Activity Resource Elasticity: A New Approach to Project Crashing

Transcription

1 Activity Resource Elasticity: A New Approach to Project Crashing Dr. Ronald S. Tibben-Lembke MGRS / 028 University of Nevada Reno, NV (775) Fax: (775) rtl@unr.edu Dr. Ted Mitchell MGRS / 028 University of Nevada Reno, NV (775) Fax: (775) mitchjt@unr.edu 1

2 Activity Resource Elasticity: A New Approach to Project Crashing August 21, 2007 Abstract Traditional project crashing methods assume that, within a given range, the time saved is a linear function of the additional resources spent. We assume the time saved is a linear function of the increase in resources used per day. As a result, we can compute how sensitive an activity s duration is to the resources made available, which we call the resource elasticity of the activity, the percentage reduction in time, divided by the percentage increase in resources. This elasticity measure is a practical, easily implemented metric that allows the manager to identify which activities duration times are more or less sensitive to daily spending rates. This metric can be used for determining which activities should be crashed, which should be slowed (if the money for crashing has to come from another activity), and for determining which activities need to be most closely monitored. Keywords: Project management, project crashing, heuristic methods 1 Introduction If a project must be sped up to meet a deadline, crashing the program or project is to speed up various activities to reduce the total time of the project, by allocating more resources to their completion. To shorten the critical path (those activities whose durations determine the total project length), it is necessary to increase the resources allocated to activities on it. This increases both the cost per day and the total cost of the activities. The duration of some activities may be 2

3 very sensitive to changes in the resources allocated to them (e.g., dollars-per-day), while others may be relatively insensitive. In the past, project managers have had metrics to help judge the total change in costs due to crashing a project They have not had a metric for judging the relative sensitivity of a duration change due to an increase in resources or cost-per-day. This paper presents resource elasticity, a metric which allows a manager to easily see which activities durations are most sensitive to increases or reductions in resources. This is useful in determining which activities are most elastic, and would be sped up the most, with increased resources, if the project needs to be crashed. Many project managers are likely faced with the reality of a finite amount of resources, whether those resources be money, programmers, engineers, welders, or something else. If one activity is to be sped up, it may result in another activity being slowed. Resource elasticity can be used to see which activities are least elastic, and the resources should be taken from them, because they will be slowed the least from the loss of resources. Finally, if an activity is very inelastic, should it fall behind schedule, a great deal of additional resources will be required to get it sped up, so these activities should be monitored closely. For example, a long project may consist of several large activities that are being executed concurrently: some of the design phase may be being finished on a software module while coding proceeds on another module. In order to help one part of the project meet a deadline, the manager may need to steal programmers from one activity to help the other. Using our results, the manager can easily determine which activities will be most affected by a loss of resources. 2 Literature Project management has received a lot of attention in the operations research literature, since the initial papers by Malcolm et al. (1959), Kelly and Walker (1959) and Kelly (1961). In their seminal paper, Kelly and Walker (1959) considered a time-cost tradeoff, with a linear time-cost relationship. They show a convex cost function, which has minimal cost at a particular duration, but as the duration increases beyond this point, costs again rise. However, as a practical matter, they are only interested in times equal to or lesser than this time, and can assume the cost function is piecewise linear and non-increasing in the time. Berman (1964) considered an upward-concave time-cost function for each activity, under certainty or uncertainty. Petrovic 3

4 (1968) approached the problem as a dynamic program, and Lamberson and Hocking (1970) applied the techniques of linear programming to the time-cost tradeoff. Elmaghraby (1977) presented methods for strictly convex and strictly concave time-cost functions. See Tavares (2002) and Herroelen (2005) for an overview of other contributions of operations research to project scheduling. Since the beginning, a great deal of research on project crashing has focused on linear programming methods, as the references in Kolisch (1995) and Hartmann (1999) indicate. LP and IP formulations assume the various time durations are known, and the cost of those durations (e.g. Boctor 1996, Hartmann 1999). Because linear programming methods can easily incorporate various resource constraints, these are the methods of preference for resource-constrained projects, as well. In the first paper on PERT, Malcolm et al. (1959) presented their well-known method for estimating the probability of completing a project at its scheduled date using the beta distribution. As Elmaghraby (1977) explains, shortcomings in the assumptions behind PERT has lead to other techniques for dealing with uncertainty, including generalized activity networks (GAN) and graphical evaluation and review technique (GERT). Most of the literature has been focused on methods for determining a schedule for the project, a priori. That is to say, given the range of times in which each activity could be completed, find the most cost-effective way to complete the project in the desired time frame. However, considerable attention has also been paid to methods for assessing and dealing with risk and uncertainty (Pich, et al. 2002). 3 Resource Elasticity The traditional time-crashing approach looks at the time-cost tradeoff from the position that completing an activity in less time will require additional resources, that is, cost is a function of time. We look at that same relationship from the opposite perspective that duration time is a function of the applied resource rate. That is to say, the higher the daily spending rate, the shorter the duration time. We approach the problem somewhat differently from much of the literature. We are interested in simple methods, which could quickly be applied by a project manager with mud on his boots. An IP formulation of the problem assumes that the decision maker has information regarding each of the possible levels of tradeoffs between time and cost for each activity. As Kelly and Walker 4

5 Activity A D C C/D Normal 200 $11,550 $57.75 Crashed 160 $12,800 $80.00 Difference 40 $1,250 $31.25 Table 1: Costs and Durations for Activities A and B. (1959, p. 164) observed, Usually in practice insufficient data is available to make more than a linear approximation. Many managers may not be able to create a full menu of all of the possible time and cost combinations, because calculating each combination requires a certain amount of time, and thus, comes at some cost. For a project manager in the field, collecting all the necessary information to formulate and solve the IP model is likely a luxury he cannot afford. 3.1 Difference of Crashing Assumptions Traditional project crashing assumes that an activity can normally be completed in D N days for a total cost C N, or crashed for a total cost C C, when it will be completed in D C days. For example, suppose activity A (summarized in Table 3.1) can be normally completed in D N = 200 days, for a cost of C N = $11,550, or crashed to D C = 160 days, for a cost of C C = $12,800. Thus, for an additional cost of C C C N = $1,250, the project can be completed D = D N D C = 40 days faster. The project manager must decide between spending an additional $1,250 to save 40 days, or $0 to save 0 days. Additionally, it is typically assumed that points in-between are possible because the time saved is linear in the amount of additional spending. Assumption 1 The time saved, D, by additional additional spending C C N, is linear in the amount of additional spending, for C N C C C. Thus, the amount of additional cost, per day to be saved, the crash cost per day (CCPD) can be computed: CCPD = C C C N D N D C. Given the CCPD for a number of activities, it would seem logical to crash first the activity on the critical path with the lowest CCPD, and when its crashing opportunities are exhausted, move to the next cheapest, etc., until the crash time reduction has been achieved, or the cost per day crashed exceeds the benefit per day. Many textbooks present some version of this. For example, 5

6 Crash critical activities, in order of increasing costs, as long as crashing costs do not exceed benefits (Stevenson 2002, p. 794). Under assumption 1, if total spending is C 1, the time saved is D = C 1 C N CCPD. Rearranging, under Assumption 1, to save D days, we may solve for C 1 as a function of D: C 1 ( D) = C N + D CCPD. So for activity A, if D = 10 days, C 1 (10) = $11, $31.25 = $11, In some cases, assuming that the time saved is linear in the amount of additional cost may be quite reasonable. For example, if an activity can be sped up by paying a group of welders to work overtime, paying half the amount of overtime should result in roughly half of the time savings. In this paper, we make a different assumption about linearity. For any project, the resource usage per day, R = C/D can be computed. For example, when crashed, activity A has a resource usage rate of R C = C C /D C = $12,800/160 = $80/day, and the normal usage rate is R N = C N /D N = $11,550/200 = $57.75/day. We assume that the time savings are linear in the resources used per day. Assumption 2 The time savings D, from crashing, are linear in the additional resources per day used, R R N, for R N R R C. Under assumption 2, the days saved, D, as a function of the resources spent per day, R, can be written: D 2 (R) = (R R N ) D N D C R C R N. (1) Alternatively, we may solve for the resources used per day, as a function of the days to be saved, R 2 ( D), under Assumption 2. R 2 ( D) = R N + D R C R N D N D C. (2) If D = 0, R 2 ( D) = R N. If D = D N D C, then R 2 ( D) = R C. For values of D in-between, R 2 (D) is linear in D. It is easily verified that R 2 ( D) D = R C R N D N D C. (3) Under Assumption 2, the relationship between additional resource usage and time saved is clearly linear. 6

7 Activity Duration D N D C Changing Duration Impact Resource Usage Impact R N R C Resources/Day Figure 1: Resource-Duration Combinations from Assumption 2 Under Assumption 1, the resources used per day, as a function of the days to be saved, R 1 ( D), are given by Taking the derivative with respect to D, it can be shown: R 1 ( D) = C N + D CCPD. (4) D N D R 1 ( D) D = D N CCPD + C N (D N D) 2. (5) Clearly, the resource usage per day is not linear in D under Assumption Similarity of Solutions Considered Figure 1 shows the combinations of R and D that would be available under Assumption 2. The vertical rectangle shows the total cost under normal conditions. With a width R N and height D N, it has an area of R N D N = C N. The horizontal rectangle has area of R C D C = C C. By assumption, C C > C N, and the areas of the rectangles reflects this. The straight line tangent to the two rectangles represents the crashing combinations available under Assumption 2, for resource usage rates between R N and R C. Although the traditional CCPD assumption does not result in a linear relationship between resources used per day and the days saved, as a practical matter, the relationship turns out to be nearly linear, as the following graphs demonstrate. 7

8 Project Length, D Resources / Day Figure 2: Resource-Duration Combinations from Assumption 1 Figure 2 represents the cost of crashing and normal completion. The rectangle with vertical striping represents the total cost for the regular project, because its width is R N = $57.75 per day, and its height is D N = 200 days, giving it area of R N D N = C N. The horizontal rectangle with horizontal striping represents the total cost for the crashed project, because its width is R C = $80 per day, and its height is D C = 160 days, giving it area of R C D C = C C > C N. The line connecting the upper-right corners of the two rectangles traces out the possible R, D combinations, under Assumption 1. Unfortunately, given the scale of the picture, it is difficult to see much of the tradeoff between crashing and normal operation. In Figure 3, the same points are shown, but the origin has been shifted, to better show detail. As a reference, the straight line of combinations under Assumption 2 has been drawn through the extreme points of crashing and normal. As it can be seen, although the CCPD line is not linear, it is very nearly linear. The conclusion we may draw from this is that although the two assumptions differ in their relationship between resources and project completion time, they end up considering a nearly identical set of crashing possibilities. 3.3 Elasticity as a Measure of Sensitivity Without crashing, an activity would be completed in time D N, using R N resources per day, for total cost of D N R N. With crashing, it is completed in D C, using R C resources per day, for total cost D C R C. We break the change in total costs into two factors: a savings due to faster completion, and an increase due to higher resource usage per day. 8

9 Project Length, D Resources / Day Figure 3: Assumption 1 Resource-Duration Combinations, Magnified If an activity is completed in less time, it could be argued there should be some savings, because fewer days of expense are incurred. However, more money is being spent per day, which likely more than offsets the savings from the shorter completion time. For example, if we could complete the activity in the crash time D C, but using normal resource levels, R N, the impact on total costs of changing the duration would be R N (D C D N ). However, for each of the D C days, the cost impact of increasing the resource usage rate is D C (R C R N ). Figure 1 shows these impacts graphically. For activity A, crashing increases costs by D C R C D N R N = $1,250. The impact from changing the duration is R N (D C D N ) = $57.75( ) = -$2,310. The impact from greater resource usage per day is D C (R C R N ) = 160($80 - $57.75) = $3,560, so the net impact of the change = $3,560 - $2,310 = $1,250, exactly the total cost difference. Because crashing is more expensive than the normal completion time, we know R N (D C D N ) < D C (R C R N ). If the crashing cost were the same as normal completion, the terms would be equal. If we have several activities to consider crashing, we would like to choose the activity with the least net impact, which will be the activity where the two terms are closest to being equal. We argue that taking the ratio of these two terms can tell us which activities should be crashed, and we define this ratio as E: E = Changing Duration Impact Resource Usage Impact = R N (D C D N ) D C (R C R N ). (6) This ratio is, in fact, the elasticity of completion time, with respect to resource usage. Because we have only two estimates of the relationship between resources and completion time, we must derive an arc elasticity term, instead of point elasticity. Arc elasticity has been discussed in the 9

10 economics literature by Sheldon (1986), following Gould and Ferguson (1980), Machlup (1952), and Lerner (1933). Following them, we may define: E = D/D R/R = R D D R = R N (D C D N ) D C (R C R N ). (7) From the first expression, we see that E is the ratio of percentage changes in D to percentage changes in R. By assumption, D C < D N, and R C > R N, so E < 0. For activity A, we obtain E = 2,310/3,560 = , which means a 1% decrease in resource expenditure leads to a 0.65% increase in activity duration. For any two activities, the one with the lower E value is less sensitive to changes in resource usage. If we have resources to give, the greatest percentage impact will be seen by giving resources to the activity with the highest E. If activities need to be taken away, they should be taken from the activity with the lowest E, since it is least sensitive to changes in R. Also, activities with lower elasticities must be monitored more carefully, because if they fall behind, more resources will be required to bring them back up to schedule. 3.4 Total Cost of Crashing Because of Assumption 2, within the range of R values from the crash and normal options R [R N,R C ], the relationship of duration to resource usage can be given by a straight line, shown in Figure 1: D 2 (R) = a br. (8) The term a represents a hypothetical project duration if zero resources are assigned, and is an artifact of the linearity assumption. The term b shows how duration decreases in R, a term previously derived in (1): b = (D N D C )/(R C R N ). We can then find a = D N + R N b. For activity A, b = ( )/( ) = 40/22.25 = so a = = Thus, total cost D(R) = R, and it is easily verified it returns the proper values for R = and R = 80. The total cost is given by C = R D. Substituting in D 2 (R) from (8), we obtain: C 2 (R) = R(a br) = ar br 2. (9) For A, C = 303.8R 1.798R 2, which is quadratic, as shown in Figure 4. The vertical line is at the lower range of R, R = $57.75, and the dashed vertical line is at the upper range of R, R = $80. 10

11 20,000 18,000 16,000 14,000 Total Cost, C 12,000 10,000 8,000 6,000 4,000 2, Resources per Day, R Figure 4: Total Cost of Activity A as function of Resources per Day, R From first-order conditions, C(R) can be shown to reach a maximum value at R = a/2b. We can see from Figure 4, C obtains a maximum value at R = 303.8/( ) = 84.5, and a minimum value at R = 0, both of which are well outside the [70,80] range. The downward sloping portion of C for R > a/2b is also an artifact of our linearity assumption. Given a total cost amount C, we can use the quadratic formula to the find the two values of R will yield the desired value of C(R). One solution is R > a/2b, and therefore not feasible. Substituting the relevant solution into (8), we can find the resulting duration. For example, if we have $12,000 to spend on activity A, we need to solve $12,000 = 303.8R 1.798R 2 for R. From the quadratic formula, we obtain R = and R = The second, > 84.5 = a/2b, and > 80 = R max. The other value, < a/2b, and it is easily verified that C 2 (62.93) = (62.93) 1.798(62.93) 2 = 19, ,119.4 = 12,000, as desired. 4 Crashing Decisions We can use our elasticity formulas to either figure out which activities to speed up, or which activities to slow down, should that be needed. 11

12 Activity D N D C C N C C E R N R C B $11,550 $12, C $16,800 $19, Table 2: Costs and Durations for Activities A and B. 4.1 Which Activitiy to Crash We can use equation (7) to to decide which activity to crash. For example, suppose we have two activities, B and C, (listed in Table 2) and we need to decide which to crash. Activity C is more elastic, so spending on C will have a greater bang for the buck than spending on B, and should be crashed first. Suppose 4 days needed to be saved from either B or C. We can use equation (2) to see how many resources would need to be used per day to gain 4 days. For activity B, R2 B (4) = (80 70)/( ) = /5 = 78. The new cost of activity B is now C B = R D = 78 (165 4) = $12, 558. Crashing increased costs by $12,558 - $11,550= $1,008. For activity A, R2 A (4) = (90 60)/( ) = /60 = 62. The new cost of activity C is now C C = R D = 62 (280 4) = $17,112. Crashing increased costs by $17,112 - $16,800 = $312. As predicted by the elasticity, C is the better activity to crash. Another way to arrive at this same result is to look at the change in R. For activity B, to gain 4 days is a reduction of 2.42% from 165. An elasticity of means that daily resource usage must be increased by 2.42% / = 11.1% of $70, for R = $ Which Activity to Slow Down We can also use elasticity to decide which activities to slow down, and equations (8) and (9) to determine the impact of adding or withholding resources. For example, suppose that we had already decided to crash both B and C, and activity A has become delayed, and we are going to have to dedicate an additional $1,000 in resources to A, so it can be crashed. When $1,000 less is spent on B or C, it will be in a slow crash mode, where we will not spend enough to achieve the full crash benefits, but the activity will be faster than normal. B has a smaller elasticity, and is thus less sensitive to reductions in resources, so the $1,000 should be taken from it, if that is 12

13 feasible. We were planning to spend $1,250 to crash B, so it would be feasible to spend $1,000 less, giving CC B = $11,800. Solving for a and b, the cost of B is CB (R) = 200R 0.5R 2. Solving $11,800 = 200R 0.5R 2, we obtain R B = $ as the resources to devote to B in a slow-crash situation. Substituting into equation (8), the length of the slow-crash is found: D B S = (71.94) = = 164 days. Taking $1,000 from B increases its completion time by 4 days. We originally planned to spend $3,000 crashing C, so taking $1,000 from it is feasible. Computing the parameter values for B, a = 400,b = 2. Given a slow-crash cost of $18,800, we solve $18,800 = 400R 2R 2, we obtain RS C = $ Substituting into equation (9), the slow-crash length is found: DS C = 400 2(75.50) = 249 days. Taking $1,000 from C increases its completion time by 29 days. Clearly, the elasticity gave the right decision. Compared to normal, daily resource usage of $71.94 is a 2.77% increase. Multiplied by the elasticity of , the percentage change in duration is = 0.606,% compared to the normal time of 165, a reduction of 1 day. 4.3 Other Duration Times In many cases, the normal and crash times represent the extreme possibilities, and nothing faster or slower than them may be considered. In some cases, however, it may be realistic to also speak about a slow normal time, which is slower than the normal time, or a fast crash time, faster than the crash time. If the normal and crash times are the project manager s best guess, and not the result of weeks of study by a fleet of analysts, it might be quite reasonable to consider times faster than the crash time, or slower than normal. If the linear relationship between daily resource usage and duration holds for the range [R N,R C ], it is possible this relationship may also hold for points close to, but outside of this range. Figure 1 shows what the linear relationship would look like if it held for R (0,a/2b). The relationship clearly could not hold for R = 0, but it could hold for values smaller than R N, or larger than R C. If the linear assumption holds for larger or smaller R values, we can consider the possibility of further crashing of an already-crashed activity, for a fast crash time, or of slowing down a normal activity for a slow normal time. 13

14 5 Summary and Conclusions In this paper, we have presented a new method for determining which project activities to crash. The calculation of resource elasticity requires no new information, and is computed using the project manager s conventional estimates of normal and crash duration times as well as normal and crash costs. Instead of assuming a linear relationship between crash spending and time saved, we assume a linear relationship between the additional resources used per day and the time saved. This assumption allows us to compute a resource elasticity measurement that tells us which activities will respond most significantly to increases in the resources allocated to them. Resources that are highly resource elastic can easily be brought back onto schedule, should they fall behind, and highly inelastic activities would require large amounts of additional resources to speed up, should they fall behind. Therefore, highly inelastic activities need to be monitored carefully in order to make sure that they do not fall behind. If resources need to be taken away from an activity, they should be taken from the activity that will be affected the least. 6 References Berman, S Resource allocation in a PERT network under continuous time-cost functions. Management Science, 10(4) Boctor, F A new and efficient heuristic for scheduling projects with resource restrictions and multiple execution modes. European Journal of Operational Research, 90, Elmaghraby, S Activity networks: Project planning and control by network models. John Wiley Interscience: New York. Gould J., and C. Ferguson Microeconomic theory (5th ed.). Irwin: Homewood, IL. Hartmann, S Project scheduling under limited resources. Springer: Berlin. Herroelen, W Project scheduling - Theory and practice. Production and Operations Management, 14(4) Kelly, J Critical-path planning and scheduling: Mathematical basis. Operations Research, 9(3) Kelly, J., and M.R. Walker Critical path planning and scheduling. in Proc. of the eastern joint computer conference. Boston, MA, December 1-3, Kolisch, R Project scheduling under resource constraints. Physica Verlag: Kiel, Ger- 14

15 many. Lamberson L., and R.R. Hocking Optimum time compression in project scheduling. Management Science, 16(10) Learner, A The diagrammatical representation of elasticity of demand. Review of Economic Studies, I(1) Machlup, F The economics of sellers competition: Model analysis of sellers conduct. Baltimore: Johns Hopkins press. Malcolm, D., J. Rosenbloom, C. Clark Application of a technique for research and development program evaluation. Operations Research, 7(5) Petrovic, R Optimization of resource allocation in project planning. Operations Research, 16(3) Pich, M., C. Loch, and A. De Meyer On uncertainty, ambiguity, and complexity in project management. Management Science, 48, (8) Sheldon, J A note on the teaching of arc elasticity. Journal of Economic Education, (3) Stevenson, W Production operations management. Irwin McGraw-Hill: Burr Ridge, IL. Tavares, L A review of the contribution of operational research to project management. European Journal of Operational Research, 136, Williams, T Allocation of contingency in activity duration networks. Construction Management and Economics, 17,