- Email: [email protected]
Strategic Project Planning: Optimal Contracting and Resource Allocation
6th IFAC Conference on Management and Control of Production and Logistics The International Federation of Automatic Control September 11-13, 2013. Fortaleza, Brazil
Strategic Project Planning: Optimal Contracting and Resource Allocation Bruno Flach ∗ Carlos Raoni Mendes ∗∗ ∗ IBM Research – Brazil Av. Pasteur 138-146, Rio de Janeiro, CEP 22290-240, Brazil e-mail: [email protected] ∗∗ IBM Research – Brazil, Pontifical Catholic University – Rio de Janeiro Av. Pasteur 138-146, Rio de Janeiro, CEP 22290-240, Brazil e-mail: [email protected]
Abstract: In this work, we describe the development of a robust optimization model aimed at determining an optimal strategy for contracting and allocating resources within the context of development and implementation of capital projects. The proposed model selects a set of activity execution modes – which differ from each other in terms of related costs and expected durations – and risk-mitigation measures – associated with uncertainties due to incomplete knowledge of the activities’ exact scope – so as to balance the trade-off between incurred costs and the project’s completion date. Computational results are provided to illustrate the proposed approach. 1. INTRODUCTION Huge projects are often conducted by governments or large enterprises for the construction of the massive infrastructure required for a diverse set of economic activities – for example, those related with energy, mining, transportation and industrial applications. A number of potential risk sources coupled with resource constraints, intricate precedence relationships and multiple inter-dependencies among different sets of activities and various areas render planning, monitoring and managing such endeavors into a very complicated task. Given the large variety of involved disciplines and required skills – activities may involve expertise and know-how related with engineering, procurement, legal, environmental and logistics issues, for example – such projects are rarely conducted by a single entity. Often, either several third-party suppliers and contractors are hired to provide required materials or carry out different sets of activities – such as the construction of buildings, installation of equipment, acquisition of appropriate licenses, etc. – or they are performed under an EPC (an acronym meaning Engineering, Procurement and Construction) agreement, in which a single contractor is held responsible for designing, procuring and building the project according to the specifications defined by the owner. At times of uncertain economic conditions, pressures mount as successful project delivery might be ultimately the key to guaranteeing the economic feasibility of a company’s long-term strategic plans. Delays and cost overruns may lead to cascading impacts in the form of huge opportunity costs or even threats to the financial stability of a company, seriously damaging its ability to honor commitments and engage in revenue-generating activities. Given the circumstances, those responsible for the implementa978-3-902823-50-2/2013 © IFAC
438
tion strategy of such projects are constantly challenged to define an optimal arrangement for the allocation of available resources and third-party contracting in such a way as to ensure that projects are completed within limits of time, cost and scope. The fact that it is reasonable to assume that activities might be completed within a shorter time frame for an increased capital expenditure – naturally, up to a certain extent – related, for example, with the assignment of extra workers to a task or the use of more efficient machinery and that one is usually able to not only identify the total work required to complete a project but also to split it into different components that might be separately executed by different agents – obviously, taking into consideration eventual predecessor relationships that possibly exist among various tasks – allows for a degree of flexibility that can be exploited by managers. Within this context, the strategic project planning problem refers to the definition of the allocation of resources and selection of contractors, suppliers and vendors so as to optimize a given metric – e.g., Net Present Value (NPV) – so as to balance the project’s total cost and completion date. Note that such a decision process may be faced by either someone from the owner company who wishes to determine an implementation and outsourcing strategy to accomplish the project or by an EPC company who itself also relies on third-party suppliers and sub-contractors and aims at estimating associated costs in order to devise a more competitive tender bid. In this work, we describe the development of a robust optimization model aimed at determining an optimal project implementation strategy by allocating resources in such a way as to properly assess the trade-off between costs involved in speeding up activity durations and those that may result from delays in project completion and, pos-
10.3182/20130911-3-BR-3021.00085
IFAC MCPL 2013 September 11-13, 2013. Fortaleza, Brazil
sibly, associated penalties. The remaining of the paper is organized as follows: section 2 analyzes previous work on related problems and highlights our contributions, section 3 describes the mathematical model of the problem by first introducing a deterministic version the problem and subsequently extending it to account for the inherent existence of associated uncertainty. Section 4 presents computational results that illustrate the application of the proposed methodology. Finally, section 5 details the conclusions and directions for future work.
The assumption that activities durations are deterministic values is known to be a weak simplification of reality. The time required to complete an activity is only known with certainty after its termination, Gutjahr et al. [2000]. One of the main sources of this uncertainty are the risks associated with the project’s execution. The occurrence of a risk can impact an activity duration, if this activity is on the critical path or the impact is higher than the activity float, the project’s duration is also impacted. Even if neither of the previously described conditions happens, the project’s duration can be impacted by certain combinations of multiple risk occurrences.
2. CONTEXT
Risk management is a complete discipline per se. There are several methods to deal with project’s risks, from qualitative analytics methods (see Wang et al. [2004] and Raz and Michael [2001]) to mathematical optimization models (see Pate-cornell and Elisabeth [1996]). The risk management approach depends on how the risks impacts and mitigation strategies are modeled. In this work, we focus on risks that have impact on activities durations and that extra resources can be spent in order to minimize these impacts.
The first research works on project management methods were related with two commonly used techniques, the critical path method (CPM), and the Program (or Project) Evaluation and Review Technique (PERT), Kelley and Walker [1959] and Fazar [1959] respectively. On both techniques, the project is represented as a set of activities to be perform, and each activity has a fixed duration and a set of precedence relationships. A precedence relationship is a constraint which dictates that activity a cannot start until activity b has finished – in this case, activity b is named a predecessor of activity a. The main role of CPM and PERT is the identification of the project’s critical path, the set of activities that prevent the project from finishing earlier and for which any delay translates into a delay on the project’s completion date, Kelley [1961]. Activities not on the critical path are said to have nonnegative float, which is the maximum delay on an activity that doesn’t impact the project’s completion date. Another concept introduced by the cited methods was the representation of the activities and its precedence relations in a network (or graph) named activity-on-arc. In the activity-on-arc (AoA) network, each arc represents an activity, and each node represents an event (ex., the start or the end of an activity). An arc associated with an activity a also links two nodes (events) (i, j) with a weight equals to the duration of a. Given that circular precedence relationships are not allowed, the AoA network is a directed acyclic graph, and the critical path is the longest path on this graph, wich can be determined by a linear time algorithm (see Cormen et al. [2001]). In section 4, we show an example of an AoA network. The development of project management research gave rise to the study of different problems. An important class of these problems is the one related with the study of time-cost trade-offs when extra allocation of resources can decrease activities durations (activity crashing). One of the first investigated problems of this class was the deterministic discrete time-cost problem (DDTCP): given the AoA network of a project, a set of crashing measures (i.e., discrete 0-1 activity crashing opportunities) with its costs, a project due date and a penalty loss function for project delays, what is the optimal crashing plan that minimizes the project cost? DDTCP is known to be a NP-Hard problem, as proved by Walter J. Gutjahr [2000]. Traditional techniques such as Dynamic Programming and Branch and Bound have been applied to produce practical solutions to the DDTCP (see Panagiotakopoulos [1977] and Hindelang and Muth [1979]). 439
The main contribution of our work is to provide a robust mathematical optimization approach that deals both with crashing opportunities (or selection of activities modes) and the uncertainty associated with risks impacts on activities durations. The combined decision approach is stronger than solving the problems separately. The case study presented in section 4 reveals the complexity of a combined decision, proving the validity of our method. We describe the method on the next section. 3. MATHEMATICAL MODEL In a deterministic setting in which there are no uncertainties associated with the duration of activities, the problem might be formulated by assuming we are given all the alternative execution modes for each activity and an activity network which describes the precedence relationships among different tasks of the project. As previously mentioned, different execution modes might be related with the allocation of extra workers or more efficient machinery in order to complete a given activity within a shorter time interval, obviously associated with an increase in costs. The formulation provided below aims at minimizing the project’s completion date subject to an upper limit on incurred costs: Min Tc s.t. Tc − ti ≥ 0 td(a) − to(a) − da ≥ 0 X da + αak · xka ≥ d¯a
∀i ∈ I ∀a ∈ A
(1) (2) (3) (4)
∀a ∈ A
(5)
k∈K
X
fak · xka ≤ C
k∈K
where: • Tc : project’s completion date; • ti : completion date of milestone i;
(6)
IFAC MCPL 2013 September 11-13, 2013. Fortaleza, Brazil
• o(·) : milestone associated with the start of activity a; • d(·) : milestone associated with the completion of activity a; • d¯a : longest duration of activity a, associated with its least expensive execution mode; • αak : reduction in duration of activity a associated with execution mode k; • xka : decision variable that indicates whether execution mode k of activity a is selected; • fak : cost associated with execution mode k of activity a; • C : available budget to be invested in the selection of activities’ execution modes. Though it might be useful to provide a manager with insights regarding the allocation of capital and resources to different activities, the model presented above does not incorporate the potential existence of uncertainties associated with estimated activities’ durations. Though one may never claim to be absolutely confident on a given duration estimate, this is particularly true in a project’s strategic planning phase during which there are likely many yet-unknown sources of disruptions, delays and changes in scope. In that sense, one might be willing to assess the potential risks to a project’s actual performance and delivery at a given deadline. The model presented below mimics the behavior of an adversary who, given the optimal solution suggested by the deterministic model and an aggregate delay B (in this case meaning the total number of delayed activities across the project), allocates it to specific activities so as to inflict the largest damage to the project – in this case, as indicated by the latest possible completion date.
Results of the model presented above provide an indication of the potential delay to which the project may be subject to under the assumption of a total aggregate delay B and may thus also be useful in order to evaluate whether the chosen resource allocation strategy might have to be revised. That process, however, would lead to an iterative and time-consuming search for a strategy that leads to a desired performance under deterministic assumptions but also to an acceptable level of risk considering existing sources of uncertainty. In order to circumvent such difficulty, one may formulate it as a min − max problem whose objective would then be to define a capital and resource allocation strategy such that it minimizes the project’s completion date under the worst possible event of unfolding uncertainty (i.e., that which leads to a maximum delay).
Min Max
Max
(d¯a −
a∈A
s.t. X
αak
·
xka )
k∈K
· ua +
X
∆a · za
(7)
a∈A
(8) ua = 1
(9)
ua = 1
(10)
X
αak · xka ) · ua +
k∈K
X
(∆a −
a∈A
s.t. X
X
βal · yal ) · za
(14)
l∈L
(15) ua = 1
(16)
ua = 1
(17)
a∈out(0)
X a∈in(n)
X X
X
(d¯a −
a∈A
ua −
a∈in(i)
X
X
X
ua = 0
∀i ∈ {1, ..., n − 1}
(18)
a∈out(i)
za ≤ B
(19)
a∈A
za ≤ ua ∀a ∈ A X X k k fa · xa + gal · yal ≤ C k∈K
(20) (21)
l∈L
a∈out(0)
X a∈in(n)
X
ua −
a∈in(i)
X
X
ua = 0
∀i ∈ {1, ..., n − 1}
(11)
a∈out(i)
za ≤ B
(12)
a∈A
za ≤ ua
∀a ∈ A
(13)
where: • ∆a : possible increase in duration of activity a associated with the uncertainty in its original duration estimate; • za : decision variable that indicates the realization (za = 1) of an increase in the duration of activity a; • ua : decision variable that indicates the selection of activity a for the path with maximum duration. • out(i): set of arcs (activities) with source on node i. • in(i): set of arcs (activities) with destiny on node i. 440
where: • βal : decrease in uncertainty in duration of activity a related with risk-mitigating measure l with an associated cost equal to gal ; • yal : decision variable that indicates the selection (yal = 1) of risk-mitigation measure l associated with activity a; There might still be significant difficulties in solving the problem above in its current form. An alternative formulation – which allows for the application of the techniques proposed in Bertsimas and Sim [2004] – might be obtained by letting P denote the set of all paths from the start of each independent activity (i.e., those for which there are no requirements in terms of preceding activities) to the final milestone associated with project completion:
IFAC MCPL 2013 September 11-13, 2013. Fortaleza, Brazil
Min Tc s.t. Tc ≥ tp
∀p ∈ P
(22) (23) (24)
∀p ∈ P
(25)
( tp ≥ Max
X
X
d¯a −
a∈p
αak · xka +
k∈K
!) (∆a −
X
βal
·
yal )
· za
l∈L
X
fak · xka +
k∈K
X
gal · yal ≤ C
(26)
l∈L
The maximization problem on the right-hand side of constraint (24) can now be replaced by the objective function of its dual, along with the incorporation of dual feasibility constraints into the original problem, leading to a (mixed integer) linear programming problem that can be solved by commercially available solvers. 4. COMPUTATIONAL RESULTS We performed a computational test to analyze the quality of the proposed method. The test was conducted on an Intel Core i5-3360M PC with 4 cores of 2.80GHz and 8 GB of RAM. Models and algorithms were implemented using Python programming language and solved by IBM(R) ILOG(R) CPLEX(R) 12.5.0.0. The case study was based on an AoA network found on [Beasley, 2013]. The table 1 describes the network used, having the following columns: • • • • •
Activity: identifier of the activity (or arc). i: identifier of the arc source node. j: identifier of the arc destiny node. Duration: duration of the activity. Predecessors: the set of identifiers of predecessors activities. Activity
i
j
Duration
Predecessors
0 1 2 3 4 5 6 7 8 9 10 11
0 0 1 1 2 2 3 3 3 4 5 7
1 2 3 7 3 7 4 5 6 5 6 4
2 4 7 3 3 1 3 6 7 4 2 7
0 0 1 1 2, 2, 2, 3, 8, 7,
4 4 4 5 9 10
Table 1. Case study AoA network. The Figure 1 gives a graphical representation of the network, where each identifier (activity or node) is placed according with table 1. The network described has three different critical paths, each one with total duration of 18. The critical paths (sequence of activities) are: {0, 2, 8, 10, 11}, {0, 3, 9, 10, 11} and {1, 5, 9, 10, 11}. To complete the case study description, and be able to test our approach, we proposed a set of crashing measures 441
Fig. 1. Graphical representation of the AoA case study network. (or activity execution modes) and a set of risk mitigation impacts. For each measure we set a cost of one unit. The table 2 describes the measure’s parameters on each activity. Activity
α1a
0 1 6 8 9 10
1 1 1 5 4
∆a
βa1
2 5
2 3
1
1
Table 2. Case study measures. The total budget (C) was set to four units, and the total number of delayed activities across the project (B) was set to two units. The optimal solution found by our method reached a project completion time (Tc ) of 17 units, due to the following four investment decisions: faster execution modes for activities 0 and 10 (i.e., x10 = 1 and x110 = 1), and risk-mitigation of activities 1 and 6 (i.e, y11 = 1 and y61 = 1). A close look at the solution shows interesting decisions. The first one to be noticed is the investment on the risk-mitigation of activity 6, which is not part of any critical path on the original network. The second interesting decision was the no investment on a faster execution mode for the activity 9, even with the activity been part of two critical paths on the original network and with a possible decrease of 5 units on its duration. These two decisions are somewhat counterintuitive if we analyze with a traditional perspective. The key decision on the optimal solution was the investment on a faster execution mode for the activity 10. This activity is part of the three critical paths on the original network. The key decision changed the critical path to the activities 0, 2 and 6. Given this new critical path, the investment decisions on a faster execution mode for the activity 0 and on a risk-mitigation on the activity 6 are just avoiding a greater critical path duration. The no investment decision on the activity 9 is also explained by the fact that this activity is not part of the new critical path after the key decision.
IFAC MCPL 2013 September 11-13, 2013. Fortaleza, Brazil
The analysis of the case study optimal solution shows the importance of the combined decisions. The counterintuitive decisions are expected to be more frequent in large scale networks, so the importance of our method is even higher on these cases. 5. CONCLUSION AND FUTURE WORK The definition of an implementation and resource allocation strategy for large capital projects is a complex and challenging task. In this work, we discussed the application of a robust optimization approach to the strategic project planning problem which simultaneously determines activity execution modes and uncertainty-reduction measures. Results from the illustrative example provided indicate the importance of the combined approach as the optimal strategy might not always match that which would be intuitively expected or obtained under assumptions of deterministic activity durations. Current / future work focuses on incorporating the possibility of decisions to be taken along the project’s duration within an adjustable robust optimization framework and on a separation algorithm that progressively determines the paths that should be added to problem (20)-(24) instead of relying on their complete enumeration in order to allow for the solution of larger instances of the problem. REFERENCES J. E. Beasley. Network analysis: activity on arc examples, 2013. URL http://people.brunel.ac.uk/~mastjjb/ jeb/or/netmore.html. Dimitris Bertsimas and Melvyn Sim. The price of robustness. Operations Research, 52(1):35–53, 2004. Thomas H Cormen, Charles E Leiserson, Ronald L Rivest, and Clifford Stein. Introduction to algorithms. MIT press, 2001. Willard Fazar. Program evaluation and review technique. The American Statistician, 13(2):10, 1959. W. J. Gutjahr, C. Strauss, and E. Wagner. A stochastic branch-and-bound approach to activity crashing in project management. INFORMS Journal on Computing, 12(2):125–135, 2000. Thomas J. Hindelang and John F. Muth. A dynamic programming algorithm for decision cpm networks. Operations Research, 27(2):225–241, 1979. James E. Kelley. Critical-path planning and scheduling: Mathematical basis. Operations Research, 9(3):296–320, 1961. James E. Kelley, Jr and Morgan R. Walker. Criticalpath planning and scheduling. In Papers presented at the December 1-3, 1959, eastern joint IRE-AIEE-ACM computer conference, IRE-AIEE-ACM ’59 (Eastern), pages 160–173, New York, NY, USA, 1959. ACM. D. Panagiotakopoulos. A cpm time-cost computational algorithm for arbitrary activity cost functions. Infor, 15 (2):183–195, 1977. Pate-cornell and M. Elisabeth. Global risk management. Journal of Risk and Uncertainty, 12(2-3):239–255, 1996. ISSN 0895-5646. T. Raz and E. Michael. Use and benefits of tools for project risk management. International Journal of Project Management, 19(1):9 – 17, 2001. ISSN 02637863. 442
Martin Toth Walter J. Gutjahr, Christine Strauss. Crashing of stochastic processes by sampling and optimisation. Business Process Management Journal, 6(1):65 – 83, 2000. Shou Qing Wang, Mohammed Fadhil Dulaimi, and Muhammad Yousuf Aguria. Risk management framework for construction projects in developing countries. Construction Management and Economics, 22(3):237– 252, 2004.
Strategic Project Planning: Optimal Contracting and Resource Allocation Bruno Flach ∗ Carlos Raoni Mendes ∗∗ ∗ IBM Research – Brazil Av. Pasteur 138-146, Rio de Janeiro, CEP 22290-240, Brazil e-mail: [email protected] ∗∗ IBM Research – Brazil, Pontifical Catholic University – Rio de Janeiro Av. Pasteur 138-146, Rio de Janeiro, CEP 22290-240, Brazil e-mail: [email protected]
Abstract: In this work, we describe the development of a robust optimization model aimed at determining an optimal strategy for contracting and allocating resources within the context of development and implementation of capital projects. The proposed model selects a set of activity execution modes – which differ from each other in terms of related costs and expected durations – and risk-mitigation measures – associated with uncertainties due to incomplete knowledge of the activities’ exact scope – so as to balance the trade-off between incurred costs and the project’s completion date. Computational results are provided to illustrate the proposed approach. 1. INTRODUCTION Huge projects are often conducted by governments or large enterprises for the construction of the massive infrastructure required for a diverse set of economic activities – for example, those related with energy, mining, transportation and industrial applications. A number of potential risk sources coupled with resource constraints, intricate precedence relationships and multiple inter-dependencies among different sets of activities and various areas render planning, monitoring and managing such endeavors into a very complicated task. Given the large variety of involved disciplines and required skills – activities may involve expertise and know-how related with engineering, procurement, legal, environmental and logistics issues, for example – such projects are rarely conducted by a single entity. Often, either several third-party suppliers and contractors are hired to provide required materials or carry out different sets of activities – such as the construction of buildings, installation of equipment, acquisition of appropriate licenses, etc. – or they are performed under an EPC (an acronym meaning Engineering, Procurement and Construction) agreement, in which a single contractor is held responsible for designing, procuring and building the project according to the specifications defined by the owner. At times of uncertain economic conditions, pressures mount as successful project delivery might be ultimately the key to guaranteeing the economic feasibility of a company’s long-term strategic plans. Delays and cost overruns may lead to cascading impacts in the form of huge opportunity costs or even threats to the financial stability of a company, seriously damaging its ability to honor commitments and engage in revenue-generating activities. Given the circumstances, those responsible for the implementa978-3-902823-50-2/2013 © IFAC
438
tion strategy of such projects are constantly challenged to define an optimal arrangement for the allocation of available resources and third-party contracting in such a way as to ensure that projects are completed within limits of time, cost and scope. The fact that it is reasonable to assume that activities might be completed within a shorter time frame for an increased capital expenditure – naturally, up to a certain extent – related, for example, with the assignment of extra workers to a task or the use of more efficient machinery and that one is usually able to not only identify the total work required to complete a project but also to split it into different components that might be separately executed by different agents – obviously, taking into consideration eventual predecessor relationships that possibly exist among various tasks – allows for a degree of flexibility that can be exploited by managers. Within this context, the strategic project planning problem refers to the definition of the allocation of resources and selection of contractors, suppliers and vendors so as to optimize a given metric – e.g., Net Present Value (NPV) – so as to balance the project’s total cost and completion date. Note that such a decision process may be faced by either someone from the owner company who wishes to determine an implementation and outsourcing strategy to accomplish the project or by an EPC company who itself also relies on third-party suppliers and sub-contractors and aims at estimating associated costs in order to devise a more competitive tender bid. In this work, we describe the development of a robust optimization model aimed at determining an optimal project implementation strategy by allocating resources in such a way as to properly assess the trade-off between costs involved in speeding up activity durations and those that may result from delays in project completion and, pos-
10.3182/20130911-3-BR-3021.00085
IFAC MCPL 2013 September 11-13, 2013. Fortaleza, Brazil
sibly, associated penalties. The remaining of the paper is organized as follows: section 2 analyzes previous work on related problems and highlights our contributions, section 3 describes the mathematical model of the problem by first introducing a deterministic version the problem and subsequently extending it to account for the inherent existence of associated uncertainty. Section 4 presents computational results that illustrate the application of the proposed methodology. Finally, section 5 details the conclusions and directions for future work.
The assumption that activities durations are deterministic values is known to be a weak simplification of reality. The time required to complete an activity is only known with certainty after its termination, Gutjahr et al. [2000]. One of the main sources of this uncertainty are the risks associated with the project’s execution. The occurrence of a risk can impact an activity duration, if this activity is on the critical path or the impact is higher than the activity float, the project’s duration is also impacted. Even if neither of the previously described conditions happens, the project’s duration can be impacted by certain combinations of multiple risk occurrences.
2. CONTEXT
Risk management is a complete discipline per se. There are several methods to deal with project’s risks, from qualitative analytics methods (see Wang et al. [2004] and Raz and Michael [2001]) to mathematical optimization models (see Pate-cornell and Elisabeth [1996]). The risk management approach depends on how the risks impacts and mitigation strategies are modeled. In this work, we focus on risks that have impact on activities durations and that extra resources can be spent in order to minimize these impacts.
The first research works on project management methods were related with two commonly used techniques, the critical path method (CPM), and the Program (or Project) Evaluation and Review Technique (PERT), Kelley and Walker [1959] and Fazar [1959] respectively. On both techniques, the project is represented as a set of activities to be perform, and each activity has a fixed duration and a set of precedence relationships. A precedence relationship is a constraint which dictates that activity a cannot start until activity b has finished – in this case, activity b is named a predecessor of activity a. The main role of CPM and PERT is the identification of the project’s critical path, the set of activities that prevent the project from finishing earlier and for which any delay translates into a delay on the project’s completion date, Kelley [1961]. Activities not on the critical path are said to have nonnegative float, which is the maximum delay on an activity that doesn’t impact the project’s completion date. Another concept introduced by the cited methods was the representation of the activities and its precedence relations in a network (or graph) named activity-on-arc. In the activity-on-arc (AoA) network, each arc represents an activity, and each node represents an event (ex., the start or the end of an activity). An arc associated with an activity a also links two nodes (events) (i, j) with a weight equals to the duration of a. Given that circular precedence relationships are not allowed, the AoA network is a directed acyclic graph, and the critical path is the longest path on this graph, wich can be determined by a linear time algorithm (see Cormen et al. [2001]). In section 4, we show an example of an AoA network. The development of project management research gave rise to the study of different problems. An important class of these problems is the one related with the study of time-cost trade-offs when extra allocation of resources can decrease activities durations (activity crashing). One of the first investigated problems of this class was the deterministic discrete time-cost problem (DDTCP): given the AoA network of a project, a set of crashing measures (i.e., discrete 0-1 activity crashing opportunities) with its costs, a project due date and a penalty loss function for project delays, what is the optimal crashing plan that minimizes the project cost? DDTCP is known to be a NP-Hard problem, as proved by Walter J. Gutjahr [2000]. Traditional techniques such as Dynamic Programming and Branch and Bound have been applied to produce practical solutions to the DDTCP (see Panagiotakopoulos [1977] and Hindelang and Muth [1979]). 439
The main contribution of our work is to provide a robust mathematical optimization approach that deals both with crashing opportunities (or selection of activities modes) and the uncertainty associated with risks impacts on activities durations. The combined decision approach is stronger than solving the problems separately. The case study presented in section 4 reveals the complexity of a combined decision, proving the validity of our method. We describe the method on the next section. 3. MATHEMATICAL MODEL In a deterministic setting in which there are no uncertainties associated with the duration of activities, the problem might be formulated by assuming we are given all the alternative execution modes for each activity and an activity network which describes the precedence relationships among different tasks of the project. As previously mentioned, different execution modes might be related with the allocation of extra workers or more efficient machinery in order to complete a given activity within a shorter time interval, obviously associated with an increase in costs. The formulation provided below aims at minimizing the project’s completion date subject to an upper limit on incurred costs: Min Tc s.t. Tc − ti ≥ 0 td(a) − to(a) − da ≥ 0 X da + αak · xka ≥ d¯a
∀i ∈ I ∀a ∈ A
(1) (2) (3) (4)
∀a ∈ A
(5)
k∈K
X
fak · xka ≤ C
k∈K
where: • Tc : project’s completion date; • ti : completion date of milestone i;
(6)
IFAC MCPL 2013 September 11-13, 2013. Fortaleza, Brazil
• o(·) : milestone associated with the start of activity a; • d(·) : milestone associated with the completion of activity a; • d¯a : longest duration of activity a, associated with its least expensive execution mode; • αak : reduction in duration of activity a associated with execution mode k; • xka : decision variable that indicates whether execution mode k of activity a is selected; • fak : cost associated with execution mode k of activity a; • C : available budget to be invested in the selection of activities’ execution modes. Though it might be useful to provide a manager with insights regarding the allocation of capital and resources to different activities, the model presented above does not incorporate the potential existence of uncertainties associated with estimated activities’ durations. Though one may never claim to be absolutely confident on a given duration estimate, this is particularly true in a project’s strategic planning phase during which there are likely many yet-unknown sources of disruptions, delays and changes in scope. In that sense, one might be willing to assess the potential risks to a project’s actual performance and delivery at a given deadline. The model presented below mimics the behavior of an adversary who, given the optimal solution suggested by the deterministic model and an aggregate delay B (in this case meaning the total number of delayed activities across the project), allocates it to specific activities so as to inflict the largest damage to the project – in this case, as indicated by the latest possible completion date.
Results of the model presented above provide an indication of the potential delay to which the project may be subject to under the assumption of a total aggregate delay B and may thus also be useful in order to evaluate whether the chosen resource allocation strategy might have to be revised. That process, however, would lead to an iterative and time-consuming search for a strategy that leads to a desired performance under deterministic assumptions but also to an acceptable level of risk considering existing sources of uncertainty. In order to circumvent such difficulty, one may formulate it as a min − max problem whose objective would then be to define a capital and resource allocation strategy such that it minimizes the project’s completion date under the worst possible event of unfolding uncertainty (i.e., that which leads to a maximum delay).
Min Max
Max
(d¯a −
a∈A
s.t. X
αak
·
xka )
k∈K
· ua +
X
∆a · za
(7)
a∈A
(8) ua = 1
(9)
ua = 1
(10)
X
αak · xka ) · ua +
k∈K
X
(∆a −
a∈A
s.t. X
X
βal · yal ) · za
(14)
l∈L
(15) ua = 1
(16)
ua = 1
(17)
a∈out(0)
X a∈in(n)
X X
X
(d¯a −
a∈A
ua −
a∈in(i)
X
X
X
ua = 0
∀i ∈ {1, ..., n − 1}
(18)
a∈out(i)
za ≤ B
(19)
a∈A
za ≤ ua ∀a ∈ A X X k k fa · xa + gal · yal ≤ C k∈K
(20) (21)
l∈L
a∈out(0)
X a∈in(n)
X
ua −
a∈in(i)
X
X
ua = 0
∀i ∈ {1, ..., n − 1}
(11)
a∈out(i)
za ≤ B
(12)
a∈A
za ≤ ua
∀a ∈ A
(13)
where: • ∆a : possible increase in duration of activity a associated with the uncertainty in its original duration estimate; • za : decision variable that indicates the realization (za = 1) of an increase in the duration of activity a; • ua : decision variable that indicates the selection of activity a for the path with maximum duration. • out(i): set of arcs (activities) with source on node i. • in(i): set of arcs (activities) with destiny on node i. 440
where: • βal : decrease in uncertainty in duration of activity a related with risk-mitigating measure l with an associated cost equal to gal ; • yal : decision variable that indicates the selection (yal = 1) of risk-mitigation measure l associated with activity a; There might still be significant difficulties in solving the problem above in its current form. An alternative formulation – which allows for the application of the techniques proposed in Bertsimas and Sim [2004] – might be obtained by letting P denote the set of all paths from the start of each independent activity (i.e., those for which there are no requirements in terms of preceding activities) to the final milestone associated with project completion:
IFAC MCPL 2013 September 11-13, 2013. Fortaleza, Brazil
Min Tc s.t. Tc ≥ tp
∀p ∈ P
(22) (23) (24)
∀p ∈ P
(25)
( tp ≥ Max
X
X
d¯a −
a∈p
αak · xka +
k∈K
!) (∆a −
X
βal
·
yal )
· za
l∈L
X
fak · xka +
k∈K
X
gal · yal ≤ C
(26)
l∈L
The maximization problem on the right-hand side of constraint (24) can now be replaced by the objective function of its dual, along with the incorporation of dual feasibility constraints into the original problem, leading to a (mixed integer) linear programming problem that can be solved by commercially available solvers. 4. COMPUTATIONAL RESULTS We performed a computational test to analyze the quality of the proposed method. The test was conducted on an Intel Core i5-3360M PC with 4 cores of 2.80GHz and 8 GB of RAM. Models and algorithms were implemented using Python programming language and solved by IBM(R) ILOG(R) CPLEX(R) 12.5.0.0. The case study was based on an AoA network found on [Beasley, 2013]. The table 1 describes the network used, having the following columns: • • • • •
Activity: identifier of the activity (or arc). i: identifier of the arc source node. j: identifier of the arc destiny node. Duration: duration of the activity. Predecessors: the set of identifiers of predecessors activities. Activity
i
j
Duration
Predecessors
0 1 2 3 4 5 6 7 8 9 10 11
0 0 1 1 2 2 3 3 3 4 5 7
1 2 3 7 3 7 4 5 6 5 6 4
2 4 7 3 3 1 3 6 7 4 2 7
0 0 1 1 2, 2, 2, 3, 8, 7,
4 4 4 5 9 10
Table 1. Case study AoA network. The Figure 1 gives a graphical representation of the network, where each identifier (activity or node) is placed according with table 1. The network described has three different critical paths, each one with total duration of 18. The critical paths (sequence of activities) are: {0, 2, 8, 10, 11}, {0, 3, 9, 10, 11} and {1, 5, 9, 10, 11}. To complete the case study description, and be able to test our approach, we proposed a set of crashing measures 441
Fig. 1. Graphical representation of the AoA case study network. (or activity execution modes) and a set of risk mitigation impacts. For each measure we set a cost of one unit. The table 2 describes the measure’s parameters on each activity. Activity
α1a
0 1 6 8 9 10
1 1 1 5 4
∆a
βa1
2 5
2 3
1
1
Table 2. Case study measures. The total budget (C) was set to four units, and the total number of delayed activities across the project (B) was set to two units. The optimal solution found by our method reached a project completion time (Tc ) of 17 units, due to the following four investment decisions: faster execution modes for activities 0 and 10 (i.e., x10 = 1 and x110 = 1), and risk-mitigation of activities 1 and 6 (i.e, y11 = 1 and y61 = 1). A close look at the solution shows interesting decisions. The first one to be noticed is the investment on the risk-mitigation of activity 6, which is not part of any critical path on the original network. The second interesting decision was the no investment on a faster execution mode for the activity 9, even with the activity been part of two critical paths on the original network and with a possible decrease of 5 units on its duration. These two decisions are somewhat counterintuitive if we analyze with a traditional perspective. The key decision on the optimal solution was the investment on a faster execution mode for the activity 10. This activity is part of the three critical paths on the original network. The key decision changed the critical path to the activities 0, 2 and 6. Given this new critical path, the investment decisions on a faster execution mode for the activity 0 and on a risk-mitigation on the activity 6 are just avoiding a greater critical path duration. The no investment decision on the activity 9 is also explained by the fact that this activity is not part of the new critical path after the key decision.
IFAC MCPL 2013 September 11-13, 2013. Fortaleza, Brazil
The analysis of the case study optimal solution shows the importance of the combined decisions. The counterintuitive decisions are expected to be more frequent in large scale networks, so the importance of our method is even higher on these cases. 5. CONCLUSION AND FUTURE WORK The definition of an implementation and resource allocation strategy for large capital projects is a complex and challenging task. In this work, we discussed the application of a robust optimization approach to the strategic project planning problem which simultaneously determines activity execution modes and uncertainty-reduction measures. Results from the illustrative example provided indicate the importance of the combined approach as the optimal strategy might not always match that which would be intuitively expected or obtained under assumptions of deterministic activity durations. Current / future work focuses on incorporating the possibility of decisions to be taken along the project’s duration within an adjustable robust optimization framework and on a separation algorithm that progressively determines the paths that should be added to problem (20)-(24) instead of relying on their complete enumeration in order to allow for the solution of larger instances of the problem. REFERENCES J. E. Beasley. Network analysis: activity on arc examples, 2013. URL http://people.brunel.ac.uk/~mastjjb/ jeb/or/netmore.html. Dimitris Bertsimas and Melvyn Sim. The price of robustness. Operations Research, 52(1):35–53, 2004. Thomas H Cormen, Charles E Leiserson, Ronald L Rivest, and Clifford Stein. Introduction to algorithms. MIT press, 2001. Willard Fazar. Program evaluation and review technique. The American Statistician, 13(2):10, 1959. W. J. Gutjahr, C. Strauss, and E. Wagner. A stochastic branch-and-bound approach to activity crashing in project management. INFORMS Journal on Computing, 12(2):125–135, 2000. Thomas J. Hindelang and John F. Muth. A dynamic programming algorithm for decision cpm networks. Operations Research, 27(2):225–241, 1979. James E. Kelley. Critical-path planning and scheduling: Mathematical basis. Operations Research, 9(3):296–320, 1961. James E. Kelley, Jr and Morgan R. Walker. Criticalpath planning and scheduling. In Papers presented at the December 1-3, 1959, eastern joint IRE-AIEE-ACM computer conference, IRE-AIEE-ACM ’59 (Eastern), pages 160–173, New York, NY, USA, 1959. ACM. D. Panagiotakopoulos. A cpm time-cost computational algorithm for arbitrary activity cost functions. Infor, 15 (2):183–195, 1977. Pate-cornell and M. Elisabeth. Global risk management. Journal of Risk and Uncertainty, 12(2-3):239–255, 1996. ISSN 0895-5646. T. Raz and E. Michael. Use and benefits of tools for project risk management. International Journal of Project Management, 19(1):9 – 17, 2001. ISSN 02637863. 442
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