Critical chain construction with multi-resource constraints based on portfolio technology in South-to-North Water Diversion Project

Jing-chun FENG*, Lei LI, Nan YANG, Yu-zhen HONG, Min PANG, Xiong YAO, Li-cheng WANG

1. Business School, Hohai University, Nanjing 210098, P. R. China

2. Construction and Administration Bureau of South-to-North Water Diversion Middle Route Project, Beijing 100053, P. R. China

Critical chain construction with multi-resource constraints based on portfolio technology in South-to-North Water Diversion Project

Jing-chun FENG*1, Lei LI1, Nan YANG1, Yu-zhen HONG1, Min PANG2, Xiong YAO2, Li-cheng WANG2

1. Business School, Hohai University, Nanjing 210098, P. R. China

2. Construction and Administration Bureau of South-to-North Water Diversion Middle Route Project, Beijing 100053, P. R. China

Recently, the critical chain study has become a hot issue in the project management research field. The construction of the critical chain with multi-resource constraints is a new research subject. According to the system analysis theory and project portfolio theory, this paper discusses the creation of project portfolios based on the similarity principle and gives the definition of priority in multi-resource allocation based on quantitative analysis. A model with multi-resource constraints, which can be applied to the critical chain construction of the A-bid section in the South-to-North Water Diversion Project, was proposed. Contrast analysis with the comprehensive treatment construction method and aggressive treatment construction method was carried out. This paper also makes suggestions for further research directions and subjects, which will be useful in improving the theories in relevant research fields.

multi-resource constraints; South-to-North Water Diversion Project; project portfolio; critical chain

1 Introduction

The critical chain has been applied in many areas since it was proposed by Goldratt (1997), an Israeli scientist. It not only enables the optimized schedule plan to satisfy the resource demand so as to shorten a construction period, but can ensure the completion rate to make the schedule plan more economical and practical. Much progress has been made on the research on the critical chain and its application under single-resource constraint (Leach 1995; Herroelen and Leus 2005; Tukel et al. 2006; Xing and Ming 2008). However, the relevant theoretical research on the critical chain with multi-resource constraints has some deficienciesand problems. At present, the problem of critical chain construction constrained by a single resource has been resolved effectively. However, projects nowadays, especially large-scale projects, often have multi-resource constraints. The existing theories and methods for critical chain construction with single-resource constraint cannot resolve the problems in the construction work. Thus, the research on and proposal of the model for critical chain construction with multi-resource constraints are necessary. Currently, there have been the comprehensive treatment construction method and aggressive treatment construction method for the critical chain construction with multi-resource constraints. Based on contrast analysis of the methods above, this paper analyzes the relevant mechanism according to the similarity principle, examines the project portfolio with multi-resource constraints, and proposes the relevant model.

2 Principle of critical chain construction

The theoretical basis of the critical chain is the theory of constraints (TOC), which insists that the weakest link (bottle-neck link) determines production efficiency at large, that is, the capacity of the bottle-neck link is the core factor affecting the whole production efficiency. Construction works consume manpower, equipment, material, and financial resources, which will constrain and affect the project schedule. According to TOC, the critical chain is the longest path created from resource balances on the basis of comprehensive consideration of the logical constraints and resource constraints existing between the projects (activities). To ensure the project schedule proceeds according to the established critical chain, buffers are supposed to be set. Meanwhile, through the buffer management scheme of the critical chain, the consumption condition of constrained resources can be inspected and the critical chain can be adjusted and controlled to a great extent (Ma et al. 2004; Long and Ohsato 2008; Liu et al. 2006; Steyn 2002; Song 2006). The framework of the critical chain technology with multi-resource constraints can be seen in Fig. 1.

Fig. 1 Framework of critical chain technique with multi-resource constraints

The critical chain construction is the first step in applying the critical chain. Theoretically,there are two kinds of project resource constraints: single-resource constraints and multi-resource constraints. Accordingly, there are two situations in critical chain construction:

(1) The critical chain construction constrained by a single resource: In this situation, according to the resource similarity, we put the works under the same resource limits into the critical chain based on their logical constraints. This type of critical chain is relatively simple.

(2) The critical chain construction constrained by multiple resources: In this situation, works or activities have the characteristics of more resource categories and more complicated logical relations. Goldratt (1997) did not propose an effective resolution to this problem. Based on the comprehensive treatment construction method and aggressive treatment construction method, constructing a critical chain generally requires resource identity, buffer setting, planning and adjusting, schedule control, etc. As discussed above, the current methods cannot resolve the critical chain construction problem constrained by multiple resources. Thus, we need to consider the similarity of all the resources needed, and then we can form the project portfolio according to logical and resource dependencies by the portfolio technology, thus constructing the critical chain. In the process of building such a chain, we need to introduce the project portfolio technology based on resource similarity for constituting the critical link, and enable the process to be divided into two steps: Step one is to make the overall plan and coordinate all the works (activities). Through analysis of the resource constraints, project portfolios are created with the portfolio technology according to the resource similarity. Step two is to realize all the critical chains in each project portfolio, to manage and execute each portfolio as one project system, and to realize reasonable resource allocation using mathematical methods. A specific illustration can be found in Fig. 2.

Fig. 2 Principle of critical chain construction with multi-resource constraints

3 Creation of project portfolio

3.1 Meaning of project portfolio

All of the definitions of project portfolio in PMI (2003) and Association for Project Management (APM) (Morris et al. 2000) do not take into consideration whether the projects contained in project portfolios have the same targets. The definition in APM underlines the target of management efficiency and (or) that the resources commonly used are general factors of project portfolio. Archer and Ghasemzadeh (1999) argued that project portfolio is a complex of projects incurred or managed by a certain organization, whose target is to fight for the scarce resources (labor, finances, time, and so on ) commonly used. Feng et al. (2009) think that project portfolio is a complex of projects of similar nature in some aspects. According to project management demand, the resource similarity may mean the same capital resource, project resource, project scale, economic nature, project complexity, constraint condition, owner, management theory and method, target, nature of deliverability, and so on. In constructing the critical chain, this study adopted the relevant definitions in Feng’s theory (Feng et al. 2009).

3.2 Principle of portfolio creation

Resource constraint is one of the major factors constraining project (activity) schedules. Not all resources will affect the critical chain period, nor will each project (activity) be constrained by any or all resources. According to TOC, we should make systematic analysis of all resources to find the constrained resources, which are needed in many projects (activities) and employed in the same project during the same period. In this study, the authors carried out the management process analysis to discuss the portfolio creation based on resource constraints. The management process analysis, a method applied in project portfolios, starts from analysis of the project management process, then moves to scientific design, merger, and simplification, and is used to realize the reasonable use of various resources in many projects based on the resource similarity. The core of this method is to create a project portfolio by assembling the projects that are similar or related in terms of resources, according to their resource constraints, consuming conditions, and logical relations, as well as the benefit maximization principle.

3.3 Model of portfolio creation

Project portfolio methods consist of fuzzy clustering analysis, clustering analysis based on characteristic index comparison on an item-by-item basis, single-chain clustering analysis, and the level ordering method. The level ordering method regards each row or each column in the vector as a binary value input, and rearranges them according to their values, so as to obtain the combinatorial matrix for the resource. The process of portfolio creation is as follows:

(1) To create a resource matrix: The number of projects (activities) and categories of resources are supposed to beIandK, respectively. Resource-consuming conditions are listed in Table 1.

Table 1 Resource matrix of projects (activities)

(2) To compute the decimal values: According to Table 1, we take each row as a binary value input, which is then converted into a decimal value as follows:

(3) To construct a project (activity)-resource matrix based on the decimal values: According to the values ofy1,y2,…,yI, we place the larger ones at the top position, so as to form the project (activity)-resource matrix.

(4) To construct the project (activity)-resource matrix of row-column rearrangement: We take each column of the resource matrix as a binary value input, which is then converted into decimal values on a sequencing basis. The larger ones are supposed to be placed at the left position, so as to form another project (activity)-resource matrix, and so forth. In the last matrix, the sequence of either the row vector or the column vector is determined by the values. If the outcome fails to meet the ordering demand, the method above is supposed to be applied continuously until a matrix with a small-to-large character is obtained.

(5) To create project portfolios according to the final project (activity)-resource matrix: The projects consuming resources in a diagonal line belong to a portfolio.

4 Mathematical model for critical chain construction

4.1 Objective function

The construction of a critical chain with multi-resource constraints is generally based on the multi-project management in static enterprises and in more complicated project environments with dynamic multi-resource constraints. Therefore, its establishment needs to meet both the logical and resource constraints, on the premise of which we replace the order ofall projects (activities) so as to define their start time and finish time to ensure the shortest construction period (critical chain length). In this study, with reference to relevant studies of the static critical chain model (Bai 1995; Rivera and Duran 2004; Ma and You 2007), the author established the mathematical model to construct the critical chain with multi-resource constraints, aiming to ensure the shortest critical chain, which is expressed as minTI, whereTIrepresents the total construction period:

whereiis the order number of the project (activity), andi=1,2,…,I;xirepresents whether project (activity)iis constrained by resources (whenxi=0, the project is not constrained by resources and belongs to the non-critical chain; whenxi=1, the project is constrained by resources and belongs to the critical chain); andDiis the duration time of project (activity)iwithout consideration of safety time.

Eq. (2) and Eq. (3) are the constraint conditions for the objective function. Eq. (2) represents the logical constraints between projects (activities) and Eq. (3) represents the resource constraints, which means that the demand for a resource in projects (activities) carried out at the same time is less than or equal to the aggregate supply.

whereRiis the assembling of predecessor activities of project (activity)i,Tfzis the finish time of predecessor activities of project (activity)i,Tsiis the start time of project (activity)i(Tfi=Tsi+Di, whereTfiis the finish time of project (activity)i),Itis the assembling of projects (activities) that consume resources at timet,Qkis the aggregate supply quantity of resourcek, andqikis the aggregate supply quantity of resourcekneeded in project (activity)i.

4.2 Solution method

The branch-and-bound method, mathematical rule method, enumerative method, and elicitation method can all be employed as solutions to the proposed mathematical model (Shou 2004; Wang et al. 2008; Wan and Cai 2003; Wang et al. 2005). This study adopted the elicitation method. First, we identified the bottlenecks of projects in the project portfolio, and divided multiple resources intoKresources, that is to identify the critical chain under the restraint of a single resource in accordance with the heuristic algorithm. Similarly, we identified the critical chains under other resource constraints, at this time, there areLcritical chains in total, and the shortest critical chain could be taken as the initial critical chain. Then, based on that, we found a conflicting project (activity) under the restraint of one resource in the remainingK?1 resources, and, according to the priority sequence determined by the importance of different projects (activities) in the construction project, added the projects(activities) that could acquire the resource into the critical chain to improve it. Then the same process was carried out to deal with this critical chain for determining conflicting projects (activities) under the restraint of one resource in the remainingK?2 resources, in order to improve the critical chain. This process would carry on forK?1 times until all the resources had been effectively allocated and resource conflicts did not occur.

Through computing, we could get the schedule chart with consideration of both logical and resource constraints. In this chart, the chain with the longest duration is the critical chain. In reality, the resource conflicts caused by uncertainty will urge us to improve the critical chain with resource allocation accord ing to their precedence. The process involves setting three buffer areas to improve its capacity of dealing with risks to ensure the expected complete rate (Goldratt 1997).

5 Critical chain construction with multi-resource constraints in South-to-North Water Diversion Project

5.1 Case condition

The case study was carried out at the A-bid section of the middle route of the South-to-North Water Diversion Project, involving major works, such as preparation of a temporary construction project (A1), the Bailianyugou drainage and inverted siphon project (A2), the Zaoyuangou drainage and inverted siphon project (A3), the Shataogou drainage and culvert construction project (A4), the Liuzhuanggou drainage and flume construction project (A5), the prefabrication project of a pre-stressed beam slab (A6), the Bailianyu Bridge project (A7), the Baibao Easten Bridge project (A8), the Liuzhuang Bridge project (A9), the Longmen second branch of the eastern main canal inverted siphon project (A10), major parts of the canal project (A11), ancillary works of the canal project (A12), the Liuzhuang bleeder project (A13), and the check and acceptance work (A14). These projects are characterized by short construction periods, intensive work, uncertainty about the start time, difficulty in facility control, and multi-resource constraints. Based on CPM/PERT and other traditional schedule planning techniques, the critical path in the case study was defined as follows: earth-rock excavation→earthwork backfill→concrete lining of major works of canal, with a total construction period of 514 days.

5.2 Critical chain construction with multi-resource constraints

According to the principles of critical chain construction with multi-resource constraints and portfolio production, the construction steps of the A-bid section critical chain are as follows: (1) to identify the resource constraints, based on which to produce project portfolios; (2) to compute the construction period on a 50% co mplete rate basis, and to define the longest path as the critical chain considering the logical relationsand resource constraints of works; and (3) to work out the project buffer PB and feeding buffer FB, and place PB at the end of theproject, and FB to the point where the non-critical chain affluxes into the critical chain with resource buffer RB added at the same time.

5.2.1 Project portfolio

Analysis shows that there are resource conflicts between six kinds of resources: labor force (B1), concrete (B2), steel (B3), excavators (B4), blenders (B5), and conveyors (B6). Resource demand conditions can be seen in Table 2.

Table 2 Resource demand analysis

According to the similarity principle, with the help of the project portfolio production model, the project-resource matrix in Table 2 is to be processed with level reordering, with the results seen in Table 3.

From Table 3, we can find that the project portfolio is｛A11, A5, A10, A2, A3, A4, A7, A9, A8｝.

5.2.2 Determination of critical chain

(1) We identified the bottle-neck resources of the project by regarding a portfolio as a project system to arrange the schedule. The bottle-neck resources are labor force, concrete, and steel.

(2) According to the 1/2 rule, without consideration of safety time, we used the elicitation method to determine the critical chain in the project. Based on the method proposed by Goldratt (1997), there are three critical chains, which are produced separately with the labor force, concrete, and steel constraints. The critical chain constrained by the labor force is CCB1= (A2, A3, A5, A10, A11); the critical chain constrained by concrete is CCB2= (A2, A5, A7, A11); and the critical chain constrained by steel is CCB3= (A2, A5, A7, A11). We recognized thelongest chain CCB1as the initial critical chain to make necessary optimization.

Table 3 Reordered project-resource matrix

(3) Based on CCB1, we determined the activities causing conflict between concrete and steel one by one, and added them to CCB1in order to improve the initial critical chain until there were no resource conflicts. Thus, the initial critical chain becomes the critical chain expected to be constructed, i.e., CC = (A2, A3, A5, A7, A10, A11).

(4) We determined whether there was an obvious method to advance the project life cycle and to insert the project buffer and resource buffer. The project buffer is at the end of the critical chain. The longest chain in this project occupies 296 days with a project buffer of 198 days, and the planning period of the whole project is 494 days. Generally, the resource buffer can be added after identification of the critical chain. If we want to make sure the chart is clear, the add work of the resource buffer can also be done last.

5.2.3 Optimization of resource conflicts

Supposing the resources needed in A10 were added, the project could start when the resources were relatively sufficient, without interference of the total time of the project. After removing the safety factor, adjusting resource conflicts, and adding buffer areas into the critical chain, we could obtain the schedule plan with a higher complete rate. The schedule with buffer added can be seen in Table 4. According to Table 4, the finish time after resolution of resource conflicts was October, 2006, and the finish time of the project after PB was added was May, 2007.

Table 4 Schedule with buffer added

5.2.4 Contrast analysis with current methods

The present construction methods mainly include the comprehensive treatment construction method and aggressive treatment construction method. Due to the absence of systematic analysis, the former cannot resolve the conflicts of major resources, nor can it find the bottle-neck resource affecting the project schedule. Thus, this method cannot resolve the construction problem. As to the aggressive treatment construction method, its outcome of at least six critical chains may cause huge amount of work. Besides, it will produce conflicts among those six critical chains. This method therefore has some defects, such as a huge time and labor cost, the phenomenon of catching one and losing another, new conflicts, and construction delay. What’s worse, the complexity of the projects’ logical relations and environmental uncertainties may cause frequent adjustments, and thus delays the construction period.

The critical chain model based on the project portfolio could construct the critical chain with multi-resource constraints more effectively and better resolve the resource conflicts, which overcomes the defects of the methods above in a better manner. Compared with the critical chain model, the traditional network schedule diagram, a status management, is so rigid that it cannot fully recognize constraint factors influencing the schedule. Thus, it requires frequent adjustments. This model added with the portfolio link could identify the most critical constraint resources. With the help of relevant computing methods, the model could also deal with the added work caused by the added resource categories, and improve work efficiency and management flexibility. In this case, the more categories of resources involved, the moreadvantages the model will display.

6 Conclusions

(1) The application of this model to the A-bid section of the middle route of the South-to-North Water Diversion Project shows its feasibility. Meanwhile, from the simulation process of this model at the A-bid section, we can see that the proposed model could reduce the work in building the critical chain to some extent.

(2) The proposed critical chain model takes full consideration of the logical relations of projects and the influences of various factors on the allocation process, and resolves the resource conflicts by computing their precedence. The resource allocation process could more accurately and reasonably display the key point of schedule management and the very point where the constraints lie, which gives more pertinence to the critical chain, so as to ensure a more efficient and scientific allocation of resources.

(3) In order to improve the construction theory of critical chains and to extend the application range of critical chains, future research needs to be carried out, including the quantitative study of critical chain buffers, and research on the progress control based on the critical chain, organizational structures matching the critical chain, and human motivation and deployment systems.

Acknowledgements

The authors are grateful to the Office of the South-to-North Water Diversion Project Construction Committee under the State Council.

Archer, N. P., and Ghasemzadeh, F. 1999. An integrated framework for project portfolio selection.International Journal of Project Management, 17(4), 207-216. [doi:10.1016/S0263-7863(98)00032-5]

Bai, S. J. 1995. Heuristic method for multiple resource-constrained in PERT/CPM network.Systems Engineering-Theory & Practice, 15(7), 42-47. (in Chinese)

Feng, J. C., Cao, W. M., Cheng, D. H., Li, J., and Hong, Y. Z. 2009. Research on the whole life cycle of project portfolio.Project Management Technology, (10), 18-23. (in Chinese)

Goldratt, E. M. 1997.Critical Chain. Great Barrington: The North Ricer Press.

Herroelen, W., and Leus, R. 2005. Project scheduling under uncertainty: Survey and research potentials.European Journal of Operational Research, 165(2), 289-306. [doi:10.1016/j.ejor.2004.04.002]

Leach, L. P. 1995. Critical chain project management improves project performance.Project Management Journal, 30(2), 39-51.

Liu, S. X., Song, J. H., and Tang, J. F. 2006. Critical chain based approach for resource-constrained project scheduling.Acta Automatica Sinica, 32(1), 60-66. (in Chinese)

Long, L. D., and Ohsato, A. 2008. Fuzzy critical chain method for project scheduling under resource constraints and uncertainty.International Journal of Project Management, 26(6), 688-698. [doi:10.1016/ j.ijproman.2007.09.012]

Ma, G. F., Tu, M. Z., Shi, Z. Z., Wu, W. 2004. Design and implementation of project scheduling management system based on critical chain method.Journal of Shanghai Jiaotong University, 38(3), 377-381. (in Chinese)

Ma, G. F., and You, J. X. 2007. Quantitative analysis of critical chain multiple project scheduling management.Systems Engineering-Theory & Practice, 27(9), 54-60. (in Chinese)

Morris, P. W. G., Patel, M. B., and Wearne, S. H. 2000. Research into revising the APM project management body of knowledge.International Journal of Project Management, 18(3), 155-164. [doi:10.1016/S0263-7863(99)00068-X]

Project Management Institute (PMI). 2003.PMI Research Program Overview. Newtown Square: The Project Management Institute, Inc.

Rivera, F. A., and Duran, A. 2004. Critical clouds and critical sets in resource-constrained projects.International Journal of Project Management, 22(6), 489-497. [doi:10.1016/j.ijproman.2003.11.004]

Shou, Y. Y. 2004. Iterative technique for scheduling resource constrained multiple projects.Journal of Zhejiang University, 38(8), 1095-1099. (in Chinese)

Song, L. H. 2006.Research on Multi-Project Collaborative Management of Software Project. Tianjin: Tianjin University. (in Chinese)

Steyn, H. 2002. Project management applications of the theory of constraints beyond critical chain scheduling.International Journal of Project Management, 20(3), 75-80. [doi:10.1016/S0263-7863(00)00054-5]

Tukel, O. I., Rom, W. O., and Eksioglu, S. D. 2006. An investigation of buffer sizing techniques in critical chain scheduling.European Journal of Operational Research, 172(2), 401-416. [doi:10.1016/j.ejor. 2004.10.019]

Wan, W., and Cai, C. 2003. Critical chain techniques in single-resource constraint project.Management Science of China, 11(2), 70-74. (in Chinese)

Wang, C. F., Lin, Z. F., and Ma, M. M. 2008.Preface of Modern Project Management. Beijing: China Machine Press. (in Chinese)

Wang, X. Q., Guo, L. Y., and Fu, T. 2005. Construction project scheduling problem research based on critical chain technique.Journal of Hebei University of Technology, 34(6), 21-23. (in Chinese)

Xing, C., and Ming, X. G. 2008. Optimization on critical chain planning with multi-resource constraints.Project Management Technology, (s1), 338-342. (in Chinese)

This work was supported by the National Science and Technology Plan (Major Project of the Eleventh Five-Year Plan, Grant No. 2006BAB04A13), the Philosophy and Social Science Fund of the Education Department of Jiangsu Province (Grant No. 07SJD630006), the Third Key Discipline (Techno-Economics and Management) of the 211 Project, and the Key Discipline of Jiangsu Province (Engineering and Project Management)．

*Corresponding author (e-mail:feng.jingchun@163.com)

Received Jul. 4, 2010; accepted Dec. 14, 2010