enforcing sensible contract design The preceding two optimisation criteria can be extended by adding constraints enforcing the contracts to adhere to certain rules for good incentive contract design. Such guidelines have been regularly proposed in literature and often include advice on minimal sharing ratios [56], and a bias towards positive incentives rather than disincentives [60]. The remainder of this section presents two constraints which can be used to enforce certain principles when evaluating contracts. The first constraint guarantees that the maximal incentive earned by the contractor is larger than or equal to a certain minimal value. If this would not be enforced, the optimal solution could potentially gravitate towards contracts which impose large disincentives on contractors. Several authors have already indicated that such contracts are undesirable [1,60]. The level at which this constraint is set is of course strongly project-specific, and several authors have quoted different values depending on the industry in which they operated. Abu-Hijleh and Ibbs [1] examined projects in industrial construction and found incentive amounts up to 2.62% of total project cost. A study Veld and Peeters [83] found that this percentage was approximately 0.7% in the aerospace industry. A much higher fraction is apparently used in road construction in the US, where an incentive cap of approximately 5% of the total project cost is recommended [77]. By defining the maximal incentive which can be earned by the contractor as I tot , this max and the lower bound for this value as LBI tot max constraint can be formulated as I tot max Z LBI tot max ð28Þ The second constraint is introduced to guarantee that there is sufficient variability in the payoff of the contractor. Insufficient variability will impede the contractor's motivation to locate the optimal payoff region. Hence, RItot is defined as the minimal range required for adequate contractor motivation. The difference tot between the maximal (I tot max ) and minimal (I min ) incentive awarded to the contractor should then be at least as great as this minimal range: I tot max I tot min Z RI tot ð29Þ 7. Evaluating the performance of contract sets One of the key objectives of this research is to evaluate the performance of different types of contracts. However, due to the presence of multiple objectives, comparing performance is not Please cite this article as: Kerkhove LP, Vanhoucke M. Incentive contract design for projects: The owner's perspective. Omega (2015), http://dx.doi.org/10.1016/j.omega.2015.09.002i 12 L.P. Kerkhove, M. Vanhoucke / Omega ∎ (∎∎∎∎) ∎∎∎–∎∎∎ trivial. In order to evaluate how different contract types perform, all Pareto-efficient contracts of a specific type are considered jointly. These local Pareto frontiers are then compared to a global Pareto frontier, which can include all contracts regardless of their type. This section discusses the methods used to make this comparison. As was stated in Section 6, the performance of a contract of a contract is measured on two key dimensions: expected owner gain (E½NOGŠ), and the alignment of the relative payoffs of the contractor and the owner, as measured by the MD and MAD metrics. Since the problem at hand has multiple objectives, the performance has to be evaluated accordingly. This is done by constructing a global three dimensional (E½NOGŠ; MAD; MD) Pareto frontier, containing all the contracts for a problem environment which are not dominated. The performance of different contract sets (i.e. the contracts belonging to a specific contract group, can then be compared to this global Pareto frontier. When making such comparisons, three aspects are of importance [66,54]: (i) the distance to the global Pareto front, (ii) the spread of the solutions found, and (iii) the number of elements found. (i) Distance to the global Pareto front. Frequently used metrics to measure the distance between the global Pareto front and approximations thereof include the error rate (ER) , the generational distance (GD), the inverted generational distance (IGD) as well as the Hausdorff distance (dH) [81,66]. However, each of the aforementioned metrics displays certain shortcomings: the error rate fails to measure the precise distance to the global Pareto front, both the GD and IGD tend to zero as the number of elements on the tested or global frontier is increased, and the Hausdorff distance only measures the worst case scenario. Hence, the averaged Hausdorff distance (Δp), which was introduced by [67] to mitigate these shortcomings, is used as a metric for the distance from the tested set to the best known Pareto front, this metric is calculated as follows: 2 !1p N 1X Δp ðX; YÞ ¼ max4 distðxi ; YÞp ; Ni¼1 !1p 3 M 1 X ð30Þ distðyi ; XÞp 5 Mi¼1 Where X and Y are the two sets between which the distance is to be measured (i.e. the global Pareto front and the tested front). These sets contain N and M points respectively, which are represented as xi and yi. The function dist measures the distance from a point to the closest point in the other set, using simple euclidean distances. Finally, p is equal to the number of dimensions in the Pareto front (i.e. p ¼3 in this study). The coordinate system used to calculate this metric in the experiments below has been standardised to the range ½0; 1Š for each of the dimensions measured, to avoid one of the dimensions having a more substantial impact than the others merely due its measurement scale. For more information regarding this metric, the reader is referred to [67]. The calculation of the Δp metric is illustrated by Fig. 7. To allow for a simple visual representation, two rather than three evaluation dimensions are considered in this example. By visual inspection alone, it is hard to determine which of these two sets is closest to the global Pareto front. The Δp metric can be used to compare both of these sets (S1 and S2) to the global frontier (G). The first step is to calculate the distance from each point on S1 to Fig. 7. Graphical illustration of the metrics used to measure the performance of certain contract sets. The top left panel shows the Pareto frontier of two contract sets S1 and S2, as well as the global Pareto frontier G, which is constructed using all the best known contracts regardless of their type. The top right and bottom left panels illustrate components of the averaged Hausdorff distance (Δp) metric used to measure the distance to the global front. The bottom right panel illustrates the measurement of the E½NOGŠ spread associated with the respective sets. Please cite this article as: Kerkhove LP, Vanhoucke M. Incentive contract design for projects: The owner's perspective. Omega (2015), http://dx.doi.org/10.1016/j.omega.2015.09.002i 13 L.P. Kerkhove, M. Vanhoucke / Omega ∎ (∎∎∎∎) ∎∎∎–∎∎∎ Table 2 Contract dataset. Base type Accountability Code Downside Upside Linear Contractor Contractor Shared Owner Owner Contractor Shared Shared Shared Owner 3-Piecewise Contractor Contractor Owner Owner 4-Piecewise Nonlinear Contracts tested Cost Duration Scope LCC LCS LSS LOS LOO 7 56 63 56 7 168 1344 1512 1344 168 9 72 63 72 9 Firm fixed price Guaranteed maximum price Fixed price incentive Cost plus incentive fee Cost plus fixed fee Contractor Shared Shared Owner 3PCC 3PCS 3POS 3POO 336 1176 1512 336 8064 36,288 36,288 8064 432 3888 1944 432 – – – – Contractor Contractor Owner Owner Contractor Shared Shared Owner 4PCC 4PCS 4POS 4POO 980 2940 4200 1260 0 0 0 0 1260 2520 3780 1260 – – – – Shared Shared – N 28,672 1,102,248 59,049 Total 41,601 1,195,488 74,790 the global Pareto front, which is simply the euclidian distance to the closest point on the global front. This is illustrated for the first set (S1) in the top right panel of Fig. 7. By taking the power mean of all these distances, the first of the two expressions within the square brackets of formula 30 is found. The calculation of the second expression is simply the inverse of the first and is done by taking the power mean of the distances from every point on the global front to the first set of points (S1). This is illustrated for the first set S1 by the bottom left panel of Fig. 7. By taking the maximum of these two expressions, the “worst case”-averaged distance is acquired. When calculating the result for this small example, it appears that the second set (S2) is somewhat closer to the global Pareto front with a Δp value of 0.48, compared to a Δp value of 0.57 for the first set (S1) (ii) Spread of solutions. To determine the quality of the spread of the tested Pareto front, a very simple metric is proposed which compares the distance between the extreme outcomes for each dimension of the tested front to the distance between the extreme outcomes for the global Pareto front. For E½NOGŠ this can be calculated as follows: sprðE½NOGŠÞ ¼ In literature E½NOGŠtest max E½NOGŠtest min E½NOGŠglob max E½NOGŠglob min ð31Þ test In Eq. 31, E½NOGŠtest max and E½NOGŠmin represent the highest and lowest values for the E½NOGŠ metric in the Pareto front which is glob compared to the global front. Similarly, E½NOGŠglob max and E½NOGŠmin represent the highest and lowest values for E½NOGŠ which can be observed on the global Pareto front. Note that since the global Pareto front will always include the best known solutions the following test glob r E½NOGŠtest is always true: E½NOGŠglob min r E½NOGŠmax r E½NOGŠmax . min Hence, the spread will always be a value in the range ½0; 1Š. The graphical interpretation of this metric is illustrated in the bottom right panel of Fig. 7. For this example, it is clear that the spread of the second set (S2) with regard to the E½NOGŠ metric is superior to the spread of the first set (S1). Since the spread for the MD and MAD dimensions is calculated analogously, the respective equations for these dimensions will not be repeated here. (iii) Number of elements. The number of elements found on the tested Pareto front will not be considered as a metric in this paper, since this number will of course be strongly dependent on the number of contracts which were tested. Which is an aspect of the experimental setup, rather than a property of the contract structure itself. 8. Computational experiments The formal models presented in the preceding sections make it possible to carry out computational experiments using carefully constructed datasets. In order to structure the analysis, the following two concepts are defined: Contract type: A collection of contracts which adhere to certain design principles with respect to the dimensions which are incentivised, the formulae used to calculate the incentive and the owner and contractor accountability in the case of extremely positive or negative outcomes. Environment: The project setting in which a contract operates, which is formed by combining a tradeoff model (see Section 5) and an evaluation model (see Section 6). The goal of these experiments is to compare the performance of different contract types, as well as the impact of differences in the environments in which contracts operate. Differences in environments can of course be due to differences in the trade-off model as well as differences in the evaluation model. The computational experiments provide answers to four key research questions: (1) How do different contract types perform in practice? (2) How do environmental factors (i.e. the properties of the project trade-off and the evaluation models) influence the performance of these different contract types? (3) How substantial are the added benefits of multi-dimensional contract forms when compared to uni-dimensional contract forms? (4) How are these multi-dimensional contract forms influenced by environmental factors? The answers to the former two research questions are provided by the uni-dimensional experiment design (Section 8.2), whereas the latter two research questions are answered through the multidimensional experiment design (Section 8.3). Before going into further detail on these two experimental designs, more information on the datasets is provided in Section 8.1. 8.1. Datasets In accordance with the methodology which was introduced when modelling the problem, three separate datasets have been created. Each of these datasets corresponds to one of the components used to model the problem: the contract, the trade-off and the evaluation model (see Section 3). All these datasets are freely Please cite this article as: Kerkhove LP, Vanhoucke M. Incentive contract design for projects: The owner's perspective. Omega (2015), http://dx.doi.org/10.1016/j.omega.2015.09.002i 14 L.P. Kerkhove, M. Vanhoucke / Omega ∎ (∎∎∎∎) ∎∎∎–∎∎∎ available and can be downloaded from the following website: http://www.projectmanagement.ugent.be/?q¼ research/ contracting. 8.1.1. Contract models The goal of the contract model dataset is to present a comprehensive cross section of contracting techniques used in practice. By testing this cross section of contract structures on a wide range of possible environments, the effectiveness of these techniques can be determined. As was mentioned in Section 7, the contract dataset consists of several contract (sub)sets, grouping contracts belonging to a specific type. An overview of this is given in Table 2. Contract types discussed in literature are usually defined based on which party is accountable in case of extremely positive or negative outcomes (see Section 2.1). This allocation of accountability for the different contract types is shown by the accountability columns in Table 2. Accountability can either be shared or allocated to the contractor/owner. Allocating the downside accountability to the contractor means that all overruns beyond a certain point will be paid in full by the contractor. Where this specific point lies depends on the other parameters set in the contract. Contrarily, when the upside risk is allocated to the contractor, (s)he can fully benefit from all the gains of the performance (e.g. cost savings) beyond a certain point. Alternatively, the risk can also be shared between the two parties according to a certain ratio. This accountability in the extrema is translated into a shortened contract code which will be used from now on to identify the different contract types (see the code column in Table 2). The final column of Table 2 also mentions the nomenclature used in existing literature for these specific types of contracts, insofar as this nomenclature exists. Practically, allocating the accountability to the owner or contractor is equivalent to setting the sharing ratio for the associated region to 0 or 1 respectively. For the simple linear contracts, this means that the sharing ratio above and/or below the target is set to 0 or 1. For three- and four-piecewise linear contracts, the upside and downside accountability refer to the first and final segment. For each of these types, the parameters which are not fixed due to the nature of the contract type are varied, resulting in a varying number of available instances per contract type, as listed in Table 2. These parameters include the respective targets (C t ; Dt ; St ), S the bounds for the various regions (BCr ; BD r ; Br ), the sharing ratios C D S (sr ), the region valuation parameters (vr ; vr ), the nonlinear S C D S C incentive amounts and bounds (I Cmax , I D max , I max , I min , I min , I min , UB , D S C D S UB , UB , LB , LB , LB ), and lump sum duration incentive targets and incentive amounts (Dtls ; I D ls ). Each of these parameters is varied across the complete relevant range, with a step size equal to 18 of the total size of the range, including or excluding the bounds of the range as is relevant for practical implementation. This results in a total of 41,601 cost contract options, 1,195,488 duration contract options, and 74,790 scope contract options. Note that the number of duration contract options is substantially greater due to the possibility of including lump-sum incentives related to a specific deadline. Readers who desire more information on these datasets are referred to the online resources accompanying this paper. 8.1.2. Trade-off models In order to create instances of the trade-off model, the independent parameters (D; S; E) are spread across a range of size 1: D A ½1; 2Š, S A ½1; 2Š and E A ½0; 1Š (i.e. The shortest possible duration for the project is defined as 1, and the longest possible duration for the project is defined as 2). For each of these dimensions, 11 equidistant discrete options were created within this range (n ¼10, see Section 5). This results in a total of 1331ð ¼ ðn þ 1Þ3 Þ trade-off points for each instance. The actual difference between these instances is represented by the variation in the costs associated with each of these trade-off points, which depends on the parameter settings used for the specific instance. These settings are summarised in the first part of Table 3. The generation procedure for the trade-off problems requires a value for each of the parameter values listed in the first part of Table 3. The generation procedure starts by simply selecting the values for the three first parameter types: the slopes (j SD j , j SS j , j SE j ), the multiplicators (mDS, …, mES), and the minimal cost value (C min ). Given these parameter values, a linear approximation of the relation between each of the independent dimensions (duration, scope and effort) can be made, for every possible value of the other two independent dimensions (e.g. the relation between duration and cost given a certain scope and effort level). These linear approximations are then used in combination with the value for the convexity magnitude metrics (CMD, CMS, CME) to determine a convex curve which represents the impact of each separate dimension on the cost. Practically, this convex curve is created using a linear programming (LP) model which uses the linear approximation as well as the relevant CM metric as an input. The workings of this model are detailed in Appendix C. This means that a total of 3
ðn þ 1Þ2 LP models are solved in order to model the impact of the individual dimensions. The implementation of this model was done using Gurobi 5.6 optimisation software, which solved each of the instances in under a minute on a 2.5 GHz Intel Core i5 machine. The result for the individual dimensions is then combined by summing the cost increases (ΔC) for the separate dimensions, resulting in ðn þ 1Þ3 costs (Cijk) corresponding to every possible duration-scope-effort combination. Using this methodology and the parameters listed in Table 3, a total of 380 trade-off models have been created. 8.1.3. Project evaluation models The generation of evaluation model instances requires the selection of the parameters which are shown in the bottom half of Table 3. The majority of these parameters are expressed relative to the total cost spread of the trade-off problem used, expressed as the difference between the lowest attainable cost (C000) and the Table 3 Environment dataset. Model Parameter Symbol Values Trade-off Basic slopes Multiplicators Minimal cost j S D j , j SS j , j S E j mDS, …, mES C min ¼ C 000 Convexity CMD, CMS, CME {0.5, 0.75, 1, 1.25, 1.5} {0, 0.1, 0.2, 0.3, 0.4}  1 1 1 1 1 ΔC nnn
0:10 ; 0:20; 0:30; 0:40; 0:50 {0, 0.0625, 0.125, 0.1875, 0.25} 125 ΔC nnn
f0:25; 0:375; 0:5; 0:625; 0:75g f0; 0:05; 0:1; 0:15; 0:2g ΔC nnn
f0:5; 0:75; 1; 1:25; 1:5g ΔC nnn
f0; 0:05; 0:1; 0:15; 0:2g ΔC nnn
f0:5; 0:75; 1; 1:25; 1:5g 5 5 5 5 5 Evaluation Max effort cost Effort ROI Time value Deadline value Scope valuation cost Em ROIE μD μls μS Instances 125 125 5 Total 380 25 Please cite this article as: Kerkhove LP, Vanhoucke M. Incentive contract design for projects: The owner's perspective. Omega (2015), http://dx.doi.org/10.1016/j.omega.2015.09.002i L.P. Kerkhove, M. Vanhoucke / Omega ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 15 Fig. 8. Distance to the Pareto front for uni-dimensional experiments. Fig. 9. The average spread obtained by different contract types for the cost, duration and scope dimensions. These are the outcomes for the situation where both a minimal earning and a minimal range for the contractor is required. highest possible cost (Cnnn): ΔC nnn ¼ C nnn C 000 . This is done to ensure a consistent variation of the different parameters over different trade-off models. As shown in Table 3, a total of 25 different evaluation model instances have been created, allowing for a total of 9500 (25
380) different environments on which contract performance can be tested. 8.2. Uni-dimensional experiments The aim of the uni-dimensional experiment is to investigate the relative performance of different contract types for the different dimensions, as well as the impact of environmental factors on the performance of these contracts. To do this, the 25 evaluation model instances are combined with the 380 trade-off model instances (see Table 3), resulting in 9500 problem environments on which the different contract type instances (see Table 2) can be tested. The nature of this experiment is uni-dimensional, which signifies that only one type of incentive is tested per environment (i.e. an environment is never subjected to both a cost and scope incentive simultaneously). Due to the difference in the number of parameter settings between the contracts, the time needed to evaluate a single contract differs depending on the contract type. On a 2.26 GHz processor the evaluation of a single environment (i.e. a combination of a trade-off and an evaluation instance) takes 27, 793 and 135 s on average for the cost, duration and scope contract sets respectively. Hence, the complete experiment takes approximately 2500 singlecore computation hours. For every environment, the performance of the different contract types is judged by creating a number of efficient frontiers containing only the contracts of a certain types. These local efficient frontiers are then compared to the global efficient frontier, which includes contracts of all available types. The comparison itself is made by calculating the averaged Hausdorff distance (Δp) Please cite this article as: Kerkhove LP, Vanhoucke M. Incentive contract design for projects: The owner's perspective. Omega (2015), http://dx.doi.org/10.1016/j.omega.2015.09.002i 16 L.P. Kerkhove, M. Vanhoucke / Omega ∎ (∎∎∎∎) ∎∎∎–∎∎∎ Fig. 10. The average impact of variation of environment parameters on contract performance, measured by the slope coefficient of the linear regression. These are the outcomes for the situation where both a minimal earning and a minimal range for the contractor's earnings is required. between the local and global frontiers as well as the spread of the local frontiers. This process is repeated for different types of sensible contract design strategies (see Section 6.3). Whenever a specific contract does not satisfy the criteria for sensible contract design, it is no longer included in any of the local or global efficient frontiers. Four different sensible contract design strategies have been used. For the first type no restrictions whatsoever are included, all contracts are considered in the efficient frontiers. The second type imposes the restriction that the maximal incentive which can be earned by the contractor cannot be negative. The third type requires that the maximal incentive which can be earned by the contractor is at least 2% of the average project cost.2 The fourth criterium extends the third by also requiring that the range between the highest and lowest incentive amounts which can be earned is at least 4% of the average project cost. These constraints are similar to the values observed in practice by other authors [1,83,77], see Section 6.3. The results of the experiments will be presented using visual representations rather than raw data tables to facilitate interpretation. Readers who would like the raw data behind these graphs can download them online from the following website: http://www.projectmanagement.ugent.be/?q¼ research/ contracting. 8.2.1. Performance of contract types The first step in analysing the results of the uni-dimensional experiment is measuring the (relative) performance of the contract types. Fig. 8 visualises the averaged Hausdorff distance Δp performance measure, which represents the proximity to the global efficient frontier. To facilitate comparing results the averaged Hausdorff distance was standardised within the range ½0; 1Š. The second performance measure – the spread of the solutions for the different evaluation dimensions – is shown in Fig. 9. Naturally these results were also standardised within the range ½0; 1Š, based on the best and worst values for each specific environment. For the majority of contract types the fraction of contracts which did not satisfy imposed restrictions for sensible contract design (see Section 6.3) remained limited: in less than 1.5% of cases violations were observed. Only the LOO contract, which transfers all risk to the owner, was unable to satisfy any of the imposed constraints, as can be seen from the results in Figs. 8 and 9. Nonlinear and piecewise linear contracts outperform linear types: Nonlinear and piecewise linear contract types generally outperform linear contract types in terms of proximity to the efficient frontier, whilst showing average to good performance in terms of spread for all metrics. Fig. 8 shows that the best performance in terms of proximity to the efficient frontier (Δp) is always observed to be either a nonlinear (N) or piecewise linear contract type. It can also be noted that the best performing piecewise linear contracts always share the upside potential, rather than allocating it completely to either party. The spread of these best performing types is also consistently good, as shown by Fig. 9. Avoid linear contracts which transfer all risk to a single party: Linear contracts which transfer all the risk to either the contractor3 (LCC) or the owner4 (LOO) perform worse in terms both proximity and spread when compared to other linear contract types. Moreover, due to the complete absence of value transfer from the owner to the contractor when using the LOO contract type, the minimum incentive and range conditions for sensible contract design cannot be satisfied. The relevance of these observations is validated by the continued extensive use of this type of contract, as can be seen from Table 1. Preferred types are robust to changing restrictions: Adding restrictions to enforce sensible contract design does not change the preferred contract type for any of the contract dimensions. For other contract types imposing additional constraints can have a significant impact on the relative performance. Simple linear contracts seem particularly sensitive to the imposition of constraints for sensible contract design. Skewness persists within contract types: The alignment skewness as measured by the MD persists within a contract type, as indicated by the consistently low values for the standardised spread for this metric (spr(MD)). Fig. 9 shows that the average spread for the MD metric is low in both absolute and relative terms for all types of contracts. Hence, it is likely that other contract types will have to be considered in order to remove an undesired skew from a contract. Based on these observations, project owners are advised to use piecewise linear or nonlinear contracts, rather than the traditional linear contract archetypes. Especially linear contracts which transfer all risk to the contractor or the owner, such as the traditional firm fixed price (FFP) or cost plus fixed fee (CPFF) contract types, should be avoided. Moreover, when an undesirable skew of the alignment (as measured by the MD indicator) is observed, it may be beneficial to include other contract types in the analysis. 8.2.2. Impact of environmental factors on contract type performance The properties of the environment in which the contract operates can also have an important influence on its effectiveness. Hence, the experiment design uses a large set of parameter settings for both trade-off and evaluation models. The impact of these environmental factors on the average Hausdorff distance Δp is measured by the slope of the linear regression line which estimates Δp as a function of the respective environment parameter value: Δp  parameter). The key considerations with respect to the influence of environment parameters are the following: Firstly, which of the environmental parameters has the largest average impact on contract performance? And secondly, which contract types are influenced most by changes in the environment in which they operate (i.e. the robustness of different contract types)? Note that impact in this case can mean either a significantly positive or a significantly negative slope of the linear regression line. 3 2 Measured as the average over all possible outcomes of the project. 4 Often dubbed a firm fixed price (FFP) contract in traditional literature. Often described as a cost plus fixed fee (CPFF) contract in contracting literature. Please cite this article as: Kerkhove LP, Vanhoucke M. Incentive contract design for projects: The owner's perspective. Omega (2015), http://dx.doi.org/10.1016/j.omega.2015.09.002i L.P. Kerkhove, M. Vanhoucke / Omega ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 17 Fig. 11. The average impact of variations in the environment for the different contract types. These are the outcomes for the situation where both a minimal earning and a minimal range for the contractor's earnings is required. Fig. 12. Average distance to the global Pareto front for multi-dimensional contract types (top panels), and the average spread obtained using different contract types for the situation where both minimal earnings and range of the contractor's earnings is required (bottom panels). Fig. 10 provides an answer to the former question by investigating the impact of different environment parameters. An answer to the latter question is provided by Fig. 11 which plots the average absolute value of the scope coefficients for the various contract types. Strong variation of impact across environment parameters: A comparison of the impact of different environment parameters shows that there is a clear distinction between parameters with a high and low impact on contract type performance. These high-impact Please cite this article as: Kerkhove LP, Vanhoucke M. Incentive contract design for projects: The owner's perspective. Omega (2015), http://dx.doi.org/10.1016/j.omega.2015.09.002i 18 L.P. Kerkhove, M. Vanhoucke / Omega ∎ (∎∎∎∎) ∎∎∎–∎∎∎ Fig. 13. The average impact of variations in the environments for contracts of different compositions. These are the outcomes for the situation where minimal earning and range contract design rules are enforced. environment parameters can be identified in Fig. 10 by their large positive or negative deviation from zero. The results show that in situations where increasing the effort has a larger impact on the other trade-offs (i.e. a high value for the mED or mES multiplicators), the proximity to the efficient frontier worsens for all dimensions. Contrarily, an increase in the convexity of the relations between the cost and any of its drivers (duration, scope and effort) has a generally positive influence on the proximity of contract sets of different types to the efficient frontier, again for all dimensions. The impact of environmental parameters on different contract dimensions (cost, duration and scope) can also be substantially different. An example of this are the convexity parameters (CMD, CMS, CME), which have a substantially larger impact on cost incentive contracts. Moreover, several other parameters show opposite influences on contract types of different dimensions. An example of is the cost of effort (Ecost m ): an increase of this parameter has a negative impact on cost incentive performance, but a positive impact on the performance of scope contracts. After analysing which environment parameters have the largest influence on contract types in general, the inverse can also be tested: which specific contract types are most susceptible to changes in environment parameters in general? This impact is expressed in relative terms for the various contract types in Fig. 11. The vertical axis on these graphs represents the absolute value of the regression slopes as they were used in Fig. 10. Duration and scope contract types are more volatile when faced with changes in the environment: Variations in the environment in which a contract type is operating have a greater average impact on duration and scope contract types than on cost contract types. Moreover, the relative impact of changes in the environment differs little between different cost contract types, whereas more significant fluctuations can be observed between duration and scope contract types. Preferred types are also robust to changing environments: The preferred contract types (see Section 8.2.1) in general, and the nonlinear contract types in particular show a remarkable robustness to changes in the environment. Hence, the superior performance of these types is likely to be valid across most types of projects. Contrarily, several contract types show an above average volatility when compared to the other contract types. Especially the LCS and 4PCS contract types for the duration and scope dimension respectively, seem to be greatly influenced by the nature of the environment. Other contract types which show above average volatility in the scope dimension include the LSS, 3PPC and 4POO types. Based on these observations, project owners are advised to pay extra attention to the nature of the environment in which they are operating when designing contracts for duration or scope dimensions. Even more so when they opt for contract types different from the ones which were identified as most effective in Section 8.2.1. Naturally, the environment parameters which were identified as being most critical should have priority over the other parameters when analysing the impact of the environment on contract type performance. 8.3. Multi-dimensional experiments The goal of the multi-dimensional experiments is to determine how the performance of multi-dimensional contracts (i.e. contracts which combine cost, duration and scope incentives) compares to that of uni-dimensional contracts. Several authors have highlighted the risk of contracts which focus on a single incentive dimension [1,11,51], yet no extensive computational experiments have been carried out to test these hypotheses. Due to the exponential increase in computational complexity when testing all possible contract combinations,5 these experiments were carried out using only a subset of the datasets which were employed for the uni-dimensional experiments (see Section 8.2). The contract set used for this experiment is limited to the linear contract types which are listed in Table 2, resulting in a total of 189 cost contracts, 4536 duration contracts, and 225 scope contracts. This means that a total of 193 million (189
4536
225) multi-dimensional contract combinations are tested for each environment. The time needed to evaluate the performance of all these combinations for a single environment is approximately 40 h using a 2.26 GHz processor. Because of this, the number of environments on which the tests are carried out is also a subset of the environments used in the uni-dimensional experiment. For both the trade-off and evaluation datasets, one instance was initialised using the average values for all parameters, plus for each parameter of the data set a high and low value instance with all other parameters still at the average value was initialised. The values used are simply the average, maximal and minimal values which were mentioned in Table 3. The number of environments was further reduced by always combining the trade-off and evaluation models with the default instance of the evaluation and trade-off 5 Testing all possible combinations would take approximately 90,000 years on a single core. Please cite this article as: Kerkhove LP, Vanhoucke M. Incentive contract design for projects: The owner's perspective. Omega (2015), http://dx.doi.org/10.1016/j.omega.2015.09.002i L.P. Kerkhove, M. Vanhoucke / Omega ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 19 Table 4 Overview of model parameters. Model Contract model Variable Description C; D; S The observed cost, duration and scope at the end of the project Cost, duration and scope incentive awarded IC ; I D ; I S Trade-off C t ; Dt ; St sCr S vD r ; vr Targets for cost, duration and scope as specified in the contract S BCr ; BD r ; Br The lower bound of cost, duration or scope region r in a (piecewise) linear contract UBC ; UBD ; UBS Upper bound of the region over which a cost, duration or scope incentive is spread using a nonlinear contract LBC ; LBD ; LBS Lower bound of the region over which a cost, duration or scope incentive is spread using a nonlinear contract S ICmax ; I D max ; I max Greatest incentive amount which can be awarded in a nonlinear cost, duration or scope contract S ICmin ; I D min ; I min Dlst IlsD Di ; Sj ; Ek ΔDi ; ΔSj ; ΔEk Greatest disincentive amount which can be awarded in a nonlinear cost, duration or scope contract S E ΔC D i ; ΔC j ; ΔC k C min SD ; S S ; S E mab Contract evaluation Sharing ratio for (piecewise) linear cost incentive contract in region r Region valuation parameter for piecewise linear duration/scope incentive CM D ; CM S ; CM E E½NOGŠ Elcost Itot l NCGl ROIE NCGladj NOGl C l ; Dl ; Sl ; E l μD ; μls ; μS RNCGladj RNOGl MD MAD Itot max LBItot max RItot DL Target date associated with a lump-sum duration incentive amount Incentive amount for a lump-sum duration incentive clause The attainable values for duration, scope and effort as defined by a trade-off model Attainable trade-off points for duration, scope and effort, defined as a distance to their lowest cost option The impact on the cost of selecting duration, scope and effort mode i, j and k respectively The lowest attainable cost for the project. The slope of the relationship between the cost (dependent) and duration, scope and effort (independents) Slope multiplicator, max impact of dimension b on the slope of the relationship between dimension a and the cost The convexity magnitude for the relation between C (dependent) and D, S and E (independent) respectively Expected net owner gain Cost of effort Total incentive awarded in scenario l Net contractor gain for scenario l Average ROI for effort investments by the contractor Net contractor gain for scenario l taking opportunity cost into account The value the owner derives from scenario l The cost, duration, scope an effort associated with scenario l The monetary valuation of time, deadline and scope of the owner Relative net contractor gain: NCGladj rescaled to ½0; 1Š Relative net owner gain: NOGl rescaled to ½0; 1Š Mean deviation, a measure for the payoff alignment Mean absolute deviation, a measure for the payoff alignment The maximal attainable incentive amount for the contractor A lower bound for the maximal incentive amount which can be earned by the contractor The range of the incentive earnings by the contractor External deadline relevant to the project owner model respectively. This results in a total of 38 different project environments, based on 27 trade-off models and 11 evaluation models. Hence, a total of approximately 1520 single-core computation hours are needed to carry out this experiment. The performance of contracts is evaluated by categorising them into different types, and then comparing the Pareto front formed by contracts of a certain type to the global Pareto front, which is constructed by taking all possible contracts into consideration. Similar to the uni-dimensional experiments, this is done for four different strategies regarding sensible contract design: the unrestricted case, the non-negative case, the minimum earnings case and the minimal earnings and range case (see Section 8.2). The results of this experiment will first be analysed in the context of the relative performance of the different types of contracts in Section 8.3.1. Next, the impact of variations in the environment is discussed in Section 8.3.2. 8.3.1. Performance of contract types The relative performance of different combinations of contract types is visualised in Fig.12. To analyse the performance of these multidimensional contracts, an number of new contract types are defined, as can be seen from the horizontal axes on the figure. These contract types can be interpreted as follows: A first set of contract types focusses on the number of dimensions used in a contract, as well as the nature of these dimensions. The n-dim nomenclature simply indicates that the contract set contains incentives affecting n contract dimensions out of a maximum of three (cost C, duration D and scope S). When this name is followed by dimension names, this indicates that the contracts belonging to this type have incentive clauses exclusively for these specific types. For example the 2-dim, C&S contract type includes contracts which have both a cost and scope incentive, but no duration incentive clause. The second set of contract types refers to the inclusion of the simple contract types as listed in Table 3 into a multi-dimensional contract. The nomenclature is similar to the one used in Section 8.2. For example, the ( C LCC contract type represents all contracts containing an LCC contract for the cost dimension, regardless of any other incentives which may or may not be included. This enables a comparison of the average effectiveness of different types of contract clauses. More dimensions yields better performance: Increasing the number of dimensions of a contract has a positive effect on all metrics for contract set performance. Looking at the performance of the contract types which only specify the number of dimensions, it is clear that the performance both in terms of proximity and spread increases substantially when moving from a 1-dim contract to a 2dim contract, and from a 2-dim contract to a 3-dim contract (see Fig. 12). Similarly, when looking at the types which also specify certain contract dimensions to be used, an improvement can always be seen when adding a contract dimension (e.g. moving from a “1-dim, C” contract to a “2-dim, C&S” contract). Duration 4 Scope 4 Cost: In case a 3-dimensional contract cannot be implemented, the owner should opt to incentivise the duration and scope dimensions in a bi-dimensional setup or the duration dimension in a uni-dimensional setup, preferring the former over the latter. The top left panel of Fig. 12 indicates that the performance in terms of proximity to the global frontier of a one and two-dimensional contract is strongly dependent on the chosen Please cite this article as: Kerkhove LP, Vanhoucke M. Incentive contract design for projects: The owner's perspective. Omega (2015), http://dx.doi.org/10.1016/j.omega.2015.09.002i 20 L.P. Kerkhove, M. Vanhoucke / Omega ∎ (∎∎∎∎) ∎∎∎–∎∎∎ incentive dimensions. Specifically, the owner should prefer to attach incentives to the duration and scope dimensions – in that order. If the right incentive dimensions are chosen, the average performance gap between contracts including more or less incentives can be greatly reduced, although it still remains present. Avoid contract components which transfer all risk to a single party: Multi-dimensional contracts which include components which transfer all risk to one of the two parties (LCC or LOO) result in suboptimal performance. The increased distance to the global frontier when using such contracts (see the right panels of Fig. 12) corroborates the undesirable behaviour of these contract types which was observed in the uni-dimensional experiments. 8.3.2. Impact of environmental factors on performance Fig. 13 shows how variations in the environment impact the performance of the tested multidimensional contracts. This performance is measured as the slope of the linear regression line which estimates the Δp as a function of the respective environment parameter. Environmental robustness increases with dimensions: The robustness to changes in the environment of contract types increases as the number of incentivised dimensions in the contract increases (see the left panel of Fig. 13). This is true for all but one case: when moving from a one dimensional cost contract (1-dim, C) to a twodimensional cost and scope contract (2-dim, C&S), there is a slight increase in the average sensitivity to changes in the environment. This is due to the fact that incentivising the scope appears to be highly sensitive to changes in the problem environment, as was already observed in Section 8.2.2. Preferred multidimensional types are relatively robust to environmental changes: The most desirable multidimensional contract types for each dimension identified in Section 8.3.1 are not heavily influenced by changes in the environment. This confirms that these managerial guidelines are valid regardless of the type of project the owner is facing. Fig. 13 also shows that unidimensional scope contracts are relatively unstable with respect to changes in the environment parameters. Project owners should therefore be cautious and verify the adequacy of such contracts in the specific environment where they will be implemented. Including certain components increases the environmental sensitivity: The right pane of Fig. 13 reveals that using certain components in multi-dimensional contracts can negatively influence the robustness of the contract type. The greatest peak can be observed for the ( C LCC contract type, which indicates that a traditional firm fixed price contract is used for the cost dimension. Again confirming preceding observations of the risks involved when implementing this type of contract. Based on these observations, a contract owner wishing to construct a contract structure which is robust to changes in the environment in which it operates is advised to use as many dimensions as possible in the contract. In case it is impossible to construct a three-dimensional contract, the combination of scope and duration is on average most robust for a two-dimensional contract. The best uni-dimensional alternative from a robustness perspective is the introduction of a cost incentive, however given that a uni-dimensional duration contract only slightly more sensitive and yields better performance on average (see Section 8.3.1), we advise a uni-dimensional duration contract instead. design of incentivised contracts. This model was also used to conduct computational experiments, from which a set of managerial contract design guidelines have been derived (see summary in the following table). The results of these experiments indicate that multidimensional contracts with piecewise linear and/or nonlinear incentive clauses are superior on all performance metrics. Moreover, good performing contract structures are generally more robust to changes in the environment in which they operate, and can therefore be applied in structurally different project environments. Linear contracts which transfer all the risk to one of the two parties, such as the traditional firm fixed price (FFP) and cost plus fixed fee (CPFF) contracts should be avoided in both uni-dimensional and multi-dimensional incentive contracts. These contracts produce inferior results in terms of both solution quality and diversity, whilst also being less robust to changes in the environment in which they operate, when compared to other contract alternatives. Furthermore, when incentivising all dimensions is impossible from a practical point of view, the owner's first aim should be to implement a duration incentive, then a scope incentive in that order. This opposes the traditional focus on cost incentives in both literature and practice. Several extensions of the problem can be considered relevant for future research. A first extension would be to implement more advanced optimisation techniques to optimise the contract parameters, rather than the full factorial approach taken in this study. Secondly, the realism of the model could also be improved by lifting the assumption of full information as well as the deterministic nature of the trade-offs. These conditions could be tested within a game-theoretical setting which analyses the optimal behaviour of both owner and contractor in both single and repeated games. A final area for future research would be the implementation of this model in real life case studies. Experiment Uni-dimensional Contract type Nonlinear and piecewise contract outperform linear contracts. Avoid transfer of all risk to one party. Preferred types are robust to imposed restrictions. Skewness persists within contract types. Environment Strong variation of impact across environment parameters. Duration and scope contracts are more volatile. Preferred types are more robust. Multi-dimensional More dimensions improve performance. Duration 4 Scope 4 Cost. Avoid contract components transferring risk to a single party. More incentivised dimensions improves robustness. Preferred multidimensional types are more robust. Including certain components increases sensitivity. 9. Conclusions This paper discussed incentive contract design for projects from the perspective of the project owner. Based on axioms derived from an extensive literature review, a model consisting of three subcomponents has been constructed. The model serves as a framework which can be used by project owners to improve the Acknowledgements We acknowledge the support provided by the Bijzonder Onderzoeksfonds (BOF) for the project with Contract no. BOF12GOA021. This work was carried out using the STEVIN Supercomputer Please cite this article as: Kerkhove LP, Vanhoucke M. Incentive contract design for projects: The owner's perspective. Omega (2015), http://dx.doi.org/10.1016/j.omega.2015.09.002i 21 L.P. Kerkhove, M. Vanhoucke / Omega ∎ (∎∎∎∎) ∎∎∎–∎∎∎ Infrastructure at Ghent University, funded by Ghent University, the Flemish Supercomputer Center (VSC), the Hercules Foundation and the Flemish Government department EWI. We would also like to express our thanks to the anonymous reviewers and the editor of Omega for their helpful comments. can be determined by the linear approximation of the curve, and the desired convexity of the curve (CM). max ΔΔΔC min s:t: 0 r ΔΔC 1 ; ΔΔC i Appendix A. Overview of notation Table 4 presents an overview of all the notation introduced in this paper. ðC:1Þ n X 1r i ¼ 1; …; n ΔΔC i ; ðC:2Þ i ¼ 2; …; n ðC:3Þ ΔΔC i ¼ ΔC n ðC:4Þ i¼1 ΔΔΔC min r ΔΔC 1 Appendix B. Derivation of the CM metric ΔΔΔC min r ΔΔC i ΔΔC i The basic definition of the CM metric is as follows: CM ¼ A A þB ðB:1Þ The variable A can be eliminated by calculating the surface of the triangle as follows: AþB ¼ A¼ ΔD n
Δ C n 2 ΔDn
ΔC n 2 B ðB:2Þ And rewriting CM as CM ¼ ΔDn
ΔC n B A 2 ¼ ¼1 A þB ΔDn
ΔC n BþB 2 2
B ðB:3Þ ΔDn
ΔC n The value of B can be calculated as n  X 1 ðΔDi ΔDi 1 ÞΔC i 1 þ ðΔDi ΔDi B¼ 2 i¼1 ðC:5Þ CM ¼ 1 2 1; i ¼ 2; …; n ðC:6Þ 3 ðC:7Þ n i 1 X X 2 1 4 ΔΔC j þ ΔΔC i 5 n
ΔC n i ¼ 1 j ¼ 1 2 Eq. (C.2) stipulates that the ΔC i curve should always be an increasing function (see axioms 1–3). The convexity of the tradeoff curve is guaranteed by Eq. (C.3) which states that the consecutive steps have to be of nondecreasing magnitude. Eq. (C.4) ensures that the final point of the convex trade-off curve coincides with the linear approximation on which this curve is based, the latter is defined as the parameter ΔC n . The value of the smallest difference between the consecutive steps is assigned by Eqs. (C.5) and (C.6). Finally, Eq. (C.7) guarantees that the resulting curve has the desired convexity magnitude specified by the parameter CM. References Δ C i ΔC i 1 Þð 1Þ  ðB:4Þ In case of generated convex curves it can be assumed that the step size of the duration is always identical and therefore all equal to ΔDn =n. This means the formula can be simplified to  n  ΔDn X ΔC ΔC i 1 ðB:5Þ ΔC i 1 þ i B¼ n i¼1 2 Appendix C. Linear model for trade-off model generation The objective of this model is to create the smoothest possible convex curve which starts and ends at the start and end points of the linear approximation curve respectively. The model uses the variable ΔΔC i to signify the change in cost between two consecutive points on the trade off curve: ΔC i ¼ ΔC i 1 þ ΔΔC i . The graphical interpretation of these variables is illustrated in the right panel of Fig. 5. Hence, this variable is indexed i¼1,…,n. Since the curve is nondecreasing and convex it is known that each of these variables is positive (see Eq. (C.2)) and at least as large as the preceding variable (see Eq. (C.3)). The smoothness of the curve can be measured by the difference between two consecutive ΔΔC i variables, which indicate the change in cost caused by a change of an independent variable (duration, scope or effort). 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