New PM World Journal Project Management Mathematical Models … · 2019. 8. 15. · Complex master plan Business plan and milestone plan Taxation plan Main requirement s for the project

PM World Journal Project Management Mathematical Models for the Customer Vol. II, Issue III – March 2013 Vladimir I. Voropajev & www.pmworldjournal.net Featured Paper Yan D. Gelrud

© 2013 Vladimir Voropajev & Yan Gelrud www.pmworldlibrary.net Page 1 of 16

PROJECT MANAGEMENT MATHEMATICAL MODELS FOR THE CUSTOMER

Vladimir I. Voropaev

Yan D. Gelrud

SUMMARY

The article considers the mathematical models intended for managing project activity at all

stages having one interested party, project customer involved. For the first time, the problem of

managing project and product configuration is explicitly formulated and stated. Use of these

models is aimed at increasing the efficiency of customer's activity; using these models ensures

implementation of relevant competences and attainment of the objectives set under various

conditions of project environment.

KEYWORDS: Stakeholder, project management mathematical models, project management

competences.

INTRODUCTION

In [1] an attempt is taken to structure the features of the main interested parties (stakeholders)

and taking those into consideration to construct project management mathematical models.

Examples of such models have been built for investor, customer, project team, main contractors,

suppliers and regulating bodies.

We have also pointed out that the choice of methods and tools of project management is

determined to a large degree by management of which interested party we are looking at in each

case. Different interested parties in a project differ in their expectations, roles, degree of

responsibility and actions. This is due to the different goals, success criteria and self-evaluation of

reaching their own goals, different values and strategies to achieve the goals. These differences

significantly impact setting the project tasks, the management problem-solving methods, tools and

technologies used oriented towards theirs specific needs. But when modeling the activity of an

individual interested party there may exist different alternatives of formulating these tasks

connected with different conditions in which the project is carried out. Other than that, methods of

implementation of optimal decision-making problems are also of considerably multivariate nature.



The article proposes the mathematical models intended for managing project activity at all

stages having one interested party, project customer involved. For every suggested alternative

particular condition which the given model is adequate to and at the same time, methods of finding

solution, which could also be multivariate, are proposed and analyzed. Use of these models is

aimed at increasing the efficiency of customer's activity; using these models ensures

implementation of relevant competences and attainment of the objectives set under various

conditions of project environment.

1. KEY DEFINITIONS:

1.1. The concept "customer"

Let's stop in more detail on the concept "customer".

Customer — legal entity or individual in whose interest the project is being carried out,

usually, future owner of the project product.

Customers are subjects of investment activity that are entitled by the investors to carry out the

investment project. To reach this goal the investor provides the customer with the right of

ownership, the right to use and the right to dispose the investments during the time frame and

within the limits as set by the investment contract and in accordance with the law. Customer should

not interfere into entrepreneurial or other activity of other participants of the investment process.

Investors may act as customers.

The main function of customer is to carry out a complex of activities final goal of which is to

provide for, in co-operation with other participants of the investment process, handing over the

final product.

The main concern of the customer when managing the project is managing the scope of the

project. At the same time the main objects of customer's attention are:

Requirements management. The procedures to manage special requirements of the

customer for the project results, as well as for the equipment, materials, services and management

procedures, including quantitative and qualitative characteristics.

Product and project configuration management. The procedures used for technical and

administrative governance of the works connected with creating, maintaining and controlling

changes in project and product configurations over a period of the project life cycle.

Scope management. The processes necessary to include within the project all of the

required works and only the works that are necessary for the successful completion of the project.



1.2. The list and the content of the project management competences

The competences for interested parties are split into two groups - basic competences and

special ones[5]:

■ basic competences define the requirements for the list, the content and the level of abilities,

knowledge, skills and personal qualities common for all interested parties;

■ special competences define the requirements for the list, the content and the level of abilities,

knowledge, skills and personal qualities specific to particular interested party taking in account its

role, interests and the functions performed;

Below is given an example of specific project management characteristics and parameters for

the selected stakeholder, project customer.

Expectations - Final profit-producing product.

Project vision - The process of creating the product.

Project goal - Competitive product, bringing in certain profit.

Criteria - Minimum number of deviations in configuration of the product and its quality,

obtaining the product on time with minimal costs.

Constraints - Product configuration and quality, technical requirements, budget..

Strategy - Allowing for the customer's functions to be carried out while obtaining his

benefits within the project.

Main risks - Low quality of the product, technical requirements, schedule and cost creep.

Main PM tools - Complex master plan, monitoring, configuration and change management,

ongoing reporting, tax optimization.

On the Fig.1 the scheme of interconnections between project management mathematical

models for customer and other interested parties is shown.



Mathematical model of the

customer’s activity

Mathematical model of the

project manager’s and project

management team’s activity

Mathematical model of

investor’s activity

Mathematical model of the regulating bodies’

activity

Mathematical model of general designer’s

activity

Mathematical model of the supplier’s

activity

Tehcnical and economic case for

the projectComplex master plan

Business plan and milestone

plan

Taxation plan

Main requirement

s for the project per milestone and stage

Requirements and constraints imposed by regulating bodies and

the supplier

Financing plan

Fig.1. The scheme of interconnections between project management mathematical models for

customer and other interested parties is shown.

2. PROJECT MANAGEMENT MATHEMATICAL MODELS FOR CUSTOMER

2.1. Customer's activity mathematical model minimizing the degree of deviations in project

configuration

Given: complex project master plan as a generalized network diagram as developed by the

project team based on detailed project model [2] which in turn is developed by and used by the

project manager, his team and chief contractor.

Let Ttp, Ti

n — early and late finishes of i activities in master network diagram;

aij, bij — minimal and maximal estimates for the duration of the tasks in the master plan;

rij — costs of completing the tasks in the master plan;

It — budget constraints for the t time period (year, quarter, month);

EECij — expert estimates of the maximum permissible degree of deviations in task configuration.

By deviations in project task configuration we mean missing the deadlines for the tasks,

omitting the tasks and replacing one task with another. For every task an expert estimate of

deviations in configuration is given using a ten-point scale:



αij(t) — missing the deadline for the task by t days;

βij — omitting the task;

γij — replacing the task with another or changing the characteristics of the task,

where values of α, γ, β, close to 0 show insignificant changes, close to 5 — rather important

changes, close to 10 — impermissible changes. Other values are used for the intermediary states.

Let us define the degree of change in project configuration as an integrated indicator (IIN),

calculated using certain function (as set by experts) FCD for the quality indicators for individual

tasks of the master plan QIij, also set by experts. Statistical analysis of the large number of projects

conducted by the authors shows that these functions are increasing, concave ones, i.e. they have

positive first and second order derivatives (both the value of the function and its growth rate are

increasing). For small values of the argument (small deviations in project task configuration) FCD

functions may be viewed as linear ones. Therefore the model for the problem will look as follows:

find such deadlines for the tasks in master plan Ti and durations of the aggregate tasks tij, for which:

, (2.1.1)

ij ij ij ija t b , (2.1.2)

where δij — missing deadlines for completion of the task (i, j);

,),(

tji

tijij Ir

t

(2.1.3)

— financing constraint during the time period t,

where Ωt — a set of tasks carried out during the time period t;

λtij — share of the task (i, j), completed during the time period t.

αij( ) + βij+ γij ≤ EECij; (2.1.4)

Within this model aggregate project risk - a risk of not reaching the goals set that defines the

reliability of the project - should be accounted for. For these purposes under the reliability of the

project we should understand, on the one side, project characteristic that becomes apparent through

the project's ability to be implemented under certain conditions of interaction with project

environment, and, on the other side, quantitative estimate of the project, unambiguously linking

probability of completion with time or other parameters characterizing implementation process

within the given conditions.

For the quantitative estimate of reliability the so called singular indicators (characterizing

only one reliability property) and complex reliability indicators (characterizing several reliability

properties).



As such indicators the following ones may be used: estimate of the probability of building the

object in time or before the deadline, good reputation of the customer and its ability to meet the

financial obligations for all stakeholders, degree of availability, compliance of the construction

phase with the schedule, accreditation from banks, insurance guarantees, real and grid burdens,

competitive environment, sales scheme, initial documentation and permits condition, etc.

In [3] a model of network project reliability as a complex technical system is suggested,

where as a reliability indicator for a task - a project component - is used the probability of its

failure-free execution Рrel, and as a project reliability indicator – received guaranteed infinimum

Оrel. Using the above scheduling problem is formulated using not only the resource constraints but

the project reliability indicator as well. It seems reasonable to use the suggested method of

calculating project reliability for the reviewed model of customer's activity. Only in this case, as a

base network model a complex master plan of the project is used in the form of generalized

network model. Thus, let us demand:

Рrel ≥ Оrel, (2.1.5) ( ) minijFCD QI , (2.1.6)

where the argument of the FCD function is the vector of the quality of all the works in the master

plan performed.

As a result of using this model, a project master plan optimal in quality, with adequate

financing, showing necessary degree of reliability is being formed.

The specific nature of this model comes from involving experts to define the quality

indicators (degree of deviations in configuration per each aggregate task and estimates of their

maximal permissible values) and to set the type of target function used to calculate the integral

indicator of the project quality. It seems unreasonable to build a universal system of quantitative

indicators due to the unique nature of many projects and various importances of the works included

in them, therefore employing experts (customer's representatives) to set indicators for the quality of

the works performed is logical and methodologically sound.

2.2. Customer's activity mathematical model minimizing the degree of deviations in project

financing plan

Given: complex master plan of the project in the form of generalized network model (as in

the model 2.1).



Let — be early and late deadlines for completion of the events i in aggregated network

model;

aij, bij — minimal and maximal estimates for the duration of the tasks in the master plan;

rij — costs of completing the tasks in the master plan;

It — budget constraints for the t time period (year, quarter, month);

EECij — expert estimates of the maximum permissible degree of deviations in task configuration.

By deviations in project configuration we will understand, as in the model 2.1, missing the

deadlines for the tasks, omitting the tasks and replacing one task with another. Let us use the same

notation:

αij(t) — missing the deadline for the task by t days;

βij — omitting the task;

γij — replacing the task with another or changing the characteristics of the task,

where values of α, γ, β, close to 0 show insignificant changes, close to 5 — rather important

changes, close to 10 — permissible changes. Other values are used for the intermediary states.

Therefore the model for the problem will look as follows: find such deadlines for the tasks in

master plan Ti and durations of the aggregate tasks tij, for which:

, (2.1.2)

ij ij ij ija t b , (2.2.2)

where δij — missing deadlines for completion of the task (i, j);

αij( ) + βij+ γij ≤ EECij, (2.2.3)

the degree of deviation in configuration of tasks should not exceed its maximal permissible

expert estimate.

Let's use as a target function weighted sum of deviations from the project financing plan:

t ji

ttijijt

t

IrG min,),(

(2.2.4)

where Ωt — a set of tasks carried out during the time period t;

λtij — share of the task (i, j), completed during the time period t.

– weight coefficient for the period of time t, set by the experts (it is reasonable to assume

that earlier periods have greater weight in comparison with the later ones because it is harder to

alter project financing during the nearest periods).



As a result of using this model, a project master plan having minimal deviations from the

project financing plan approved by the investor, the deviations in configuration of which lie within

the set limits, is being formed.

2.3. Customer's activity mathematical model maximizing the degree of project quality

We can review a multitude of factors defining project quality - these are ecological factors,

social, technical, financial ones, and others. Specific character of the project defines dependence of

project quality on its defining factors. Within this model we will use the following factors as

defining the quality: volume of financing of individual stages and of the project as a whole, the

deadline for the project as a whole and for rolling out its individual complexes.

Let F(K, T) be the function showing dependence of project quality from the vector К

(volumes of financing the whole project and its individual stages) and the vector Т(deadlines for

the implementation of the whole project and roll-out of its individual complexes). This function is

set by the experts, often on a discrete set of its defining factors Р(K, T) (the most probable

alternatives of the financing plan for the project as a whole and its individual parts along with

corresponding deadlines are being set). Each of alternatives from the set of possible (permissible)

alternatives of the project plan рР(K, T) must meet the constraints of the model 2.2. In addition,

extra time constraints arising from deadlines of individual complexes of tasks must be taken into

consideration.

Target function in this model is F(K, T), i.e. project quality must be maximized while

meeting time and resources constraints.

2.4. Multi-criteria customer's activity mathematical model

Let us add to the model reviewed in 2.3 target functions 2.1.6 and 2.2.4. Thus, a mathematical

model for the problem of finding the project implementation plan meeting time and resource

constraints and at the same time maximizing project quality, minimizing degree of deviations in

project configuration and showing minimal deviations in project financing plan approved by the

investor, is being formed. A set of alternatives Р(K, T) is formed by the project team, at the same

time, in parallel alternatives for the plans of providing the necessary resources and taxation

alternative taxation plans are formed. These alternative plans when approved the supplier and

regulating bodies may in their turn receive quality estimations (related to the irregular nature of



supplies, supplier's capabilities, and the requirements from regulating bodies). In these cases it is

reasonable to include these criteria in the model, too. There comes into existence multi-criteria

problem of forming project implementation plan. Generally, this problem does not have a solution,

i.e. there is no project implementation plan satisfying all of the criteria. mentioned above But in

theory and practice of solving similar problems there exist methodological approaches, ensuring a

selection of acceptable alternatives [6]. Let us look at the main approaches among these. We have

consciously omitted from the following list various alternatives of criteria folding (additive,

multiplicative, mixed) due difficulties, frequently to the point of non-applicability in practice,

associated with their normalization (i.e. ensuring equal dimensionality and importance). Definition

of normalizing weight coefficients in this problem is of a rather subjective character.

2.4.1. Pareto optimization.

We have a problem with k criteria F1, F2,…, Fk. Let there be within the set of possible

solutions two such solutions р1, р2 that the values of all criteria F1, F2,…, Fk for the first solution

are better than the values of corresponding criteria for the second solution. Then from set Р(K, T)

the solution р2 is forced out (or «dominated by») the solution р1.

As a result of such procedure apparently unfavorably solutions within the set Р(K, T) there

are left only effective («Pareto-effective») solutions, characterized by the fact that for none of them

there exists a dominating solution. The set of Pareto-effective solutions is easier to review than the

whole set Р(K, T). As for making the final decision, it is still a human privilege. Only human with

his unsurpassed ability to solve informal tasks, make compromise solutions (not strictly optimal but

acceptable in a set of criteria) may take the responsibility to make the final choice.

However, the procedure of decision-making itself, having been repeated multiple times, may

serve as a basis for developing certain formal rules then used without human participation. We are

talking about the so called «heuristic» methods of decision-making. Suppose the customer is

choosing compromise solution multiple times within the multi-criteria task of finding project

implementation plan, solved in different conditions . By collecting the statistics of choice results,

an algorithm may be developed, in general depending from the conditions and the indicators F1,

F2,…, and we can employ such algorithm to select a decision, but this time automatically, without

involving a human.

In some cases it is highly useful to employ a decision-making procedure in an interactive

dialog mode [8], when computer consecutively shows on the screen a series of questions regarding

the values of controlled project parameters and alternative answers (or permissible ranges of the



values) and the customer selects one of them. Each consecutive question and alternative answers

are selected by the customer depending on the answer to the previous question and the content of

the system's knowledge base. The result of such dialog may be a project alternative as well as the

description of the recommended activities.

Interactive mode allows the user: to select among the options for setting the optimization

problem (from the set of optimality criteria and corresponding sets of optimizing variables); to

select among the alternative ways of computation (by level of the model's detail); to select the

method of optimization among those in the library and to set the parameters of the method; to select

among the alternative ways to print modeling results in the beginning and in the ending point of the

search as well as the intermediate optimization results.

The importance of the dialog mode of interaction during the decision-making process is hard

to overestimate. Quick dialog between the customer and PC is necessary because usually real tasks

when selecting optimal project parameters include the stages that are hard to formalize and require

human interference, making certain decisions on his part. Other than that, prompt connection with

PC allows to review within a short time from a multitude of technical solutions and find an optimal

one, as well as speeds up information search processes and creates the conditions for its effective

use.

2.4.2. Method of consecutive reductions

Let us place indicators F1, F2,… in the order of diminishing importance. First, we search for

the solution turning into an extremum the first (most important) indicator F1=F1*. Then we assign,

judging from practical assumption, taking in consideration the accuracy level at which we know the

input data, certain «reduction» F1, which we agree to make (by changing in amount of the

reduction the extremal value F1*) in order to achieve extremum for the second indicator F2. This

way, the indicator F1 taking the reduction in consideration is turned into a constraint and under this

constraint we search for a solution, turning into extremum the indicator F2. Then again we assign a

«reduction» F2, using which we may achieve extremum for F3, and so forth. Such way of

constructing a compromise solution is good in that it immediately shows what «reduction» in one

of the indicators leads to gains in the other one and what the prices of such gains is.

Within the reviewed multi-criteria mathematical model of the customer's activity the criteria

may have different priorities but it seems most reasonable to employ the following order:



F1=F(K, T) the function showing dependence of project quality from the vector К (volumes

of financing the project and its individual stages) and the vector Т(deadlines for implementation of

the whole project and roll-out of its individual complexes) from the model 2.3;

F2= FCD – the degree of deviations in project configuration from the model 2.1

F3= G – weighted sum of deviations from the project financing plan from the model 2.2;

F4 – quality of the supply plan (alternatives are assessed in cooperation with the supplier);

F5 – quality of the taxation plan (alternatives are assessed in cooperation with the regulating

bodies).

2.4.3. Hierarchy method

Analytical hierarchy process (AHP) suggested by T. Saaty in late 70s of the last century [7] is

also a multi-criteria decision-making method. Its advantage comes from the simplicity of the

expertise used that assumes decomposing the existing problem into simpler components. As a result

of such procedure a relative importance of the alternatives analyzed is defined for all the criteria in

the hierarchy, quantitatively expressed in the form of priority vectors.

The hierarchy includes the goal situated at the top, intermediate levels (criteria) and

alternatives, placed at the lowest level.

To figure out relative importance of the hierarchy elements vij a comparison scale (Table 1) is

used that allows, using a pair-by-bair comparison method to quantitatively estimate the degree of

preference of one object over another.

Table 1

Degree of importance vij

Definition Explanation

1 Equal importance Two actions provide equal input for reaching the goal

3 Certain dominance of the importance of one action over others (weak importance)

There exist certain arguments in favor of one of the actions but these arguments are not persuasive

5 Significant or strong importance

There exist reliable judgments or logical conclusions to prefer one of the actions

7 Very strong importance There exist persuasive proofs in favor of one of the actions over others

9 Absolute importance The degree of preference is set at the absolute level

2,4,6,8 Intermediate values between two adjacent judgments

For situations where a compromise judgment is needed

Inverse values 1/vi j

Action j in comparison with i the inverse value is given

When comparing two actions in reverse order the value of the scale vi j is turned into an inverse value of 1/vij



As a result a set of pair comparison matrices is built. Each matrix А has the following

appearance:

А={aij}, where aij = vij and aji = 1/aij, n – the order of the pair comparison matrix.

For each pair comparison matrix own vectors (WE), priority vectors are calculated using the

following algorithm:

matrix А is normalized by dividing all of its elements by the sum of elements of the respective

column. The components of the vector WE are calculated as averages of arithmetic elements of the

normalized matrix..

Therefore, for each comparison matrix we should estimate:

maximal own value max using formula

(matrix А on the right is multiplied by the vector WE and then all components of the resulting

vector are added together).

consistency of the judgment by means of calculation

- the index of consistency IC = (max – n)/(n–1);

- relation of consistency СR = IC/AC, where AС – expected value of consistency index for

randomly constructed pair comparison matrix. Approximately, the value of AC can be

calculate using the following formula

AС=nn )2(98.1 .

The value of CR must be about 10% or less, to be acceptable (up to 15% in rare cases).

Otherwise, the judgments provided should be checked again.

Example. The problem of selecting the project among the three alternatives using the following

four criteria is being reviewed: Quality, Configuration change, Deviation in financing, and Costs.

Criteria

Alternatives

Fig.2. Criteria and alternatives' hierarchy when selecting the project

Project

ЕT WАe ][max

Configuration change

Deviation in financing plan

Quality Costs

Alternative Alternative Alternative



First, a pair comparison matrix for criteria is constructed and a criteria priority vector is

formed. Then for each criterion a pair comparison matrix of alternatives is constructed and the

corresponding priority vectors are formed. The obtained alternatives priority vectors for each

criterion are scalarly multiplied by criteria priority vectors and therefore we find out the resulting

alternatives priority vector in relation to the final goal - the selection of the project.

In case during the calculations discordance in experts' judgments either for criteria or for the

alternatives is discovered, the provided judgments should be re-checked and the calculations should

be performed again.

CONCLUSIONS

The suggested models implement mathematical programming models with linear and non-

linear constraints and target functions. At the moment there exists a wide range of software tools to

solve similar tasks [4], it suffices to point out the Solver add-on for Microsoft Excel.

Within the article we have reviewed new scientific and practical directions in organizational

management in general and in project management in particular. The degree of the project

participant's - customer's - interest has been analyzed: what are the customer's values, concerns,

role and responsibility in project activity, how money, power and other valuables are distributed

between the customer and other interested parties. As usual, these may be multilateral and often

intersect.

For the first time, the problem of managing project and product configuration is explicitly

formulated and stated. The suggested examples of setting tasks for customer may serve as a basis

for the development of objectively multivariate PM system. At the same time, the mathematical

models shown above implement many of the customer's competences during the implementation of

the project. These models may already serve as a methodological basis for the development of the

applied software suites (automated systems) to accomplish the project management tasks described

above by the customer during all project stages.

Further improvement of project management and increase of its effectiveness requires a more

complete description of the mathematical models for other interested parties.



SOURCES

1. Voropaev V., Gelrud Y. Project management mathematical models for interested parties // PM World Journal Vol. I, Issue III – October 2012. www.pmworldjournal.net 2. Voropaev V., Y. Gelrud Y. Generalized stochastic network models for complex project

management // Project and Program Management. — 2008. — №1–2.

3 Topka V. Time and cost minimization under reliability constraint within the disjunctive project

model. Automat. and telemech., 2012, № 7, 78–88.

4. D. Miloshevich Project Management Toolbox // Translated from English. Chief editor S. I.

Neizvestniy.— Moscow: DMK Press, 2006. — 729 pp.

5. Project management: professional knowledge foundations. National competence baseline for

project management specialists. Version 3.0. / Ed. V. I. Voropaev. V. I. — Moscow: Proyektnaya

Praktika, 2010.

6. Lotov A. V., Pospelova I. I. Multicriteria optimization - theory and methods. (Published by MSU

Dept of computational mathematics and cybernetics, 2006.

7. Saaty T. Decision Making with Dependence and Feedback: The Analytic Network Process. —

Moscow: LKI Publishing house, 2008. — 360 pp.

8. A.V. Ilyin, V.D. Ilyin. Interactive resource converter with variable behavior rules. Information

technologies and computation systems, №2, 2004, pp. 67-77.



About the Authors

VLADIMIR VOROPAJEV Author, Professor, International PM Expert Founder, Former President, Chair – SOVNET Former Vice President – IPMA Full Member, Russian Academy of Natural Sciences Moscow, Russia Professor Vladimir Voropajev, PhD. is Founder and former President and Chairman of the Board of the Russian Association

of Project Management, SOVNET. Dr. Voropajev is professor of Project Management at the State University of Management, Moscow, Russia. He is also Head of the Program and Project Management Faculty for the Russian State Academy’s Program for Professional Retraining and Professional Skill Development for Executives and Specialists in Investment Fields. He is a full member of the Russian Academy of Natural Sciences on Information Science and Cybernetics, and of the International Academy of Investments and Economy in Construction. From 1991 to 2001, he was Vice-president and a member of the Executive Board of the International Project Management Association (IPMA), the global federation of national PM associations based in Zurich, Switzerland. He is the First Assessor for several IPMA certification bodies. In 2005 he was awarded IPMA Honorary Fellowship Award. He is also an honorary Fellow of the Indian Project Management Association and a past member of the Global Project Management Forum Steering Committee. During his 40 years of engineering, scientific, teaching and consulting activities, he has published over 250 scientific research works including 7 monographs and 5 textbooks about the organization and planning of construction, information systems, and project management. Vladimir serves on the editorial boards of several international project management journals, is a frequent participant in PM conferences worldwide, and provides ongoing counsel and support to PM professional leaders in Azerbaijan, Kazakhstan, Ukraine, Yugoslavia and several other countries. Professor Voropajev can be reached at [email protected]



Yan D. Gelrud South Ural State University Chelyabinsk, Russia

Mr. Yan Gelrud was born in 1947 in Birobidjan (Khabarovsk Territory). In 1965 he finished a school of

mathematics and physics at Novosibirsk. In 1970 he graduated from the mathematical faculty of university at Novosibirsk on "Mathematics" speciality. From 1970 to 1991 Yakov was working in the Research Institute of automated control systems as a head of mathematical division. He took part in creation and adoption of more than 100 automated control systems in different branches of industry. From 1991 to 1997 Mr. Gelrud was doing business, being director general of "URAL-ASCО-SERVICE". Since the 1st of September 1997 till now he works as a professor of the "Enterprise and management" department in South Ural State University. He teaches a multitude of disciplines, such as "Mathematics", "Theory of probability and mathematical statistics", "Econometrics", "Economic and mathematical methods", "Mathematical methods of decision-making", "Bases of decision-making methodology", "Economical evaluation of investments", "Mathematical methods and models of project management", "Studies of managerial systems." Yan Gelrud has more than 100 publications and speeches on seminars and conferences of different level. His monograph "Project management in conditions of risk and uncertainty" was published recently. He can be contacted at [email protected]

Documents

New PM World Journal Project Management Mathematical Models … · 2019. 8. 15. · Complex master plan Business plan and milestone plan Taxation plan Main requirement s for the project