Stochastic Programming in Finance and Business management

Stochastic Programming in Finance and Business management

Lecture 7

Leonidas SakalauskasInstitute of Mathematics and InformaticsVilnius, 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 ClearingMAKER (DRAWER)

DATE CHECKNUMBERAMOUNT

CURRENCY

AUTHORIZED SIGNATURE OF

MAKER’S AGENT

DRAWEEBANK

PAYEE

DRAWEE BANKNUMBER

The paper check is justa carrier of information.

Electronic transmission isbetter.

We dematerialize the check(remove the paper).

0 6 1 3 0 0 1 8 1 8 4 3 1 0 1 4 3 7 0 0 0 0 0 0 0 0 0 1 0 0 0 0 U S D 0 6 5 2 0 0 3 5 6 4 2 5 0 2 0 0 1 0 1 3 0 DRAWER

ACCOUNTNUMBER

DRAWEEBANK

NUMBER

CHECKNUMBER

AMOUNT CURRENCY PAYEEBANK

NUMBER

PAYEEACCOUNTNUMBER

DATE

Only the information is sent to the clearing house

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

Annual Annual Percentage Country Paper-based Electronic Electronic Payments

Switzerland 2 65 97%Netherlands 19 128 87Belgium 16 85 84Denmark 24 100 81Japan 9 31 78Germany 36 103 74Sweden 24 68 74Finland 40 81 67United Kingdom 7 58 50France 86 71 45Canada 76 53 41Norway 58 40 41Italy 23 6 20United States 234 59 20

Noncash Transactions Per Person

ELECTRONIC PAYMENT SYSTEMS 20-763 SPRING

2004 COPYRIGHT © 2004 MICHAEL I. SHAMOS

ACH Credit Transaction

BUYER SELLER

SELLER’SBANK

BUYER’SBANK

CLEARINGHOUSESETTLEMENTBANK

1. BUYER SENDS AN ORDER TO BUYER’S BANK TO CREDIT $X TO SELLER’S ACCOUNT IN SELLER’S BANK 6. SELLER’S BANK

CREDITS SELLER’S ACCOUNT WITH $X

3. CLEARINGHOUSE DETERMINES THAT BUYER’S BANK OWES SELLER’S BANK $Y (ALL TRANSACTIONS ARE NETTED)

4. BUYER’S BANK PAYS $Y TO SETTLEMENT BANK

5. SETTLEMENT BANK PAYS $YTO SELLER’S BANK

2. BUYER’S BANK SENDS TRANSACTION TO AUTOMATED CLEARINGHOUSE

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

Generation of transaction flow

Simulation of settlement process

Analysis of costs and liquidity

Stochastic optimization of settlement system

Calibration of transaction flow

Input of settlement flow data

Analisys

Simulation and Optimization

Simulation of settlement process(BoF-PSS2 simulator)

Data input Simulation of execution Analysis

Įvesties duomenų

bazė

Input module

CSV data of input

Data editor

systemsagentsordersbalansesreserves

Simulation parameters

Imitavimo process

Settlement algorithms

User’s choices

Output data

siytemsagentsordersbalancesreserves

Statistics

Output analyzer

Export module

CSV files

BoF-PSS simulation parameters

1)

2)

3)

DATA FLOWS AND MODEL CALIBRATION

INPUT DATA

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)

C lk,ij

1

,

lijz

k

lkij

lij P

ij – payment flow ith to jth agent;k – payment numberPij –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.

Poisson-lognormal model of transaction flow

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.

,i ip

,ij i ijp r

Transaction flow is Poisson with intensity ;,

Transaction value is lognormal logN(,σ2).


Intensities of interbank flows

Estimation of parameters of Poisson lognormal model

,z

zp i

i

where:

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.

1 ,i j J


,

ln1

z

Pz

ll

2

2

12ln

z

Pz

ll

,zt

z ,

i

ijij z

zr

Generation of transactions

,,,1, lkij

lkij

lkij tt

where:

– random U[0,1];– standard normal;t – transaction application time;Pij – transaction value;l – day of settlement, ;k – payment number;T – period.

, lnk lij

ij

1 l T

SIMULATION OF TRANSACTION FLOW

exp,lkijP

Tt lkij

,,

Costs of ith agent

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.

i i i i i iD RE F B TT AC

COSTS OF PERIOD OF SETTLEMENTS


where:

Kil-1 – correspondent account of ith agent at l-1 day ;

i – day balance:

Gi – deposit (or withdrawal).

1max(0, )l l l li i i iK K G

Correspondent Account

);(1

ji

J

jiji

Premium of satisfaction of reserve requirements

LR – interest rate of refinansing;RRi – reserve requirements.

1max ,

100 360

Tl

i il

i

RR K r

RE

1

lT

l

LRr

T


Penalty of violation of reserve requirements

p – penalty percent points (usually 2.5) .

1

max 0,

360 100

Tl

i il

i

RR K r p

F


Costs of short time loans

i

l– day balance;STL – overnight loan interest rate Xi – deposit (or withdrawal).

1

1min 0,

Tl l l

i i i il

B STL K G

1 1max , max ,0l l l lG X Ki i i i


Losses due to freeze of finance

IBR – interbank interest rate .

0

T li it

TT IBR G


– cost of performance of one transaction

,1 1

T Jl

i i jt j

AC z


Operational costs

Probability of system liquidity

H – Heavyside function;T – period.

1min 0,

1 1

T Jl l lH K Gi i i

l iPlikv T

Condition of unliquidity

1 0l l lK Gi i i


Statement of optimization of settlement costs

Agent costs during period: iiii XDD ,

),,,( 21 Tiiii Vector of day balances:

Expected costs of agent: ,i i i i iL X ED X

Objective function (total expected costs):0

( ) minX

L X

where:

1

J

i ii

L X L X

STATISTICAL OPTIMIZATION OF SETTLEMENT COSTS

Analytical example (one agent, one day period)5.0 5.0 LBR=5 %, IBR=9%, STL=10%

.

dseRR

dseRRxsdseRRxsxgRRXLxRR

sxRR

x

sx

s

2

2

2

05,02

2

075,0125,02

12

2

075,01,02

109,0, 2

)(

2

)(

15

2

)(


dsedsexgRRXQxRR

x

sx

s

2

2

2

125,02

2

2

1,009,0, 2

)(

2

)(

1)

2)

Statistical simulation of expected costs function and its gradient

niiii XDD ,,

where:

- statistical estimate of expected agent costs;

,1

1,

N

i i i i ninL X D X

N

- estimator of gradient of cost function Li(Xi ); ,1

1,

N

i i x i i i nn

Q X D XN


N – number of simulated periods;

- costs during period;

iiix XD , - generalized gradient of cost function;

Stochastic optimization

- step multiplier, >0.

1 1t ti iX X Q X

A (.) – sampling covariance matrix

( , , )11( ) ( ( )) ( ( )) '

tJ Fish J N JtNt t tQ X A X Q X

( , , )tFish J N J ( , )tJ N J


-quantile with degrees of freedom- Fisher

Sakalauskas (2000)

Statistical termination criterion

Testing of optimality hypothesis with significance according to Hotelling criteria:

Confidence interval of estimator of objective function:

- standart normal quantile;

1( ) ( ( )) ( ( )) ( ( ))( , , )

t t t ttN J Q X A X Q X

Fish J N JJ

,)(

t

t

N

N

Xtd


td N - sampling standard deviation of the objective function.

1)

2)

COMPUTER MODELLING

Initial data:

- interest rate of short time loans; - interbank interest rate; - premium interest rate; - penalty percent points on violation of reserve requirements; - period length J=11 – number of agents.

0.10STL 0.08IBR 0.05LBR

2.5r

30T

COMPUTER MODELLING

18080

18100

18120

18140

18160

18180

18200

18220

18240

18260

2710000 2720000 2730000 2740000 2750000

Deposited sum

Su

m o

f sett

lem

en

ts c

osts

-0.004

-0.003

-0.002

-0.001

0

0.001

0.002

0.003

0.004

0.005

2710000 2720000 2730000 2740000 2750000

Deposited sum

Valu

e o

f g

rad

ien

t

Dependencies of expected costs L(X) and gradient Q(X) on deposit (1 ir 10 agents)

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

1E+06 1E+06 2E+06 2E+06 2E+06 2E+06 2E+06 2E+06 2E+06 2E+06

Deposited sum

Su

m o

f sett

lem

en

ts c

osts

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

1E+06 1E+06 2E+06 2E+06 2E+06 2E+06 2E+06 2E+06 2E+06 2E+06

Deposited sum

Valu

e o

f g

rad

ien

t

COMPUTER MODELLING

Total expected costs and deposit under number of iterations

72000 73000 74000 75000 76000 77000 78000

0 10 20 30 40 50 60 Number of iteration

Su

m o

f se

ttle

men

ts c

ost

s

118000000 118500000 119000000 119500000 120000000 120500000 121000000 121500000 122000000 122500000 123000000

0 21 41 61 Number of iteration

De

po

sit

ed

su

m

COMPUTER MODELLINGTotal expected costs and deposited sum under number of iterations (1 ir 10 agents)

17950 18000 18050 18100 18150 18200 18250 18300 18350


Su

m o

f s

ett

lem

en

ts c

os

ts

2710000 2715000 2720000 2725000 2730000 2735000


De

po

sit

ed

su

m

0 5000

10000 15000 20000 25000 30000 35000


Su

m o

f sett

lem

en

ts c

osts

0 200000 400000 600000 800000

1000000 1200000 1400000 1600000 1800000


De

po

sit

ed

su

m

COMPUTER MODELLING

Agent Nr, Expected costs(Iter=0), Lt

Expected costs (Iter=62), Lt

Deposit (Iter=0), Lt106

Deposit(Iter=62), Lt106

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 Planningand 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.


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.


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 – line of credit (x1t),2 – pledging of accounts receivable (factoring) (x2t), 3 – stretching of accounts payable (x3t),4 – 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 ≤ β1 x4 ≤ β4vx21 ≥ 0.7 · AR0 x11 + x4 ≤ β41x21 ≤ 0.9 · AR0 x21 + x4 ≤ β42x31 ≤ 0.8 · AP0 x51 ≤ AR0 ·rvx4 ≥ β4a x51 + L1 ≥ LR

Second stage constraints:x12 - L1 ≤ 0 x11 + x12 + x4 ≤ β41x22 ≥ 0.7 · AR1 x21 + x22 + x4 ≤ β42x22 ≤ 0.9 · AR1 x52 ≤ AR0 ·rvx32 ≤ 0.8 · AP1 x51 + x52 + L1 ≥ LRx31 + x32 ≤ β3

Model details (5)

Initial balance:

First stage balance:

Second stage balance:

Objective function:

Model details (6)

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

+

The objective function includes costs of financial instruments:

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:

5900

5950

6000

6050

6100

6150

6200

6250

0 50 100 150 200 250

Iteration number

Ob

ject

ive

fun

ctio

n v

alu

e

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

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- ]

the investment management problem differs from considered above only by the objective function that involves the surplus at the second stage as a profit

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 $110 million

Coal Hydroelectric $180 million

Nuclear power $450 million

Hydroelectric $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

Demand (μ,GW)Duration (hours)

#1 26.0 490

#2 21.5 730

#3 17.3 2190

#4 13.9 3260

#5 11.1 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 Cost per KWH

Gas Turbine 3.92 ¢

Coal 2.44 ¢

Nuclear 1.40 ¢

Hydroelectric 0.40 ¢

External Source 15.0 ¢

Each year costs grow with rate 1%.

The problem can be modeled as a two-stage stochastic linear optimization model

4 5 5 15

1 1 1 1

min min,i i i j k ijkyi i j k

c x E q h r y

4

1

4

5

1

10000,

5.0,

, 1,2,3,4, , , ,

, , , ,

0, 0,

i ii

ijk i

ijk jki

c x

x

y x i j k

y D j k

x y

subject to

where

x=(x1,x2,x3,x4)vector, representing the gigawatts of capacity to be built for each type of plant

yijkωamount 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

PlantOptimal Construction

Decision based on stochastic data

Optimal Construction Decision based on

expected data

Gas Turbine 4.65 GW 1.87 GW

Coal 4.56 GW 3.29 GW

Nuclear 4.70 GW 4.10 GW

Hydroelectric 5.00 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


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


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.


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. )


Uncertain factors: investments maintenance / transportation costs supply demand


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;


ic

kijx

kijq

iy

investment cost for building facility

per-unit cost of processing product k

flow of product k from node i to node j

1 if a processing facility i is built or machine i is procured, and 0 otherwise


min)(

Kk Aij

kij

kij

Piii xqyc

||1,0 PYy

KkPjxxNl

kil

Ni

kij

,0

KkCjdx kj

Ni

kij

,

KkSisx ki

Nj

kij

,

Kkjj

Ni

kij

kj Pjymxr

|||| KARx

conservation of product k across each processing node j

demand satisfaction

supply restriction

capacity constraints of the processing nodes, denotes per-unit processing requirement,

denotes capacity

kjrkjm

Supply chain management

min],([:)( yQEycyf T ||1,0 PYy

Q(y,ξ)=qTx+hTz ―>min

Nx = 0

dzDx

sSx

MyRx |||| KARx

conservation of products

demand satisfaction

supply restriction

capacity constraints of the processing nodes

Second Stage:

First Stage:

),,,( Msdq - uncertain parameters

Supply chain management

NN1100 WN7342

LQ8811

AJ8172

RN0098

11

21

AJ8172 is the final product

NN1100, WN7342, LQ8811, RN0098 are components

Example of materials planning requirement

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 developedComputer 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.

Documents

Stochastic Programming in Finance and Business management