Project Management and Resource Constrained Scheduling Using An Integer Programming Approach Héctor R. Sandino and Viviana I. Cesaní Department of Industrial Engineering University of Puerto Rico Mayagüez, Puerto Rico 0068 Mario Padrón and Gustavo Barrantes Lucent echnologies CA South & Caribbean Abstract Earned Value Project Management is a dynamic methodology that brings early warning signals related with the project performance. his allows managers to take advantage and make project adjustments that will reduce the variance with respect to the original budget. Under this approach, it is very important to develop an explicit methodology that helps project managers to assign additional resources in enough quantity to complete the critical tasks that are delayed, thus, keeping the project profitable and satisfying customer requirements. he present research proposes an integrated technique that includes Integer Programming and the Earned Value approach. Keywords Project Management, Resource-Constrained Scheduling, Earned Value, and Integer Programming.. Introduction Since the initiation of the industrial revolution, the world has seen an amazing expansion of the organizations in size and complexity. Small businesses have evolved into big corporations. he difficulty of the old problems allowed solving them relying on experience and knowledge. Nowadays operational issues are so complex that problem solving requires usually software tools and other types of support systems. he traditional management techniques usually explore and control short-term projects and do not have the appropriate criteria to ensure a successful project completion. Project management techniques on the other hand offer companies the opportunity to achieve long-term goals in order to reach a continuing growth and competitiveness. Project management is required if projects are complex and under demanding constraints of time, cost, resources and environmental regulations. Project management is a viable approach if several activities or disciplines should be integrated, and there is the need to coordinate different functional departments. Earned Value project management arises as a result of the evolution of project management techniques. his dynamic methodology follows the project development and brings early warning signals related to project performance. It allows managers to take decisions that can reduce the variances with respect to the original budget. Usually the best way to adjust a project is by assigning additional resources to delayed tasks. Under this approach, it is very important to develop an explicit methodology to help managers assign additional resources while keeping the project profitable and satisfying customer requirements. he resource constraint project scheduling problem considered in this paper is an approach to address the difficulty of allocating the optimum number of specialized resources to delayed critical tasks in order to minimize the total project cost while keeping the project profitable and fulfilling the contracted requirements and policies.
2. Literature Review 2. Project Management Nowadays, projects are becoming progressively larger and complex in terms of physical size and cost. In the modern world, the execution of a project requires the management of scarce resources like: manpower, materials, money and machines throughout the project s life from conception to completion [Ahuja 994]. According with the Project Management Institute, there is a three-dimensional matrix that illustrates the project management model (Figure ). Each dimension represents the functions, processes and stages of project management. Every one of these dimensions is dynamically linked to the others and all are applied to the project. Project Management Functions i.e. What We Manage Scope ime Cost Quality Communications Human Resources Risk Figure. Project management model Management Processes i.e. How We Manage Plan, Organize, Execute, Monitor & Control. Project Stages (Life Cycle) i.e. When We Manage Conceive Development Execution Finish 2.2 Earned Value Project Management Earned Value is a management technique that relates resource planning with technical, cost, and schedule requirements. All tasks are planned, budgeted and scheduled in the initial phase of the project, this becoming the cost and scheduling measurement baseline. here are two major objectives in an earned value system: to encourage contractors to use effective internal cost and schedule management control systems; and to allow customers to rely on timely data produced by those systems to determine the contract development status. Using this methodology, project management can assess the schedule development and performance costs. As result project manager can forecast with some degree of confidence when the project shall be finished and how much could be the total bill, in order to make project adjustments. he first step of a project is the planning activity. It is very important to establish a baseline that allows assessing the project s development. his is a very critical step because it will affect the forecast and the decisions that will be taking to ensure a successful development. he process to outline the project baseline is:. Define the project scope. his is the step where the limits and the reach of the project are defined. 2. Plan and schedule the contracted work. 3. Estimate and authorize the required resources to accomplish each task. It implies an authorized budget. 4. Define each Control Account Plan (CAP) and its tasks using the key milestones. he Control Account Plan (CAP) is the single most important feature that distinguishes a project that employs earned value from projects that do not. his corresponds to a group of tasks and is the lowest element of the project to be controlled. Each CAP must include at least the next elements:. A discrete scope of work, typically expressed with work package tasks. 2. A time frame to complete each work package. 3. he authorized project resources. 4. he approved budget. 5. A designated team leader to manage the effort, typically called Control Account Manager.
Finally, each CAP has its own measuring method that should be selected during the first stage of the project. he specific measuring method for each CAP must be defined in according with its characteristics. Once the project is started, periodic assessments are made by managers to evaluate performance and make the necessary adjustments. Next, control accounts should be evaluated:. Budgeted cost of work scheduled (BCWS): What was the budgeted value of the work scheduled? 2. Budgeted cost of work performed (BCWP): What was the budgeted value of the work performed? 3. Actual cost of work performed (ACWP): What was the incurred cost to perform the work? When the control accounts have been calculated, the next stage is to analyse results to determine the project s variances. Projects with a negative schedule performance require a through analysis to identify possible solutions. Relevant questions in the process of analysis are: Are delayed tasks on the critical path or near to the critical path? Are delayed tasks considered a high risk against achieving the project objectives? If at least one of the answers is yes, the delayed critical tasks should be completed promptly. One approach to reach this objective is assigning overtime and/or allocating additional resources to complete these tasks at the earliest possible time. Resource assignments should be made in a responsible manner to ensure project performance while keeping the project profitable. An excessive allocation of resources will increase the project s cost. One of the possible solutions to improve the process of estimating the required resources to add to the delayed tasks is linear programming. his technique offers the best possible solution while keep the constraints related with the project. 2.3 Linear Programming Nowadays, due to the size and complexity of the companies, the efficient allocation of scarce resource becomes a difficult job. his is a suitable environment for the application of Linear Programming (LP). A linear programming problem is a special case of a mathematical programming. From an analytical perspective, a mathematical program tries to identify an extreme (i.e., minimum or maximum) point of a function, which furthermore satisfies a set of constraints like money, time, materials, etc. LP is the area of mathematical programming where both objective function and problem constraints are linear. One of its applications is the problem of assigning limited resources to complete a group of activities in the best possible way. Integer Programming is a specific class of LP, where variables are integer and the number of feasible solutions is finite. 2.4 Integer Programming Project Scheduling Model he objective of this kind of model is to assign the optimum quantity of resources to complete the activities that are part of a project. his can be one option to assigning in a responsible manner the resources that are needed to get the project closer to its baseline. Next model is based on the exact algorithm presented by albot (982). his general multi-mode resource constrained project scheduling problem was formulated as an integer programming problem where binary decision variables X jtm assume a value of if activity j operating in mode m is completed at the end of time period t; otherwise X jtm assumes a value of 0 (Constraint ). x jtm If activity j is completed at period t using mode m = 0 Otherwise () he model considers that for any activity there are a known number of different ways to do it and d jm is the processing time for activity j in mode m. Each activity must be done in one form (Constraint 2). Mj Lj = m= t= Ej x jtm j (2) Precedence constraints between pairs of activities i and j are described through a group of restrictions. If activity i is a precedent of activity j then the time when j start is at least the time after activity i is finished. For all activities i part of the precedence of activity j (P j ) (Constraint 3).
Mi Li Mj Lj txitm + ( t d jm ) x jtm 0 j, i Pj m= t= Ei m= t= Ej he quantity of resources allocated to a specific activity must be at most equal to the number of available resources. o ensure that, it is necessary to classify the resources in two groups: renewable (R) and non-renewable (n). Renewable resources are in each time interval t fixed and none depends of the resources allocated in the previous time intervals (Constraint 4). Non-renewable resources are over the project duration fixed and cannot been renewed (Constraint 5). n Mj t+ d jm r jkm x jqm R kt j= m= q= t (3) k R ; t=... (4) n Mj L j w jkm x jtm W k j= m= t= E j k n (5) Finally, the objective function will allow manager to guide the model to achieve his/her goals. he objective function could be to minimize the total project execution time (Equation 6). Mn Ln ntm m= t= En Minimize tx (6) 3. Proposed Model he proposed multi-mode integer programming model could be a useful tool to be used during the project planning and development phases. he objective of this model is to achieve on term and on budget completion of the project. he model will bring information to programming of resources and when will be executed each project s activity. 3. Variables related with the model c = Contracted time period when the project should be finished. = ime period when the project finishes (Obtained from the model execution). x jt = Number of activities of type j completed at the end of time period t. X jt = Number of total activities of type j completed until the end of time period t. 0 A = If c ' c Otherwise 0 If c B = c ' Otherwise b = If the resource r is assigned to the project r 0 Otherwise If the resource r is assigned to activity j at period t. y rjt = 0 Otherwise 3.2 Parameters included in the model = Maximum number of time periods to complete the project. CA = Penalty cost per time period incurred if the project ends after contracted time period. CB = Savings per time period obtained if the project ends before contracted time period. CL r = Cost of assigning the resource r to the project. CD r = Cost per time period of using resource r in the project. d j = Duration in time periods of the activity j. n jk = Number of resources of type k needed to complete each activity j. P i = Group of precedence activities of activity i. Q ij = Number of precedence activities i required to begin an activity j. Z j = Number of activities of type j to be completed during the project. V rt = Resource r availability at time period t.
3.3 Model equations he objective of this model is to minimize the total project cost, keeping in mind the different penalties and costs that involves a project execution. Objective function: Minimize Z = CA* A CB * B + CL * b + CD * y (7) Subject to: R r r r= t= j= r= All activities must be completed: x jt = Z j j (8) t= otal activities j completed at time period t: X jt = x j t, j (9) τ τ = ht Qij * X i( t+ D i ) Job s precedence must be fulfilled: X 0 t, i, h P (0) t+ D j Resources of type k assigned at time t: y ( n * x ) = 0 t, k, j () r k rjt τ = t J Schedule of resource r: yrjt Vrt r, t (2) j= Assignment of resource r to the project: M * b r y rjt 0 r (3) t J jk j= t= Project length: t * x + B A = (4) t= jτ last, t c 0 he proposed methodology is shown in Figure 2 and is currently been implemented in a telecommunications project with a major client. he goal is to develop tools that monitor progress, and assign resources at minimum cost, with the objective of achieving on term and on budget completion of the project. Begin J R r rjt i IP model execution Information Gathering Baseline development Software development Resource scheduling Project adjustment Project assessment IP model execution No Yes End Yes Is the end of the project? Project needs to adjust? No Figure 2. Proposed methodology
As a final step, this research will contribute with a complete evaluation of the Earned Value methodology applied to the telecommunications industry and the integer programming suitability to identify the project adjustments in a project considering the constrained resources, the magnitude of the project, and the need for integrating the company in one synergistic group that work together to achieve their goals. he way that will be used to integrate the proposed methodology is by developing computer software that executes the mathematical model bringing as a result enough information related with the project in order to make the best decisions. For modeling the project and its constraints Microsoft Project could be used since it allows using its Visual Basic for Applications module to develop the algorithms that manage the information related with the project in order to obtain the best possible solution from the integer programming model. 4. Conclusions he proposed model will be a useful tool that helps project oriented companies to achieve their goals. he appropriate scarce resource allocation is a critical activity into the project execution that ensures a successful completion. Its importance makes the resource assignment, an activity that must be made with the help from a specialized tool such as the one proposed in this paper. he next step will be to continue with the integer programming model improvement to achieve the expected goals and identify the best earned value indexes to be used as data source to run the model. In this way, project performance measurement will be used to adjust the resource schedule to obtain adjusted resource requirements for the pending activities keeping in mind the objective function of minimizing the project execution time and/or the total project cost. References. Ahuja, Hira; Dozzi, S. and Abourizk, S. (994) Project Management. echniques in Planning and Controlling Construction Projects. John Wiley & Sons, Inc. 2. Fleming, Quentin and Koppelman, Joel. (2000) Earned Value Project Management. Project Management Institute Inc. 3. Georgia Institute of echnology. (200) Industrial and Systems Engineering. http://www.isye.gatech.edu/~spyros/lp/lp.html 4. Hiller, Frederick and Lieberman, Gerald. (990). Introduction to Operations Research. McGraw-Hill Publishing Company 5. Military Standards. (200) http://www.acq.osd.mil/pm/evbasics.htm 6. NASA. (200). Earned Value Management Website. http://evm.nasa.gov/index.html 7. Project Management Institute (986). Project Management Body of Knowledge. Special Summer Issue. Vol. XVII, No. 3, Drexill Hill, NJ. 8. aha, Hamdy A. (975). Integer Programming: heory, Applications and Computations. Academic Press, Inc. 9. albot, Frederick (982) Resource-Constrained Project Scheduling with ime Resource radeoffs: the non Preemptive Case. Management Science 28, 997-20 0. albot, Frederick and Patterson J. (978). An efficient Integer Programming Algorithm with Network Cuts for Solving Resource Constrained Scheduling Problems. Management Science 24, 63-74. United States Defense. (996). Earned Value Management Implementation Guide. Washington. 2. Weglarz, Jan. (999). Project Scheduling: Recent Models, Algorithms and Applications. Kluwer s International Series.53-76 3. Wideman, R. M. (983). Cost Control of Capital Projects and the Project Cost Management System Requirements. AEW services, Vancouver, Canada.