=Paper=
{{Paper
|id=Vol-494/paper-45
|storemode=property
|title=Agent based Modeling and Simulation of Multi-project Scheduling
|pdfUrl=https://ceur-ws.org/Vol-494/masspaper8.pdf
|volume=Vol-494
|dblpUrl=https://dblp.org/rec/conf/mallow/ArauzoPLP09
}}
==Agent based Modeling and Simulation of Multi-project Scheduling==
https://ceur-ws.org/Vol-494/masspaper8.pdf
Agent-based modeling and simulation of multi-
project scheduling
José Alberto Araúzo, Javier Pajares, Adolfo Lopez- Juan Pavón
Paredes Facultad de Informática
Social Systems Engineering Centre (INSISOC) Universidad Complutense de Madrid
University of Valladolid Madrid (Spain)
Valladolid (Spain) jpavon@fdi.ucm.es
{arauzo,pajares,adolfo}insisoc.es
Abstract—There are no analytical solutions for the problem of Classical methods are based on mathematical programming
dynamic scheduling of resources for multiple projects in real- and can solve this problem when the complexity is low. And
time. Mathematical approaches, like integer programming or there are some heuristics and meta-heuristics that are able to
network based techniques, cannot describe complexity of real provide good schedules for more complex problems [9]. The
problems (multi-projects environments have many interrelated traditional scheduling and control systems propose hierarchical
elements), and have difficulties to adapt the analysis to dynamics and centralized architectures, where a classical scheduler
changes. However, this complex problem can be modeled as a system that has a global model of the multi-project
multi-agent system, where agents negotiate resources through an environment makes schedules according to the current state of
auction inspired mechanism. Agents can be used to represent
the system. Hans et al. [4] review existing literature in
projects and resources. Projects demand resources for fulfilling
their scheduled planned work, whereas resources offer their
hierarchical approaches and propose a generic project planning
capabilities and workforce. An auction inspired mechanism is and control framework for helping management to choose
used to allocate resources to projects and the price of resources between planning methods, depending on organisational issues.
emerges and changes over time depending on supply and demand But these techniques are not flexible or robust enough, and
levels in each time slot. By means of this multi-agent system, it is have difficulties to consider many real factors. In addition, real
possible to overcome most of the problems faced in multi-project environments undergo frequent changes (new resources, new
scheduling such as changes in resources capabilities, allocation
technologies) that force to modify the scheduling system. The
flexibility, changes in project strategic importance, etc.
traditional scheduling and control systems, which are based on
Keywords—agent-based modelling; agent-based simulation; hierarchical and centralized architectures, have not enough
multi-project environments; auction based resources allocation; flexibility to adapt themselves to the dynamism and complexity
project scheduling. of multi-project environments.
These issues have motivated, in last years, successive
I. INTRODUCTION proposals are appearing to improve the scheduling and control
The problem of allocating resources for multiple concurrent in a multi-project environment. The paradigm of Multi-agent
projects appears in large cases of service and manufacturing Systems (MAS) can help to find solutions, especially in cases
organizations. A paradigmatic example can be an engineering where some social behaviour emerges. This paper shows an
projects office. This organization makes different kinds of agent-based approach for online dynamic scheduling and
projects that are proposed at any time, which must be handled control in multi-project environments that takes advantage of
in a given time frame. Each project consists of a number of the ability of agents to negotiate and adapt to changing
activities (calculations, design, checks, budgeting, etc.) that are conditions. The MAS has basically two types of agents:
performed by workers and with some precedence relationships. projects managers and resources managers.
The workers can perform one or several activities according to Projects have scheduled work to be done by different
their skills. Decision makers have to reject inadvisable projects resources. Resources are endowed with some capabilities
and decide which resources will be allocated to which projects (knowledge, work force, etc.) that are needed to do the work.
and when. Projects demand resources over time and resources offer their
Previous decisions have high impact in the office’s profit. capabilities and time availability. There is an auction process,
In order to achieve strategic goals it is important to give and the price of resource-time slots emerges endogenously as a
priority to projects, and to allocate activities to the most result of supply and demand. The design of the auction process
efficient workers at the appropriate time. Because of this, uses a technique that has been proposed for distributed
before executing projects it is advisable to make a schedule that scheduling in the literature [8], [14], [11].
optimizes the allocation of resources. This agent-based approach has two distinctive aspects with
respect to other works: the integration of strategic decisions
(accept or reject new projects) and operative aspects (resource system to create new agents and monitoring the global
allocation), and the ability to manage resource flexibility. This behavior.
allows mangers to study the advisability of increasing the
flexibility of resources. A. Project Manager Agents
The next section introduces the role of agent-based Each project is associated to a Project Manager Agent. The
modeling and simulation in project scheduling. Section 3 system is considered dynamic: while some projects are being
presents the MAS for the real-time scheduling problem, which developed other projects can be included or rejected in real-
has been specified with an agent-oriented modeling language, time, which implies the creation and deletion of the
INGENIAS [10]. This has been the basis for implementing a corresponding agents.
simulation, which is described in section 4, and whose results At any instant t there are I projects in the system, each one
are discussed in section 5. Finally, section 6 presents main denoted by i. Each one is characterized by a value Vi, that can
conclusions of using this agent-based modeling and simulation be interpreted as the revenue obtained for the project, a weight
approach. wi representing the strategic importance given to the specific
project, a desirable delivery date Di, a limit delivery date Di*,
II. AGENT ORIENTED MODELING AND SIMULATION FOR which cannot be exceeded, an arrival date of the project to the
REAL-TIME SCHEDULING OF MULTIPLE PROJECTS system, Bi , and a limit answer date Ri that represents the latest
Multi-projects environments are complex and dynamic date to decide whether to accept or reject the project.
systems. They include many components and dependencies, Each project i consists of Ji activities, each one denoted by
and many changes may occur in the execution of projects. ij, where i∈{1, 2,…, I} and j∈{1, 2,…, Ji}. Every activity j of a
Moreover, projects are inherently distributed; each task may be project i is associated with a competence h(i,j). Any activity ij
completed by different resources or in different geographical with a given competence h(i,j) can be performed by a resource
locations and each project manager may be in different places. m just if m is endowed with the competence h(i,j). The duration
MAS have been shown to deal with problems of of the activity ij depends on the resource assigned to perform it.
complexity, openness (components of the system are not known The duration of activity ij in resource m is denoted as dijm. It is
in advance, can change over time, and are highly calculated according to dijm=dij/em,h(ij), where dij is the standard
heterogeneous, dynamic in project management terms), with duration of activity j of project i and em,h(ij) is the efficiency of
dynamical and unknown environments changing over time resource m to perform the competence h(i,j).
(uncertainty) and ubiquity (the activity is distributed over the This first simplified model assumes that the activities of
complete structure) [5] [12]. any project should be performed sequentially in the order
In the particular case of multi-project systems, the agents defined by j and only one resource can be assigned to an
can be abstracted as tasks, resources, project managers, etc. activity. There is also the assumption that once some resource
This design enables to distribute the management system in has begun a task, the activity cannot be interrupted; the
elemental components directly identifiable in the target system, resource needs to finish it to be assigned to any other activity.
and hence giving the opportunity to create systems easier to
design, to adapt and to maintain. Moreover, since the system is
distributed according to its structure, any change in the
structure can be easily translated to the management system.
This decentralized approach facilitates the design of market
mechanisms to solve the scheduling problem by means of
distributed approximations [2]. Recently, Lee, Kumara and
Chatterjee [7] have proposed an agent-based dynamic resource
scheduling for multiple distributed projects using market
mechanisms. Following the same research line, Confessore et
al. propose in [3] another iterative combinatorial auction
mechanism. Other examples of agent-based approaches in
project management can be found in the works of Kim and
colleagues [6], Wu and Kotak [13], and Cabac [1].
III. A MAS MODEL FOR MULTIPLE PROJECT SCHEDULING
The system can be modeled with two types of agents
representing project and resource managers. Agents have the
ability to interact with each other. In this case, it is important to
define an auction protocol for project agents to compete for the
use of resources. Resource Manager Agents interact with Figure 1. MAS organization model (with INGENIAS notation [10]). The
project agents to inform on the status, capabilities and cost at diagram shows an organization (Engineering Company), which has several
departments (Projects Office and Production Unit). The Projects Office has
each specific time. A third type of agent is included in the one Monitor Agent and several Project Manager Agents. The Production Unit
has Resource Managers that take care of the use of Resources.
B. Resource Manager Agents project beyond Di*, it will be rejected. If not, but the
A resource is modelled as a Resource Manager Agent. inclusion of the new project increases the delay costs
There are M resources, which can be assigned simultaneously of the other projects more than the direct benefit
to one activity. Each resource is endowed with a given cost rate obtained for the project, it will also be rejected.
per unit of time, cm (m ∈{1, 2, 3…M}), and a subset Hm of
competences that can be performed (H={h1, h2, ... hK} is the set A. Auction Interactions
of competences that are necessary to complete the projects). At any time, the system has as many Project Manager
Each resource has a certain grade or ability to perform a Agents as projects are ordered. Each one represents a particular
competence. Therefore, the work capacity of resources can be project characterized by its tasks, precedence relationships, due
symbolized by means of a vector of abilities per resource date, value, local programs and their execution state. Their goal
em=(em1, em2,…,emk), where emf ≥ 0 shows the ability degree of is to look for contracts with resources that can perform the
resource m to perform the competence hf. If emf = 0 then the required activities and hence completing successfully the
resource m has not the competence hf, if 0 (local schedule).
1 it will do it efficiently. The decision-making process is decentralized as it emerges
from interactions among the agents in an auction process. Each
C. Monitoring Agent project manager creates its own schedule (local schedule) by
A Monitoring Agent has the responsibility to visualize the taking into account its own project goals and its own
current state of the system to the user. Moreover, this agent knowledge. This procedure can bring incompatible local
allows the user to create new Project Manager Agentsm, as schedules (several projects try to use the same resource at the
shown in Figure 1. same moment). Moreover, the local schedules can be globally
inefficient (profitable projects are rejected; most important
projects have delays; etc). These difficulties that arise from the
IV. AGENT WORKFLOWS AND INTERACTIONS
autonomy of each agent are solved with a market mechanism
The agent workflows and interactions must be designed in that ensures that local schedules are nearly compatible and
order to maximize the global efficiency of the system, which globally efficient according to the expression (1). This auction
will be evaluated by the average benefit obtained in a certain based multi-project scheduling approach is founded on
time interval T according to: Lagrangian Relaxation [8][11][14], a decomposition technique
for mathematical programming problems.
∑ (Vi − Cost (i )) In order to apply the market metaphor, the periods when
B (1) resources are available are subdivided in a set of small time
Efficiency = T = i
T T intervals or time slots. Each time slot on each resource is
modelled as a good that can be sold in an auction, where each
for all projects i that are finished in T, Cost(i) is the cost to resource acts as a seller. Thus, a local schedule will be a bundle
complete the project i. This cost has two components, the direct of time slots that has been allocated to a project.
resource cost and the delay cost: The number of sellers is equal to the number of resources in
the system. Each resource proposes a price for the time slots
from the current time to the end of the scheduling horizon. The
d ij scheduling horizon changes dynamically by coinciding with
Cost (i ) = ∑ C m(j) ⋅ + wi ⋅ ( Di − Fi ) 2 (2)
the latest time slot that some project has asked at any moment.
j eijm
Each project agent plays the role of a bidder that
The first addend corresponds to the direct resource cost to participates in auctions by asking the Resource Manager
finish each activity j. m( j) denotes the resource selected to Agents for the set time slots that it requires to execute its
comply with activity j. The second addend is the delay cost pending tasks at the current time. It will try to find a set of time
associated with the project, where Fi is the real delivery date. slots (Zi) through the resource pool while incurring the
minimum possible local cost (LCi). This cost has two
The problem considers the decision to reject projects. This components, the sum of the price of the selected time slots and
could happen in any of the following cases: the delay cost (expression 3):
• The revenue obtained from the project does not
compensate the costs.
• The scheduling exceeds the Di* of the project. LCi = ∑ pmt + wi ⋅ ( Di − Fi ) 2 (3)
mt∈Z i
• The impact on the scheduling of the rest of the projects
is not acceptable. This may happen for two causes. where pmt is the price of the time slot (t) of the resource
First, if the new project obliges to delay a committed (m).
To select the set of time slots (Zi) that minimizes their local these agreements are obtained, project agents will never
cost, Project Manager Agents use a dynamical programming consider the tasks included as firm contracts as pending.
algorithm where all possible combinations of time slots and
resources are considered [13]. In their decision, they take into The global efficiency and the compatibility of local
account that only those resources endowed with the necessary schedules depend on the degree of convergence of market
competences can carry out a certain activity. Moreover, the prices to the equilibrium prices. If the prices get closer to the
number of time slots necessary to complete a task (duration) equilibrium price, they will be representative of the system
are determined according to the ability degree of the resource state; they will have information about any system feature and
in the competence. Each project agent will regard as scheduling local schedules will be compatible and globally efficient. If
horizon the time slot that goes from the current time to the limit agents are making firm contracts when prices are not
delivery date (Di*). If some project agent cannot find a set of representative of the system state, then incompatibilities could
time slots in such a manner that it allows to schedule tasks take part. In these cases, the agents resolve incompatibilities by
before Di*, with a smaller cost than its value (Vi), then it will means of local schedule based heuristics rules. More exactly,
not ask for any set of time slots. This implies that the project is when several activities use the same resource at the same
unprofitable at the correspondent round of bidding and must be moment, the activity that has been earliest programmed in local
schedule will have priority to be contracted in firm agreements.
rejected.
Although this heuristic does not ensure global efficiency, it will
Each Resource Manager Agent determines the price achieve perfect compatibility in final decisions.
charged for the time slots with the purpose of reducing
resource conflicts and maximizing their revenue. In order to get V. SIMULATION AND RESULTS
this goal a subgradient optimization algorithm is used to adjust
prices at each round of bidding. By means of this algorithm the The system has been implemented and simulated with
Resource Manager Agents increase the price of the time slots different scenarios. Here the analysis focuses on the role of
where there is conflict (more than one project manager has resource capabilities and the option of project rejection. The
asked for this time slot) and reduce the price of the time slots first scenario shows a simple case to illustrate the main features
that have not been demanded. The process of price adjustment of the system, in the next subsection. This is followed by a
and bid calculation continues indefinitely. At each round of dynamic scenario in order to evaluate the system performance
bidding the resource conflicts will be reduced. in evolving complex environments.
At the first round of bidding, the time slots prices for the A. Simple Case Study
resource (m) are equal to the resource cost rate (cm). At the rest
of bidding round, the prices will be updated by means of the Consider three different resources (R1, R2 and R3),
endowed with the competences C1, C2 and C3 respectively.
expression 4. αn is calculated according to [8].
TABLE I. shows a portfolio of five projects, and the tasks
needed to complete each project. Each task is defined by means
pmt
n +1
{
= max cm , pmt + α n ⋅ g mt
n n
} (4)
of the pertaining competence and expected standard time to be
completed.
Where:
TABLE I. SIMPLE CASE STUDY
n +1
• p mt is the price of the time slot (t) of resource
(m) at the round (n+1) Proj. Tasks Arrival Starting DD1 DD2 Value
n
• p mt is the price of the time slot (t) of resource (m) Task Task Task date Date
1 2 3
at the round (n) P1 C1 50 C2 25 C3 30 0 0 120 180 10000
• α n is the step at the round (n). It decreases when (n) P2 C3 40 C1 45 C3 10 0 0 180 240 12000
increases. P3 C2 35 C1 40 C2 25 0 0 120 180 30000
• And ( g
n
mt = a − 1 ) is the subgradient, where a
n
mt
n
mt
P4 C3 30 C1 50 C2 10 50 90 150 270 15000
P5 C1 45 C3 20 C1 50 50 90 150 270 30000
is the demand of slot (t) of resource (m)
The arrival date is the date when the project is included in
B. Contract Interactions the system. Projects can start-up in the starting date; otherwise,
By means of the auction mechanism described above, they should have been rejected before this date. Due Date 1
project agents build compatible and globally efficient local (DD1) is the most desirable duration whereas Due Date 2
schedules for their pending activities. Moreover, at the same (DD2) is the maximum allowed. All the projects have a weight
time, agents interact through a complementary process to make of 1.
firm agreements based on the local schedules that have been Figures 2 and 3 show the system state at a given time
created by means of the auction process. These agreements (current time). In the upper area of the figures the relative
determine fixed programs for earliest scheduled tasks. When duality gap evolution is presented. The prices of time slots are
the solution of the dual problem and the duality gap is a
measure of the difference between the primal and dual Note that project P5 has been rejected although it has a high
objective function, so it quantifies the quality of the solution value, because its value was not available at time 0, when
[8]. The relative duality gap is calculated as the duality gap projects P1, P2, P3 were waiting to start-up. The calculus of the
divided by the dual solution. A small relative duality gap payment that projects have done for time slots (TABLE II. )
means that the prices are representative of the system state, shows that the same projects do a payment higher than their
thus, a good solution is achieved. The lower part of the figures values. When projects P4 and P5 arrive at the system the prices
present charts of resources. These charts show the tasks that of time slots of the resource R1 grow because P5 is able to pay
each resource has performed until the current time (lower area higher prices to be performed. Although P5 accepts higher
of the resource charts) and the time slot prices (upper area of prices than other projects, P1, P2 and P3 cannot be rejected and
the resource chart). The time slots prices previous to current finally they must pay the market prices. The final total value
time are the prices when agents were doing firm agreements for (BT=total values of performed projects minus total delay cost)
those time slots. The prices later than current time are the is 55700.
estimated prices in the current round of the auction.
Figure 2. shows the system state at the moment 45, just TABLE II. PAYMENT FOR TIME SLOTS PER PROJECT
before projects P4 and P5 arrive at the system. When the first Project Value Total payment
projects (P1, P2 and P3) are included in the system, the duality P1 10000 12719
gap is high, indication of a bad solution. But then, the price P2 12000 11414
formation mechanism makes the prices to stabilise, and the gap P3 30000 8298
becomes smaller. This means that prices are close to P4 15000 6887
equilibrium. P5 0 0
The simulation not only gives the dynamic schedule and the
refused projects, but the value of each resource as well. For
instance, in Figure 3. the prices of resource R1 are very high
during all time slots. This means that the resource competence
is very valuable (bottleneck), so if the firm is going to be
engaged in similar projects in the nearby future, it would be
useful to include more resources with the same competences.
On the other hand, prices of resources R2 and R3 are small,
although they are working on different tasks during the
simulation.
So, the possibility of enhancing the range of capabilities of
Figure 2. System state at the moment 45
resources R2 and R3 should be considered; for instance, in the
case of human resources, this can be done by means of
Figure 3. shows the evolution of the tasks performed by training.
each resource and the prices of the time slots after finishing the
simulation. This figure shows how the duality gap increases
when the projects P4 and P5 arrive to the system. At this
moment previous prices did not reflect the new system state
(new projects are in the system). After some time, the prices
change to adapt themselves to the new system conditions, and
the gap decreases again. It can be observed that the new prices
are very different from previous prices. This happens especially
in resource R1 where prices are very high.
Figure 4. Tasks performed by resources (competences of resource R2
increased).
Figure 4. shows the evolution of the system when the
resource R2 is also endowed with the competence C1 with
efficiency 0.8. Compared with the previous case, now the price
range is lower for resource R1 and higher for R2. Although the
duration of task T22 of project P2 is smaller in resource R1
than in R2, now the system have to reallocate in real-time this
activity to R2. So, R1 can perform in time the task T51 of the
project P5. In this experiment, the project P5 is accepted and
Figure 3. Tasks performed by resources. Tij denotes Task j of project Pi. executed, and the total value has been increased from 55700 to
69369. This shows that the system is capable to use the
flexibility of resource R2 to improve in real-time the global
performance.
B. Complex Dynamic Scenario
In order to check the system performance in very dynamic
environments, consider 12 projects (table 3) that arrive at the
system every 20 units of time (first P1, second P2, …, and
finally P12). In TABLE III. DD1 and DD2 are relating to
starting date. Resources and competences are similar to the
previous case study.
Figure 5. Total value in complex experiments
TABLE III. DYNAMIC PORTFOLIO OF PROJECTS (COMPLEX SCENARIO)
Proj. Tasks DD1 DD2 Value
Task 1 Task 2 Task 3 VI. CONCLUSIONS
P1 C1 50 C2 25 C3 30 60 150 10000 Although project management literature has been mainly
P2 C3 40 C1 45 C3 10 60 120 15000 concerned with managing individual projects, in practice firms
P3 C2 35 C1 40 C2 25 60 120 6000 usually work in dynamic and complex multi-project
P4 C3 30 C1 50 C2 10 90 120 7000 environments.
P5 C1 45 C3 20 C1 50 60 120 8000 We propose a multi-agent system and an auction
P6 C3 10 C2 45 C1 20 120 150 7000 mechanism for online dynamic scheduling in multi-project
P7 C1 20 C2 10 C3 30 60 150 15000 environments. Projects have tasks to be completed, so they
P8 C3 40 C1 45 C3 50 90 120 10000 compete for the resources endowed with the capabilities
P9 C2 35 C1 10 C2 45 60 120 15000 required to do some pieces of work. The prices of resources
P10 C3 30 C1 15 C2 10 60 120 10000 emerge endogenously by means of an auction process.
P11 C1 15 C3 50 C1 10 90 150 7000 We show some of the possibilities of this multi-agent
P12 C3 35 C2 50 C1 20 60 120 12000 approach to deal with some of the decisions that managers need
to take within multi-project environments. The system allocates
dynamically resources to projects, and decides what projects to
We have done several simulations by changing two types of accept or reject taking into account project value, profitability
parameters: the response period and the set of competences of and (feedback) operational information. We also show how it is
resources. The response period is the time interval between the possible to discover which resources are the most valuable, so
arrival date and the starting date. During this period, projects they should be added to the firm.
wait in the system for rejection or acceptance decision. If this
period is long, more projects are waiting for decision This approach contributes to fill the gap between the
simultaneously, so decisions will be more efficient. literature in portfolio project management (usually focused on
corporate strategy and finance) with the work in multi-project
We have simulated three competence distribution cases: management (mainly concerned with operational issues,
case A (R1 has the competence C1, R2 the C2, R3 the C3), scheduling and resource allocation).
case B (R1 C1, R2 C1 and C2, R3 C3), and case C (R1 C1, R2
C1 and C2, R3 C2 and C3).
Figure 5. shows the total values obtained in different ACKNOWLEDGMENTS
experiments. Each curve represents the value variation for
cases (A, B and C) when the response period increases. Note This work has been done in the context of the following
that the system efficiency is higher when the response period projects: (1) “Agent-based Modelling and Simulation of
increases and when the resources are more flexible (they have Complex Social Systems (SiCoSSys)”, supported by Spanish
more competences). This shows that in this scenario the system Council for Science and Innovation, with grants TIN2008-
performance is suitable; the software is able to manage 06464-C03-01 and TIN2008-06464-C03-02; (2) “ABACO
complexity to improve the global efficiency. VA006A09”, (3) the Programa de Creación y Consolidación de
Grupos de Investigación UCM-BSCH GR58-08, and (4)
GR251/09 supported by the “Junta de Castilla y Leon”.
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