Multiobjective evolutionary finance-based scheduling

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The most proactive operating strategy contractors can follow for financial planning is ... financing costs into the project total cost as well as scheduling under cash ...
icccbe 2010

© Nottingham University Press Proceedings of the International Conference on  Computing in Civil and Building Engineering  W Tizani (Editor)

Multiobjective evolutionary finance-based scheduling: the entire projects' portfolio M. Abido & A. Elazouni

King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia

Abstract A Strength Pareto Evolutionary Algorithm (SPEA) is proposed and employed to devise a set of optimum finance-based schedules of multiple projects being implemented simultaneously by a construction corporate. The problem involves the minimization of the conflicting objectives of financing costs, duration of the group of projects, and the required credit. The SPEA was employed to obtain the Pareto-optimal fronts for different two-objective combinations. In addition, a fuzzy-based technique was utilized to help the corporate decision makers select the best compromise solution over the Pareto-optimal solutions. The SPEA algorithm was validated using a multiple objective time/cost trade-off algorithm from the literature. The proposed approach has been developed and implemented on two simultaneous projects of 25 and 30 activities. The results obtained by the proposed SPEA and the developed fuzzy-based approach demonstrate its potential and effectiveness in finance-based scheduling of multiple projects from the corporate perspective. Keywords: multiobjective optimization, scheduling, financing, construction project

1

Introduction

A crucial challenge for construction contractors to run a sustained business represents the ability to timely procure adequate cash to execute construction operations. Alongside payments from their customers, contractors often procure additional funds from external sources, including banks. Typically, such cash incurs financing charges. Given the facts that the customers actually pay after the accomplishment of the work while retaining some money, and the cash that contractors can withdraw from banks is limited by credit limits, contractors often operate under cash-constrained conditions. The most proactive operating strategy contractors can follow for financial planning is to devise project schedules based on cash availability. The concept and technique of finance-based scheduling achieves the sought integration between the functions of scheduling and financing by incorporating financing costs into the project total cost as well as scheduling under cash constraints. Being an aspect of the whole corporate rather than the individual projects, contractors manage the financing aspect at the corporate level. The contractors' concern is to timely procure cash for all ongoing projects. Finance-based scheduling in this context ensures that the expenditures of the individual projects do not add up to top the credit limit, whereas the surplus cash amounts that occur in some projects are utilized to schedule activities of some other projects. This ensures that scheduling concurrent projects is carried out in view of the overall liquidity situation of the corporate. Financebased scheduling techniques schedule projects' activities while fulfilling the financial constraint of

credit limits. One notable study (Elazouni, 2009) employed a heuristic technique to implement the concept of finance-based scheduling on multiple concurrent projects. Generally, in the last few years, developing evolutionary multiobjective optimization techniques has attracted researchers in many areas (Abido 2003; Abido 2006; Coello 1999; Deb et al. 2002; Ritzel et al. 1994). In construction, the evolutionary multiobjective optimization technique was employed to solve scheduling, resource management, and time/cost trade-off problems. A parallel Multiobjective genetic algorithm framework was developed (Kandil and El-Rayes, 2006) to enable an efficient and effective optimization of resource utilization in large-scale construction projects. A Multiobjective optimization model (Hyari and El-Rayes, 2006) was presented for the planning and scheduling of repetitive construction projects. A Genetic-Algorithm Multiobjective time/cost optimization model (Zheng et al. 2005) was developed. A Multiobjective deterministic and stochastic modeling and task-optimization scheduling technique was developed (Kasprowicz, 1994). In this paper, the main objective is to implement SPEA (Zitzler and Thiele, 1998) to devise Paretooptimal finance-based schedules of corporate simultaneous multiple construction projects. The devised schedules are referred to as the Pareto-optimal solutions. Another objective is to develop and implement a fuzzy-based technique (Dhillon et al., 1993) to help decision makers select the best compromise solution over the trade-off curve of Pareto-optimal solutions.

2

Multiobjective finance-based scheduling

Finance-based scheduling of corporate simultaneous projects incorporates the objectives of financing costs, duration of the group of projects, and the required credit. For profitable business, the efforts of the corporate management should be focused on the minimization of the three objectives. However, reducing the financing costs definitely increases the profit but entails shortening the duration by increasing the required credit. Second, minimizing the required credit increases the possibility of being approved by bankers and offers the corporate more leverage to negotiate better interest rates and terms of payment but definitely results in an inevitable increase in the duration and financing costs. Third, shortening the duration increases the profit by reducing the overhead costs and the financing costs but requires high credit. Therefore, the objectives of financing costs, duration of the group of projects, and the required credit constitute a set of multiple corporate conflicting objectives.

3

Cash flow model

The cash-flow profile, as shown in Figure 1, is developed from the contractors’ perspective. Contractors often procure funds from banks by establishing credit-line accounts. A project cash outflow during a typical project period t is represented by Et and encompasses costs of overheads and taxes in addition to the direct costs including the costs of materials, equipment, labor, and subcontractors. In case of multiple simultaneous projects, the contractors' cash outflow at the end of a given period includes the Et components of the projects ongoing during the same period. On the other hand, the contractors' cash inflow, represented by Pt, includes the payments contractors receive, at the ends of periods, as an earned value of the accomplished works calculated based on the unit prices. In case of multiple simultaneous projects, the contractors' cash inflow at the end of a given period includes the Pt components collected of the projects at this time. Contractors normally deposit the payments into the credit-line accounts to continually reduce the outstanding debit (cumulative negative balance). The cumulative balance at the end of period t is defined by Ft where;

Ft = N t −1 + Et

(1)

Figure 1. Definition of floor plan and usage

The cumulative net balance at the end of period t after receiving payment Pt is defined as Nt. At the end of period t-1, Ft-1 is the cumulative balance, Pt-1 is the payment received, and Nt-1 is the cumulative net balance where;

Nt −1 = Ft −1 + Pt −1

(2)

Typically, cash procurement through the banks' credit lines incurs financing costs. The financing cost charged by the bank at the end of period t is It which is calculated using Eqs 3 till 5. For period t, if the cumulative net balance of the previous period Nt-1 is positive, this implies that the contractor debit is null and the contractor can use the surplus cash to finance activities during the current period. If the surplus cash completely cover the amount of Et, the contractor borrows no cash and Eq.5 applies, otherwise, the contractor will pay financing costs only for the amount of borrowed money in excess of the surplus cash as in Eq.4. In case Nt-1 is negative, Eq.3 applies to calculate the financing cost.

I t = rN t −1 + r

Et 2

if Nt-1≤ 0

(3)

It = r(

E t − N t −1 ) if Nt-1 > 0 and Nt-1-Et < 0 2

(4)

It = 0

if Nt-1-Et ≥ 0

(5)

The first term in Eq.3 represents the financing costs per period on the cumulative net balance Nt-1 at an interest rate r per period, and the second term approximates the financing costs on the cash outflow Et during period t. The summation of the values of It over the periods comprising the duration of the group of projects constitutes the value of the financing costs objective. When contractors decide to pay the financing costs at the end of the project, the periodical financing costs are compounded by applying Eq.6. t

I t′ = ∑ I l (1 + r )

t −l

(6)

l =1

Thus, the cumulative balance at the end of period t including accumulated financing costs is represented by Ft′ which is calculated as shown in Eq.7 below.

Ft′ = Ft + I t′

(7)

The corporate debit amounts at the end of the periods are represented by the values of the negative cumulative balance Ft′ . The minimum value signifies the required credit that must be procured to carry out the group of projects. The cumulative net balance including financing cost is represented by N t′ as in Eq.8.

N t′ = Ft′ + Pt

(8)

The positive value of N t′ at the end of the last period represents the corporate profit G shown in Figure 1.

4

Application of the algorithm

The application of SPEA was demonstrated using two concurrent projects of 25 and 30 activities with the networks shown in Figure 2. The durations of the 25 and 30-activity projects span over 27 and 29 working days respectively which are both equivalent to six 5-day weeks. The two projects were set up such that the start of the 30-activity project is shifted three weeks behind the start of the 25-activity project. The time data, financial data, and the contractual terms of the two projects are presented in Table 1. Table 2 presents the activities' direct cost per day for the 30-activity project. The indirect costs including the overhead costs, mobilization costs, taxes, and bond premium are calculated at the bottom of Table 2. Then, the markup and indirect costs are prorated to determine the activities' prices on daily basis. The implementation of the evolutionary multiobjective optimization technique involved coding and testing the integrated financial module and the SPEA algorithm using FORTRAN language.

Figure 2. CPM Network of the 25- and 30-activity projects Table 1. The time and financial data, and the contractual terms of the two projects

Table 2. The rates of the cash outflows and inflows of the activities of the 30-activity project

Figure 3a show the Pareto-optimal front of the profit versus required credit objectives for the 25 and 30-activity projects. The best compromise solutions are indicated in the Pareto-optimal front. Figure 3a shows clearly that the higher the credit the higher the corporate profit. This trend can be explained in view of the duration and financing costs. First, increasing the credit will definitely alleviate the conflict for cash allocation occurring when many activities are being scheduled simultaneously which in turn will allow devise short-duration schedules for both projects. This trend is very clear in the Pareto-optimal front of the objectives of duration versus the required credit shown in Figure 3b. Short-duration schedules will definitely reduce the overhead costs. Second, increasing the credit will reduce the financing cost through the reduction attained in the duration. This trend is demonstrated in the Pareto-optimal front of the objectives of financing costs versus the required credit shown in Figure 3c. Thus, increasing the credit decreases the overhead costs and financing costs which eventually will result in higher corporate profit.

Figure 3. Pareto-optimal fronts of the 25- and 30-activity projects

For the solution of the maximum profit of the Pareto-optimal solutions in Figure 3a, Table 3 presents the weekly expenditures Et and income Pt calculations of the individual projects and the two projects together. The Et and Pt values of the two projects together are used to determine the other financial parameters of the corporate as presented in Table 4. The results in Table 4 indicate that the profit is $41954 and the minimum Ft′ value which determines the required credit is $81545.

5

Validation

Figure 4 shows that the Pareto-front solutions are identical to the Pareto-front solutions obtained for a multiple-objective time/cost tradeoff scheduling problem (Leu et al., 1999) from the literature.

Table 3. Weekly expenditures and income of the individual projects and the two projects together

Table 4. The cash flow parameters of the two projects together

Figure 4. Validation results

6

Conclusion

A Strength Pareto Evolutionary Algorithm (SPEA) was proposed and successfully implemented to devise a set of Pareto optimum finance-based schedules for construction corporate with multiple projects being implemented simultaneously. The three objectives of financing costs, duration of the group of projects, and the required credit constitute a set of multiple corporate conflicting objectives. The problem of minimizing the three objectives has been formulated as a multiobjective optimization problem where the proposed approach has been employed to solve this problem. The SPEA algorithm was validated using a multiple objective time/cost trade-off algorithm from the literature. The Pareto-optimal fronts were obtained for the different two-objective combinations. In addition, a fuzzy-based technique was utilized to help the corporate decision makers select the best compromise solution over the trade-off curve of Pareto-optimal solutions. The results show the potential of the proposed approach and its effectiveness to generate well-distributed Pareto-optimal solutions for the problems under consideration.

Acknowledgement This research was a part of research project No. IN080422 supported by King Fahd University of Petroleum & Minerals which is gratefully acknowledged by the Authors.

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