Lecture slides

Transcript Lecture slides

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

MAKER (DRAWER) PAYEE DRAWEE BANK DRAWEE BANK NUMBER DATE CHECK NUMBERAMOUNT The paper check is just a carrier of information.

Electronic transmission is better.

CURRENCY AUTHORIZED SIGNATURE OF MAKER’S AGENT 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 DRAWEE BANK NUMBER DRAWER ACCOUNT NUMBER CHECK NUMBER AMOUNT CURRENCY PAYEE BANK NUMBER PAYEE ACCOUNT NUMBER DATE Only the information is sent to the clearing house

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

Annual Annual Percentage Country Paper-based Electronic Electronic Payments Switzerland Netherlands Belgium Denmark Japan Germany Sweden Finland United Kingdom France Canada Norway Italy United States 2 19 16 24 9 36 24 40 7 86 76 58 23 234 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 BUYER’S BANK 2. BUYER’S BANK SENDS TRANSACTION TO AUTOMATED CLEARINGHOUSE 4. BUYER’S BANK PAYS $Y TO SETTLEMENT BANK SELLER SELLER’S BANK 6. SELLER’S BANK CREDITS SELLER’S ACCOUNT WITH $X 5. SETTLEMENT BANK PAYS $YTO SELLER’S BANK SETTLEMENT 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)

Simulation of execution Data input Analysis Input module Įvesties duomenų bazė



systems



agents



orders



balanses



reserves Simulation parameters Imitavimo process Output data



siytems



agents



orders



balances



reserves Statistics Output analyzer Export module Settlement algorithms

User’s choices

CSV data of input Data editor CSV files

1) 2) 3)

BoF-PSS simulation parameters

DATA FLOWS AND MODEL CALIBRATION

INPUT DATA

DATA FLOWS AND MODEL CALIBRATION Transaction data      Transaction number Sender code

; Recipient code

;

; Date and time of transaction

; Transaction value

DATA FLOWS AND MODEL CALIBRATION (CNS SYSTEM)



l ij



k z l ij

  1

P ij k

 C k, ij l 

– payment flow i th to j th agent;

k – P ij

payment number

–

payment from i th to j th agent;

l – C z ij

day of payment; – indicator of settlement performance (C=1 – performed, C=0 – delayed, in CNS systems C=1) ;

–

number of payments from i th to j th agent.

DATA FLOWS AND MODEL CALIBRATION

Poisson-lognormal model of transaction flow Transaction flow is Poisson with intensity  ;, Transaction value is lognormal

logN

(  , σ 2 ).

Intensities of interbank flows 



where: 

p i

, 

p i i –

intensity of i th

– r ij

  2

– –

probability of i agent flow; th agent payment; – conditional probability of transaction from i th average of logarithms transaction value; variance of logarithms transaction value.

to j th agent;

DATA FLOWS AND MODEL CALIBRATION

Estimation of parameters of Poisson lognormal model  

 

t z l z

  1 ln

, ,

p i



z i z

r ij



z z i ij

 2 

l z

  1  ln    2

  2 , where: 1  

J z –

number of transactions;

J –

number of agents;

z i – t z ij z – –

number of transactions of time of the session end.

i th

number of transactions from

agent;

j th

agent;

SIMULATION OF TRANSACTION FLOW

t k ij



t k ij

 1 ,

 Generation of transactions 

k ij

, 

,   ln 

t k

l ij



T P ij k

 exp        where: 

–

random U[0,1];  – standard normal;

t – P ij l – k – –

transaction application time; transaction value; day of settlement, ; payment number;

T –

period.

COSTS OF PERIOD OF SETTLEMENTS

Costs of i th agent

D i



RE i



F i



B i



TT i



AC i D i RE i F i – – B i – –

total costs of

th agent; premium of satisfaction of reserve requirements; penalty of violation of reserve requirements; costs of short time loans;

TT i

– losses due to freeze of finance;

AC i –

operacional costs.

COSTS OF PERIOD OF SETTLEMENTS

Correspondent Account

K i l



max(0,

K i l

 1  

i l



G i l

)

where:

K i l-1

– correspondent account of

i th

agent at

-1 day ; 

i G i –

day balance: 



 

1 – deposit (or withdrawal).

( 

 

);

COSTS OF PERIOD OF SETTLEMENTS

Premium of satisfaction of reserve requirements

RE i



l T

  1 max 

i i l

 

r r



l T

  1

LR l T LR – RR i –

interest rate of refinansing; reserve requirements.

COSTS OF PERIOD OF SETTLEMENTS

Penalty of violation of reserve requirements

F i

  max 0, 

l T

  1

RR i



K l i

   



p –

penalty percent points (usually 2.5) .

COSTS OF PERIOD OF SETTLEMENTS

Costs of short time loans

B i

 

STL



l T

  1

K i l

 1  

i l



G i l



G i l

 max

X l i



K l i

 1  

i l

 1 , 0  

i l – STL – X i –

day balance; overnight loan interest rate deposit (or withdrawal).

COSTS OF PERIOD OF SETTLEMENTS

Losses due to freeze of finance

TT i



IBR T

 

 0

G i l IBR –

interbank interest rate .

COSTS OF PERIOD OF SETTLEMENTS

Operational costs

AC i t T J

 

z l i j



–

cost of performance of one transaction

COSTS OF PERIOD OF SETTLEMENTS

Probability of system liquidity

Plikv



l T J



H K i l T H – T –

Heavyside function; period.



i l G l i

 Condition of unliquidity

K l i

 0

STATISTICAL OPTIMIZATION OF SETTLEMENT COSTS Statement of optimization of settlement costs Agent costs during period:

D i



D i



X i

, 

 Vector of day balances: Expected costs of agent: 

 ( 

1 , 

2 ,  , 

i T

)  

, 

 Objective function (total expected costs):  min

 0 where: 

i J

  1

STATISTICAL OPTIMIZATION OF SETTLEMENT COSTS 1) 2)   0 .

5 Analytical example (one agent, one day period)   0 .

5 LBR=5 %, IBR=9%, STL=10%



  0 , 09 

g Q



     1 2     

 15  0 , 1  



  0 , 075 

 

 (

  ) 2 2   2 

   1 2   

  

x x

 0 , 125  



  0 , 075 

 

 (

  ) 2 2   2 

 0 , 05 

  2   

  

e x

 (

  ) 2   2 2 

0 , 09 

    0 , 1 2     

 

 (

  ) 2 2   2 

   0 , 125 2   

  

x x e

 (

  ) 2 2   2 

STATISTICAL OPTIMIZATION OF SETTLEMENT COSTS Statistical simulation of expected costs function and its gradient where:

L i

 1

N n N

  1

D i L i D i



D i





1 ,



n n N

   1



,   costs during period;

Q i

 1

N n N

  1 

x D i



X i

,  

X i

,   

estimator of gradient of cost function

L i (X i )

; 

x D i



X i

, 

 generalized gradient of cost function;

– number of simulated periods;

STATISTICAL OPTIMIZATION OF SETTLEMENT COSTS

X t

 1 

X t

  Stochastic optimization  

step multiplier

 >0

N t

 

 

J N



t t

))  1  

)

)) ' Sakalauskas (2000)

Fish



J N t



) - Fisher  -quantile with

(.)

–

sampling covariance matrix



) degrees of freedom

STATISTICAL OPTIMIZATION OF SETTLEMENT COSTS Statistical termination criterion 1) Testing of optimality hypothesis with significance  according to Hotelling criteria: (

N t



J t t

))  1 

)) 

Fish





)

2) Confidence interval of estimator of objective function:   

d N t

(

X t

)   ,

N t

 

standart normal  quantile;

d N t

sampling standard deviation of the objective function.

COMPUTER MODELLING

Initial data:

STL IBR LBR

   0.10

0.08

0.05

r T

 2.5

 30

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

COMPUTER MODELLING Dependencies of expected costs

L(X)

(1 ir 10 agents) and gradient

Q(X)

on deposit 18260 18240 18220 18200 18180 18160 18140 18120 18100 18080 2710000 2720000 2730000 2740000

Deposited sum

2750000 0.005

0.004

0.003

0.002

0.001

0 -0.002

-0.003

-0.004

2720000 2730000 2740000 2750000

Deposited sum

18000 16000 14000 12000 10000 8000 6000 4000 2000 0 1E+06 1E+06 2E+06 2E+06 2E+06 2E+06 2E+06

De pos ite d s um

2E+06 2E+06 2E+06 0.01

0.005

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

-0.02

De pos ite d s um

COMPUTER MODELLING Total expected costs and deposit under number of iterations 78000 77000 76000 75000 74000 73000 72000 0 10 20 30 40

Number of iteration

50 60 123000000 122500000 122000000 121500000 121000000 120500000 120000000 119500000 119000000 118500000 118000000 0 21 41

Number of iteration

COMPUTER MODELLING Total expected costs and deposited sum under number of iterations (1 ir 10 agents) 18350 18300 18250 18200 18150 18100 18050 18000 17950 0 10 20 30

Number of iteration

40 50 60 2735000 2730000 2725000 2720000 2715000 2710000 1 21

Number of iteration

41 61 35000 30000 25000 20000 15000 10000 5000 0 0 10 20 30 40

Number of iteration

50 60 1800000 1600000 1400000 1200000 1000000 800000 600000 400000 200000 0 1 21

Number of iteration

41 61

Agent Nr, Expected costs (Iter=0), Lt COMPUTER MODELLING Expected costs (Iter=62), Lt Deposit (Iter=0), Lt  10 6 Deposit (Iter=62), Lt  10 6 8 9 10 11

Total

1 2 3 4 5 6 7 18292,4±150,5 243183,9±1159,5 102995,0±909,8 37527,3±524,9 156596,1±807,2 16343,1±435,2 55480,1±675,5 48146,4±636,6 120797,1±974,3 30794,2±873,5 16841,9±1438,8

76999,7±361,2

18167,0±19,0 241641,8±144,1 100477,5±83,8 36206,6±51,6 155361,6±91,1 10510,2±17,0 52718,4±46,8 46277,6±52,3 118070,5±88,8 22294,6±34,7 13128,0±139,7

74077,6±62,0

2,71 36,20 14,80 5,25 23,20 1,40 7,70 6,80 17,50 3,00 1,30

119,86

2,73 36,40 15,10 5,45 23,50 1,57 7,93 7,00 17,80 3,33 1,71

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 liquidity risk has been created optimization of settlement costs, a view on settlement costs and

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 – 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 ≤ β4v x21 ≥ 0.7 · AR0 x21 ≤ 0.9 · AR0 x31 ≤ 0.8 · AP0 x4 ≥ β4a x11 + x4 ≤ β41 x21 + x4 ≤ β42 x51 ≤ AR0 ·rv x51 + L1 ≥ LR

Second stage constraints:

x12 - L1 ≤ 0 x11 + x12 + x4 ≤ β41 x22 ≥ 0.7 · AR1 x21 + x22 + x4 ≤ β42 x22 ≤ 0.9 · AR1 x32 ≤ 0.8 · AP1 x31 + x32 ≤ β3 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)= r + r 2 6 · x (2 · x 21 60 + r 12 · x 11 – AR 0 ) + r 3 + r 11 · x 31 +r +r 4 12 + r 2 +r 2 + r 4 + r 7 · x 4 – r 5 ·(x 11 · x 51 + r 6 + x 12 ) + r 11 (2 · x 21 (2 · x 22 – AR 1 ) + r 3 · x 4

·x 62 +

– r 5 – AR 0 )+ ·(x 51 · x 61 · (L 1 ·(x 31 + + x 52 ) + r 6 + · L 1 + – x 12 · x ) + + x 32 )+ 62 +

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 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

– [ r x 31 – rg rr 5 5 12 + r 4 · x xr 51 · (xr 51 11 · x – [ r 12 ·(x 11 AR 0 ) + r 2 4 f(x) = x 60 + – [ rtr · xg 1 + r 11 – rg 5 + x 61 + · L 1 · xg + r 51 2 - x + x – x 61 · (2 · x 21 – AR 0 ) + r 3 – (rf 5 – rg · xg 1 + rtr · xg 2 (2 · x 22 – AR 1 ) + r 3 · (xg 51 + x + xg 52 ) – (rf 5 + xr 12 52 + x 62 + ) + r 11 ) – rg · (xg 1 60 60 – – ] + + rk 5 ) · xk 51 + r 6 · x 61 - x 62 · (L 1 – – x 12 ) + r 2 · (x 31 + x 32 + rk 5 ) · (xk 51 + xg 2 ) + r 6 ] + – rr 5 ) + r 4 · (2 · x 21 – · · x 4 + xk 52 ) – · x 62 ]

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

Gas Turbine Coal Hydroelectric Nuclear power Hydroelectric

Cost per GW capacity

$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

4   1

c x i i



  min

  

5   1

5  1

15  1

q h r y i j k ijk

       min, subject to

4   1

c x i i x

4  5.0,  10000,

y ijk

 

x i

 1, 2, 3, 4,

x i

5   1 

y ijk

 0,

 

D jk

 0, ,    

where

x=(x 1 ,x 2 ,x 3 ,x 4 ) y ijk ω

vector, representing the gigawatts of capacity to be built for each type of plant amount of electricity capacity used to produce electricity by power plant type

for demand block

in year

in GW

c i

investment cost per GW of capacity for power plant type

i q i h j r k D jk ω

operating cost of power generation for power plant type

duration of the demand block

operating costs growth in year

power demand in year

at demand block

N( μ j , 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 Decision based on stochastic data Optimal Construction Decision based on expected data

4.65 GW 4.56 GW 1.87 GW 3.29 GW 4.70 GW 5.00 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

   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

c y i i

investment cost for building facility 1 if a processing facility i is built or machine i is procured, and 0 otherwise

x ij k

flow of product k from node i to node j

q ij k

per-unit cost of processing product k

Supply Chain Management





P c i y i

  



(

) 

A q ij k x ij k y



  0 , 1  |

|  min 



N x k ij

 



N x k il

 0 



, 



conservation of product k across each processing node j

 

N x ij k



d j k





, 



K j

 

N x ij k



s i k





, 



demand satisfaction supply restriction



K r j k

 



N x k ij

  

m j y j





P x



 |

|  |

| capacity constraints of the processing nodes,

r m j k k j

denotes per-unit processing requirement, denotes capacity

Supply chain management

First Stage:

(

) : 

c T y



[

(

,  ]  min Second Stage:



  0 , 1  |

Q(y,ξ)=q T x+h T z

Nx = 0





d Sx



s Rx



My x



 |

|  |

―>min

conservation of products demand satisfaction supply restriction capacity constraints of the processing nodes   (

) - uncertain parameters

Supply chain management

Example of materials planning requirement NN1100 1 LQ8811 WN7342 1 2 1 RN0098 AJ8172 AJ8172 is the final product NN1100, WN7342, LQ8811, RN0098 are components

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.