Project Management and Optimization Processes Choices to … · 2019. 9. 14. · Project management and optimization processes 851 To compute the probability P intwe consider the

Applied Mathematical Sciences, Vol. 13, 2019, no. 17, 845 - 857

HIKARI Ltd, www.m-hikari.com

https://doi.org/10.12988/ams.2019.98115

Project Management and Optimization

Processes Choices to Maximize Resource

Allocation Results

Giuseppe Caristi

Department of Economics

University of Messina, Italy

Vera Fiorani

R.F.I., Italy

Sabrina Lo Bosco

UniPegaso, Italy

Alberto Vieni

UniPegaso, Italy

This article is distributed under the Creative Commons by-nc-nd Attribution License.

Copyright c© 2019 Hikari Ltd.

Abstract

Integer Linear Programming (ILP) deals in a systematic way with

the minimization (maximization) problem of a linear function with sev-

eral variables (some also of a random nature), subject to equality con-

straints and linear inequality and to the constraint that one or more

846 Giuseppe Caristi, Vera Fiorani, Sabrina Lo Bosco and Alberto Vieni

variables has integer values. This mathematical approach is essential to

address and solve a very large number of real problems, typical of the ap-

plied sciences, when there is an indivisibility of the good to be produced

or of the resource to be used and thus the consequent need to repre-

sent the problem through models of Linear Programming with integer

variables. The applications of the method concern many applications of

modern society and extend from industrial analysis for the distribution

of goods and the sequencing of productive activities, to economic prob-

lems aimed at the optimal management of a securities portfolio, to those

of planning and optimal planning of public investments, to the problems

inherent in biology, high energy physics, engineering and robotics. in

this paper we consider a cutting plane problem for an irregular lattice

represented in fig. 2.

1 Introduction

In this work a mathematical methodology is developed that refers to the ge-

ometric analysis of the variables in a suitable Rm space, with a polyhedral

approach, useful for optimizing strategic choices in public decision-making

processes, ie in corporate ones. The criterion also lends itself to being ap-

plied in the various sectors of the business economy, also to solve problems

of resource allocation (investments, allocation of personnel to strategic areas,

etc.). Through specific geometric probability techniques elaborated below, we

tried to provide the decision maker with the formal basis of the strategic choice

aimed at optimizing the overall results of the action under consideration, tak-

ing into account both the different constraints posed by the problem, and

the context scenario, proceeding from a Life-Cycle Cost Analysis perspective.

Project Management and optimization processes choices to maximize resource

allocation results. In this work a mathematical methodology is developed that

refers to the geometric analysis of the variables in a suitable Rm space, with a

polyhedral approach, useful for optimizing strategic choices in public decision-

making processes, ie in corporate ones. The criterion also lends itself to being

applied in the various sectors of the business economy, also to solve problems

of resource allocation (investments, allocation of personnel to strategic areas,

etc.). Through specific geometric probability techniques elaborated below, we

tried to provide the decision maker with the formal basis of the strategic choice

Project management and optimization processes 847

aimed at optimizing the overall results of the action under consideration, tak-

ing into account both the different constraints posed by the problem, and the

context scenario, proceeding from a Life-Cycle Cost Analysis perspective. In

practice, it is often necessary to tackle problems in which the decision variables

represent indivisible quantities (such as the number of trains on a relationship

OD of a railway network, of people assigned to specific company tasks, etc.) or

others characterized by the need to choose between a finite number of different

alternatives. In the latter case, in particular, Linear Programming problems

01 will arise, ie problems in which the variables are binary and take the value

0 or 1. The methodology proposed here allows the construction of a math-

ematical model based on PLI integrated linear programming and studies the

so-called ”cutting planes” in Rm: the resolution of the problem is carried out

by eliminating the ”integrity” constraint of the variables. If the solution found

at a generic step of the process is continuous, then it is also excellent for the

original problem with integer variables; otherwise, the method continues by

generating characteristic cutting planes, or new inequalities to add to the re-

laxed problem, which must render the current solution inadmissible and not

discard any of the solutions of the original problem in the study (see fig. 1)

Programmazione Lineare Intera PLI.1

Programmazione Lineare Intera (PLI)

zPLI = min cTx

Ax ∏ b

x ∏ 0

x intero

• vincoli di interezza non lineari: es. sin(ºx) = 0

• Rimuovendo il vincolo di interezza

) PL (rilassamento continuo di PLI), tale che

zPL ∑ zPLI (lower bound per zPLI)

Dim. si cerca il minimo in un insieme piu grande. 2

-

6

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⇢⇢⇢⇢⇢⇢⇢⇢

⇢⇢

HHHHHHHHH

AAAAAAAAA

f� xPL (ottimo di PL)v

v

¢¢¢¢¢Æ

xPLI (ottimo di PLI)

XXXXXXy bxPLc (arrotondamento di xPL)°°°µ

@@

P = {x ∏ 0 : Ax ∏ b} X = {x 2 P : x intero}

Se la soluzione ottima xPL di PL e ammissibile per PLI allora ne e

la soluzione ottima

Dim. zPL ∑ zPLI ;

xPL ammissibile per PLI ) zPL = cTxPL ∏ zPLI 2

PLI Optimum

PL Optimum

Rounding of xPL

fig. 1

The problems of the cutting plan are increasingly important in these economic-

productive sectors, because they allow an appropriate choice of the truncation

plan in each iteration of the whole Linear Programming algorithm, ensuring in

the mathematical treatment of the specific cases-study the rapidity of conver-

gence of the characteristic functions. Cutting plan problems are increasingly


important for economic applications, in particular the choice of the cutting

plane in each iteration of the related ILP algorithm is very important in terms

of convergence speediness. Several studies have been done on the matter, based

on the study of the feasible cutting planes polynomial structures. Ralph Ed-

ward Gomory [10] proved that the optimal cutting plane is the one that max-

imizes the number of feasible integer points the cut touches. A theorem intro-

duced by Georg Alexander Pick [13] allows calculating the area of each polygon

whose vertices belong to a bi-dimensional lattice, as a function of the number

of its internal and boundary lattice points. In 1957, John Edmund Reeve pro-

posed a generalization of Pick’s theorem to the three dimensional case. Many

empirical problems, such as those typical of the economic-financial sector, or

in studies of investment optimization, or for the optimal choice of alternatives

for the construction of network infrastructures (tangible and intangible), can

be represented by means of particular programming models linear integer, in

which the objective function and the constraints are integer variables. In this

case, however, the constraints of wholeness must be added to the constraints

of non-negativity, representing the problem in the m-dimensional space of the

decision variables (see fig. 2). Motivated by above facts in the present pa-

per starting from results obtained in recent years [1], [2],..., [8] we consider a

cutting plane problems for an irregular lattice (see in fig. 3), we compute the

probability that a segment of random position and constant length intersects

(cut) a side of lattice.


fig. 2 – Logical phases of the project management process to identify the optimal decision.

Optimum Solution

Definitionproblem

(Life-Cycle Cost Analysis)

Objective Function

and constraint analysis

Mathematics models

(Cutting plane in Rm)

Evaluation and analysis of

possible alternatives

Analysis and consistency of

results

2 Cutting probabilities

Considering the fundamental results obtained by Poincare [14], Stoka [15],

we want to consider a Laplace type problem for a particular irregular lattice

composed by three triangles, showing the geometric point of view also.

Let < (a;α, β), the lattice with fundamental cell C0 rappresented in figure 2


A

B H L G C

fig.3

where α and β are angles with

0 < β < α ≤ π

2. (1)

By this figure we obtain the following relations

|BE| = a sinα

sin β, |CF | = |EG| = a sinα,

|BG| = a sinαctgβ, |GC| = a (sinαctgβ − 2 cosα) ,

AE = a (sinαctgβ − cosα) ; (2)

b = 2a (sinαctgβ − cosα) ; (3)

ctgα = ctgβ − 2ctgα; (4)

|CE| = a sinα

sinα(5)

areaC0 = 2a2 sinα (sinαctgβ − cosα) . (6)

We want to compute the probability that a segment s with random position

and of constant length l < a2

intersects a side of lattice <, i.e. the probability

Pint that a segment s intersects a side of the fundamental cell C0.

The position of the segment s is determinated by its middle point and by

the angle ϕ that it formed with the line BC (o AD).


To compute the probability Pint we consider the limiting positions of seg-

ment s, for a specified value of ϕ, of segment s .

Thus we obtain the fig. 4

A

B C

DE

ϕ

C01(ϕ )

A1

A2

A3

B1

B2

B3

B4

C1

C2

C3

C5

C4

C6

D1

E1

E3

E2

E4

E5

E6

E7

a1

a2

a3

a4a

5

c1

c2 c

3

c4

c5

b1

b2

b3

b4

b5

C03(ϕ )

C04(ϕ )

fig. 4

and the relations

areaC01 (ϕ) = areaC01 −5∑i=1

areaai (ϕ) , (7)


areabi (ϕ) , (8)


areaci (ϕ) . (9)

By fig. 2 we have that:

areaa1 (ϕ) =l2 sinϕ sin (α− ϕ)

2 sinα,

areaa2 (ϕ) =al

2sin (α− ϕ)− l2 sinϕ sin (α− ϕ)

2 sinα,

areaa4 (ϕ) =l2 sinϕ sin (ϕ− β)

2 sinα,

areaa3 (ϕ) =al sinα

2 sin βsin (ϕ− β)− l2 sinϕ sin (ϕ− β)

2 sin β,

areaa5 (ϕ) =al

2(sinαctgβ − cosα) sinϕ− l2

4(ctgβ − ctgα) (1− cos 2ϕ) .


We obtain that:

A1 (ϕ) =5∑i=1

areaai (ϕ) =al

2(sinαctgβ − cosα)

sinϕ− l2

4(ctgβ − ctgα) (1− cos 2ϕ) , (10)

with this value the relation (7) is written

areaC01 (ϕ) = areaC01 − A1 (ϕ) . (11)

In order to compute areaC02 (ϕ) we have that:

areab1 (ϕ) =l2 sinϕ sin (ϕ+ α)

2 sinα,

areab5 (ϕ) =al

2(sinαctgβ − cosα) sinϕ− l2 sinϕ sin (ϕ− β)

2 sin β,

areab3 (ϕ) =l2 sin (α− ϕ) sin (ϕ+ α)

2 sin (α + α),

areab4 (ϕ) =al

2sin (α− ϕ)− l2 sin (α− ϕ) sin (ϕ+ α)

2 sin (α + α),

areab2 (ϕ) =al sinα

2 sinαsin (ϕ+ α)− l2 sinϕ sin (ϕ+ α)

2 sinα

− l2 sin (α− ϕ) sin (ϕ+ α)

2 sin (α + α).

We obtain that:

A2 (ϕ) =5∑i=1

areabi (ϕ) = al sinα [cosϕ+ (ctgβ − ctgα) sinϕ]−

l2

4 (ctgβ − ctgα){(ctgβ − 3ctgα) sin 2ϕ+ [1 + ctgα (ctgβ − 2ctgα)] (12)

cos 2ϕ+ 1− ctgα (ctgβ − 2ctgα)} .


With this value the formula (8) become

areaC02 (ϕ) = areaC02 − A2 (ϕ) . (13)

In order to compute areaC03 (ϕ) we have that:

areac1 (ϕ) =l2 sin (ϕ− β) sinϕ

2 sin β, (14)

areac3 (ϕ) =l2 sinϕ sin (ϕ+ α)

2 sinα,

areac2 (ϕ) = al (sinαctgβ − cosα) sinϕ− l2 sin (ϕ− β) sinϕ

2 sin β

− l2 sinϕ sin (ϕ+ α)

2 sinα,

areac5 (ϕ) =al sinα

2(ctgβ sinϕ− cosϕ)− l2

4

[ctgβ (1− cos 2ϕ)− sin 2ϕ] ,

areac4 (ϕ) =al sinα

2[(ctgβ − 2ctgα) sinϕ+ cosϕ]− l2

4

[(ctgβ − 2ctgα) (1− cos 2ϕ) + sin 2ϕ] .

We obtain that:

A3 (ϕ) =5∑i=1

areaci (ϕ) =3al

2sinα (ctgβ − ctgα) sinϕ

− l2

2(ctgβ − ctgα) (1− cos 2ϕ) . (15)

With this value the (9) is written

areaC03 (ϕ) = areaC03 − A3 (ϕ) . (16)

By relations (10), (12) and (15) we obtain


3∑i=1

Ai (ϕ) = al sinα [3 (ctgβ − ctgα) sinϕ+ cosϕ]− l2

4 (ctgβ − ctgα)

[(ctgβ − 3ctgα) sin 2ϕ+

(1− ctg2α + 7ctgαctgβ − 3ctg2β

)cos 2ϕ+

1 + 5ctg2α− 7ctgαctgβ + 3ctg2β]

. (17)

Denoting by Mi (i = 1, 2, 3) the set of segments s which have their center

in C0i and with Ni the set of segments s all contained in the cell C0i we have

[15]:

Pint = 1−∑3

i=1 µ (Ni)∑3i=1 µ (Mi)

, (18)

where µ is the Lebesgue measure in the Euclidean plane.

To compute the above measure µ (Mi) and µ (Ni) we use the Poincare

kinematic measure [14]:

dk = dx ∧ dy ∧ dϕ,

where x, y are the coordinate of middle point of s and ϕ the fixed angle.

3 Main Result

Taking in account that ϕ ∈ [β, α] , we have that:

µ (Mi) =

∫ α

β

dϕ

∫ ∫{(x,y)εC0i}

dxdy =

∫ α

β

(areaC0i) dϕ = (α− β) areaC0i,

then

3∑i=1

µ (Mi) = (α− β) areaC0. (19)

Then, considering relation (11), (13) and (16) we can write


µ (Ni) =

∫ α

β

dϕ

∫ ∫{(x,y)εC0i(ϕ)}

dxdy =

∫ α

β

(areaC0i (ϕ)

)dϕ =

∫ α

β

[areaC0i − Ai (ϕ)] dϕ =

(α− β) areaC0i −∫ α

β

[Ai (ϕ)] dϕ,

therefore

3∑i=1

µ (Ni) = (α− β) areaC0 −∫ α

β

[3∑i=1

A1 (ϕ)

]dϕ. (20)

Replacing relations (19) and (20) in the (18) we obtain

Pint =1

(α− β) areaC0

∫ α

β

[3∑i=1

Ai (ϕ)

]dϕ. (21)

By formula (17) we have

∫ α

β

[3∑i=1

Ai (ϕ)

]dϕ = al sin [sin β − sinα + 3 (cos β − cosα) (ctgβ − ctgα)]−

l2

8 (ctgβ − ctgα)

[(1 + ctg2α + 7ctgαctgβ − 3ctgβ

)(sin 2α− sin 2β)−

(ctgβ − 3ctgα) (cos 2α− cos 2β) + 2 (α− β)(1 + 5ctg2α− 7ctgαctgβ + 3ctg2β

)].

(22)

The relations (6), (21) and (22) give us

Pint =1

2a2 (α− β) sin2 α (ctgβ − ctgα)

{al sinα [sin β − sinα + 3 (cos β − cosα) (ctgβ − ctgα)]− l2

8 (ctgβ − ctgα)

[(1− ctg2α + 7ctgα− ctgβ − 3ctg2β

)− (ctgβ − 3ctgα) (cos 2α− cos 2β)


2 (α− β)(1 + 5ctg2α− 7ctgα− ctgβ − 3ctg2β

)]}.

In particular for α = π2, β = π

4, l = a

3we obtain

Pint = 0, 7.

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Received: August 27, 2019; Published: September 11, 2019

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Project Management and Optimization Processes Choices to … · 2019. 9. 14. · Project management and optimization processes 851 To compute the probability P intwe consider the