Product development resource allocation with foresight q


 Loren Lane
 2 years ago
 Views:
Transcription
1 European Journal of Operational Research 160 (2005) Product development resource allocation with foresight q Nitin R. Joglekar a, David N. Ford b, * a Boston University School of Management, 595 Commonwealth Avenue, Boston, MA 02215, USA b Department of Civil Engineering, Texas A&M University, College Station, TX , 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 q Both authors contributed equally to this article. * Corresponding author. Tel.: ; fax: (N.R. Joglekar), tel.: ; fax: (D.N. Ford). addresses: (N.R. Joglekar), (D.N. Ford). 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 identified 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 /$  see front matter Ó 2003 Elsevier B.V. All rights reserved. doi: /j.ejor
2 N.R. Joglekar, D.N. Ford / European Journal of Operational Research 160 (2005) 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; AbdelHamid 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. 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
3 74 N.R. Joglekar, D.N. Ford / European Journal of Operational Research 160 (2005) 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 development 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 reentrant 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 lookahead 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
4 N.R. Joglekar, D.N. Ford / European Journal of Operational Research 160 (2005) product development various tasks traverse between different subgroups 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 longterm 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 (W c ), work ready for quality assurance after completion (W qa ), and work ready for iteration (W it ) as critical elements in managing product development rework cycles. In our model 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 ðf i Þ for any given set of backlog conditions are specified by summing for each process the products of each allocation weight ðc B;i Þ and the appropriate backlog size ðw B Þ and then proportioning the sums so their total is 100%: f i ¼ X B, X ðw B C B;i Þ i! X ðw B C B;i Þ ; for B and i 2fcompletion; quality assurance; iterationg where f i is the fraction of total resources applied to activity i, W B is the work backlog of process B, C B;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 nondiagonal terms specify that only the backlog reduced by B Fig. 1. A Resource Allocation Policy Matrix describing a directly proportional policy.
5 76 N.R. Joglekar, D.N. Ford / European Journal of Operational Research 160 (2005) 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 nonzero 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 decisionmaking 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 (AbdelHamid 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 (W qa ). Work that passes quality assurance is approved and adds to the stock of approved work (W a ). The fraction of the work that is discovered to require changes moves into a backlog of tasks requiring iteration (W it ). 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.
6 N.R. Joglekar, D.N. Ford / European Journal of Operational Research 160 (2005) p(disc Iteration) Wit Iteration Discovery rate Quality Assurance rate Quality Assurance resource effect Quality Assurance process time Iteration process time Iteration rate Wqa Approval & Release rate Wa Iteration resource effect Wc Completion rate 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 {W c ðtþ, W qa ðtþ, W it 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.
7 78 N.R. Joglekar, D.N. Ford / European Journal of Operational Research 160 (2005) Work Backlog (Wb) Wc Wqa Wit Completion Rate (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 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 ðs i Þ. The off diagonal terms represent the rework and approvals rates. 2 The variables s c, s qa, and s it 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 fullstate 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 ðtþþbuðtþ: 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 u 0 Ru þ 2X 0 Nugdt: 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 A 0 S ðsb þ NÞR 1 ðb 0 S þ N 0 ÞþQ ¼ 0; where F ¼ R 1 B 0 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
8 N.R. Joglekar, D.N. Ford / European Journal of Operational Research 160 (2005) Development Work Backlog Weight Net Allocation Fraction With Activity Wc Wqa Wit Uniform Backlogs Completion Quality Assurance Iteration Total 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 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 W c, W qa and W it 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 in Fig. 1) and toward quality assurance (0.409 versus the average row sum of 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 dynamics 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 (W qa 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
9 80 N.R. Joglekar, D.N. Ford / European Journal of Operational Research 160 (2005) Quality 2 2 Assurance 15 Backlog Time (days) Simple Project work packages Complex Project 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 (W c ) 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 (W na, work to be completed but not available) is added. The outflow from this stock feeds into the stock of work to be completed (W c ). Partial concurrence is described by starting with all the work in W na and controlling its outflow based on the amount of work approved (W a ) 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 W a, nor W na 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 Resource Allocation Simple project (p ¼ 25%) Complex project (p ¼ 62:5%) Policy Duration (days) Reduction Duration (days) Reduction With foresight % % Directly proportional Table 2 Project duration with partial concurrence Resource Allocation Simple project (p ¼ 25%) Complex project (p ¼ 62:5%) Policy Duration (days) Reduction Duration (days) Reduction With foresight )0.92% )15.62% Directly proportional
10 N.R. Joglekar, D.N. Ford / European Journal of Operational Research 160 (2005) % Change in Duration 30 0 Foresight Increases Duration Foresight Reduces Duration Rework Concurrence 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
11 82 N.R. Joglekar, D.N. Ford / European Journal of Operational Research 160 (2005) 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 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 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 )34.04% )26.82% Quality assurance % 7.1% Iteration % 19.72% Total % 0%
12 N.R. Joglekar, D.N. Ford / European Journal of Operational Research 160 (2005) Table 4 Yanking for fully concurrent complex project (p ¼ 62:5%) Development activity Amount of yanking % Reduction Directly proportional policy Foresighted policy Initial completion Quality assurance Iteration 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 (W c ) through exogenous task creation, or both are best modelled with a KalmanBusy 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. 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
13 84 N.R. Joglekar, D.N. Ford / European Journal of Operational Research 160 (2005) 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 counterintuitively, 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 lefthand 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 (W c ), available for quality assurance or being tested (W qa ), available for iteration or being iterated (W it ), and approved (W a ). 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 W a ). For example in Eq. (A.3) the activity of testing that discovers a required change ðdþ takes work out of the quality assurance backlog (W qa ) and in Eq. (A.4) the same activity deposits that work into the iteration backlog (W it ). ðd=dtþw na ¼ r; ða:1þ ðd=dtþw c ¼ r c; ðd=dtþw qa ¼ c þ it d a; ðd=dtþw it ¼ d it; ðd=dtþw a ¼ a; where ða:2þ ða:3þ ða:4þ ða:5þ AppendixA. System dynamics model equations Units follow parameter descriptions in square brackets (i.e., [ ]) along with initial values of W na W c W a work not available for initial completion [S, work packages] work needing initial completion [0, work packages] work approved [0, work packages]
14 N.R. Joglekar, D.N. Ford / European Journal of Operational Research 160 (2005) W qa W it 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] A.2. Flows and auxiliaries 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 ðp c Þ, tested ðp qa Þ, discovered to require a change ðdþ or approved ðaþ, and iterated ðp it 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 ðr i 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)). f a;i;t ¼ f a;i;t 1 þðf t;i;t f i;a;t 1 Þ=s r ; r ¼ MaxðW na ; ðw c SÞ ðs ðw a þ W qa þ W it ÞÞÞ=s na ; wc ¼ oðw a =SÞ; where for activity i and i 2fc; qa; itg: ða:10þ ða:11þ ða:12þ P i progress of activity i; soc ¼ P c,qa¼ P qa, and it ¼ P it [work packages/day] w i work available for completion by activity i [work packages] s i time required to perform activity i on a work package [1, 2, 1 days] s na time required to release a work package for initial completion [time step days] s r time required to adjust resources from target resource allocation fraction to applied resource allocation fraction [35 days] R i rate of activity i allowed by resources [work packages/day] R 0 i maximum rate of activity i allowed by resources [2000 work packages/day] R T total quantity of resources [2000 persons] f a;i;t fraction of total resources applied to activity i at time t [dimensionless] f a;i;t 1 fraction of total resources applied to activity i at time t 1 [dimensionless] f t;i;t targeted fraction of total resources to activity i at time t as generated by the Resource Allocation Matrix [dimensionless] u i productivity of resource applied to activity i [1 work package/personday] p probability discovering work requiring iteration [dimensionless] P i ¼ðw i =s i ÞðR i =R 0 i Þ; d ¼ qa p; a ¼ qa ð1 pþ; R i ¼ R T f a;i u i ; ða:6þ ða:7þ ða:8þ ða:9þ A.3. Concurrence relationships Under fully concurrent condition, Eq. (A.9) releases all work from W na to W c at the beginning of the project, making all work available for initial completion. Therefore with full concurrency:
15 86 N.R. Joglekar, D.N. Ford / European Journal of Operational Research 160 (2005) w c ¼ 1: ða:13aþ Under partial concurrence the value of W c =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) w c ¼IF ðw a =SÞ<alpha THEN ðc1þðc2 C1Þ W a =S alphaþ ELSE ðif ððw a =SÞ<betaÞ THEN ðc2þðc3 C2Þ ððw a =SÞ alphaþ=ðbeta alphaþþ ELSE ðc3þð1 C3Þ ððw a =SÞ betaþ=ð1 betaþþþ; ða:13bþ where alpha Fraction of scope approved at which the rate of concurrence with respect to the fraction approved increases [20%] beta Fraction of scope approved at which the rate of concurrence with respect to the fraction approved decreases [90%] C1 Minimum concurrence (at start of project) [5%] C2 Concurrence at progress described by alpha [25%] C3 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. References AbdelHamid, T., Madnick, S., Software Project Dynamics, An Integrated Approach. PrenticeHall, Englewood Cliffs, NJ. Anderson, E.G., The nonstationary staff planning problem with business cycle and learning effects. Management Science 47 (6), Backhouse, C.J., Brookes, N.J., Concurrent Engineering, WhatÕs Working Where. The Design Council. Gower, Brookfield, VT. Cooper, K.G., Naval ship production: A claim settled and a framework built. Interfaces 10 (6), Cooper, K.G., The rework cycle: Benchmarks for the project manager. Project Management Journal 24 (1), Cooper, K.G., The $2,000 hour: How managers influence project performance through the rework cycle. Project Management Journal 25 (1), Cooper, K.G., Mullen, T.W., Swords and Plowshares: The rework cycle of defense and commercial software development projects. American Programmer 6 (5), Eppinger, S.D., Whitney, D.E., Smith, R.P., Gebala, D.A., A modelbased method for organising tasks in product development. Research in Engineering Design 6 (1), Ford, D., Achieving multiple project objectives through contingency management. ASCE Journal of Construction Engineering and Management 128 (1), Ford, D., Sterman, J., 1998a. Expert knowledge elicitation for improving formal and mental models. System Dynamics Review 14 (4), Ford, D., Sterman, J., 1998b. Modeling dynamic development processes. System Dynamics Review 14 (1), Ford, D., Sterman, J., 2003a. Iteration management for reduced cycle time in concurrent development projects, Concurrent Engineering Research and Applications, in press. Ford, D., Sterman, J., 2003b. The LiarÕs Club: Impacts of concealment in concurrent development projects, Concurrent Engineering Research and Applications, in press. Forrester, J.W., Industrial Dynamics. Productivity Press, Cambridge, MA. Concurrence 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% C C C
16 N.R. Joglekar, D.N. Ford / European Journal of Operational Research 160 (2005) Gaimon, C., Planning information technology knowledge worker systems. Management Science 43 (9), Graham, A.K., Beyond PM 101: Lessons for managing large development programs. Project Management Journal 31 (4), Graves, S.B., The timecost trade off in research and development: A review. Engineering Costs and Production Economics 16, 1 9. Graves, S.C., A tactical planning model for a job shop. Operations Research 34, Holstein, W.K., Berry, W.L., The labour assignment decisions: An application of work flow structure information. Management Science 18 (7), Holt, C.C., Modigliani, F., Simon, H.A., A linear decision rule for production and employment scheduling. Management Science 2 (1), Joglekar, N.R., Whitney, D.E., Automation Usage Pattern during Complex Electro Mechanical Product Development, MIT Center for Technology Policy and Industrial Development report, prepared under contract number F C4429 from the USAir Force Research Laboratory (USAFRL), Cambridge, MA. Joglekar, N.R., Yassine, A., Eppinger, S.D., Whitney, D.E., Performance of coupled product development activities with a deadline. Management Science 47 (12), Jones, A., Rabelo, L.C., Survey of job shop scheduling techniques, NISTIR National Institute of Standards and Technology, Gaithersburg, MD. Jones, C.H., An economic evaluation of job shop dispatching rules. Management Science 20 (3), Krishnan, V., Managing the simultaneous execution of coupled phases in concurrent product development. IEEE Transactions on Engineering Management 43 (2), Krishnan, V., Eppinger, S.D., Whitney, D.E., A modelbased framework to overlap product development activities. Management Science 43, McDaniel, C.D., A Linear Systems Framework for Analyzing the Automotive Appearance Design Process, Unpublished MasterÕs Thesis (Mgmt./EE), MIT Cambridge, MA. McNulty, R., Reducing DOD Product Development Cycle Time: The Role of Schedule in Development Process, Unpublished Doctoral Dissertation (TMP/Aero), MIT Cambridge, MA. Meyer, C., Fast Cycle Time, How to Align Purpose, Strategy, and Structure for Speed. The Free Press, New York. Moffatt, L.K., Tools and teams: Competing models of integrated product development project performance. Journal of Engineering Technology and Management 15, Patterson, M.L., Accelerating Innovation, Improving the Process of Product Development. Van Nostrand Reinhold, New York. Rodrigues, A., Williams, T.M., System dynamics in project management: Assessing the impacts of client behavior on project performance. Journal of the Operational Research Society 49, Roemer, T.A., Ahmadi, R.H., Wang, R.H., Timecost trade off in overlapped product development. Operations Research 48 (6), Rosenthal, S.R., Effective Product Design and Development. Business One Irwin, Homewood, IL. Smith, R.P., Eppinger, S.D., 1997a. A predictive model of sequential iteration in engineering design. Management Science 43 (8), Smith, R.P., Eppinger, S.D., 1997b. Identifying controlling features of engineering design iteration. Management Science 43 (3), Sterman, J.S., Business Dynamics: Systems Thinking and Modeling for a Complex World. Irwin McGrawHill, New York. Wheelwright, S., Clark, K., Revolutionizing Product Development. Free Press, New York.
Product Development Resource Allocation with Foresight
Product Development Resource Allocation with Foresight Nitin R. Joglekar and David N. Ford Contact author: David N. Ford, Department of Civil Engineering, Texas A&M University, College Station, Tx 778433136,
More informationEffects of Resource Allocation Policies on Project Durations
Effects of Resource Allocation Policies on Project Durations Zee Woon Lee 1, David N. Ford 2, and Nitin Joglekar 3 200 International System Dynamics Conference July 2529, 200 Keble College, Oxford, England
More informationEffects of Resource Allocation Policies for Reducing Project Durations: A Systems Modelling Approach
Systems Research and Behavioral Science Syst. Res. 24, 551^566 (2007) Published online in Wiley InterScience (www.interscience.wiley.com).809 & Research Paper Effects of Resource Allocation Policies for
More informationProject Controls to Minimize Cost and Schedule Overruns: A Model, Research Agenda, and Initial Results
Project Controls to Minimize Cost and Schedule Overruns: A Model, Research Agenda, and Initial Results David N. Ford 1, James M. Lyneis 2, and Timothy R.B. Taylor 3 Abstract System dynamics has been successfully
More informationMaintenance performance improvement with System Dynamics:
Maintenance performance improvement with System Dynamics: A Corrective Maintenance showcase R.E.M. Deenen 1,2, C.E. van Daalen 1 and E.G.C. Koene 2 1 Delft University of Technology, Faculty of Technology,
More informationAnalysis of a Production/Inventory System with Multiple Retailers
Analysis of a Production/Inventory System with Multiple Retailers Ann M. Noblesse 1, Robert N. Boute 1,2, Marc R. Lambrecht 1, Benny Van Houdt 3 1 Research Center for Operations Management, University
More informationGeneral Framework for an Iterative Solution of Ax b. Jacobi s Method
2.6 Iterative Solutions of Linear Systems 143 2.6 Iterative Solutions of Linear Systems Consistent linear systems in real life are solved in one of two ways: by direct calculation (using a matrix factorization,
More information3.4. Solving Simultaneous Linear Equations. Introduction. Prerequisites. Learning Outcomes
Solving Simultaneous Linear Equations 3.4 Introduction Equations often arise in which there is more than one unknown quantity. When this is the case there will usually be more than one equation involved.
More informationThe application of linear programming to management accounting
The application of linear programming to management accounting Solutions to Chapter 26 questions Question 26.16 (a) M F Contribution per unit 96 110 Litres of material P required 8 10 Contribution per
More informationIntegrate Optimise Sustain
Prepared by James Horton  james.horton@  +61 8 9324 8400 August 2012 Overview The intent of this paper is to describe how rail automation can be integrated with logistics scheduling. It provides general
More information8.3. Solution by Gauss Elimination. Introduction. Prerequisites. Learning Outcomes
Solution by Gauss Elimination 8.3 Introduction Engineers often need to solve large systems of linear equations; for example in determining the forces in a large framework or finding currents in a complicated
More information8.3. Gauss elimination. Introduction. Prerequisites. Learning Outcomes
Gauss elimination 8.3 Introduction Engineers often need to solve large systems of linear equations; for example in determining the forces in a large framework or finding currents in a complicated electrical
More informationTEACHING AGGREGATE PLANNING IN AN OPERATIONS MANAGEMENT COURSE
TEACHING AGGREGATE PLANNING IN AN OPERATIONS MANAGEMENT COURSE Johnny C. Ho, Turner College of Business, Columbus State University, Columbus, GA 31907 David Ang, School of Business, Auburn University Montgomery,
More informationAlgebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 201213 school year.
This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra
More informationAlabama Course of Study Mathematics Algebra 2 with Trigonometry
A Correlation of Prentice Hall Algebra 2 to the Alabama Course of Study Mathematics THE COMPLEX NUMBER SYSTEM NUMBER AND QUANTITY Perform arithmetic operations with complex numbers. 1. Know there is a
More informationChapter 10: Network Flow Programming
Chapter 10: Network Flow Programming Linear programming, that amazingly useful technique, is about to resurface: many network problems are actually just special forms of linear programs! This includes,
More informationThe standards in this course continue the work of modeling situations and solving equations.
Algebra II Overview View unit yearlong overview here Building on the understanding of linear, quadratic and exponential functions from Algebra I, this course will extend function concepts to include polynomial,
More informationA Comparison of System Dynamics (SD) and Discrete Event Simulation (DES) Al Sweetser Overview.
A Comparison of System Dynamics (SD) and Discrete Event Simulation (DES) Al Sweetser Andersen Consultng 1600 K Street, N.W., Washington, DC 200062873 (202) 8628080 (voice), (202) 7854689 (fax) albert.sweetser@ac.com
More informationOverview of Math Standards
Algebra 2 Welcome to math curriculum design maps for Manhattan Ogden USD 383, striving to produce learners who are: Effective Communicators who clearly express ideas and effectively communicate with diverse
More informationDecisionmaking with the AHP: Why is the principal eigenvector necessary
European Journal of Operational Research 145 (2003) 85 91 Decision Aiding Decisionmaking with the AHP: Why is the principal eigenvector necessary Thomas L. Saaty * University of Pittsburgh, Pittsburgh,
More informationSimple Linear Regression Chapter 11
Simple Linear Regression Chapter 11 Rationale Frequently decisionmaking situations require modeling of relationships among business variables. For instance, the amount of sale of a product may be related
More informationOptimal Scheduling for Dependent Details Processing Using MS Excel Solver
BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 8, No 2 Sofia 2008 Optimal Scheduling for Dependent Details Processing Using MS Excel Solver Daniela Borissova Institute of
More information6.2 The Economics of Ideas. 6.1 Introduction. Growth and Ideas. Ideas. The Romer model divides the world into objects and ideas: Chapter 6
The Romer model divides the world into objects and ideas: Chapter 6 and Ideas By Charles I. Jones Objects capital and labor from the Solow model Ideas items used in making objects Media Slides Created
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationMATHEMATICS FOR ENGINEERING BASIC ALGEBRA
MATHEMATICS FOR ENGINEERING BASIC ALGEBRA TUTORIAL 3 EQUATIONS This is the one of a series of basic tutorials in mathematics aimed at beginners or anyone wanting to refresh themselves on fundamentals.
More informationA Review of the Impact of Requirements on Software Project Development Using a Control Theoretic Model
J. Software Engineering & Applications, 2010, 3, 852857 doi:10.4236/jsea.2010.39099 Published Online September 2010 (http://www.scirp.org/journal/jsea) A Review of the Impact of Requirements on Software
More informationSystem dynamics applied to oversight of ongoing project: a case of Clinical Trials Programming
System dynamics applied to oversight of ongoing project: a case of Clinical Trials Programming Yosuke NAKAJIMA *1, Yutaka TAKAHASHI *2, Naohiko KOTAKE *3 Keio University Graduate School of Systems Design
More informationAutomated Scheduling Methods. Advanced Planning and Scheduling Techniques
Advanced Planning and Scheduling Techniques Table of Contents Introduction 3 The Basic Theories 3 Constrained and Unconstrained Planning 4 Forward, Backward, and other methods 5 Rules for Sequencing Tasks
More informationSensitivity Analysis 3.1 AN EXAMPLE FOR ANALYSIS
Sensitivity Analysis 3 We have already been introduced to sensitivity analysis in Chapter via the geometry of a simple example. We saw that the values of the decision variables and those of the slack and
More informationMarketing Mix Modelling and Big Data P. M Cain
1) Introduction Marketing Mix Modelling and Big Data P. M Cain Big data is generally defined in terms of the volume and variety of structured and unstructured information. Whereas structured data is stored
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationECRP No AN INDIRECT DERIVATION OF INPUTOUTPUT COEFFTnTFNT.q MAKERERE INSTITUTE OF SOCIAL RESEARCH
ECRP No. 163 AN INDIRECT DERIVATION OF INPUTOUTPUT COEFFTnTFNT.q B. Jensen MAKERERE INSTITUTE OF SOCIAL RESEARCH These papers are prepared as a basis for Seminar discussion. They are not publications
More informationA formulation for measuring the bullwhip effect within spreadsheets
6 th International Conference on Industrial Engineering and Industrial Management. XVI Congreso de Ingeniería de Organización. Vigo, July 180, 01 A formulation for measuring the bullwhip effect within
More information14TH INTERNATIONAL CONFERENCE ON ENGINEERING DESIGN 1921 AUGUST 2003
14TH INTERNATIONAL CONFERENCE ON ENGINEERING DESIGN 1921 AUGUST 2003 A CASE STUDY OF THE IMPACTS OF PRELIMINARY DESIGN DATA EXCHANGE ON NETWORKED PRODUCT DEVELOPMENT PROJECT CONTROLLABILITY Jukka Borgman,
More informationOPTIMAL DESIGN OF A MULTITIER REWARD SCHEME. Amir Gandomi *, Saeed Zolfaghari **
OPTIMAL DESIGN OF A MULTITIER REWARD SCHEME Amir Gandomi *, Saeed Zolfaghari ** Department of Mechanical and Industrial Engineering, Ryerson University, Toronto, Ontario * Tel.: + 46 979 5000x7702, Email:
More informationMAT Solving Linear Systems Using Matrices and Row Operations
MAT 171 8.5 Solving Linear Systems Using Matrices and Row Operations A. Introduction to Matrices Identifying the Size and Entries of a Matrix B. The Augmented Matrix of a System of Equations Forming Augmented
More informationChapter 15 Introduction to Linear Programming
Chapter 15 Introduction to Linear Programming An Introduction to Optimization Spring, 2014 WeiTa Chu 1 Brief History of Linear Programming The goal of linear programming is to determine the values of
More informationEgo network betweenness
Social Networks 27 (2005) 31 38 Ego network betweenness Martin Everett a,, Stephen P. Borgatti b a University of Westminster, 35 Marylebone Road, London NW1 5LS, UK b Department of Organization Studies,
More informationUNIT 2 MATRICES  I 2.0 INTRODUCTION. Structure
UNIT 2 MATRICES  I Matrices  I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress
More informationVALUE STREAM MAPPING FOR SOFTWARE DEVELOPMENT PROCESS. Ganesh S Thummala. A Research Paper. Submitted in Partial Fulfillment of the
VALUE STREAM MAPPING FOR SOFTWARE DEVELOPMENT PROCESS by Ganesh S Thummala A Research Paper Submitted in Partial Fulfillment of the Requirements for the Master of Science Degree In Management Technology
More informationAlgebra 2 Chapter 1 Vocabulary. identity  A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity  A statement that equates two equivalent expressions. verbal model A word equation that represents a reallife problem. algebraic expression  An expression with variables.
More information5STEPS. Patrick Weaver PMP, FAICD. Brian Doyle MAIPM. Originally Presented 1995, Updated and Revised
Project Services Pty Ltd 5STEPS ( 5 Steps To Ensure Project Success ) By Patrick Weaver PMP, FAICD. Brian Doyle MAIPM. Originally Presented 1995, Updated and Revised For more scheduling papers see: www.mosaicprojects.com.au/resources_papers.html#scheduling
More informationSUPPLEMENT TO CHAPTER
SUPPLEMENT TO CHAPTER 6 Linear Programming SUPPLEMENT OUTLINE Introduction and Linear Programming Model, 2 Graphical Solution Method, 5 Computer Solutions, 14 Sensitivity Analysis, 17 Key Terms, 22 Solved
More informationChapter 4  Systems of Equations and Inequalities
Math 233  Spring 2009 Chapter 4  Systems of Equations and Inequalities 4.1 Solving Systems of equations in Two Variables Definition 1. A system of linear equations is two or more linear equations to
More informationCreating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities
Algebra 1, Quarter 2, Unit 2.1 Creating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned
More informationDefinition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.
8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent
More informationOperation Count; Numerical Linear Algebra
10 Operation Count; Numerical Linear Algebra 10.1 Introduction Many computations are limited simply by the sheer number of required additions, multiplications, or function evaluations. If floatingpoint
More informationMontana Common Core Standard
Algebra I Grade Level: 9, 10, 11, 12 Length: 1 Year Period(s) Per Day: 1 Credit: 1 Credit Requirement Fulfilled: A must pass course Course Description This course covers the real number system, solving
More informationObjectives. By the time the student is finished with this section of the workbook, he/she should be able
QUADRATIC FUNCTIONS Completing the Square..95 The Quadratic Formula....99 The Discriminant... 0 Equations in Quadratic Form.. 04 The Standard Form of a Parabola...06 Working with the Standard Form of a
More informationNonlinear Programming Methods.S2 Quadratic Programming
Nonlinear Programming Methods.S2 Quadratic Programming Operations Research Models and Methods Paul A. Jensen and Jonathan F. Bard A linearly constrained optimization problem with a quadratic objective
More information3.4. Solving simultaneous linear equations. Introduction. Prerequisites. Learning Outcomes
Solving simultaneous linear equations 3.4 Introduction Equations often arise in which there is more than one unknown quantity. When this is the case there will usually be more than one equation involved.
More informationFAST SENSITIVITY COMPUTATIONS FOR MONTE CARLO VALUATION OF PENSION FUNDS
FAST SENSITIVITY COMPUTATIONS FOR MONTE CARLO VALUATION OF PENSION FUNDS MARK JOSHI AND DAVID PITT Abstract. Sensitivity analysis, or socalled stresstesting, has long been part of the actuarial contribution
More informationLINEAR PROGRAMMING P V Ram B. Sc., ACA, ACMA Hyderabad
LINEAR PROGRAMMING P V Ram B. Sc., ACA, ACMA 98481 85073 Hyderabad Page 1 of 19 Question: Explain LPP. Answer: Linear programming is a mathematical technique for determining the optimal allocation of resources
More informationPortable Assisted Study Sequence ALGEBRA IIA
SCOPE This course is divided into two semesters of study (A & B) comprised of five units each. Each unit teaches concepts and strategies recommended for intermediate algebra students. The first half of
More information1 Algebra  Partial Fractions
1 Algebra  Partial Fractions Definition 1.1 A polynomial P (x) in x is a function that may be written in the form P (x) a n x n + a n 1 x n 1 +... + a 1 x + a 0. a n 0 and n is the degree of the polynomial,
More informationMath 018 Review Sheet v.3
Math 018 Review Sheet v.3 Tyrone Crisp Spring 007 1.1  Slopes and Equations of Lines Slopes: Find slopes of lines using the slope formula m y y 1 x x 1. Positive slope the line slopes up to the right.
More information19 LINEAR QUADRATIC REGULATOR
19 LINEAR QUADRATIC REGULATOR 19.1 Introduction The simple form of loopshaping in scalar systems does not extend directly to multivariable (MIMO) plants, which are characterized by transfer matrices instead
More informationIntroduction to Matrix Algebra
Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra  1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary
More informationCurrent Standard: Mathematical Concepts and Applications Shape, Space, and Measurement Primary
Shape, Space, and Measurement Primary A student shall apply concepts of shape, space, and measurement to solve problems involving two and threedimensional shapes by demonstrating an understanding of:
More informationThe Basics of FEA Procedure
CHAPTER 2 The Basics of FEA Procedure 2.1 Introduction This chapter discusses the spring element, especially for the purpose of introducing various concepts involved in use of the FEA technique. A spring
More informationA MANAGER S ROADMAP GUIDE FOR LATERAL TRANSSHIPMENT IN SUPPLY CHAIN INVENTORY MANAGEMENT
A MANAGER S ROADMAP GUIDE FOR LATERAL TRANSSHIPMENT IN SUPPLY CHAIN INVENTORY MANAGEMENT By implementing the proposed five decision rules for lateral transshipment decision support, professional inventory
More informationSchool of Engineering Department of Electrical and Computer Engineering
1 School of Engineering Department of Electrical and Computer Engineering 332:223 Principles of Electrical Engineering I Laboratory Experiment #4 Title: Operational Amplifiers 1 Introduction Objectives
More informationID Class MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Exam: Operations Management Name Total Scors: ID Class MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Which of the following incorporates the
More informationReused Product Pricing and Stocking Decisions in ClosedLoop Supply Chain
Reused Product Pricing and Stocking Decisions in ClosedLoop Supply Chain Yuanjie He * California State Polytechnic University, Pomona, USA Abolhassan Halati California State Polytechnic University, Pomona,
More informationIncreasing for all. Convex for all. ( ) Increasing for all (remember that the log function is only defined for ). ( ) Concave for all.
1. Differentiation The first derivative of a function measures by how much changes in reaction to an infinitesimal shift in its argument. The largest the derivative (in absolute value), the faster is evolving.
More informationTexas Christian University. Department of Economics. Working Paper Series. Modeling Interest Rate Parity: A System Dynamics Approach
Texas Christian University Department of Economics Working Paper Series Modeling Interest Rate Parity: A System Dynamics Approach John T. Harvey Department of Economics Texas Christian University Working
More informationmax cx s.t. Ax c where the matrix A, cost vector c and right hand side b are given and x is a vector of variables. For this example we have x
Linear Programming Linear programming refers to problems stated as maximization or minimization of a linear function subject to constraints that are linear equalities and inequalities. Although the study
More informationHigh School Algebra 1 Common Core Standards & Learning Targets
High School Algebra 1 Common Core Standards & Learning Targets Unit 1: Relationships between Quantities and Reasoning with Equations CCS Standards: Quantities NQ.1. Use units as a way to understand problems
More informationPowerTeaching i3: Algebra I Mathematics
PowerTeaching i3: Algebra I Mathematics Alignment to the Common Core State Standards for Mathematics Standards for Mathematical Practice and Standards for Mathematical Content for Algebra I Key Ideas and
More informationMincost flow problems and network simplex algorithm
Mincost flow problems and network simplex algorithm The particular structure of some LP problems can be sometimes used for the design of solution techniques more efficient than the simplex algorithm.
More informationBenefits of a critical chain a System Dynamics based study
Benefits of a critical chain a System Dynamics based study Abstract Number: 0020351 Second World Conference on POM and 15th Annual POM Conference Cancun, Mexico, April 30  May 3, 2004. Jan Jürging Industrieseminar,
More informationWarranty Designs and Brand Reputation Analysis in a Duopoly
Warranty Designs and Brand Reputation Analysis in a Duopoly Kunpeng Li * Sam Houston State University, Huntsville, TX, U.S.A. Qin Geng Kutztown University of Pennsylvania, Kutztown, PA, U.S.A. Bin Shao
More informationA linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form
Section 1.3 Matrix Products A linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form (scalar #1)(quantity #1) + (scalar #2)(quantity #2) +...
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationPA Common Core Standards Standards for Mathematical Practice Grade Level Emphasis*
Habits of Mind of a Productive Thinker Make sense of problems and persevere in solving them. Attend to precision. PA Common Core Standards The Pennsylvania Common Core Standards cannot be viewed and addressed
More informationSouth Carolina College and CareerReady (SCCCR) Algebra 1
South Carolina College and CareerReady (SCCCR) Algebra 1 South Carolina College and CareerReady Mathematical Process Standards The South Carolina College and CareerReady (SCCCR) Mathematical Process
More informationSTRATEGIC CAPACITY PLANNING USING STOCK CONTROL MODEL
Session 6. Applications of Mathematical Methods to Logistics and Business Proceedings of the 9th International Conference Reliability and Statistics in Transportation and Communication (RelStat 09), 21
More informationSolving Simultaneous Equations and Matrices
Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering
More informationJoint models for classification and comparison of mortality in different countries.
Joint models for classification and comparison of mortality in different countries. Viani D. Biatat 1 and Iain D. Currie 1 1 Department of Actuarial Mathematics and Statistics, and the Maxwell Institute
More informationSIMPLY stated, concurrent engineering (CE) is the incorporation
144 IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, VOL. 46, NO. 2, MAY 1999 A Decision Analytic Framework for Evaluating Concurrent Engineering Ali A. Yassine, Kenneth R. Chelst, and Donald R. Falkenburg
More informationNPTEL STRUCTURAL RELIABILITY
NPTEL Course On STRUCTURAL RELIABILITY Module # 02 Lecture 6 Course Format: Web Instructor: Dr. Arunasis Chakraborty Department of Civil Engineering Indian Institute of Technology Guwahati 6. Lecture 06:
More informationNonlinear Iterative Partial Least Squares Method
Numerical Methods for Determining Principal Component Analysis Abstract Factors Béchu, S., RichardPlouet, M., Fernandez, V., Walton, J., and Fairley, N. (2016) Developments in numerical treatments for
More information1111: Linear Algebra I
1111: Linear Algebra I Dr. Vladimir Dotsenko (Vlad) Lecture 3 Dr. Vladimir Dotsenko (Vlad) 1111: Linear Algebra I Lecture 3 1 / 12 Vector product and volumes Theorem. For three 3D vectors u, v, and w,
More informationLinear Programming. March 14, 2014
Linear Programming March 1, 01 Parts of this introduction to linear programming were adapted from Chapter 9 of Introduction to Algorithms, Second Edition, by Cormen, Leiserson, Rivest and Stein [1]. 1
More informationDynamic Change Management for Fasttracking Construction Projects
Dynamic Change Management for Fasttracking Construction Projects by Moonseo Park 1 ABSTRACT: Uncertainties make construction dynamic and unstable, mostly by creating non valueadding change iterations
More informationThe Solution of Linear Simultaneous Equations
Appendix A The Solution of Linear Simultaneous Equations Circuit analysis frequently involves the solution of linear simultaneous equations. Our purpose here is to review the use of determinants to solve
More informationOverview. Essential Questions. Precalculus, Quarter 4, Unit 4.5 Build Arithmetic and Geometric Sequences and Series
Sequences and Series Overview Number of instruction days: 4 6 (1 day = 53 minutes) Content to Be Learned Write arithmetic and geometric sequences both recursively and with an explicit formula, use them
More informationEigenvalues and eigenvectors of a matrix
Eigenvalues and eigenvectors of a matrix Definition: If A is an n n matrix and there exists a real number λ and a nonzero column vector V such that AV = λv then λ is called an eigenvalue of A and V is
More informationDynamic Modeling of Product Development Processes
D4672 2/27/97 Dynamic Modeling of Product Development Processes David N. Ford 1 and John D. Sterman 2 January 1997 The authors thank the Organizational Learning Center and the System Dynamics Group at
More informationExamples of Functions
Examples of Functions In this document is provided examples of a variety of functions. The purpose is to convince the beginning student that functions are something quite different than polynomial equations.
More informationA Matlab / Simulink Based Tool for Power Electronic Circuits
A Matlab / Simulink Based Tool for Power Electronic Circuits Abdulatif A M Shaban Abstract Transient simulation of power electronic circuits is of considerable interest to the designer The switching nature
More informationKirchhoff's Rules and Applying Them
[ Assignment View ] [ Eðlisfræði 2, vor 2007 26. DC Circuits Assignment is due at 2:00am on Wednesday, February 21, 2007 Credit for problems submitted late will decrease to 0% after the deadline has passed.
More informationChap 4 The Simplex Method
The Essence of the Simplex Method Recall the Wyndor problem Max Z = 3x 1 + 5x 2 S.T. x 1 4 2x 2 12 3x 1 + 2x 2 18 x 1, x 2 0 Chap 4 The Simplex Method 8 corner point solutions. 5 out of them are CPF solutions.
More informationDRAFT. Further mathematics. GCE AS and A level subject content
Further mathematics GCE AS and A level subject content July 2014 s Introduction Purpose Aims and objectives Subject content Structure Background knowledge Overarching themes Use of technology Detailed
More informationContent Emphases by ClusterKindergarten *
Content Emphases by ClusterKindergarten * Counting and Cardinality Know number names and the count sequence. Count to tell the number of objects. Compare numbers. Operations and Algebraic Thinking Understand
More informationThis unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.
Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course
More informationCommon Core State Standards. Standards for Mathematical Practices Progression through Grade Levels
Standard for Mathematical Practice 1: Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for
More informationLecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization
Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization 2.1. Introduction Suppose that an economic relationship can be described by a realvalued
More information6 Scalar, Stochastic, Discrete Dynamic Systems
47 6 Scalar, Stochastic, Discrete Dynamic Systems Consider modeling a population of sandhill cranes in year n by the firstorder, deterministic recurrence equation y(n + 1) = Ry(n) where R = 1 + r = 1
More informationMATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix.
MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix. Matrices Definition. An mbyn matrix is a rectangular array of numbers that has m rows and n columns: a 11
More information