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European Journal of Operational Research 160 (2005) 72–87 www.elsevier.com/locate/dsw

Product development resource allocation with foresight q Nitin R. Joglekar a, David N. Ford a b

b,*

Boston University School of Management, 595 Commonwealth Avenue, Boston, MA 02215, USA Department of Civil Engineering, Texas A&M University, College Station, TX 77843-3136, USA Received 1 September 2002; accepted 1 June 2003 Available online 27 October 2003

Abstract Shortening project duration is critical to product development project success in many industries. As a primary driver of progress and an effective management tool, resource allocation among development activities can strongly influence project duration. Effective allocation is difficult due to the inherent closed loop flow of development work and the dynamic demand patterns of work backlogs. The Resource Allocation Policy Matrix is proposed as a means of describing resource allocation policies in dynamic systems. Simple system dynamics and control theoretic models of resource allocation in a product development context are developed. The control theory model is used to specify a foresighted policy, which is tested with the system dynamics model. The benefits of foresight are found to reduce with increasing complexity. Process concurrence is found to potentially reverse the impact of foresight on project duration. The model structure is used to explain these results and future research topics are discussed. Ó 2003 Elsevier B.V. All rights reserved. Keywords: Project management and scheduling; Control theory; Concurrent development; Product development; Rework cycle; Resource allocation; System dynamics

1. Introduction Failing to complete projects by their deadlines can lead to staggering levels of cost escalation and technology obsolescence in defense related product development projects (McNulty, 1998). Meeting development schedule deadlines has been identi-

q

Both authors contributed equally to this article. Corresponding author. Tel.: +1-617-353-4290; fax: +1-617353-4098 (N.R. Joglekar), tel.: +1-979-845-3759; fax: +1-979845-6554 (D.N. Ford). E-mail addresses: [email protected] (N.R. Joglekar), [email protected] (D.N. Ford). *

fied as the most important concern for product development project performance in many fields ranging from construction to software development (Patterson, 1993; Meyer, 1993; Wheelwright and Clark, 1992). Good schedule performance is particularly difficult in development projects due to the iterative nature of the process that creates closed loop flows of work. Two primary approaches to improving schedule performance are process improvements and resource management. A variety of process improvement approaches to improving schedule performance have been explored including, the use of information technology tools (Joglekar and Whitney, 1999), crossfunctional development teams (Moffatt, 1998), and

0377-2217/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2003.06.021

N.R. Joglekar, D.N. Ford / European Journal of Operational Research 160 (2005) 72–87

increased levels of concurrence 1 (Backhouse and Brookes, 1996). Increasing concurrence in particular has become a common form of process improvement. Research directed at concurrent development process design has addressed several aspects of iteration, including information evolution (Krishnan, 1996; Krishnan et al., 1997) and activity order and grouping (Smith and Eppinger, 1997a,b). Studies that build formal models of the concurrent development approach have been addressed in previous system dynamics work by the authors and others (Ford and Sterman, 1998a,b; Cooper, 1980, 1993; Cooper and Mullen, 1993; Abdel-Hamid and Madnick, 1991). The nature of process concurrence relationships is often governed by the product architecture. For example in semiconductor chip development the design phase specifies the locations of components and electronic pathways on the chip. Prototype building requires all of the design to be completed before prototype construction can begin, thereby requiring a sequential concurrence relationship between the two phases. These constraints on process concurrence often leave managers with few degrees of freedom with which to alter the process once the architecture is defined (Joglekar et al., 2001). Because of the difficulty in reducing project duration through process improvement, effective resource management is important to the timely completion of projects. Managers can have a large effect on resource utilization even when the total quantity of resources (e.g., the number of developers) is fixed. Both resource productivity improvement and policies for allocating resources among specific development activities can be used to speed up projects. The current work focuses on resource allocation policies as a means of reducing project duration. In contrast to process improvements, the resource allocation policies associated with the dynamics of product development in general and the rework cycle in specific have not been extensively researched. Within a concurrent

1 Consistent with the literature, we conceptually define concurrence as the overlapping of development activities or phases in time. We further discuss this concept in the next section and provide an operational definition in the appendix.

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development context two resource allocation questions must be addressed to improve policy design and implementation: 1. How should product development project managers allocate resources to shorten project duration? 2. How does concurrence in a product development process impact the effectiveness of resource allocation policies? A simple approach is taken here to develop initial but valuable insights on these issues. First, foresight, an omnipresent and potentially powerful form of information processing used by managers, is chosen as a focus. As discussed here, foresight is the use of resource demand forecasts as the basis for allocation. Second, models that are very simple relative to actual product development practice are used to expose the relationships between policy structure and project behavior. For example, we assume that the total resource quantities and productivity are fixed, and that concurrence relationships are known a priori and are immutable. Projects are described with two important development project characteristics: complexity and concurrence. These assumptions (as well as others made in modeling) allow us to integrate system dynamics and control theory to derive insights that are not available by either alone, but at the cost of limiting the range of applicability of our results. We describe model extensions to partially address these limitations in our conclusions. Here a system dynamics model and control theoretic model of a simple product development rework cycle are used in combination to investigate the effectiveness of foresighted resource allocation policies in reducing project duration.

2. The challenges of resource allocation in product development projects Development project resource management can improve schedule performance by increasing the quantity of resources, productivity, and other means (Wheelwright and Clark, 1992). Total resource quantities and their productivity are often

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limited and difficult to improve. In contrast, the allocation of scarce resources among development phases and activities is a realistic management opportunity for improving schedule performance. A complete dynamic view of product development project resource management must include delays in making allocation decisions, the desire of the workforce to minimise the number and frequency of reassignments, and delays in productivity rampup during the implementation of allocation decisions. Effective resource allocation is made more difficult by the challenges in accurately predicting the amounts of work to be initially completed, inspected or tested to discover change requirements, and reworked, and by the different types of concurrence relationships among these activities. We define project complexity as the probability that a task will have to be reworked. A projectÕs concurrence is defined by the amount of work that is released for completion at any phase of the project. In general, earlier releases of work translate into higher concurrency because they allow simultaneous development of a larger fraction of the scope. A mathematical definition of concurrence is provided in the appendix. Resource allocation practice in many projects is based on a simple approach. Managers look at the current backlogs of work to be serviced by the different development activities and allocate resources to each activity in the same proportion that the activityÕs backlog contributes to the total amount of work waiting to be done. Such a directly proportional policy is an example of how managers use relatively simple information structures and heuristics to bridge the gap between high project complexity and their bounded rationality (see Ford (2002) for discussion and examples). But, as will be shown, the dynamic structure of development causes directly proportional policies to be relatively ineffective in minimizing project duration. As described by Ford and Sterman (1998b), that dynamic structure includes important sequencing and concurrence constraints. The development of any given piece of work can only occur in a particular sequential order within a single development phase (e.g., initially complete, inspect and correct errors) and within individual projects (e.g., plan, design and build). These de-

velopment activities occur over characteristic durations and establish the minimum phase duration. A cascade of such relationships amount to an ‘‘ageing chain structure’’ (Sterman, 2000). Ageing delays the development of backlogs in downstream activities and phases, distorting optimal resource allocations away from those described by current backlogs. In addition, development activities and phases are coupled through closed, conserved rework cycles that change future backlogs in ways that are not reflected in current backlogs. By delaying the evolution of backlogs, development processes create additional work for some activities through rework cycles. Directly proportional policies are myopic in that they do not fully capture future resource needs in current resource allocations. The delays and closed conserved flows inherent in development and resource allocation suggest that the allocation targets set by managers should be based on future resource needs. This requires that managers make adjustments by looking ahead to the evolution of demands for various development activities. We term policies that incorporate future conditions in current allocations ‘‘foresighted’’ and propose them as a potentially improved type of resource allocation policy. Practising managers include some foresight into policies intuitively. For example, Graham (2000) describes an aerospace company where managers foresaw the detrimental effects of rework and responded by shifting interaction with customers early in development projects. Our work in accelerating product development through foresight is informed by the literature addressing accelerating production processes, particularly in classic job shop settings (Holstein and Berry, 1972; Graves, 1986). This literature accounts for the re-entrant flows within the job shop problem structure while allocating resources. The problem is typically formulated to optimally complete a stream of incoming jobs subject to a tardiness penalty (see Jones and Rabelo (1998) for a recent survey of job shop scheduling). In these settings, a foresighted or look-ahead policy performs better than a myopic policy, i.e., a policy that ignores the feedback flows (Jones, 1973). These studies inform our work because during


product development various tasks traverse between different sub-groups within the process (Eppinger et al., 1994) in an iterative manner similar to the flow of tasks in a job shop. However, the differences between scheduling a product development project and a production job shop lie in the finite amount of jobs associated with a product development project and the level of concurrence between the releases of these jobs. The rest of this article focuses on these differences. Foresighted resource allocation policies are intuitively attractive for dynamic analysis because they can potentially balance short- and long-term objectives, which can be critical in improving the performance of dynamic systems (Forrester, 1961; Sterman, 2000). But the tacit nature of these policies and their design prevent their complete description, evaluation, and improvement. Questions of managerial relevance are––Do foresighted resource allocation policies always improve schedule performance compared to myopic resource allocation policies? How should managers anticipate the impacts of development processes and rework on resource demands and adjust resource allocations accordingly?

3. The Resource Allocation Policy Matrix Describing policies for allocating resources to development activities that are based on the backlogs of work to be processed requires a means of relating each work backlog to each process. Prior research (Ford and Sterman, 1998b) has established the backlogs of work not yet completed the first time (Wc ), work ready for quality assurance after completion (Wqa ), and work ready for iteration (Wit ) as critical elements in managing product development rework cycles. In our model

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each backlog is reduced by the development activity with the same name (i.e., completion, quality assurance, or iteration). We introduce the Resource Allocation Policy Matrix (Fig. 1) as a tool that describes how individual backlogs influence, separately and in combination, the activities that serve those backlogs. Each cell of the matrix quantifies the influence of one work backlog on one development activity. When these relative influences, i.e., allocation weights, are combined for each backlog and compared across activities, they specify how resource allocations respond to work backlogs. Generally, the demand for an activity is the weighted sum of the demands created by each backlog for that activity. Mathematically, allocation fractions to the development activities ðfi Þ for any given set of backlog conditions are specified by summing for each process the products of each allocation weight ðCB;i Þ and the appropriate backlog size ðWB Þ and then proportioning the sums so their total is 100%: fi ¼

X B

, ðWB CB;i Þ

X

X

i

B

! ðWB CB;i Þ ;

for B and i 2 fcompletion; quality assurance; iterationg where fi is the fraction of total resources applied to activity i, WB is the work backlog of process B, CB;i is the weight of work backlog, B on development activity, i, from the Resource Allocation Policy Matrix. A directly proportional policy (e.g., Fig. 1) illustrates a simple resource allocation policy. A verbal form of this policy is ‘‘Allocate the same fraction of the available resources to each activity as the fraction of the current total work backlog that the activityÕs backlog contributes to the total work backlog.’’ The zeros in the non-diagonal terms specify that only the backlog reduced by

Fig. 1. A Resource Allocation Policy Matrix describing a directly proportional policy.

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each activity impact the amount of resources received by that activity. The equal values of the diagonal terms specify that no backlog receives more or less than indicated by the size of its backlog. Note that in this and all Resource Allocation Policy Matrices the resource allocation is dynamic even though the weights are constant (except for a null matrix) because the backlog sizes vary dynamically, thereby constantly changing the resource fractions. A directly proportional resource allocation policy is attractive to managers because it is relatively easy to design and implement. Ford (2002) describes and models how development project managers design and use resource allocation policies that are significantly simpler than the system being managed. A directly proportional policy requires very little information processing because each activity is impacted only by its own backlog. However, a directly proportional allocation policy might be myopic in that it does not adjust for the previously described delays and backlog changes created by processing work. Alternatively, a foresighted policy that accounts for future resource needs is derived using a linear model of a system with closed loop feedback control (based on McDaniel, 1996). Foresighted resource allocation policies use more information, in the form of additional work backlogs, to determine allocation fractions. Therefore, for typical development scenarios, foresighted matrices have non-zero off diagonal terms. Resource Allocation Policy Matrices facilitate the design of improved policies and the explanation of the factors that influence these policies by explicitly and clearly describing the individual relationships between each work backlog: development activity pair. We use Resource Allocation Policy Matrices to describe the two types of policy described above: proportional and foresighted. In a subsequent section we derive a foresighted matrix and show that comparing these matrices can develop an intuitive understanding of the resource allocation choices. This comparison helps explain how the foresighted policy allocates more resources to a bottleneck task identified in previous research (Cooper, 1993, 1980) than the directly proportional policy allocates.

4. Basic system model Our work builds on a project model developed with the system dynamics approach by Ford and Sterman (1998b). System dynamics is a methodology for studying the management of dynamically complex systems by building and applying simulation models. Forrester (1961) develops the methodologyÕs philosophy and Sterman (2000) specifies the modeling process with examples and describes numerous applications. When applied to project management problems, system dynamics focuses on how performance evolves in response to interactions between managerial decision-making and development processes. This methodology has been applied to study the impact of a variety of management issues on project performance, including changes in project scope (Cooper, 1980; Rodrigues and Williams, 1997); rework (Cooper, 1980, 1993, 1994); poor scheduling (Abdel-Hamid and Madnick, 1991); failures in project fast track implementation (Ford and Sterman, 2003a); and concealing rework requirements (Ford and Sterman, 2003b). Our study focuses on the impact of foresight on project duration. Therefore, only those features that describe basic resource allocation policies and the fundamental processes they impact are included in our study. We elaborate on the limitations of our approach at the end of this article. The model consists of two sectors: a workflow sector (Fig. 2) and a resource allocation sector. The structure of the workflow sector is consistent with a standard rework cycle from the system dynamics and project management literature (Cooper, 1993, 1994; Cooper and Mullen, 1993). In this structure as work is first completed it enters the backlog of work requiring quality assurance (Wqa ). Work that passes quality assurance is approved and adds to the stock of approved work (Wa ). The fraction of the work that is discovered to require changes moves into a backlog of tasks requiring iteration (Wit ). After rework these tasks return to the quality assurance backlog for reinspection because rework can generate or expose new change requirements. The completion, quality assurance, and iteration activities are constrained by both processes and available resources.

N.R. Joglekar, D.N. Ford / European Journal of Operational Research 160 (2005) 72–87 p(Disc Iteration) Iteration Discovery rate

Quality Assurance resource effect

Quality Assurance rate

Wit

77

Quality Assurance process time

Iteration process time

Wa

Wqa Approval & Release rate

Iteration rate

Iteration resource effect

Completion rate

Wc

Completion resource effect Completion process time

Fig. 2. Workflow during a capacitated product development project rework cycle (based on Ford and Sterman, 1998b).

In the resource allocation sector resources are allocated among the completion, quality assurance, and iteration activities based proportionally on the indicated demand for each. The indicated demands for resources are adjusted from the levels indicated purely by the current size of activity backlogs according to the Resource Allocation Policy Matrix to reflect specific managerial policies, as previously described. The resource impact is modeled as a fraction that reduces the uncapacitated rate of progress of each activity. The fraction is the ratio of the progress based on the amount of resources provided to the maximum rate allowed by resources. The progress based on provided resources is the product of total quantity of resources, resource productivity, and the fraction allocated to the specific activity. It is through the allocated fraction that the resource allocation policy impacts system behaviour. See Appendix A.2 for a listing of the model equations. Next we use a control theoretic model from linear systems theory for the closed loop control of resources to derive the coefficient for the foresighted policies.

formulation of the system dynamics model described above that included a closed loop optimal controller. This work builds on a discrete time formulation developed by McDaniel (1996). We define the system behaviour as X ðtÞ, a vector of the three stocks of work to be processed {Wc ðtÞ, Wqa ðtÞ, Wit ðtÞ} that describes the sizes of these stocks over time. This linear closed loop control mechanism (Fig. 3) is congruent with the control policies for the system dynamics model shown in Fig. 2. The formulation in Fig. 3 includes two feedback loops. The small loop represents the directly proportional allocation of resources (with identity matrix A) in response to the systemÕs behaviour ðX ðtÞÞ. Therefore matrix A shifts resources toward larger backlogs without regard to targets. In contrast, the large feedback loop in Fig. 3 represents the purposeful use of resource allocation to impact system behaviour. Y ðtÞ is the vector of stock values that describes the desired schedule of backlogs. F transforms the error signal into a control signal. B is the rate transformation matrix (see Fig. 4) that

5. Linear optimal control of resources The strength of the close loop linear control model lies in its ability to specify the resource allocation policy associated with an optimal controller. We developed a continuous time (CT)

Fig. 3. Resource allocation with closed loop control.

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N.R. Joglekar, D.N. Ford / European Journal of Operational Research 160 (2005) 72–87 Work Backlog (Wb) Wc

Wqa

Wit

(-1)/T c

0

0

Quality Ass. Rate

1/T c

(-1)/T qa

1/T it

Iteration Rate

0

p/T qa

(-1)/T it

Completion Rate

Fig. 4. The rate transformation matrix (B).

computes the various rates based on the sizes of the backlogs and the characteristic processing duration. In this formulation the traditional system dynamics equations are replaced by the coefficients in matrices that relate the state variables (the stocks) to the changes in state variables (the rates) as shown in rate transformation matrix (Fig. 4). The diagonal terms of this matrix, with negative signs, are the reciprocals of the completion times ðsi Þ. The off diagonal terms represent the rework and approvals rates. 2 The variables sc , sqa , and sit are the times for completion, and p is the probability of rework. The influence of resource capacity constraints and allocation policies are described in matrix F . This matrix is the Resource Allocation Policy Matrix. It is applied to the difference between the actual behaviour ðX ðtÞÞ and the desired behaviour ðY ðtÞÞ in a manner similar to the application of an adjustment to a gap between desired and actual conditions in many system dynamics models (Sterman, 2000). Therefore the control signal sent to the system is given by uðtÞ ¼ F fX ðtÞ Y ðtÞg: Just like most system dynamics models, this control theory model uses the state conditions to determine the rates of change. In contrast to many system dynamics models, this particular control theory model uses full-state feedback, meaning

2 The discrete time equivalent formulation is called the work transformation matrix (Smith and Eppinger, 1997a,b). We have added the negative sign for algebraic convenience in setting up the state equation.

that the sizes of all the backlogs (not a subset of the backlogs) are used to determine the control signal. Substituting the notation above into the usual control theory notation, the state equation is dX ðtÞ=dt ¼ AX ðtÞ þ BuðtÞ: The optimisation seeks a policy that will minimize the cost J over the duration of the project where the cost is defined as Z J ¼ fX 0 QX þ u0 Ru þ 2X 0 Nug dt: The three terms within the integrand represent the three contributors to quadratic costs: system conditions ðX Þ, the control signal ðuÞ, and the interaction between them. Q, R, N are weights on these three types of costs. The nature of the cost function forces the optimisation process to reduce both X (the stock of work), and u (the effort needed) at once. Following Holt et al. (1955), it is customary for production and employment models to use a quadratic cost structure in order to derive linear decision rules (Gaimon, 1997; Anderson, 2001). Empirical evidence in the product development literature also confirms that development efforts scale up, and their associated cost structure can be represented by quadratic formulations, when firms try to reduce the development time (Graves, 1989; Rosenthal, 1992; Roemer et al., 2000). In the example that follows Q and R are the identity matrix and N is null, representing equal cost of system performance and control and no cost interaction between performance and control. Later we discuss how we test the impact of different cost structures by changing Q, R, and N . This formulation yields the linear time invariant Riccati Equation: SA þ A0 S ðSB þ N ÞR1 ðB0 S þ N 0 Þ þ Q ¼ 0; where F ¼ R1 B0 S and S is the solution to the Riccati equation. This equation was solved numerically using Matlab. For example, the model generated the following Resource Allocation Policy Matrix for the time constants shown in the model equations and p ¼ 25%. A comparison of the Resource Allocation Policy Matrices in Fig. 1 (directly proportional) and Fig. 5 (foresighted) provides an intuitive

N.R. Joglekar, D.N. Ford / European Journal of Operational Research 160 (2005) 72–87 Development Activity Completion Quality Assurance Iteration Total

Work Backlog Weight Wc Wqa Wit 0.208 -0.009 -0.001 0.132 0.150 0.127 0.065 0.057 0.272

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Net Allocation Fraction With Uniform Backlogs 0.198 0.409 0.394 1.000

Fig. 5. Resource allocation matrix derived from the control theory model for p ¼ 25%.

understanding of the impacts of foresight on resource allocation. The average weight of the three diagonal terms in Fig. 5 is 0.21. Thus, a foresighted policy increases the influence of iteration on the amount of resources used to process that backlog (0.272 versus the average diagonal value of 0.21). At the same time, it decreases the influence of the quality assurance backlog on the resources applied to process that backlog (0.15 versus the average diagonal value of 0.21). Foresighted policies also add important impacts of backlogs on activities that do not process the work in that backlog (off diagonal terms). These impacts can result in net shifts in allocations that are different in direction and size than shifts indicated by changes in the diagonal terms alone. For ease of further comparison, assume that the three backlog stocks associated with Wc , Wqa and Wit are equal. Under this assumption, the values from the matrix can be been summed across the rows and those sums proportioned to determine the net allocation fractions for each of the three activities (right column in Fig. 5). Notice in particular that, resources are shifted away from initial completion (0.198 versus the average row sum of 0.333 in Fig. 1) and toward quality assurance (0.409 versus the average row sum of 0.333 in Fig. 1). These shifts, derived through the control theory model, are consistent with the results of previous system dynamics research on means of reducing project duration (Cooper, 1993) and the previously described generation of future backlogs by development processes that are not captured in myopic resource allocation policies. The shift in allocation due to foresight in Fig. 5 is for a project with complexity described by a rework fraction of 25%. Later we describe the impacts of other rework fractions on shifts in allocation. The coefficients in Fig. 5 and similar matrices for other conditions were used in the system dy-

namics model as weightings to alter the indicated sizes of work backlogs and thereby the relative demands for resources for each activity. In this way the policies described by the Resource Allocation Matrix generated by the control theory model were implemented in the system dynamics model, which simulated project duration.

6. Simulation results Resource Allocation Matrices were calculated and projects simulated over a range of project complexity and concurrence conditions. Results for two complexities and two levels of concurrence are used to illustrate the method applied before complete results are described. Initially the project is assumed to be as concurrent as the conserved closed loop structure (Fig. 2) allows. Project duration using directly proportional and foresighted policies for a simple and a complex project were simulated in the system dynamics model. Simple and complex are described with the fraction of inspected work that is discovered to require rework. A simple project is assumed to have 25% rework probability and a complex project is assumed to have a 62.5% rework probability. Evolution of the quality assurance backlog (Wqa in Fig. 2) for a simple and complex project with foresighted resource allocation are illustrated in Fig. 6. This evolution illustrates the characteristic increase in the quality assurance backlog as work packages are initially completed in days 0–10, followed by slower reductions as resources shift to quality assurance and iteration in response to the growth in those backlogs. The longer duration of the complex project reflects its larger volume of work that is discovered to require iteration and remains trapped in the rework cycle longer. Simple and complex projects managed with directly

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N.R. Joglekar, D.N. Ford / European Journal of Operational Research 160 (2005) 72–87 30

Quality Assurance 15 Backlog

2 2 1

2

1

0

2 1

0

25

1

2 1

1

50

2

12

75

12

12

12

100

125

Time (days) Simple Project 1 work packages 1 1 Complex Project 2 2 2 work packages

Fig. 6. Quality assurance backlog for a simple and complex project.

proportional and foresighted resource allocation policies were performed. The simulation results for project duration, i.e., elapsed time in days for reaching 99% completion, are shown in Table 1. The results in Table 1 support our basic premise that foresighted policies improve schedule performance with full concurrence. The simple project finishes 18.8% faster and the complex project finishes 8.94% faster with foresighted policies than with directly proportional policies. More complex and therefore difficult projects are traditionally expected to benefit most from foresight. However, the simulations show a counterintuitive result: more complex projects appear to benefit less from foresighted policy than simple projects. This result will be further described and an explanation provided in Section 7. In the case of full concurrence all the work to be accomplished is assigned to the initial condition of the stock for work to be completed (Wc ) and is

therefore available for initial completion. We enhance the basic structure of the system dynamics model to allow for partial concurrence as follows. An additional stock (Wna , work to be completed but not available) is added. The outflow from this stock feeds into the stock of work to be completed (Wc ). Partial concurrence is described by starting with all the work in Wna and controlling its outflow based on the amount of work approved (Wa ) through a look up function. Specification of the partial concurrence relationship is based on the table function described in Ford and Sterman (1998a). The concurrence relationship used consists of three linear relationships (see Appendix A) that approximate an ‘‘S’’ shaped concurrence relationship used by practitioners (Ford and Sterman, 1998b). We assume that the act of releasing the work based on the stock of the work approved does not require any development resources. Hence, neither Wa , nor Wna figure in the derivation of the governing resource allocation matrix. The simulation results for project duration, i.e., elapsed time in weeks for reaching 99% completion, with a partial and relatively low (10%) concurrence are shown in Table 2. A comparison of Tables 1 and 2 indicates that a closed loop foresighted policy, which effectively improves schedule performance in a concurrent project, can degrade schedule performance in a partially concurrent project. These results will further discussed in the next section.

Table 1 Project duration with full concurrence Simple project (p ¼ 25%)

Complex project (p ¼ 62:5%)

Resource Allocation Policy

Duration (days)

Reduction

Duration (days)

Reduction

With foresight Directly proportional

44.12 54.37

18.8%

89.12 97.87

8.94%

Table 2 Project duration with partial concurrence Simple project (p ¼ 25%)

Complex project (p ¼ 62:5%)

Resource Allocation Policy

Duration (days)

Reduction

Duration (days)

Reduction

With foresight Directly proportional

287.25 284.62

)0.92%

648.75 561.12

)15.62%


30

% Change in Duration

Foresight Increases Duration Foresight Reduces Duration

0

-30

81

0.75

1.

Concurrence

0.0 0.00

Rework Fraction

Fig. 7. Percent change in duration due to foresighted Resource Allocation Policy.

To more thoroughly evaluate the impacts of foresight in resource allocation policies on project performance Resource Allocation Matrices were generated using the control theory model and project performance simulated using the system dynamics model across a range of complexity and concurrence conditions for the example conditions described above. Fig. 7 shows the percent changes in project duration due to the use of a foresighted resource allocation policy when compared to a directly proportional policy for project complexities ranging from low (12.5% rework) to high (75% rework) and for concurrence from 0.1 (low) to 1.00 (full). Concurrence was quantified as the fraction of the maximum concurrence possible within the process flows (Fig. 2) as described by Ford and Sterman (1998a). 3 In Fig. 7 negative values indicate reduced durations and therefore a benefit of using foresight. Positive duration changes indicate longer project durations and therefore no benefit to the use of foresight. The maximum benefit of foresight is 18.8% reduction in project duration for a project that is fully concurrent with 37.5% rework. The size of this reduction supports our hypothesis that

3

Very simple projects that would have almost no rework are considered a special case beyond the scope of the current work. The completion time of very complex projects, with rework greater than 75%, increases approximately exponentially, following the pattern shown in Fig. 7. For ease of presentation we have chosen to simulate a range of concurrence from 0.1 to 1.0 in ten uniformly distributed steps.

foresight in resource allocation policies can have important impacts on development project performance. Fig. 7 indicates that the benefits of foresight generally increase with project concurrence and project simplicity. This shows that the counterintuitive results described in the example above are consistent across a range of project conditions. Similar simulations were performed reflecting two other project scenarios: (I) a project in which costs are also impacted by interactions between backlogs and control (by setting the interaction cost matrix N as the identity matrix) (II) a project in which managers only respond to targets and not to backlog sizes directly (by setting matrix A as the null matrix). The surface of changes in project duration with backlog and control cost interaction (scenario I) has the same shape as Fig. 7. The maximum benefit of foresight is about the same as in the example above (17.4% versus 18.8% reduction) for a fully concurrent project, but the maximum occurs with slightly more rework (50% versus 37.5%). Duration changes become positive (indicating no benefit to foresight) with higher concurrence and less rework, effectively shrinking the project conditions under which foresight is beneficial. This suggests that if backlog size and control costs are positively correlated (e.g., project management staff increases with larger backlogs and the resulting predictions of longer duration) then projects may require less

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in columns 4 and 5. The biases, or distances (in allocation fractions) that the average allocations with a directly proportional policy differ from the optimal allocations, are shown in columns 6 and 7. With a directly proportional policy increasing complexity shifts resources away from completion and toward the rework loop (quality assurance and iteration), as shown by the directions of the changes from columns 2 to 3. Foresight has the same effect, as shown by the directions of the changes from columns 2 to 4 for a simple project and from columns 3 to 5 for a complex project. But columns 6 and 7 are not zero, indicating that increasing complexity shifts resources toward (but not completely to) the optimal allocations. Notice that the size of the gap between the allocations with a directly proportional policy and the optimal allocations decreases with increasing complexity, as shown by the reduced bias from completion to the rework loop (26.82% < 34.04%). The structure of the development processes (Fig. 2) provides an explanation for this reduction in the marginal improvement in schedules with increasing complexity. Complexity increases the fraction of total effort needed away from initial completion and toward the rework loop by increasing the fraction of work ‘‘trapped’’ in the closed flow of the rework loop. This increases the size of the quality assurance and iteration backlogs, thereby increasing the resource allocations to those activities. Foresight shifts resources toward the rework loop by taking the future of the project into account and shifting resources to accommodate future quality assurance and iteration backlogs. Therefore, increasing project complexity and rework have similar effects on resource allocation, more complex projects without foresight are closer

rework and more concurrence to benefit from foresight. Scenario II yields similar results with maximum benefit of foresight shown to be 15% for a fully concurrent project with 50% rework.

7. Discussion Consistent with the system dynamics tradition (Forrester, 1961; Sterman, 2000), our goal is neither to show how optimisation can improve existing modelling practice nor to present the best optimisation techniques for a set of resource allocation problems. Instead, we use the optimal results as a benchmark to discuss how conventional managerial intuition might be informed through policy changes and some related issues. We first explain the counter intuitive results presented in Tables 1 and 2 and further described in Fig. 7. We then discuss one challenge to successfully implementing foresighted resource allocation policies to reduce project duration, the ‘‘yanking’’ of labour. We use this discussion to identify some of the limitations of our control theory model and suggest opportunities for over coming these limitations by developing more sophisticated system dynamics models. The counterintuitive decrease in foresight benefit with increasing complexity can be explained by comparing the impacts of complexity and foresight on resource allocation. Table 3 compares the project average allocation fraction for each of the three development activities for a simple and complex project with and without foresight. The average fractions using a directly proportional policy are shown in columns 2 and 3. The foresighted allocations are described by the fractions Table 3 Average allocation fractions Development activity

Average fraction: directly proportional policy

Average fraction: foresighted policy

Bias of allocation: foresight over proportional policy

p ¼ 25%

p ¼ 62:5%

p ¼ 25%

p ¼ 62:5%

p ¼ 25%

p ¼ 62:5%

Completion Quality assurance Iteration

0.54 0.25 0.21

0.36 0.29 0.35

0.20 0.41 0.39

0.09 0.37 0.54

)34.04% 15.9% 18.14%

)26.82% 7.1% 19.72%

Total

1.00

1.00

1.00

1.00

0%

0%


83

Table 4 Yanking for fully concurrent complex project (p ¼ 62:5%) Development activity Initial completion Quality assurance Iteration

Amount of yanking

% Reduction

Directly proportional policy

Foresighted policy

0.93 0.41 0.53

0.20 0.04 0.16

to optimal allocations than simple projects, and therefore more complex projects are less able to take advantage of foresighted policies. The Resource Allocation Policy Matrices also help explain the degradation of schedule performance under partially concurrent conditions. As modelled here, foresighted policies cannot account for work that becomes available for initial completion only after the beginning of the project. The policy therefore is based on the work available at the project start. For the concurrence modelled here, and many experienced in practice (see Ford and Sterman, 1998a) this represents only a small fraction of the total scope. This is a limitation of our control theory model formulation. We speculate that partial concurrence, increases in work available but not completed (Wc ) through exogenous task creation, or both are best modelled with a Kalman-Busy type ‘‘look ahead’’ controller that dynamically predicts work loads. This could effectively generate coefficients in our Resource Allocation Policy Matrix that evolve dynamically during projects. Such an analysis lends itself to system dynamics modelling (Anderson, 2001), but the implementation lies beyond the scope of this paper. The Resource Allocation Policy Matrix and our combined use of control theory and system dynamics models can provide insight into other important resource management issues. One example is ‘‘yanking’’ in which workers are moved frequently from one activity to another. Workers involved in product development settings dislike frequent reassignment, which results in a feeling of being ‘‘yanked around’’ across activities or being given different work priorities. The switching of assignments across activities also affects organisational learning and thus work force productivity.

78 90 70

We measured yanking with the cumulative absolute value of changes in the resource allocation fraction for each development activity. As shown in Table 4, the foresighted policy reduces yanking dramatically, by 78%, 90%, and 70% for the completion, quality assurance, and iteration activities, respectively. This suggests that, in addition to potentially reducing duration, foresighted policies provide an added benefit of less yanking, which can improve labour productivity. But reducing yanking with foresight may not occur even though it would improve project performance. Although managers may want to reduce yanking, they are also pressured by their bosses to allocate resources based on the existing and very visible backlogs of unfinished work. Because the sizes of future backlogs (especially rework) that any given set of development processes, project characteristics, and managerial policies will create is difficult to forecast a priori, managers may tend to shy away from foresighted policies or keep their policy design processes tacit and intuitive.

8. Conclusions In this work we integrate a traditional control theory based derivation of optimal resource allocation and a system dynamics model. We use the control theory model to derive an optimal allocation policy, which we describe with a Resource Allocation Policy Matrix. The matrix is useful in explaining differences in project performance and developing an intuitive understanding of the characteristics and impacts of different allocation policies. Resource allocation policies were then used in a system dynamics model of the system to test performance. Our results show that and how

84


foresighted policies can improve schedule performance (i.e., reduce cycle time), without increasing the total amount of resources. While preliminary, our results could have far reaching impacts on resource management through allocation policies because the schedule improvements are essentially free, requiring no additional resources. However, not all projects benefit from foresight. Gains in schedule performance increase with project concurrence, and counter-intuitively, generally decrease with increases in project complexity. The use of foresight in some project conditions (e.g., complex sequential development) can degrade schedule performance. These results suggest that an improved understanding and heuristics for the design of foresighted resource allocation policies are needed to use foresight to improve product development project schedule performance. Our choice of model structure and parameters is dictated by our desire to explore efficacy of foresighted (i.e., look ahead) policies over proportional policies during a product development project in the presence of rework and concurrency. We have built parsimonious models that provide managerial insights around these efficacy issues. This approach brings with it several limiting assumptions with regards to the control rules, quadratic cost structure, time needed to adjust the deployment of resources, constant probability of rework, a priori knowledge of concurrence relationship etc. Improved models that relax assumptions made here are required to fully develop and test foresighted resource allocation policies. These models can include dynamic Resource Allocation Matrices, more complex control structures, more realistic cost functions, improved partial concurrence structures, and other impacts of resource allocation and allocation control efforts. Through such work the combined application of control theory and system dynamics can improve product development project planning and management.

stocks and values for constants used in the example in text. The default values for concurrence correspond to low partial concurrence. A.1. Stocks The stocks represented by the variables on the left-hand side of Eq. (A.2)–(A.5) are the same as those described graphically in Fig. 2 with boxes. They represent the sizes of the backlogs of work that are available for initial completion or being completed (Wc ), available for quality assurance or being tested (Wqa ), available for iteration or being iterated (Wit ), and approved (Wa ). The stock in Eq. (A.1) represents the backlog of work that must be initially completed but is not yet available to the initial completion activity due to the projectÕs concurrence constraint. The sizes of the stocks change due to the rates of the development activities, as represented by the variables on the right hand sides of Eqs. (A.1)–(A.5). These can be viewed as the actions of developers that withdraw work from their upstream backlog, complete their development activity, and deposit the work in a downstream stock, which is also a backlog of another development activity (except Wa ). For example in Eq. (A.3) the activity of testing that discovers a required change ðdÞ takes work out of the quality assurance backlog (Wqa ) and in Eq. (A.4) the same activity deposits that work into the iteration backlog (Wit ). ðd=dtÞWna ¼ r;

ðA:1Þ

ðd=dtÞWc ¼ r c;

ðA:2Þ

ðd=dtÞWqa ¼ c þ it d a;

ðA:3Þ

ðd=dtÞWit ¼ d it;

ðA:4Þ

ðd=dtÞWa ¼ a;

ðA:5Þ

where Wna

Appendix A. System dynamics model equations Wc Units follow parameter descriptions in square brackets (i.e., [ ]) along with initial values of

Wa

work not available for initial completion [S, work packages] work needing initial completion [0, work packages] work approved [0, work packages]


Wqa Wit r c it d a S

work needing quality assurance [0, work packages] work known to need iteration (changes) [0, work packages] release work for initial completion [work packages/day] initial completion rate [work packages/ day] iteration rate [work packages/day] discovery of need for iteration rate [work packages/day] approval rate [work packages/day] scope of work [100 work packages]

fa;i;t ¼ fa;i;t1 þ ðft;i;t fi;a;t1 Þ=sr ;

85

ðA:10Þ

r ¼ MaxðWna ; ðwc SÞ ðS ðWa þ Wqa þ Wit ÞÞÞ=sna ; wc ¼ oðWa =SÞ;

ðA:11Þ ðA:12Þ

where for activity i and i 2 fc; qa; itg: Pi wi si

A.2. Flows and auxiliaries

sna

The flows represented by the variables on the left hand side of Eqs. (A.6)–(A.9) are the rates at which development work is initially completed ðPc Þ, tested ðPqa Þ, discovered to require a change ðdÞ or approved ðaÞ, and iterated ðPit Þ. Resource allocation influences these rates by constraining completion, quality assurance, and iteration rates to less than those allowed by uncapacitated processes (A.6). Work that is tested (qa) is split between the work discovered to require rework ðdÞ and the work approved ðaÞ in Eqs. (A.7) and (A.8) based on a probability that reflects project complexity (p). Resource constrained progress ðRi Þ is based on the total resources, productivity, and allocation fraction, as determined by from the Resource Allocation Matrix (Eq. (A.9)). Consistent with practice, the fraction applied is delayed (A.10). Concurrence constrains progress by limiting the release of work to the difference between the total work available (Eq. (A.12) and below) and the work already released to the completion backlog (Eq. (A.11)).

sr

Ri R0i RT fa;i;t fa;i;t1 ft;i;t

ui p

progress of activity i; so c ¼ Pc , qa ¼ Pqa , and it ¼ Pit [work packages/day] work available for completion by activity i [work packages] time required to perform activity i on a work package [1, 2, 1 days] time required to release a work package for initial completion [time step days] time required to adjust resources from target resource allocation fraction to applied resource allocation fraction [35 days] rate of activity i allowed by resources [work packages/day] maximum rate of activity i allowed by resources [2000 work packages/day] total quantity of resources [2000 persons] fraction of total resources applied to activity i at time t [dimensionless] fraction of total resources applied to activity i at time t 1 [dimensionless] targeted fraction of total resources to activity i at time t as generated by the Resource Allocation Matrix [dimensionless] productivity of resource applied to activity i [1 work package/person-day] probability discovering work requiring iteration [dimensionless]

Pi ¼ ðwi =si Þ ðRi =R0i Þ;

ðA:6Þ

A.3. Concurrence relationships (@)

d ¼ qa p;

ðA:7Þ

a ¼ qa ð1 pÞ;

ðA:8Þ

Ri ¼ RT fa;i ui ;

ðA:9Þ

Under fully concurrent condition, Eq. (A.9) releases all work from Wna to Wc at the beginning of the project, making all work available for initial completion. Therefore with full concurrency:

86


wc ¼ 1:

ðA:13aÞ

Under partial concurrence the value of Wc =S increases from an initial value (C1) to 100% as the project progresses, making increasing amounts of the scope available for initial completion. Eq. (A.13b) approximates a typical ‘‘S’’ shaped concurrence graph (see Ford and Sterman, 1998b) with three lines with the following coordinates on a scope approved (0–100%) versus concurrence (0– 1.0) graph: (0, C1)-(alpha, C2), (alpha, C2)-(beta, C3), and (beta, C3)-(1.00, 1.00) wc ¼ IF ðWa =SÞ < alpha THEN ðC1þðC2C1Þ Wa =S alphaÞ ELSE ðIF ððWa =SÞ < betaÞ THEN ðC2þðC3C2Þ ððWa =SÞalphaÞ=ðbetaalphaÞÞ ELSE ðC3þð1C3Þ ððWa =SÞbetaÞ=ð1betaÞÞÞ; ðA:13bÞ

where alpha

beta

C1 C2 C3

Fraction of scope approved at which the rate of concurrence with respect to the fraction approved increases [20%] Fraction of scope approved at which the rate of concurrence with respect to the fraction approved decreases [90%] Minimum concurrence (at start of project) [5%] Concurrence at progress described by alpha [25%] Concurrence at progress described by beta [95%]

Concurrence was adjusted by increasing the values of C1, C2, and C3 as follows:

See Ford and Sterman (1998a) for the method used to quantify concurrence.

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