=Paper=
{{Paper
|id=Vol-2851/paper9
|storemode=property
|title=The Criteria for Choosing the Optimal Solution Under the Uncertainty in Project Management
|pdfUrl=https://ceur-ws.org/Vol-2851/paper9.pdf
|volume=Vol-2851
|authors=Tetiana Shestakevych,Oleksandr Volkov
|dblpUrl=https://dblp.org/rec/conf/itpm/ShestakevychV21
}}
==The Criteria for Choosing the Optimal Solution Under the Uncertainty in Project Management==
https://ceur-ws.org/Vol-2851/paper9.pdf
The Criteria for Choosing the Optimal Solution Under the
Uncertainty in Project Management
Tetiana Shestakevycha, Oleksandr Volkovb
a
Lviv Polytechnic National University, Lviv, Ukraine
b
Argon Corporation, 160 Great Neck Rd, Great Neck, NY 11021, USA
Abstract
Thoughtful, well-organized cooperation of specialists in intersectoral projects has an
undoubted synergetic effect. At the same time, the support of such a complex project
involves the thorough involvement of business analysts. To model the process of selecting
additional functions of the information technology, it is convenient to use Petri nets to
visualize both parallel and sequential processes. The use of methods of expert evaluation,
decision-making, decision-making under uncertainty to support the work of business analysts
will allow taking into account various criteria of such intersectoral cooperation. As a criteria
of decision-making under uncertainty, the Laplace, Wald, Bayes, Hermeyer, Savage,
Hurwitz, Khodg-Lehmann criteria were analyzed, as well as the criteria of extreme optimism,
extreme pessimism, compromise criterion, and product criterion.
Keywords 1
Project management, decision under uncertainty, business process, business analyst, Petri
nets, analytic hierarchy process, AHP
1 Introduction
The pandemic influenced different spheres of our life. Fighting the damages it caused in medicine,
economics, psychology, education, wellbeing, etc., we must not concentrate on negative consequences
only, but on the very reasons why our society was not ready for such cataclysm. Humanity is forced to
reconsider the management in such almost canonical processes as education, science, medicine, etc.
Public administration should shift the focus from managing each direction separately to complex
vision, and convert sporadic practices of intersectoral projects on the state level into solid politics of
interconnectional development, thoroughly planned and planned as such from the very beginning of
the cooperation [1-8]. No doubt, that information technologies should be the mean – basic and
powerful – of such intersectoral cooperation.
It seems convenient to explain the benefits and difficulties of such top-level intersectoral project
management on the example of education of disabled students. The very beginning of such
cooperation is on the stage of assessment of the psychophysical conditions of a child. In Ukraine, it is
a duty of state health institution, which is called Psychological, Medical, and Pedagogical Committee
(here and after, PMP Committee). The specialists of such committee should perform the
psychophysical diagnostics of a child, make a conclusion about its condition, suggest a form of
education (normal, inclusive, special), develop a treatment strategy (if needed) [9]. Until recently, the
only information technology that supported the activity of the PMP Committee was a website with
basic information on working hours, staff, general information on various disabilities. The internal
medical documentation was in paper. Before 2020, creating an information technology to automate
internal document flow in PMP Committee seemed to be an actual practical task, but it is not
Proceedings of the 2nd International Workshop IT Project Management (ITPM 2021), February 16-18, 2021, Slavsko, Lviv region, Ukraine
EMAIL: Tetiana.v.shestakevych@lpnu.ua (A. 1), volkov.sawa@gmail.com (А. 2)
ORCID: 0000-0002-4898-6927 (A. 1), 0000-0001-8961-679X (А.2)
©️ 2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
anymore. To support the processes in PMP Committee now, it is crucial to involve representatives
from different branches [9], that are involved in processes of support of disabled persons` lifelong
wellbeing.
Traditionally, designing a health care institution management software, developers are led by state
standards and protocols of health care. As input data, such software needs patient ID, screening and
diagnosing results, etc. These data are saved, can be looked through, and printed out in a form of a
history of a disease. To order such software (an information technology, actually), in Ukraine, the
public health care organization should use an open electronic system of public procurement ProZorro
(https://prozorro.gov.ua/). This resource is an online platform where state and municipal customers
announce tenders for the purchase of goods, works, and services, and business representatives
compete in tenders for the opportunity to supply it to the state. From the point of view of the PMP
Committee needs, in such a way the state could have ensured the cooperation between the state health
care organization and IT company. But such cooperation between these three subjects is rather
narrowly focused from the side of the state, at least. Such cooperation is rather limited in its
variations: the state health organization gets the IT to automate some processes, and the state pays for
it. But this fact can be changed by adding one more facet to such cooperation, i.e. the scientific
approach. In fig 1, there’s a concept of such cooperation, enriched with possibilities that enable an
application of data science to the cooperation of IT, state, and medicine.
Figure 1. The concept of cooperation between Data Science, IT, state, and medicine
To achieve the synergetic effect from such cooperation, it should be well organized. On the
opposite of the traditional order of IT development which ends with the IT implementation, the
proposed scheme foresees constant cooperation and constant improvement of the outcome (black-
colored edges). By participating in such cooperation, the IT company/ies should provide the project
managers of such complex process, and involve business analyst (or a group of one) to control a lot of
business processes. It is worth mentioning that the business process is understood as one that enables
some profit, expressed in money or time savings (any other kind of profit can be measured in time or
money units, as well). The main goal of this research is to suggest an approach to the management of
the projects, which should result in an IT that has two layers of functions, i.e. functions that should be
realized unconditionally, and functions that are additional, and the very set of the functions, the
sequence of its realization, and their effectiveness may vary. If we go back to the example of PMP
Committee example, the must-have function of the IT, developed to automate the processes in the
Committee, are those that enable the document flow according to state health care standards. The
additional functions are those that can the PMP Committee specialists suggest with their experience of
working with disabled children, and, of course, those functions that might suggest the data analysts,
according to diagnostics results they would be provided [10], [11]. For example, some data science
methods can analyze, which diagnosing procedures can be odd, and the decision on the diagnosis can
be made on less number of characteristics of a child.
2 Modeling the process of selecting additional functions
Let us recall that during the intersectoral cooperation, mentioned above, demands that the
developed information technology should unconditionally have basic functions, realized according to
state standards. The list of additional functions can be formed during a cooperation between all
participants. Additional functionality of such information technology might be, for example, the
following.
• Availability of methods of data and knowledge analysis extraction based on existing results of
psychophysical diagnostics.
• Availability of methods of data and knowledge analysis, based on which it is advisable to
update the protocol of psychophysical diagnosis.
• Automation of the PMP Committee activity.
• Development of a national system for consolidating the results of psychophysical diagnosis;
• Adaptation of IT to the national system of support for lifelong learning.
• Automate compliance with IT requirements used by people with special needs, etc.
Collaborators mentioned in Fig.1, can evaluate the usefulness of each of the proposed functions,
and the use of optimization methods will support the decision-making process of the business analyst.
The model of such an organized decision-making process in project management is conveniently
presented with Petri nets, which will allow visualizing sequential and parallel processes. Petri nets
have proved to be a convenient tool for modeling the learning processes of people with special needs,
maintenance, flows of an online store, basic cyber-physical attack models [12]-[14], etc.
The Petri net С=(Р, Т, І, О) models the process of decision making in choosing additional
functions (Fig. 2), where the set of positions Р={р1, р2, …, р8}, the set of transitions Т={t1, t2, t3};
initial marking μ0 is one chip in position р1.
Figure 2. The process of decision making in choosing additional functions
Transitions in the given Petri net are events and are interpreted as processes (Table 1).
Table 1
Transitions in a model of decision making in choosing additional functions
Transitions Meaning of transitions
t1 Make an expert evaluation
t2 Range the efficiency of the additional functions
t3 Choose the best strategy of the realization of additional functions
Positions in the given Petri net can be interpreted as a condition of event occurrence (Table 2).
Table 2
Transitions in a model of decision making in choosing additional functions
Positions Meaning of position
p1 The alternatives of the additional functions
p2 The criteria of efficiency evaluation of the additional functions
p3 The assessments of the alternatives according to the criteria (by data scientists)
p4 The assessments of the alternatives according to the criteria (by IT specialists)
p5 The assessments of the alternatives according to the criteria (by state)
p6 The assessments of the alternatives according to the criteria (by medics)
p7 A set of alternative strategies of the realization of additional functions
p8 A strategy of the realization of additional functions
In the process of expert evaluation of each of the additional functions that can improve basic IT,
you can use, for example, the method of analytic hierarchy process (AHP). This method has been used
for expert evaluation in various fields. For example, to model the delineation of artificial groundwater
recharge potential regions, to model the selection of riparian protected area stretches [15]-[17].
In AHP, the evaluated alternatives are additional functions (mentioned before), and the evaluation
criteria are the usefulness according to the assessments of experts (the representatives of the
intersectoral cooperation process from Fig. 1). As a result of the application of the basic stages of
AHP we will receive estimations of the usefulness of alternatives (at Fig. 3 there is an example of
such estimations, where three alternatives are estimated by four experts). To obtain such estimations,
the eigenvectors of each of the pairwise comparison matrices are multiplied by 10 and rounded to the
integers.
7 2 4 3
2 6 5 3
1 2 1 4
Figure 3. Estimations of alternatives usefulness (using AHP)
To decide which of the alternatives should be realized, we propose to apply optimization methods,
namely methods of game theory. Such a decision is made under conditions of uncertainty because
even the most experienced experts do not always rationally estimate the magnitude of expected
benefits or losses. Therefore, the problem presented by the matrix in Fig. 3 we propose to understand
as a problem of game theory, namely games with nature, where decisions are made in conditions of
uncertainty [18]. Several criteria have been developed for decision-making in conditions of
uncertainty, which we will consider below.
3 Criteria for decision-making in conditions of uncertainty
In decision-making under uncertainty, the information required for decision-making is presented in
the form of a matrix Q = qij , i = 1,..., n, j = 1,..., m with rows S = {Si }, i = 1,..., n and
columns V = {V j }, j = 1,..., m . According to the analyzed example (Fig. 3), the rows S i of such a
matrix are the strategies of implementation of additional functions in information technology. The
columns V j of the matrix correspond to the conditions under which the corresponding additional
functions will be implemented. Thus, the elements q ij of the matrix are the numerical values of the
usefulness of the corresponding strategy implementation (Fig. 4). We will consider the best strategy
as one that is the most useful. The choice of such a strategy is complicated by a large number of
alternatives and the need to take into account many conditions, which affect the usefulness of
additional IT functions. Such conditions can be, for example, to promote the development of each
particular industry, represented by the participant of the intersectoral cooperation process.
V
V1= of state medicine
analysis> development> development>
S1 7 2 4 3
S2 2 6 5 3
S3 1 2 1 4
Figure 4. Alternatives (strategies) of realization of additional functions
We will choose one of the alternatives according to different criteria C: the criteria of Laplace,
Wald, Bayes, Hermeyer, Savage, Hurwitz, Khodg-Lehmann, as well as the criteria of extreme
optimism, extreme pessimism, compromise criterion, and product criterion [19]-[24]. Each of the
criteria has certain features and characteristics.
3.1 Wald criterion СW
The criterion is considered a pessimistic criterion, because it focuses on the worst result, thus
reducing the risks of decision-making. For each of the alternatives find the values of least usefulness,
and choose the maximum of the values found. Wald criterion is formalized by formula (1):
CW = max min qij , i = 1,..., n, j = 1,..., m . (1)
i j
Finding the best alternative by Wald's criterion for the task in Fig. 4, calculations are given in Fig.
5.
V1 V2 V3 V4 min q ij
j
S1 7 2 4 3 2
S2 2 6 5 3 2
S3 1 2 1 4 1
Figure 5. Calculations for decision-making according to the Wald criterion
According to the Wald criterion, it is advisable to choose alternatives S1 or S2.
3.2 The criterion of extreme optimism CX
The criterion involves the choice of the alternative with the highest efficiency, the criterion does
not take into account possible risks. To find the best alternative by this criterion, choose the maximum
of the found maximum efficiency values for each alternative. The criterion of extreme optimism is
formalized by formula (2):
C Х = max max qij , i = 1,..., n, j = 1,..., m . (2)
i j
Finding the best alternative by the criterion of extreme optimism for the task in Fig. 4, calculations
are given in Fig. 6.
V1 V2 V3 V4 max q ij
j
S1 7 2 4 3 7
S2 2 6 5 3 6
S3 1 2 1 4 4
Figure 6. Calculations for decision-making according to the criterion of the extreme optimism
According to the criterion of extreme optimism, it is advisable to choose an alternative S1.
3.3 The criterion of extreme pessimism CP
The criterion involves choosing the alternative with the most pessimistic result. Only the worst
possible result is taken into account. To do this, choose the minimum value among the smallest for
each strategy, which is formalized by formula (3):
C Р = min min qij , i = 1,..., n, j = 1,..., m . (3)
i j
Finding the best alternative by criterion of extreme pessimism for the task in Fig. 4, calculations
are given in Fig. 7. According to the criterion of extreme pessimism, it is advisable to choose an
alternative S3.
V1 V2 V3 V4 min q ij
j
S1 7 2 4 3 2
S2 2 6 5 3 2
S3 1 2 1 4 1
Figure 7. Calculations for decision-making according to the criterion of extreme pessimism
3.4 Product criterion СM
This criterion for decision-making in conditions of uncertainty is mentioned in [19]. According to
this criterion, the strategy with the largest product of efficiency values is chosen (all values of the
efficiency must be positive). The criterion is formalized by formula (4):
n
C М = max q ij , i = 1,..., n, j = 1,..., m . (4)
i i =1
Finding the best alternative by product criterion for the task in Fig. 4, calculations are given in Fig. 8.
V1 V2 V3 V4 n
qij
i =1
S1 7 2 4 3 168
S2 2 6 5 3 180
S3 1 2 1 4 8
Figure 8. Calculations for decision-making according to product criterion
According to the product criterion, it is advisable to choose an alternative S2.
3.5 The criterion of compromise СК
It is believed that the criterion makes it possible to choose a compromise strategy between
optimistic and pessimistic. According to this criterion, the strategy with the highest value of the
average between the maximum and minimum value of utility is selected. The criterion is formalized
by formula (5):
max qij + min qij
CК = max , i = 1,..., n, j = 1,..., m .
j j
(5)
i 2
Finding the best alternative by the criterion of compromise for the task in Fig. 4, calculations are
given in Fig. 9. According to the criterion of compromise, it is advisable to choose an alternative S1.
V1 V2 V3 V4 max qij + min qij
j j
2
S1 7 2 4 3 4,5
S2 2 6 5 3 4,0
S3 1 2 1 4 2,5
Figure 9. Calculations for decision-making according to the criterion of compromise
3.6 Hurwitz criterion СН
The criterion makes it possible to indicate how close to the optimistic or pessimistic development
the decision-maker is inclined. To do this, we use a coefficient λ called the coefficient of pessimism
(realism), which is given within [0, 1]. In the case of λ = 1, the decision maker is prone to pessimism,
and when λ = 1 to optimism. The difficulty of applying this criterion is in the need to determine the
coefficient λ. The criterion is formalized by formula (6):
CН = max min qij + (1 − ) max qij , 0,1, i = 1,..., n, j = 1,..., m . (6)
i j
j
j
Finding the best alternative by Hurwitz criterion for the task in Fig. 4, calculations are given in
Fig. 10. According to the criterion of Hurwitz criterion, it is advisable to choose an alternative S1.
V1 V2 V3 V4 min qij + (1 − ) max qij
j
j
j
S1 7 2 4 3 5,5
S2 2 6 5 3 4,8
S3 1 2 1 4 3,1
Figure 10. Calculations for decision-making according to Hurwitz criterion (with λ=0,3)
3.7 Laplace criterion CL.
It is used in case of insufficient information about the event to be decided, and the probability of
occurrence of each condition Vi will be considered the same. Then the optimal alternative is
determined by the maximum average efficiency values (7).
1 m
CL = max qij , i = 1,..., n . (7)
i m j =1
Finding the best alternative by Laplace criterion for the task in Fig. 4, calculations are given in Fig.
11. According to the Laplace criterion, it is advisable to choose alternatives S1 or S2.
V1 V2 1 4V3 V4
qij
4 j =1
S1 7 2 4 3 4
S2 2 6 5 3 4
S3 1 2 1 4 2
Figure 11. Calculations for decision-making according to Laplace criterion
3.8 Bayesian criterion CB.
In contrast to the Laplace test, where the probabilities of conditions Vi are equal, the Bayesian
criterion is based on the assumption that such probabilities are different. This fact often indicates that
the conditions under which decisions are made are sufficiently studied, and it is possible to indicate
т
the corresponding probabilities рі of occurrence of each of the conditions, and p j = 1 . According
j =1
to the Bayesian criterion, the alternative for which the mathematical expectation is higher is also
considered optimal. The criterion is formalized by formula (8):
m т
CB = max p j qij , i = 1,..., n , p j [0,1] , p j = 1 . (8)
i j =1 j =1
Finding the best alternative by Wald's criterion for the task in Fig. 4, calculations are given in Fig.
12. Here, the vector of probabilities of occurrence of each of the conditions is Р=(0.2, 0.3, 0.1, 0.4).
V1 V2 V3 V4 m
p j qij
j =1
S1 7 2 4 3 3,6
S2 2 6 5 3 3,9
S3 1 2 1 4 2,5
Figure 12. Calculations for decision-making according to Bayesian criterion
According to the criterion of extreme optimism, it is advisable to choose an alternative S2.
3.9 Hermeyer criterion СG.
As in the Bayesian criterion, the Hermeyer criterion takes into account the different probabilities of
occurrence of each condition, however, the optimal alternative is chosen for which the minimum
product of utility values on the probability of occurrence of the corresponding condition is the largest.
The criterion is formalized by formula (9):
т
C G = max min p j q ij , i = 1,..., n , p j [0,1] , p j = 1 . (9)
i j j =1
Finding the best alternative by Hermeyer criterion for the task in Fig. 4, calculations are given in
Fig. 13. Here, the vector of probabilities of occurrence of each of the conditions is Р=(0.2, 0.3, 0.1,
0.4). According to Hermeyer criterion, it is advisable to choose alternatives S1 or S2.
V1 V2 V3 V4 min p j q ij
j
S1 7 2 4 3 0,4
S2 2 6 5 3 0,4
S3 1 2 1 4 0,1
Figure 13. Calculations for decision-making according to Hermeyer criterion
3.10 Hodge-Lehmann criterion СHL.
As in the Bayesian criterion, the Hodge-Lehmann criterion takes into account the different
probabilities of occurrence of each condition. Also, as in the Hurwitz criterion, you need to set the
value of the subjective variable z, which expresses the degree of confidence in the information about
the conditions under which the decision is made, z is set within [0, 1]. The criterion is formalized by
formula (10):
C HL = max z qij pi + (1 − z ) min qij , i = 1,..., n, j = 1,..., m .
n
(10)
i i =1 j
It should be mentioned that in the case of z=0 (extreme distrust of the received information about
the state of the system), the Hodge-Lehmann criterion becomes the Hermeyer criterion, and at z=1
(the information about the state of the system is reliable), the Hodge-Lehmann criterion becomes the
Bayesian criterion.
Finding the best alternative by Hodge-Lehmann criterion for the task in Fig. 4, calculations are
given in Fig. 14. Here, the vector of probabilities of occurrence of each of the conditions is Р=(0.2,
0.3, 0.1, 0.4), z=0.3.
V1 V2 V3 V4 n
qij pi min p j qij n
i =1
j z qij pi + (1 − z ) min qij
i =1 j
S1 7 2 4 3 3,6 0,4 1,36
S2 2 6 5 3 3,9 0,4 1,45
S3 1 2 1 4 2,5 0,1 0,82
Figure 14. Calculations for decision-making according to Hodge-Lehmann criterion
According to Hodge-Lehmann criterion, it is advisable to choose an alternative S2.
3.11 Savage criterion СS.
The criterion should be used to protect the decision-maker from excessive losses. To make a
decision, a risk matrix R is constructed on the basis of the initial condition matrix: the differences
between the found maximum values in columns and the values of the initial matrix
( rij = max qkj − qkj ) are put into the risk matrix. The criterion is formalized by formula (11):
k
C S = min max rij , i = 1,..., n, j = 1,..., m . (11)
i j
Finding the best alternative by Savage's criterion for the task in Fig. 4, calculations are given in
Fig. 15. According to Savage's criterion, it is advisable to choose an alternative S1.
V1 V2 V3 V4 max rij
j
S1 7 2 4 3 0 4 1 1 4
S2 2 6 5 3 R= 5 0 0 1 5
S3 1 2 1 4 6 4 4 0 6
max 7 6 5 4
Figure 15. Calculations for decision-making according to Savage's criterion
4 Discussion
According to the results of calculations based on decision-making criteria, alternative S1 was the
most useful according to four criteria (it is more than 36% of all criteria), according to three criteria -
alternative S2 (over 27% of criteria), according to one criterion – alternative S3 (over 9% of criteria),
according to three criteria – alternatives S1 or S2 are equally useful (more than 27% of the criteria).
Let's summarize the results of calculations graphically.
Figure 16. Visualization of the results of evaluating the feasibility of alternatives
5 Conclusions
To manage complex projects involving specialists from different fields, it is advisable to have a
tool that will allow you to take into account the needs of such specialists. The proposed approach
should be used by business analysts in the case of project management, which has some alternative
scenarios, each of which is more or less useful in different conditions. In this case, the combination of
AHP and decision-making under the uncertainty method will allow choosing the best alternative. The
decision can be made from a pessimistic or optimistic point of view, and the ability to adjust certain
parameters of the criteria (for example, in Hurwitz and Hodge-Lehmann criteria) will allow taking
into account the views of experts involved in intersectoral cooperation.
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