Lecture 7
Stochastic Programming in
Finance and Business management
Leonidas Sakalauskas
Institute of Mathematics and Informatics
Vilnius, Lithuania
EURO Working Group on Continuous Optimization
Content






Introduction
Interbank payment management
Rational cash planning and investment
management
Power production planning
Supply chain management
Network stochastic optimization
Introduction


The aim of stochastic programming is being applied
in a wide variety of subjects ranging from
agriculture to financial planning and from industrial
engineering to computer networks and data mining.
Our prime goal is to help to develop an intuition on
how to model uncertainty into mathematical
problems, what uncertainty changes bring to the
decision process, and what techniques help to
manage uncertainty in solving the problems.
Interbank payment management
Electronic Clearing
The paper check is just
a carrier of information.
MAKER (DRAWER)
DATE
PAYEE
DRAWEE
BANK
CHECK
NUMBERAMOUNT
CURRENCY
AUTHORIZED
SIGNATURE OF
MAKER’S AGENT
DRAWEE BANK
NUMBER
Electronic transmission is
better.
We dematerialize the check
(remove the paper).
06130018184310143700000000010000USD065200356425020010130
DRAWEE
BANK
NUMBER
DRAWER
ACCOUNT
NUMBER
CHECK
NUMBER
AMOUNT
CURRENCY
PAYEE
BANK
NUMBER
PAYEE
ACCOUNT
NUMBER
Only the information is sent to the clearing house
DATE
Interbank payment management
Interbank payment management



Active introduction of means of electronic data
transfer in banking and concentration of interbank
payments were related with creation of an automated
system of clearing (ACH).
ACH should provide the principles of stability,
efficiency, and security. The participants of a system
must meet the requirements of liquidity and capital
adequacy measures.
Sensitivity of the sector of interbank payments and
settlements to changes makes the subject of
investigation on simulation and optimization of
interbank payments topical both in theory and in
practice.
Automated Clearing House (ACH)






Nationwide wholesale electronic payments
system
Transactions not processed individually
Banks send transactions to ACH operators
Batch processing store-and-forward
Sorted and retransmitted within hours
Banks





Originating Depository Financial Institutions (ODFIs)
Receiving Depository Financial Institutions (RDFIs)
Daily settlement by RTGS
Posted to receiver’s account within 1-2
business days
Typical cost: $0.02 per transaction; fee higher
Noncash Transactions Per Person
Country
Payments
Switzerland
Netherlands
Belgium
Denmark
Japan
Germany
Sweden
Finland
United Kingdom
France
Canada
Norway
Italy
United States
Annual
Paper-based
2
19
16
24
9
36
24
40
7
86
76
58
23
234
Annual Percentage
Electronic
Electronic
65
128
85
100
31
103
68
81
58
71
53
40
6
59
97%
87
84
81
78
74
74
67
50
45
41
41
20
20
ACH Credit Transaction
1.
BUYER SENDS
AN ORDER TO
BUYER’S BANK TO
CREDIT $X TO
SELLER’S ACCOUNT
IN SELLER’S BANK
BUYER
SELLER
6.
2.
BUYER’S
BANK
4.
BUYER’S BANK
SENDS TRANSACTION
TO AUTOMATED
CLEARINGHOUSE
SELLER’S BANK
CREDITS SELLER’S
ACCOUNT WITH $X
SELLER’S
BANK
BUYER’S BANK
PAYS $Y TO
SETTLEMENT BANK
SETTLEMENT
BANK
5. SETTLEMENT BANK
PAYS $YTO
SELLER’S BANK
CLEARINGHOUSE
3. CLEARINGHOUSE DETERMINES THAT
BUYER’S BANK OWES SELLER’S BANK $Y
(ALL TRANSACTIONS ARE NETTED)
ELECTRONIC PAYMENT SYSTEMS
20-763
SPRING
2004
COPYRIGHT © 2004 MICHAEL I.
SHAMOS
Daylight Overdrafts



An overdraft that must be repaid by the close
of business the same day
U.S Federal Reserve allows daylight overdrafts
Hong Kong does not
U.S. Federal Reserve Daylight Overdraft History
SYSTEM OF SIMULATION AND OPTIMIZATION
OF INTERBANK SETTLEMENTS
Analisys
Input of settlement flow data
Calibration of transaction flow
Simulation and Optimization
Generation of transaction flow
Simulation of settlement process
Analysis of costs and liquidity
Stochastic optimization of
settlement system
Simulation of settlement process
(BoF-PSS2 simulator)
Data input
Įvesties
duomenų
bazė
Input
module
systems
agents
orders
balanses
reserves
Simulation of execution
Simulation
parameters
Imitavimo process
Settlement
algorithms
Analysis
Output
data
siytems
agents
orders
balances
reserves
Statistics
Output
analyzer
Export
module
User’s choices
CSV data
of input
Data editor
CSV files
BoF-PSS simulation parameters
1)
2)
3)
DATA FLOWS AND MODEL
CALIBRATION
INPUT DATA
DATA FLOWS AND MODEL
CALIBRATION
Transaction data





Transaction number ID;
Sender code a;
Recipient code b;
Date and time of transaction t;
Transaction value P.
DATA FLOWS AND MODEL
CALIBRATION (CNS SYSTEM)
l
zij
ijl   Pijk ,l  Ck,l
k 1
ij
ij – payment flow ith to jth agent;
k – payment number
Pij –payment from ith to jth agent;
l – day of payment;
C – indicator of settlement performance (C=1 – performed, C=0 – delayed,
in CNS systems C=1) ;
zij – number of payments from ith to jth agent.
DATA FLOWS AND MODEL
CALIBRATION
Poisson-lognormal model of transaction flow
Transaction flow is Poisson with intensity ;,
Transaction value is lognormal logN(,σ2).
Intensities of interbank flows
i    pi ,
ij    pi  rij ,
where:
i –intensity of ith agent flow;
pi – probability of ith agent payment;
rij – conditional probability of transaction from ith to jth agent;
 – average of logarithms transaction value;
2 – variance of logarithms transaction value.
DATA FLOWS AND MODEL
CALIBRATION
Estimation of parameters of Poisson lognormal model
zij
zi
,
pi  , rij 
z
zi
z
 ,
tz
z
 lnPl 
  l 1
z
2
z
 lnPl 
,
 2  l 1
z
 2
where:
1  i, j  J
z – number of transactions;
J – number of agents;
zi – number of transactions of ith agent;
zij – number of transactions from ith to jth agent;
tz – time of the session end.
SIMULATION OF TRANSACTION FLOW
Generation of transactions
tijk ,l  tijk 1,l  ijk ,l ,
 ijk ,l

 ln  
Pijk ,l  exp    
where:
 – random U[0,1];
– standard normal;
t – transaction application time;
Pij – transaction value;
l – day of settlement, 1  l  T ;
k – payment number;
T – period.
ij
,
t k ,l  T
ij
COSTS OF PERIOD OF SETTLEMENTS
Costs of ith agent
Di  REi  Fi  Bi  TTi  ACi
Di – total costs of i-th agent;
REi – premium of satisfaction of reserve requirements;
Fi – penalty of violation of reserve requirements;
Bi – costs of short time loans;
TTi – losses due to freeze of finance;
ACi – operacional costs.
COSTS OF PERIOD OF SETTLEMENTS
Correspondent Account
l
Ki
l 1
l
 max(0, Ki  i
l
 Gi )
where:
Kil-1 – correspondent account of ith agent at l-1 day ;
i – day balance:
J
 i   (ij  ji );
j 1
Gi – deposit (or withdrawal).
COSTS OF PERIOD OF SETTLEMENTS
Premium of satisfaction of reserve requirements
T


 max RRi , Kil  r
REi 
l 1
100  360
LRl
r
l 1 T
T
LR – interest rate of refinansing;
RRi – reserve requirements.
COSTS OF PERIOD OF SETTLEMENTS
Penalty of violation of reserve requirements
 T
l
max  0,  RRi  Ki    r  p 
 l 1

Fi 
360 100
p – penalty percent points (usually 2.5) .
COSTS OF PERIOD OF SETTLEMENTS
Costs of short time loans
T

Bi   STL   min 0, Kil 1   il  Gil
l 1



Gil  max  X il ,  max Kil 1  il 1,0 


il– day balance;
STL – overnight loan interest rate
Xi – deposit (or withdrawal).
COSTS OF PERIOD OF SETTLEMENTS
Losses due to freeze of finance
T
TTi  IBR   Gl
t 0 i
IBR – interbank interest rate .
COSTS OF PERIOD OF SETTLEMENTS
Operational costs
T
J
ACi      zil, j
t 1 j 1
 – cost of performance of one transaction
COSTS OF PERIOD OF SETTLEMENTS
Probability of system liquidity
T J

Plikv  l 1i 1


H  min 0, Kil 1  il  Gil 


T
H – Heavyside function;
T – period.
Condition of unliquidity
Kil 1   il  Gil  0
STATISTICAL OPTIMIZATION OF SETTLEMENT
COSTS
Statement of optimization of settlement costs
Agent costs during period:
Di  Di  X i , i 
 i  ( i1 ,  i2 , ,  iT )
Vector of day balances:
Expected costs of agent:
Li  X i   EDi  X i , i 
Objective function (total expected costs):
where:
J
L  X    Li  X i 
i 1
L( X )  min
X 0
STATISTICAL OPTIMIZATION OF SETTLEMENT
COSTS
Analytical example (one agent, one day period)
  0.5   0.5
1)
L X , RR  0,09  g x  
LBR=5 %, IBR=9%, STL=10%
x
 ( s ) 2
2
1
  0,1 s  x   0,075 RR e 2
  2   15
 ( s ) 2
2)
 ds 
 ( s ) 2
RR x
2
1
  0,125 s  x   0,075 RR e 2
 2  x
 ( s ) 2
x
2
0,1
0,125 RR x 2 2
2


Q X , RR  0,09  g x  
 e
 ds 
  e
 ds
  2   
 2  x
.
 ds 

 ( s ) 2
2
0,05 RR
  e 2 
  2   RR x
 ds
STATISTICAL OPTIMIZATION OF SETTLEMENT
COSTS
Statistical simulation of expected costs function and its gradient

where:
1 N
Li  X i  
 Di X i ,  i , n
N n 1



1 N
Li  X i   - statistical
 Di Xestimate
i ,  i ,n of expected agent costs;
N n 1
Di  Di X i ,  i ,n  - costs during period;


1 N
Qi  X i  
  x Di X i , i ,n - estimator of gradient of cost function Li(Xi );
N n 1
 x Di  X i , i  - generalized gradient of cost function;
N – number of simulated periods;
STATISTICAL OPTIMIZATION OF SETTLEMENT
COSTS
Stochastic optimization
X t 1  X t  1    Qi  X i  
 - step multiplier, >0.
N t 1 
J  Fish( , J , N t  J )
  Q( X t )  ( A( X t ))1  (Q( X t )) '
Sakalauskas (2000)
Fish( , J , N t  J ) - Fisher  -quantile with ( J , N t  J ) degrees of freedom
A (.) – sampling covariance matrix
STATISTICAL OPTIMIZATION OF SETTLEMENT
COSTS
Statistical termination criterion
1) Testing of optimality hypothesis with significance  according to Hotelling criteria:
( N t  J )  (Q( X t ))  ( A( X t ))1  (Q( X t ))
 Fish(  , J , N t  J )
J
t


d
(
X
)
2) Confidence interval of estimator of objective function:  N t
Nt

d
- standart normal  quantile;
N t - sampling standard deviation of the objective function.
 ,
COMPUTER MODELLING
Initial data:
STL  0.10 - interest rate of short time loans;
IBR  0.08 - interbank interest rate;
LBR  0.05 - premium interest rate;
r  2.5 - penalty percent points on violation of reserve
requirements;
T  30 - period length
J=11 – number of agents.
COMPUTER MODELLING
18260
0.005
18240
0.004
18220
0.003
18200
Value of gradient
Sum of settlements costs
Dependencies of expected costs L(X) and gradient Q(X) on deposit
(1 ir 10 agents)
18180
18160
18140
18120
0.001
0
2710000
-0.001
2720000
2730000
2740000
2750000
-0.002
18100
18080
2710000
0.002
-0.003
2720000
2730000
2740000
2750000
-0.004
Deposited sum
Deposited sum
18000
0.01
0.005
14000
12000
Value of gradient
Sum of settlements costs
16000
10000
8000
6000
4000
2000
0
1E+06 1E+06
2E+06 2E+06
2E+06 2E+06 2E+06 2E+06 2E+06 2E+06
-0.005
-0.01
-0.015
0
1E+06 1E+06 2E+06 2E+06 2E+06 2E+06 2E+06 2E+06 2E+06 2E+06
Deposited sum
-0.02
Deposited sum
COMPUTER MODELLING
Total expected costs and deposit under number of iterations
123000000
122500000
77000
122000000
Deposited sum
Sum of settlements costs
78000
76000
75000
74000
121500000
121000000
120500000
120000000
119500000
73000
119000000
72000
118500000
0
10
20
30
40
Number of iteration
50
60
118000000
0
21
41
Number of iteration
61
COMPUTER MODELLING
Total expected costs and deposited sum under number of iterations
(1 ir 10 agents)
2735000
18300
Deposited sum
Sum of settlements costs
18350
18250
18200
18150
18100
18050
2730000
2725000
2720000
2715000
18000
17950
2710000
0
10
20
30
40
50
60
1
21
1600000
30000
1400000
Deposited sum
Sum of settlements costs
61
1800000
35000
25000
20000
15000
10000
1200000
1000000
800000
600000
400000
5000
0
41
Number of iteration
Number of iteration
200000
0
10
20
30
40
Number of iteration
50
60
0
1
21
41
Number of iteration
61
COMPUTER MODELLING
Agent Nr,
Expected costs
(Iter=0), Lt
Expected costs
(Iter=62), Lt
Deposit
(Iter=0), Lt106
Deposit
(Iter=62), Lt106
1
18292,4±150,5
18167,0±19,0
2,71
2,73
2
243183,9±1159,5
241641,8±144,1
36,20
36,40
3
102995,0±909,8
100477,5±83,8
14,80
15,10
4
37527,3±524,9
36206,6±51,6
5,25
5,45
5
156596,1±807,2
155361,6±91,1
23,20
23,50
6
16343,1±435,2
10510,2±17,0
1,40
1,57
7
55480,1±675,5
52718,4±46,8
7,70
7,93
8
48146,4±636,6
46277,6±52,3
6,80
7,00
9
120797,1±974,3
118070,5±88,8
17,50
17,80
10
30794,2±873,5
22294,6±34,7
3,00
3,33
11
16841,9±1438,8
13128,0±139,7
1,30
1,71
Total
76999,7±361,2
74077,6±62,0
119,86
122,52
DISCUSSION AND CONCLUSIONS



The statistical Poisson-lognormal model of electronic
settlement flows has been created as well as the
methodology for calibrating the settlement flow
model has been developed and adapted to the
analysis of real time settlement data
The methodology of modeling interbank settlement
flows by the Monte-Carlo estimator has been
developed following to instructions of Central Bank
The algorithm of stochastic optimization of
settlement costs, a view on settlement costs and
liquidity risk has been created
Rational Cash Planning
and Investment Management
Introduction
Problems of cash management often arise in
public, non-profit-making or business institutions
Many companies are faced with choosing a
source of short-term funds from a set of
financing alternatives
Various constraints are placed on the financial
options. The objective is to minimize the short
run financial costs of the financial alternatives
plus expected penalty costs for shortages and
surpluses in stage’s balances.
Financial instruments (1)
Line of credit – is the maximum amount of
credit that a financial institution can give
out to a business firm. This alternative has
two interest rates – one on taken part of
the credit line and another on not taken.
So, if the firm doesn’t have needs for
additional financing, she pays only small
interest rate on limit for credit line.
Financial instruments (2)
Pledging of accounts receivable (factoring)
– is the financial transaction whereby a firm
may borrow by pledging its accounts
receivable to a third part as security for
loan. The bank will lend up 70-90% of the
face value of pledged accounts receivable
and will get remainder 30-10% accounts
receivable part then debtor pay all his debt
to the bank. For this option firm must pay
interest rate for difference from borrowed
and reminded parts of the accounts
receivable.
Financial instruments (3)
Stretching of accounts payable – the firms,
at this option, may delay payments of
accounts payable. The firm may stretch up
to 80% of the payments due in the period.
Term loan – the firm may take out a term
loan from bank at the beginning of the
initial period.
Marketable securities – the firm may invest
any cash in short term securities.
Stochastic linear model
Thus, interpreting financial instruments
described above the stochastic linear model
for this task is:
The objective is to minimize the short run
financial costs of the financial alternatives
plus expected penalty costs for shortages
and surpluses in stage’s balances.
Model details (1)
The formulation given below refers to a
short term financial planning model based
on Kallberg, White and Ziemba, 1982.
Funds are received or disbursed at the
beginning of periods.
All quantities are in thousands of dollars.
In this model xit – denote the amount
obtained from option i in period t, t = 1,2.
Model details (2)
Used financial alternatives:
1
2
3
4
–
–
–
–
line of credit (x1t),
pledging of accounts receivable (factoring) (x2t),
stretching of accounts payable (x3t),
term loan (x4),
5 – marketable securities (x5t).
Another used options:
ARj – accounts receivable, j = 0,1,2. (j – denote
planning moments),
APj – accounts payable,
LR – liquidity reserve,
L1 – contribution to liquidity reserve from credit line
option,
Model details (3)
x6j+, x6j- – surpluses and shortages respectively,
j=0,1,2.
r12, r11 – interest rate for taken/not taken part of
the credit line,
r2 – interest rate for pledging of accounts
receivable,
r3 – interest rate for stretching of accounts payable,
r4 – interest rate for term loan,
r5 – interest rate for marketable securities,
rv – norm of cash which can be invested,
β1 – upper bound of a limit of a credit line,
β3 – upper bound of stretching of accounts payable,
β4a , β4v – lower and upper bound of a term loan,
β41 , β42 – upper bound for constraints on financing
combinations,
Model details (4)
First stage constraints:
x11 + L1 ≤
x21 ≥ 0.7 ·
x21 ≤ 0.9 ·
x31 ≤ 0.8 ·
x4 ≥ β4a
β1
AR0
AR0
AP0
x4 ≤ β4v
x11 + x4 ≤
x21 + x4 ≤
x51 ≤ AR0
x51 + L1 ≥
β41
β42
·rv
LR
Second stage constraints:
x12
x22
x22
x32
x31
- L1 ≤ 0
≥ 0.7 · AR1
≤ 0.9 · AR1
≤ 0.8 · AP1
+ x32 ≤ β3
x11 + x12 + x4 ≤ β41
x21 + x22 + x4 ≤ β42
x52 ≤ AR0 ·rv
x51 + x52 + L1 ≥ LR
Model details (5)
Initial balance:
First stage balance:
Second stage balance:
Objective function:
Model details (6)
The objective function includes costs of financial
instruments:
F(x)= r6 · x60- + r12 · x11 + r11 · L1+
+ r2 (2 · x21 – AR0) + r3 · x31 +
+r4 · x4 – r5 · x51 + r6 · x61- +
+r12 ·(x11 + x12) + r11 · (L1 – x12) +
+ r2 (2 · x21 – AR0)+
+r2 (2 · x22 – AR1) + r3 ·(x31 + x32)+
+ r4 · x4 – r5 ·(x51 + x52) + r6 · x62-+
+ r7 ·x62+
Model details (7)
This model describes the case, when factoring
option was used both in first and in the second
stages of the financial planning.
Other cases with term loan and factoring
options applied or not are similar.
We analyzed two models. In the first model
(TDD) we have used term loan and used
factoring option in the first stage, in the second
stage we have used or not used factoring
option depending on objective function value.
In the second model (DD) we have not used
term loan and have used factoring option in the
first stage, in the second stage we have used or
not used factoring option depending on
objective function value.
Stochastic algorithm for optimization (1)
The Monte-Carlo method is applied for the
solving of the two-stage stochastic model
described in (1).
Generated random sample:
estimator of the objective function is:
where
sampling variance:
,
Stochastic algorithm for optimization (2)
Sampling estimator of the gradient:
The iterative stochastic procedure of gradient
search could be used further:
,
where
is a step-length multiplier, and
is projection of a gradient estimator
to the ε -feasible set.
Initial solution is obtained solving the deterministic
problem.
Results (1)
The approach described above has been
realized by means of C++ to solve two-stage
stochastic linear model and calculate the
objective function by Monte-Carlo method. Two
different initial solutions for the subsequent
optimization were applied and the software was
tested with three data files.
Results for first (TDD) and second (DD) models
are presented below.
Results (2)
Optimal solutions for 3 data sets for TDD model.
Optimal solutions for 3 data sets for DD model.
Results (3)
Objective function values during
optimization process:
6250
Objective function value
6200
6150
6100
6050
6000
5950
5900
0
50
100
150
Iteration number
200
250
Results (4)
Optimal objective function values for 3 data sets
for TDD model.
Optimal objective function values for 3 data sets
for DD model.
Investment Management
the investment management problem differs from
considered above only by the objective function that
involves the surplus at the second stage as a profit
f(x) = x60+ - x60- –
– [ rtr · xg1 + x60- ] +
+ x61+ – x61- –
– [ r12 · x11 + r11 · L1 + r2 · (2 · x21 – AR0) + r3 ·
x31 + r4 · x4 – rg5 · xg51 – (rf5 + rk5) · xk51 – rr5 ·
xr51 – rg · xg1 + rtr · xg2 + r6 · x61- ] +
+ x62+ - x62- –
– [ r12 ·(x11 + x12) + r11 · (L1 – x12) + r2 (2 · x21 –
AR0) + r2 (2 · x22 – AR1) + r3 · (x31 + x32) + r4 · x4
– rg5 · (xg51 + xg52) – (rf5 + rk5) · (xk51 + xk52) –
rr5 · (xr51 + xr52) – rg · (xg1 + xg2) + r6 · x62- ]
Conclusion
Two-stage short term financial planning model
under uncertainty was formulated and solved
using the Monte-Carlo method for estimation of
the objective function.
The results of computer experiment have been
showed that stochastic optimization allows us to
compare different financial management
decisions.
Thus, comparison of various alternatives of
financial instruments takes opportunities to
reduce costs of financial instruments and ensure
liquidity as well as optimal planning of the cash
flows.
Power production
planning
Power Plant Investment Planning
Problem
Let the plants are expected to operate
over 15 years.
The budget for construction of power
plants is $10 billion, which is to be
allocated for four different types of plants:
gas turbine,
coal,
nuclear power,
and hydroelectric.
The objective is to minimize the sum of
the investment cost and the expected value
of the operating cost over 15 years.
Power plants are priced according to their
electric capacity, measured in gigawatts
(GW).
Plant
Cost per GW
capacity
Gas Turbine
Coal Hydroelectric
Nuclear power
Hydroelectric
$110 million
$180 million
$450 million
$910 million
Expected demand for electric power is
normally distributed random value D~N(μ,
0.5).
A set of demands and durations during
the year is shown in Table:
Demand
Block
#1
#2
#3
#4
#5
Demand (μ,GW)
26.0
21.5
17.3
13.9
11.1
Duration
(hours)
490
730
2190
3260
2090


Each year costs grow with rate 1%.
Since hydroelectric energy depends
on the availability of rivers which
may be dammed, the geography of
the region constrains the
hydroelectric power capacity no
more than 5.0 GW.
Operating cost of power generation
Plant
Gas Turbine
Coal
Nuclear
Hydroelectric
External Source
Cost per KWH
3.92 ¢
2.44 ¢
1.40 ¢
0.40 ¢
15.0 ¢
Each year costs grow with rate
1%.
The problem can be modeled as a
two-stage stochastic linear optimization
model

 5 5 15



c
x

E
min
q
h
r
y
 y   i j k ijk    min,

i i
i 1

 i 1 j 1 k 1



4
subject to
4
c x
i 1
i i
 10000,
x4  5.0,
yijk  xi , i  1, 2,3, 4,  j , k ,  ,
5
y
i 1
ijk
 D jk ,  j , k ,  ,
x  0, y  0,
where
x=(x1,x2,x3,x4)
vector, representing the gigawatts of capacity to be built
for each type of plant
yijkω
amount of electricity capacity used to produce electricity
by power plant type i for demand block j in year k in GW
ci
investment cost per GW of capacity for power plant type i
qi
operating cost of power generation for power plant type i
hj
duration of the demand block j
rk
operating costs growth in year k
Djkω
power demand in year k at demand block j N(μj, 0.5)
Solving the Problem
The first stage of the problem
contains 6 variables and 4 restrictions, the
second stage contains correspondingly
375 variables and 375 restrictions.
Solving the problem the termination
conditions have been met after t=123
iterations.
Results
The optimal expected cost of power plant
investment problem is turns out to be
$16.5030.029 billion versus deterministic cost
$17.4750.053 billion.
The size of the last Monte-Carlo sample =
15638, when the total amount of calculations
(the size of all Monte-Carlo samples) 290866,
thus, the approach developed required only
18.6 times more computations as compared
with the computation of one function value.
Change of the objective function
Power plant capacity construction
decisions
Plant
Gas Turbine
Coal
Nuclear
Hydroelectric
Optimal Construction Optimal Construction
Decision based on
Decision based on
stochastic data
expected data
4.65 GW
1.87 GW
4.56 GW
4.70 GW
5.00 GW
3.29 GW
4.10 GW
5.00 GW
Conclusions


The approach presented is grounded by the
termination and the rule for adaptive
regulation of the size of Monte-Carlo
samples, taking into account statistical
modelling accuracy.
The proposed termination procedure allows
us to test the optimality hypothesis and to
evaluate reliably confidence intervals of
objective function in a statistical way.
Supply Chain
Management
Supply Chain Management



development and operating in
supply chain is the critical action of
the enterprise
the supply network structure
(number, location and capacity of
facilities) is designed on strategic
level
the amounts of stuffs and goods
are planned on operational level
Supply Chain Management


T. Santoso, S. Ahmed, M. Goetschalckx and
A. Shapiro. A stochastic programming
approach for supply chain network design
under uncertainty. European Journal of
Operational Research, 2005, vol. 67, No1,
pp. 96-115.
S. Wob, and D. L. Woodruf Introduction to
Computational Optimization Models for
Production Planning in a Supply Chain.
Springer, Berlin, 2005.
Supply Chain Management
uncertainty is a critical factor in
supply management rising due to
globalization
 the disorder of the supply chain
causes 8.6% decrease of goods
price that can grow up to 20%
(see, M.Hicks. (2002) When the chain
snaps. )

Supply Chain Management
Uncertain factors:
 investments
 maintenance / transportation costs
 supply
 demand
Supply Chain Management
Supply chain G=(N, A)
N=S ν P ν C – nodes;
S – suppliers;
P = M ν F ν W – maintenance points;
M – production units;
F – final bases;
W – warehouses;
C – customers;
A = – connections;
Supply Chain Management
ci
yi
k
ij
investment cost for building facility
1 if a processing facility i is built or machine i
is procured, and 0 otherwise
x
flow of product k from node i to node j
qijk
per-unit cost of processing product k
Supply Chain Management
 ci yi  
iP
k k
q
 ij xij  min
kK ( ij )A

y Y  0,1
x x
iN
k
ij
lN
x
x
jN
k
ij
 sik i  S , k  K

 r   x
kK
k
j
 0 j  P, k  K
 d kj j  C, k  K
k
ij
iN
k
il
|P|
iN
k
ij

  m j y j j  P

x  R
| A||K|
conservation of product k across each
processing node j
demand satisfaction
supply restriction
capacity constraints of the processing nodes,
rjk
denotes per-unit processing requirement,
mk denotes capacity
j
Supply chain management
First Stage:
f ( y) : cT y  E[Q( y,  ]  min

y Y  0,1
|P|
Second Stage:
Q(y,ξ)=qTx+hTz
Nx = 0
Dx  z  d
Sx  s
conservation of products
demand satisfaction
supply restriction
Rx  My
x  R
―>min
| A||K|
capacity constraints of the processing nodes
  (q, d , s, M )
- uncertain parameters
Supply chain management
Example of materials planning requirement
NN1100
WN7342
1
1
LQ8811
RN0098
1
2
AJ8172
AJ8172 is the final product
NN1100, WN7342, LQ8811, RN0098 are components
Network Stochastic
Optimization
Network Stochastic Optimization
Single server system
Open Queuing network
The arrival and service times are
distributed according to Poisson law in
Markov systems and networks
Criteria of optimization



Costs of service
Response time (delay)
Throughput
Wrap-Up and Conclusions
Nonlinear and linear stochastic programming models
for application in finance and business have been
considered.
The approach for stochastic optimization by the
Monte-Carlo method has been developed
Computer experiments have been shown that
stochastic optimization allows us to compare
different managerial decisions.
Stochastic comparison of various alternatives of
managerial decisions takes us opportunities to
reduce logistic costs and ensure the reliable meeting
of engagements.