Download The Optimal Working Solutions for The Specific Parts of The

The Optimal Working Solutions for The Specific Parts of The Financial Service
Systems
Vladimir Simovic, Zlatko Golubic
University of Zagreb, Police College,
Avenija Gojka Suska 1, HR - 10000 Zagreb
The Republic of Croatia
Tel: +385 1 239 1341; Fax: +385 1 239 1415
E-mail: [email protected]
Miljenko Crnjac
University of Osijek, Economic Faculty in Osijek
Gajev Trg 7, HR - 31000 Osijek
The Republic of Croatia
Tel: +385 31 211 605, +385 31 224 400/43; Fax: +385 31 211 604
E-mail: [email protected]
ABSTRACT: Objective of this work is to explain the OR modelling concept that was analytically
used for the organisation of specific parts of the financial service systems. Solution for that OR
modelling concept was the specific application of procedure well known as the transportation
simplex method. The same concept can be used as good analytical solution not only for production
and optimal distribution of usual financial documentation, but also as a part of production and
distribution service for the electronic payment duties (for Certifying Authorities, Third Trust Party,
etc.) needed for applications of digital signature or similar procedures. This analytical work was
done for development of the financial service systems of the various connected parts of the financial
and governmental institutions of the Republic of Croatia. These problems are connected also with
the development of the financial service systems of the electronic markets.
Keywords:
1
applied transportation simplex method, financial service systems
INTRODUCTION
The conventional Operations Research (OR) goal, optimal decision making, can be adequate for
achieving the best system design and developing information on behaviour of a specific parts for a
whole system of the financial analytical services. This work is short explanation of the main OR
concepts and results that are accomplished during the processes of analysis and application of
specific linear programming solutions, which are mainly based on a streamlined simplex method for
the transportation problem. Also here are some important details about “the applied analytical
algorithm of a streamlined transportation simplex method” and useful experiences about “the
analytical application of a streamlined transportation simplex method”. An illustrative example is
presented at the end of this article.
2
A STREAMLINED TRANSPORTATION SIMPLEX METHOD AND PROBLEM
2.1 Insight in the analytical application of a streamlined transportation simplex method
The analytical application of a streamlined transportation simplex method emphasizes the wide
applicability of specific linear programming methods and solutions in the world of the financial
analytical services. Transportation problem, as one specific type of linear programming problems,
received this name because a lot of its applications involve determining how to optimally transport
goods or services, but some important applications actually have nothing to do with transportation
directly. It is well known that the assignment problem is another type of linear programming
problem and it can be viewed as a special type of transportation problem. Also the transportation
and the assignment problem are actually special cases of the minimum cost flow problem. Although
applications of the transportation and the assignment problems usually requires a very large number
of coefficients in constraints (and variables), most of the coefficients in the constraints are zeros,
and other nonzero coefficients appear in a distinctive pattern. These special structures (the
coefficients in the constraints are zeros, and other nonzero coefficients appear in a distinctive
pattern) of the problem have result that it has been possible to relatively easy develop special
streamlined algorithms. This streamlined procedure can be referred as the transportation simplex
method.
2.2 Terminology and notation used with mathematically based model formulation
The following mathematically based model formulation with extension of standard terminology and
notation is usually used:
general transportation
problem =
the general model for the transportation problem is one specific
type of linear programming problems with lot of applications
involving determining how to optimally transport goods or
services, and which use less specific terms for components then
the prototype example
m sources i = any group of supply centres i (e.g., various kinds of the
financial analytical teams) from which a supply of si units (any
supply of commodity: goods or services; e.g., various financial
analytical services) are distributed with minimal total
distribution cost (e.g., sums in various currencies, like HRK) to
any group of receiving centres, where i = 1, 2, … , m
n destinations j = any group of receiving centres j (e.g., various complexity of the
financial analytical jobs) at which a demand of dj units (any
demand for commodity: goods or services; e.g., various
financial analytical services) are received with minimal total
distribution cost (e.g., sums in various currencies, like HRK)
from any group of supply centres, where j = 1, 2, … , n
distributed unit cost
cij =
denotes the cost per unit distributed, with a basic assumption
that the cost of distributing units from source i to destination j is
directly proportional to the number distributed
z = total distribution cost
xij = number (integer value) of distributed units (e.g., financial
analytical services) from source i to destination j, where
(i = 1, 2, … , m; j = 1, 2, … , n)
integer solution property = for a transportation problem where every si and dj have an
integer value, all the basic variables (allocations) in every basic
feasible (BF) solution (including an optimal one) also have
integer values
feasible solution property
=
system must be in balance
condition =
a sufficient and necessary condition for a transportation
problem to have any feasible solution is that
the constraints require that
m
n
i 1
j 1
 si   d j .
m
n
m
n
i 1
j 1
i 1 j 1
 si   d j   xij , or total
supply must equal the total demand (where either si or dj
represents a bound rather then exact requirement, and where a
factitious (dummy) source or destination can be usefully used)
liner programming
formulation of the
transportation problem
solution =
m
minimize
subject to:
i 1 j 1
n
 xij si for i = 1, 2, … , m,
j 1
and
prototype example =
n
Z   cij xij ,
xij  0 ,
m
x
i 1
ij
d j for j = 1, 2, … , n,
for all i and j.
an illustrative example, with real integer values given for
coefficients in constraints (and variables), which have practical
solution (e.g., for the applied analytical algorithm of a
streamlined transportation simplex method) and represents
useful experience (e.g., about the analytical application of a
streamlined transportation simplex method)
ui = multiple of original row i that has been (directly or indirectly)
subtracted from original row 0 by the simplex method during all
iterations leading to the current simplex matrix (table)
vj = multiple of original row m+j that has been (directly or
indirectly) subtracted from original row 0 by the simplex
method during all iterations leading to the current simplex
matrix (table)
Follows (an algorithmically) explanation how in general with (un-streamlined) simplex method the
transportation problem can be set up in tabular (matrix) form. First, after constructing the table of
constraint coefficients, converting the objective function to maximization form, and using “Big M
method” to introduce artificial variables z1, z2, … , zm+n into the m+n respective equality constraints.
Because of that we are introducing the matrix (cost and requirements table) of constraint
coefficients, where aij is the coefficient of the jth variable in the ith functional constraint, where
coefficient entries equal to zero are indicated by leaving their column partitions of the matrix blank.
The only one adjustment has to be made before the first iteration of the simplex method is to
algebraically eliminate the nonzero coefficients of the initial (artificial) basic variables in row 0.
After any subsequent iteration, row 0 then would have the original simplex matrix (table) form
when simplex method is applied to transportation problem, where ui and vj are dual variables. Also,
if xij is a non-basic variable, cij - ui - vj is interpreted as the rate at which Z (as a total distribution
cost) will change as xij is increased.
Consequently, the transportation simplex method obtains the same information in much simpler
ways. No artificial variables are needed for constructing an initial BF solution, because a simple and
convenient procedure with several variations is available. By calculating the current values of ui and
vj directly, the current row 0 can be obtained without using any other row. Each basic variable must
have a coefficient of zero in row 0, the current values of ui and vj are obtained by solving the set of
equations cij - ui - vj = 0 for each i and j such that xij is a basic variable, which can be done in
straightforward way. The leaving basic variable can be identified in a simpler way without explicitly
using the coefficients of the entering basic variable. Special structure of the problem makes it easy
to see how the solution must change as the entering basic variable is increased. The resulting new
BF solution can be obtained immediately without any algebraic manipulation on the row of the
simplex matrix (table). Almost the entire simplex matrix and the maintaining work can be
eliminated. Convention is that, since all non-basic variables are automatically zero, the current BF
solution is fully identified by recording just the values of the basic variables. Streamlined
transportation simplex method efficiency is better then the simplex is. For a transportation problem
with m sources and n destinations the simplex matrix (table) has m+n+1 rows and (m+1) (n+1)
columns (excluding all left to the xij columns), but streamlined transportation simplex matrix (table)
has only m rows and n columns (excluding the two extra informational rows and columns). It is very
significant fact for solving medium (m=10 and n=100) and larger transportation problems.
2.3 The applied analytical algorithm of a streamlined transportation simplex method
The applied analytical algorithm of a streamlined transportation simplex method has three main
steps:
- Initialisation
- Optimality test
- Iteration
As a first main step, initialisation is process of construction an initial BF solution by the two wellknown alternative criteria and one comparison procedure. A first alternative criterion is “the Vogel’s
Approximation Method” (VAM), and second is “the Russell’s Approximation Method” (RAM).
(Remark: There is also a third alternative criterion “the Northwest Corner Rule” (NCR), which is
quick and easy, but it pays no attention to unit costs cij, and to find a good initial BF solution,
consequently the NCR solutions are far for optimal.) As a first alternative criterion the VAM is very
popular, hands can easily implement it. The VAM pays attention to unit costs cij effectively, because
the difference represents the minimum extra unit cost incurred by failing to make an allocation to
the cell having the smallest unit cost in that row or column. A second alternative criterion the RAM
is another really excellent criterion, which can bee easily implemented by a computer, and which
has usually better solutions for large problems then the VAM. Also, the RAM is patterned directly
after first step for the transportation simplex method, and mainly simplifies the overall computer
code. A comparison procedure is two phase process. A first phase is process of applying both
alternative criterions (VAM and RAM). A second phase is decision process, in which decision must
be to use the alternative criterion with better results. Go to a second step, the optimality test.
As a second main step, optimality test is process based on fact that “a BF solution is optimal if and
only if cij - ui - vj  0 for every (i, j) such that xij is non-basic”. During this process derivation of ui
and vj must be done by selecting the row having the largest number of allocations, settings its ui = 0,
and then solving the set of equations cij = ui + vj for each (i, j) such that xij is basic. If cij - ui - vj  0
for every (i, j) such that xij is non-basic, then the current solution is optimal, so stop. Otherwise go to
iteration step.
As a third main step for this streamlined version, iteration is process of three phases. In first phase
iteration must determine an entering basic variable, in second leaving basic variable, and then in
third phase identify the resulting new BF solution. During first phase, since cij - ui - vj represents the
rate at which the objective function will change as the non-basic variable xij is increased, the
entering basic variable must have a negative cij - ui - vj value to decrease the total cost Z. Thus to
chose between the candidates, select the one having the larger (in absolute terms) negative value of
cij - ui - vj to be the entering basic variable. During second phase, increasing the entering basic
variable from zero sets off a chain reaction of compensating changes in other basic allocations
(variables), in order to continue satisfying the supply and demand constraints. The first basic
variable to be decreased to zero then becomes the leaving basic variable. In the case of a tie for the
donor cell having the smallest allocation, any one can be chosen arbitrary to provide the leaving
basic variable. During last-third phase, the new BF solution is identified simply by adding the value
of the leaving basic variable (before any change) to the allocation for each recipient cell and
subtracting this same amount from the allocation for each donor cell.
Shortly, these three phases of iteration are:
- 1st determine the entering basic variable: Select the non-basic variable xij having the largest
(in absolute terms) negative value of cij - ui - vj.
- 2nd determine the leaving basic variable: Identify the chain reaction required to retain feasibility
when the entering basic variable is increased. From the donor cells, select the basic variable
having the smallest value.
- 3rd determine the new BF solution: Add the value of the leaving basic variable to the
allocation for each recipient cell. Subtract this value from the allocation for each donor cell.
This way of solving the applied analytical algorithm of a streamlined transportation simplex method
was also based on the computer-based algorithm with interactive modelling design that was
prepared with the program «MathProg», which is McGraw-Hill modular developed program for
various analytical solutions (see pp. 947-950 and accompanied disks in [6]). That design can be
controlled with modular analytical programs solutions (designed in Fortran’77, Fortran’90 and
Lahey language, and supported with C++ source code for: DOS, QuickWin, Windows’95,
Windows’98, Xwindow’X11 and NT Windows) taken from Springer-Verlag Compact Disk (see pp.
595-608 and accompanied CD in [4]).
3
ILLUSTRATIVE EXAMPLE
3.1 The optimal working solutions for the financial analytical service system
In Croatian case, the financial analytical service system conventionally looks like a large “agency”
that administers distribution of various financial analytical services throughout the whole Intranet or
secure part of Internet in Croatia. Sometimes that service must be purchased from various and
different types of financial analytical teams for various analytical job destinations, which have a
different level of job complexity, also. For example, suppose that the sources are in fact five
different types of financial analytical teams (m = 5), and that we have six various customers or
analytical job destinations (n = 6) that have also a significantly different level of complexity of their
financial analytical jobs. This large “financial analytical service agency” gives various financial
analytical services from various sources to various customers at different analytical job destinations;
throughout the Intranet or Internet, or whole WAN (wide area network). Also, suppose that it is
possible to supply any of these six financial analytical service destinations from any of five sources,
with the exception that no provision has been made to supply 3rd and 4th analytical job destination
with 4th and 5th team service (or specific analytical product). Reason can be typical financial
analytical service incompatibility with large complexity of the specific analytical job at destination.
The cost of supplying service is different because it depends upon both the source of the analytical
service (type of a team) and the destination of the specific financial analytical job (level of job
complexity). Different values for the variable cost per one hour of analytical work (here are given in
HRK) for each combination of a team type and level of job complexity are given in Table 1.
Table 1. Data for the illustrative example of the financial analytical system
With five sources (m=5) and six destinations (n=6)
Cost Per Unit Distributed
(one hour of analytical work in HRK)
Destination (analytical job)
Source
Team
1
2
3
4
5
6
Supply
1
2
3
4
5
Needed
Requested
130
70
60
100
100
7
7
140
80
80
100
90
3
5
60
40
30
2
2
90
50
50
0
90
60
40
60
50
0
1
100
70
50
70
50
0
2
6
3
2
2
3
Minimum
Maximum

The analytical management is now faced with the problem of how to allocate the available
analytical service during the (suppose) whole analytical day, with these minimums of needs and also
these maximums requests. The amounts of available service from these five types of analytical
teams are given in the rightmost (supply) column of Table 1. The analytical management is
committed to provide a certain minimum of service amount to meet the essential need of each
analytical job (destination), but with the exception of 4th, 5th and 6th job, which have been almost
independent from outsourcing of analytical service (as shown, zeros in the minimum needed row of
the table). Also, requested maximum row indicates that 1st and 3rd job needs no more than minimum
service amount, but 2nd job needs 2 hours of analytical work more, 5th job as much as 1 hour more,
6th job up to 2 hours more, and 4th job needs as much at it can get (). Management wishes to
allocate all the available service from the five source analytical teams to the six analytical jobs in
such a way as to at least meet the essential needs of each analytical job while minimising the total
cost to the whole “financial analytical service agency”. Now, formulation of proper analytical form
for Table 1 is very important analytical step. The only one difficulty is that it is not quite clear what
the demands at the destination should be. The service amount to be reached at each destination
(except 1st and 3rd job) actually represents a decision variable, with both a lower bound (minimum)
and an upper bound (maximum). This upper bound is the service amount requested unless the
request exceeds the total supply remaining after the minimum needs of the other financial analytical
jobs are met, in which case this remaining supply becomes the upper bound. Thus insatiably thirsty
4th analytical job has an upper bound of:
(6 + 3 + 2 + 2 + 3) – (7 + 3 + 2 + 0 + 0) = 16 – 12 = 4.
Only one difficulty is that the demand quantities must be constants, not bounded decision variables
(just like the other members in the cost and requirements table of a transportation problem).
Supposing that the upper bounds (maximum requests) are the only constraints on service amounts to
be allocated to the analytical jobs, after one adjustment the requested allocations can be viewed as
the demand quantities for this formulation problem. In opposite to usually situations of “excess
supply capacity” here is situation of “excess demand capacity”. Consequently, the adjustment
needed here is to introduce a “dummy source” to send the unused demand capacity (rather then to
introduce a dummy destination to receive the unused supply capacity). The imaginary supply
quantity for this dummy source would be the service amount by which the sum of the demands
exceeds the sum of the real supplies:
(7 + 5 + 2 + 4 + 1 + 2) – (6 + 3 + 2 + 2 + 3) = 21 – 16 = 5.
This formulation yields the cost and requirements table shown in Table 2.
Table 2. The cost and requirements table without minimum needs
Cost Per Unit Distributed
(one hour of analytical work in HRK)
Destination (analytical job)
Source
Team
1
2
3
4
5
6
Supply
1
2
3
4
5
6 (dummy)
Demand
130
70
60
100
100
0
7
140
80
80
100
90
0
5
60
40
30
1M
1M
0
2
90
50
50
1M
1M
0
4
90
60
40
60
50
0
1
100
70
50
70
50
0
2
6
3
2
2
3
5
Maximum
The cost entries in the dummy rows are zero because there is no cost incurred by the fictional
allocations from the dummy source. A huge unit cost of 1M is assigned to the 4-3, 4-4, 5-3, 5-4
(team-job) spots, because assigning a cost of 1M will prevent any allocations that are impossible to
exist in reality. Now, it is important to take each analytical job minimum needs into account, and to
better solve problem formulated in table before (Table 2). Because 4th, 5th and 6th analytical jobs
have no minimum need it is all set for them. Also, formulations for the 1st and 3rd job minimum
need equals its requested allocation, so their entire demands must be filled from the real source
rather than from the dummy source, and this requirement calls for the big 1M method. Finally,
consider formulations for the 2nd job minimum need. The dummy source has an adequate (fictional)
supply to “provide” at least some of 2nd job minimum need in addition to its extra requested service
amount. Therefore, since 2nd job minimum need is 3, adjustment must be made to prevent the
dummy source from contributing more than 2 to 2nd job’s total demand of 5. This adjustment is
accomplished by splitting 2nd job into two destinations, one having a demand of 3 with the unit cost
of 1M for any allocation from the dummy source and the other having demand of 2 with a unit cost
of zero for the dummy source allocation. This formulation gives the final cost and requirements
table shown in Table 3.
Table 3. The final cost and requirements table with all needs
Cost Per Unit Distributed
(one hour of analytical work in HRK)
Destination (analytical job)
Source
Team
1
2
3
4
5
6
7
Supply
1
2
3
4
5
6D
Demand
130
70
60
100
100
1M
7
140
80
80
100
90
1M
3
140
80
80
100
90
0
2
60
40
30
1M
1M
1M
2
90
50
50
1M
1M
0
4
90
60
40
60
50
0
1
100
70
50
70
50
0
2
6
3
2
2
3
5
3.2 Analytical results
With plugging the parameters into the program «MathProg», which is McGraw-Hill modular
developed program for various analytical solutions (see pp. 947-950 and accompanied disks in [6]),
we must obtain the applied analytical algorithm solution of a streamlined transportation simplex
method, for all three main steps: Initialisation, Optimality test and Iteration.
In a first step, initialisation, we construct an initial BF solution by the two well-known alternative
criteria and one comparison procedure. A first alternative criterion is VAM, and second is RAM.
Because the VAM pays attention to unit costs cij effectively, accomplished results for all 12
variables (m + n - 1 = 6 + 7 – 1 = 12) are:
x11 = 2, x14 = 2, x15 = 1, x16 = 1, x21 = 3, x31 = 2,
x41 = 0, x42 = 2, x52 = 1, x57 = 2, x63 = 2, x65 = 3,
and total distribution cost Z = 1280 , because
Z = 130(2) + 60(2) + 90(1) + 90(1) + 70(3) + 60(2) + 100(0) +100(2) + 90(1) + 50(2) + 0(2) + 0(3).
As a second alternative criterion the RAM is another really excellent criterion, which can bee easily
implemented by a computer program «MathProg», and which has usually better solutions for large
problems then the VAM, accomplished results for all 12 variables are:
x11 = 2, x14 = 2, x15 = 2, x21 = 3, x31 = 2, x41 = 0,
x42 = 2, x52 = 1, x57 = 2, x63 = 2, x65 = 2, x66 = 1,
and total distribution cost Z = 1280 , because
Z = 130(2) + 60(2) + 90(2) + 70(3) + 60(2) + 100(0) +100(2) + 90(1) + 50(2) + 0(2) + 0(2) + 0(1).
Process of applying both alternative criterions (VAM and RAM) is finished. A second phase is
decision process, in which decision must be to use the alternative criterion with better results.
Useful comment:
There is also a third alternative criterion NCR, which pays no attention to unit costs cij, and to find a
good initial BF solution, consequently the NCR solutions are far for optimal. Here are NCR results
accomplished for all 12 variables:
x11 = 6, x21 = 1, x22 = 2, x32 = 1, x33 = 1, x43 = 1,
x44 = 1, x54 = 1, x55 = 2, x65 = 2, x66 = 1, x67 = 2,
and total distribution cost Z = 4M + 1270 , because
Z = 130(6) + 70(1) + 80(2) + 80(1) + 80(1) + 100(1) + 1M(1) + 1M(1) + 1M(2) + 0(2) + 0(1) + 0(2).
But we stay with VAM and RAM, and with the same total distribution cost Z = 1280, where really
accomplished resulting demands are:
by VAM: for 1st job = 7, for 2nd job = 3, for 3rd job = 2, for 4th job = 1, for 5th job = 1, for 6th job = 2;
by RAM: for 1st job = 7, for 2nd job = 3, for 3rd job = 2, for 4th job = 2, for 5th job = 0, for 6th job = 2.
We go to a second step with RAM, the optimality test (optimality test is process based on fact that
“a BF solution is optimal if and only if cij - ui - vj  0 for every (i, j) such that xij is non-basic”), and
after which we go to iteration step (because the current solution was not optimal).
As a third main step for this streamlined version, iteration is process of three phases: 1st determine
the entering basic variable, 2nd determine the leaving basic variable, 3rd determine the new BF
solution. We obtain very good results.
This example of analytical work was done for development of the financial service systems of the
various connected parts of the financial and governmental institutions of the Republic of Croatia.
These problems are connected also with the development of the financial service systems of the
electronic markets. Solution for that OR modelling concept was the specific application of
procedure well known as the transportation simplex method. The same concept can be used as good
analytical solution not only for production and optimal distribution of usual financial
documentation, but also as a part of production and distribution service for the electronic payment
duties (for Certifying Authorities, Third Trust Party, etc.) needed for applications of digital
signature or similar procedures.
4
CONCLUSION
In this work we describe how the streamlined transportation simplex method can be powerful
algorithm, which can solve surprisingly large versions of any of these financial analytical services
problems. Because some of analytical problem types have such simple formulation then streamlined
algorithms can easily solve them, by exploiting their special structure.
These streamlined algorithms can cut down tremendously on the computer time required for larger
financial analytical service problems, and they sometimes make it computationally feasible to
similarly solve huge problems. Also, these special-purpose algorithms are included in some other
linear programming software packages, also. This conclusion and this work are a solid base for
future financial analytical modelling processes that are connected with specific customers needs.
Specially, in similar situations, when customers must receive financial analytical service at several
different service facilities, and in one network.
Very much financial analytical research continues to be devoted to developing streamlined
algorithms for special types of linear programming problems. There is widespread interest in
applying linear programming to optimise the operation of complicated large-scale analytical
systems, including some economic systems. The resulting formulations usually have special
structures that can be effectively exploited. Also, being able to recognize and exploit special
financial analytical structures has become a very important factor in the successful application of
these linear programming solutions and specific methods.
5
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