Time/Cost trade-off Analysis: The missing link

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2 Prime Journal of Engineering and Technology Research (PJETR) ISSN: Vol. 1(2), pp , November 28 th, Prime Journals Full Length Research Time/Cost trade-off Analysis: The missing link Edem OP Akpan Department of Project Management Technology, Federal University of Technology P. M. B. 1526,Owerri, Imo State, Nigeria Accepted 12 th November, 2012 Time/cost trade-off analysis is mainly concerned at finding the total minimum cost of a project taking into consideration the direct (crash) and indirect costs. Despite its importance in the field of project management especially with respect to penalty clause, the solution to the problem has so far remained elusive and efforts therefore tend to go in the direction of direct cost/time reduction towards the saturation point. The solution to this problem has mostly been approached using linear programming technique but there seems to be a problem of incorporating the indirect cost into the model formulation; hence the shift in emphasis in the existing procedure. The present study is an attempt to bring in this missing link using a different approach which involves comparing the crash cost calculated at any point in time with that of the indirect cost. Since it is difficult to calculate the individual crash cost at any point within the range of normal duration and saturation point, the difference between the overall costs of two adjacent points makes this possible. If the cost is lower than the indirect cost, a lower point is chosen and if higher, a decision in the opposite direction is taken. A point of convergence will occur between these two points with the one having the lower cost being the optimum solution. An extension is also made with regards to the penalty cost. A numerical example is used to test the proposed model which indicates its potential industrial applications. It is also observed that the model is capable of handling cases where different indirect costs may be contemplated for the same project. Keywords: Saturation point, crash cost, penalty clause, linear programming, total minimum cost. INTRODUCTION A lot of materials have appeared in the open literature on time/cost trade-off but Kelley (1961) was about the first to do an extensive work in this area in which he came out with the idea of minimizing the project overall total cost. This idea is established on a tradeoff between time and cost. He realized that there is a functional relationship between project cost and duration and that work could be speeded up by the allocation of more resources in the form of direct cost for such group of activities. The aim initially was to develop a model with the overall minimum cost schedule for any given project duration bearing in mind the direct and indirect costs. Having realized that cost is not necessarily linear but sometimes convex, he developed a parametric linear programming flow algorithm to obtain project cost curve. With certain assumptions and using an approximate linear cost function instead of the convex, which could have been difficult to model, he suggested two pairs of time estimates for each activity, the normal and the crash duration with their corresponding costs. He proceeded to give a linear programming formulation in which a single objective function, that of minimizing project cost curve subject to the limits defined by the normal and crash points was envisaged. With all these assumptions, the solution to the problem was obtained. He went further to demonstrate his model with a worked example, which involved a reactor maintenance and its other accessories and later presented this in the form of network diagram. What is of interest here are the solutions obtained; the minimum time (that is the crash duration or the saturation point as it is sometimes called) in which this project may be completed is 418 hours derived from his Theorem 3 but the cost obviously is not $ as stated but rather $ derived from an improved linear programming formulation as given by Levin and Kirkpatrick (1978) or the model developed by Akpan (2001). The longest the project needs to take to attain minimum cost of $ (the product of summing all the costs at normal duration) is 475 hours derived from Theorem 2. Computationally it may be more efficient using maximal flow algorithm (based on the flow-graph theory) to derive

3 27 Prim. J. Engineer. Technol. Res. the project duration than linear programming approach. However, linear programming is required for those situations where cost is desired at certain due dates (milestones). In this case the milestone would form one of the constraints. Where the total minimum cost is required, that is taking into consideration the direct and indirect costs, which means that the date is not known, the use of basic calculus similar to what we have in inventory control model may be more appropriate but finding an appropriate model formulation poses a major problem. It must be remembered that the whole concept of time/cost trade-off analysis is built on the notion that the longer the project duration, the more one incurs in indirect cost and the lower the direct cost which is basically the cost at normal duration. Conversely the shorter the project duration, the lower the indirect cost and the more one incurs in direct cost as a result of attendant (crash) cost of those activities used for the expediting exercise which may be in the form of overtime, more resources, etc. The indirect cost is consumed equally on a periodical (daily. hourly, etc) basis throughout the project duration. The difficulty of getting that point where the indirect cost is equal to the direct cost as in the inventory control model (that is EOQ where Ordering cost = Carrying/holding cost) might have caused Kelley to probably avoid this aspect even though this was his major intention in the first place which he supported with a diagram (Figure 6 in his work). He even failed to provide data for indirect cost. At this point however, this condition (i.e. indirect cost = direct cost) may not hold even though the cost is optimal. Similarly Levin and Kirkpatrick (1978) treated those aspects relating to the saturation point and the penalty clause. A project scheduling problem compiled by Kolisch and Sprecher (1996) in which time/cost trade-off analysis featured prominently did not mention this aspect either. There have also been other approaches; the LP/IP hybrid by Burns (1996), Hegazy (1999) and Azaron (2005) using genetic algorithm in which they acknowledge the problem as combinatorial, a dynamic programming solution by Robinson (1975) and minimal cut concept by Phillips and Dessouky (1977). Tomasz (2009) uses stochastic dominance rule to compare alternatives with respect to completion time and project cost and their associated risks while Ammar (2011) using a nonlinear mathematical optimization model incorporates the discounted cash flow in his analysis. All these authors concentrate mainly on minimal direct cost/time reduction with little or no consideration of the indirect cost to achieve the original concept of project overall minimum cost and the time of occurrence. METHODOLOGY Versions of time/cost trade-off analysis While it may be important to look into the mathematical model surrounding time/cost trade-off analysis, it is assumed that such efforts have long been established and therefore, there may be no need for a repeat performance. It must be remembered that there are basically two versions of the model, the original concept of finding a point where the two cost components direct and indirect costs are minimized which forms the centrepiece of this paper and the second which concentrates on completing the project at the lowest possible cost at the saturation point. If time is the dominant factor (i.e. a situation where a penalty clause is inserted in the contract document for delivery beyond a certain date), this option offers the possibility of working out the direct cost (or extra cost) of completing the project at certain due dates. This represents an opportunity cost the direct cost versus the penalty cost. FINDINGS The Search for Optimal Solution As earlier stated, the problem is combinatorial in nature and many techniques have been used to find the optimal solution even for the direct cost/time reduction towards the saturation point. In recent years, meta-heuristics have been quite popular and useful in solving combinatorial problems which are NP-hard. Since the proposed model relies on direct cost/time reduction to establish the project overall minimum cost, a more efficient heuristic/metaheuristic therefore becomes imperative. A lot of materials seem to favour genetic algorithm (GA) as reported by Hagazy (1999), Li et al (2011), Chen and Tsai (2011), Drake and Choudhry (1997) just to mention a few in which the latter used in improving on Campbell et al s (1970) which before then was adopted by Akpan (1996) in analyzing the job-shop sequencing problems. However the random activity (job) selection used by Akpan (1996) could equally give the same result as those of Drake and Choudhry (1997) if more iteration runs are embarked upon. There is controversy as to the superiority of some heuristic approaches to one another. Many researchers such as Ponnambalm et al. (1999), Kalir and Sarin (1999), El-Gafy (2007), Mendes et al (2002) and Bokang and Sooyoung (2002) have challenged this superiority in certain situations. The intention of this paper is not to go into this controversy and since random activity can equally give very good solutions in many situations, it is the one adopted for this work. It is observed that the existing models rely on a particular time period in the search for optimal solution, there are at times that these are done beyond the saturation point leading in such situations to an inconclusive solution. In order not to get trapped in this kind of a situation, a model developed by Akpan (2001) based on implicit elimination procedure is partly used for this work. The procedure adopts the old concept of network flow with each set having all the activities as a chain from the source node to the sink node, each time summing up the activities normal and crash duration separately along each path. The path with the longest normal duration has the critical path while the

4 Akpan 28 Table 1: Data relating to the construction project Activity Duration in Preceding Direct Cost (N) Weeks Activity Normal Crash Normal Crash A , , B , , C , , D A , , E A , , F B, D , , G C , , Figure1: Network diagram for the worked example Based on this network diagram, we have: Activities Normal Duration Crash Duration AE ADF BF CG 17 8 which indicates that the path ADF is critical and also having the saturation point. path with the longest crash duration is the one with the saturation point. Within this range of normal duration and saturation point lay the optimum solution and the point in which it occurs. At this point, the period crash cost should be equal or nearly equal to the indirect cost but not greater than it. Since it is difficult to calculate the individual crash cost at any point within this range, the difference between the overall costs of two adjacent points makes this possible. If the cost is lower than the indirect cost, a lower point is chosen and if higher, a decision in the opposite direction is taken. A point of convergence will occur between these two points with the one having the lower cost being the optimum solution. Numerical illustration using a worked example It is necessary at this point for a worked example to concretize the model development. The following information relates to a construction project for which a contract is about to be signed. Seven activities are involved and the normal duration, normal cost, crash duration and crash cost have been derived from the best available sources as shown in Table 1. Each activity may be reduced to the crash duration in weekly stages at a pro rata cost. There is a fixed cost (indirect cost) of N per week. Solution First the network diagram has to be drawn for easy identification of the paths for the purpose of determining the critical path and the saturation point. This can be done using Activity on Arc (A on A) or Activity on Node (A on N). The latter is used in this case as shown in Figure1. The path CG can even be ignored in the search for optimal solution since the normal duration falls below the saturation point. Using the least cost slope concept, we

5 29 Prim. J. Engineer. Technol. Res. Table 2: Crashing towards the Saturation point from the Normal Duration Period Total Crash Weekly Crash Activities involved (Weeks) Cost Cost D* D*1, A* A*2, F* A*3, F*1, B* A*3, F*2, B*1, E* A*3, F*3, B*1, E* A*3, F*3, B*2, D*1, E*3 get the crash cost immediately after the normal duration that is Week 29 to be N After week 29, there are two critical paths and the least cost slope concept may not necessarily hold since the problem has now become combinatorial in nature. If least cost slope concept is followed, activity E with a lower cost slope of N1900 would be taken for the path of AE at Week 28 before considering A with cost slope of N2500 for the path of ADF. In that case at Week 28 we will end up with a total cost of N6800 using the least cost slope concept. The same would go in for Week 27, activity D would form a part of the solution which should have been N rather than N The optimal solution for the different weeks is shown in Table 2. The crash cost for the different periods is calculated as shown in Table 3. This is not only for the indirect cost of N as specified in the worked example which turns out to be 24 weeks but considers other possibilities as well. If the indirect cost was N , there could have been no need for the crashing exercise since the crash cost is higher than the indirect cost. The indirect cost of N6000 is not applicable in the worked example as there is no minimum project overall total cost associated with it within the range of the normal duration and the saturation point. The different indirect costs with respect to the minimum project overall total costs is shown in Table 4 and the graph of it in Figure 2 based only on the indirect cost of N The total cost at the normal duration is N , that at the saturation point is N and the one for overall minimum cost which occurs in week 24 is N a = indirect cost per period, i.e. hour, day, weeks, etc. b = period in which the total minimum cost (both crash and indirect costs) occur c = total minimum cost at period b d = total crash cost at period b e = total crash cost just before period b (i.e. period b + 1) f = cost difference between d and e g = total crash cost immediately after period b (i.e. period b-1) h = cost difference between d and g Penalty Clause/Cost It is not unusual that the project duration resulting from the work plan with its attendant critical path may not be acceptable to the client as he/she may opt for a shorter duration and even with a proviso for a penalty to be paid beyond that date. On the other hand, the project duration as given by the critical path if accepted by the client may

6 Akpan 30 Table 3: Period in which total minimum cost occurs a b c D E F G h Table 4: Total Cost (Indirect and Crash) at different Time Periods Indirect Cost (N) Duration Table 5: Total Crash Cost Versus the Combined Indirect Cost and Penalty Cost Week Total Crash Indirect Penalty Cost Cost (a) Cost (b) (a) + (b) come with the same condition of a penalty cost beyond this point. The two cases are the same and the solution to the problem is at a point where the total crash cost is higher than the combined cost of indirect and penalty costs as this will entail incurring more cost and spending less in the alternative. Based on the worked example above, suppose the project duration is given as 25 weeks attracting a penalty cost of N per week beyond the period, the analysis is as shown in Table 5. The most appropriate period is Week 26 with a combined cost (penalty and indirect costs) of N11, as against incurring a total crash cost of N11, To avoid incurring any penalty cost and the indirect cost after 25 weeks, N must be incurred as crash cost. Supposing the penalty cost is N3, and the indirect cost is not considered as always the case (see Levin and Kirkpatrick (1978)), then it will be better to pay the penalty cost for the two days, that is complete the project in 27 weeks with extra cost of N6, than incurring the total crash cost of N7, Analysis of the different indirect cost/penalty cost with respect to the expected completion date within the normal cost and saturation point can also be carried out similar to what we have in Table 4 to draw conclusion as to the optimum solution. DISCUSSION Time/cost trade-off analysis has always concentrated mostly on direct cost/time reduction using many techniques such as linear programming and of recent; the meta-heuristics for optimal solution. What is sometimes ignored is the indirect cost without which the whole concept is not complete. As the direct cost goes up with a reduction in project duration, the indirect cost moves in the opposite direction. Having observed that the existing methodologies and models relied mostly on the foundation laid down by Kelley (1961) which was not comprehensive and to some extent deficient, there was the need to extensively review his work as a basis for the present research. This paper therefore touches on four main aspects of the model; the total cost at the normal duration, at the saturation point, at the time where the total cost is minimal and finally the point of optimum solution with respect to penalty cost and crash cost incorporating indirect cost in the analysis in all cases. Even though

7 31 Prim. J. Engineer. Technol. Res. random activity is adopted for this work and found to be reasonable in providing the optimal solution for the problem under discussion, efforts at utilizing other heuristics are very much encouraged. CONCLUSIONS The model has presented a systematic approach to finding the minimum project total cost and the point in which it occurs with possible changes of indirect cost if the need to do so is desirable; a kind of sensitivity analysis. While the penalty cost is normally considered in line with crash cost in the existing models, the present study incorporates the indirect cost in determining the optimum solution, that optimum solution being at a point where the combined penalty cost and indirect costs is very close to the crash cost. REFERENCES Akpan EOP (2001). Time/cost trade-off computation using implicit elimination procedure. Technical Transactions of the J. the Nigerian Institution of Production Engineers, 6(2): Akpan EOP (1996). Job-shop sequencing problems via network scheduling technique. Int l J. Operations and Production Management, 16(3): Ammar MA (2011). Optimization of project time-cost trade-off problem with discounted cash flows. J. Construction Engineering and Management, 137(1): Azaron A (2005). A genetic algorithm approach for the time/cost trade-off in PERT network. Applied Mathematics and Computation 168(2): Bokang K, Sooyoung K (2002). Application of genetic algorithms for scheduling batch-size production system. Production Planning and Control, 13(2): Burns SA, Liu L, Feng C (1996). The LP/IP hybrid method for construction time/cost trade-off analysis. Construction Management and Economics. 14(3): Campbell HG, Smith ML, Dudek RA (1970). A heuristic algorithm for the N jobs, machine sequencing problem, Management Science, 16: Chen S, Tsai M (2011). Time-cost tradeoff analysis of project networks in fuzzy environments. Euro. J. Operat. Res., 212(2): Drake PR, Choudhry IA (1997). Adapting a genetic algorithm to schedule variants of the manufacturing shop. Http/csswww.essex.ac.uk/conferences/upssig/Liverp ool-15/drake.doc. El-Gafy MA (2007). Construction time-cost trade-off using ant system. Int l J. Construction Education and Research, 3(2): Hegazy T (1999). Optimization of construction time/cost tradeoff analysis using genetic algorithms. Canadian J. Civil Engineering 26(6): Kalir AA, Sarin SC (1999). The role of advanced start and dominance rules in simulated annealing for parallel processor scheduling problems. Production Planning and Control, 10(8): Kelley JE (1961). Critical path planning and scheduling: mathematical basis. Operations Research, 9(3): Kolisch R, Sprecher A (1996). PSPLIB a project scheduling library. European J. Operational Research, 96: Levin RI, Kirkpatrick CA (1978). Quantitative approaches to management (Tokyo: McGraw-Hill). Li H, Al-Hussein M, Le Z (2011). Incentive genetic algorithm based time-cost trade-off analysis across a build-operate-transfer project concession period. Canadian J.Engineering, 38: Mendes AS, Muller FM, Franca PM, Moscato P (2002). Comparing meta-heuristic approaches for parallel machine scheduling problems. Production Planning and Control, 13(2): Ponnambalm SG, Jawabar N, Aravindan P (1999). A simulated annealing algorithm for job shop scheduling. Production Planning and Control, 10(8): Phillip S, Dessouky MI (1977). Solving the project time/cost tradeoff problem using minimal cut concept. Management Science 24(4): Robinson DR (1975). A dynamic programming solution to cost-time tradeoff for CPM. Management Science 22(2): Tomasz B (2009). The time-cost trade-off analysis in construction project using computer simulation and interactive procedure. Technological and Economic Development Economy, 15(4):