Directory UMM :Data Elmu:jurnal:I:International Journal of Production Economics:Vol67.Issue2.Sept2000:

(1)

*Corresponding author. Tel.: 215-4669; fax: 00358-2-215-4806.

E-mail address:ralf.ostermark@abo."(R.OGstermark)

A nonlinear mixed integer multiperiod

"rm model

Ralf O

G

stermark

!

,

*

, Hans Skrifvars

"

, Tapio Westerlund

"

!Department of Business Administration, Asbo Akademi University, Henriksgatan 7, 20500 Turku, Finland

"Department of Chemical Engineering, Asbo Akademi University, Process Design Laboratory, Biskopsgatan 8, 20500 Turku, Finland Received 11 November 1996; accepted 27 January 2000

Abstract

We formulate and test an advanced multiperiod model for strategic"rm planning. This has not been previously considered as a mixed integer nonlinear problem (MINLP). Our approach is to show the di!erences between a linear and a nonlinear mixed integer approach. The key property of our model is the simplicity and e$ciency of generating optimal

"rm strategies, a cornerstone for managerial acceptance. Our purpose is to maximize the discounted value of net income and return on investment (ROI). Our model has been tested on some major Finnish"rms and it seems to give reliable results. With the data of our sample"rm for this paper, optimal ROI and optimal net income presuppose di!erent strategies. When optimizing ROI the model balances between cash and"xed assets, while optimizing net income results in an intensive investment program. Even if our sample "rm is but one case, the results are indicative of some fundamental principles governing managerial decision making. ( 2000 Elsevier Science B.V. All rights reserved.

Keywords: Strategic planning; Firm models; Linear programming; Optimization

1. Introduction

Mathematical experimenting is valuable for

planning

"

rm processes. Projections can be made

on future

"

nancial performance using a

"

nancial

analysis framework. Linear programming (LP) is

well known and widely used in business. Even

though, in many cases it is insu

$

cient to fully

capture the problem. Instead, many managerial

decision problems are of a mixed-integer character,

possibly containing various non-linearities. The

incentive for this study is to extend the framework

of So

K

derlund and O

G

stermark [1] to mixed integer

problems. We will subsequently maximize the

dis-counted net income (linear) and ROI (nonlinear) in

four cases as shown in Table 1.

Most

"

rm planning models use simulation to

project the consequences of alternative strategies

under a range of assumptions about the future.

However, these models do not provide the optimal,

i.e. the best, strategy, but only the consequences of

a strategy speci

"

ed by the user. The simulation

models primarily produce future accounting

state-ments, but there is no

"

nance theory to support

them. In this paper we are concerned with

optimiz-ing some main economic variables of the

"

rm over

a multi-year planning horizon, corresponding to

critical

"

nancing, operating and investment

deci-sions. The key property of our model is the

simpli-city in generating optimal

"

rm strategies.

The key contribution of our study is in the

derivation of a managerially convenient planning

(2)

Table 1

The objective functions in the di!erent test settings Solution space

Decision variables Linear Nonlinear Continuous Net income (LP) ROI (NLP) Discrete Net income (MILP) ROI (MINLP)

system, requiring only a few input parameters for

producing full-

#

edged multiperiod

"

rm strategies.

The imprecision inherent in the parameters may be

recognized in Monte-Carlo-type scenarios (cf. e.g.

[2]). A further justi

"

cation for parsimony is the

speed of generating alternative optimal strategies.

Applications of linear programming models in

"

rm

planning are well documented. For example,

banking models have been widely developed within

operational research (cf. e.g. [3

}

7]). Reid and

Bradford [8] produced a farm

"

rm model of

ma-chinery investment decisions. The features of this

model have in

#

uenced various details of our design.

2. The MINLP-algorithm

The MINLP problem used in the method may be

formulated as follows:

min

x

,

y

Mc

T

x

#

c

T

y

yN

,

Mx

,

yN

3 N

,

(2.1)

where

N

"

Mx

,

yD

u

(

x

,

y

)

0N

.

(2.2)

c

x

and

c

y

are vectors with constants,

x

is a vector

with continuous variables,

y

is a vector with integer

variables and

u

(

x

,

y

) is a vector with nonlinear

dif-ferentiable functions, all de

"

ned on a set

¸"

X

>

_,

_(2.3)

where

X

is an

n

-dimensional compact polyhedral

convex set,

X

"

M

x

D

Ax

)

a

,

x

3 _Rn

N

(2.4)

and

>

_{is a}

"

_{nite discrete set de}

"

_{ned by}

>

_"

_M

_y

_D

_By

₎

_b

_,

_y

₃

_Z

_m

_N

_.

_(2.5)

In the case of a nonlinear objective function

f

(

x

,

y

),

the objective function can be written as a nonlinear

constraint

f

(

x

,

y

)

!

k

)

0 (2.6)

and the corresponding variable

k

is minimized.

A sequence of points

M

(

x

_k

,

y

_k

),

k

"

0, 1,

₂

,

K

N

generated by the ECP and

a

-ECP methods [9,10],

converging to the optimal solution of the problem

in Eq. (2.1) is given by

min

x_k

_,

y_k

Mc

T

x

k

#

c

T

y

k

N

,

M

x

k

,

y

k

N

3 X

k

,

k

"

0, 1,

2 ,

K

,

(2.7)

where

X

_k

is de

"

ned by

X

_K

"¸

WMx

,

yD

l

k

(

x

,

y

)

0,

k

"

0, 1,

2 ,

K

!

1 N

,

(2.8)

where

l

k

(

x

,

y

)

"

f

k

(

x

k

,

y

k

)

#

a

k

)

AA

L

f

k

L

x

B

_,

(

x

!

x

k

)

#

A

L

f

k

L

y

B

_,

(

y

!

y

_k

)

B

(2.9)

and

f

k

(

x

,

y

) is the function

g

i

(

x

,

y

) corresponding to

max

i

M

g

i

(

x

k

,

y

k

)

N

.

From the de

"

nition of the

X

_k

set it follows from

Westerlund et al. [10] that the optimal solution

M

x

H,

y

H

N

of (2.1) is a subset of

X

_k

for convex and

quasi-convex problems, and

X

_K

L

X

_K

_~1

L2L

X

_k

L2L

X

₀

.

(2.10)

From Eq. (2.10) it also follows that the solutions

Z

k

"

min

Mc

T

x

k

#

c

T

y

k

N

form a monotonically

in-creasing sequence,

Z

K

*

Z

K

~1

2 Z

k

2 Z

0 .

(2.11)

The

convergence

of

the

sequence,

M

Z

k

,

k

"

0, 1,

2 ,

K

N

, to the optimal solution in

N

is

shown for convex problems in [9], and for

quasi-convex problems in [10].

3. Model speci

5 cation

Our

"

rm model encompasses a planning horizon

(3)

Table 2

The objective functions, restrictions, decision variables and parameters of the"rm model Discounted

objectives

Restrictions Variables Parameters

O1 Net income R1 Sales}upper bound is a function of production capacity

#inventory

SALES

t Sales mindep Minimal depreciation R2 Amortization}equals

a proportion of long-term debt

PROD

t Production tax Taxes R3 New issues}upper bound is

a proportion of stockholders' equity

NEWDEBT

t New debt cost Operating costs/Turnover R4 Dividends}upper bound is

unrestricted shareholders equity AMO

t Amortization sr Sales receivable/Turnover R5 Dividends}lower bound is a

fraction of capital stock

INV

t Investments mxiss Maximal new issue/Stockholders_equity R6 Depreciation}lower bound is

a proportion of"xed assets

NEWISS

t New issue minequ Stockholders'equity/Liabilities R7 Equity/Debt}lower bound DIVID

t Dividends r Interest rate on long-term debt R8 Nonnegativity of cash DEP

t Depreciation cl Current liabilities/Operating costs R9 Nonnegativity of debt Deviation variables o" Other"nancial items/Other

"nancial assets R10 Nonnegativity of"xed assets AMODIFF in R2 d Discount factor

EQUITYDIFF in R3 X Machine cost MAXDIVDIFF in R4 p

t Unit sales price at timet MINDIVDIFF in R5 Factor Production capacity factor DEDIFF in R7 rep Amortization/Long-term debt

mindiv Minimal dividend/Capital stock

planning, where we are dealing with aggregate

deci-sions, a planning perspective of

"

ve years or longer

is usually desirable [[11], p. 713]. The set of

vari-ables is limited to those necessary for de

"

ning an

enterprise in economic terms. In order to represent

the

"

nancial variables accounting logic is used. The

"

nancing choice for an investment is an essential

decision. Should the management borrow, issue

new equity or use internal funds? The use of

retained earnings a

!

ects the capability of paying

dividends. The level of investments is a

!

ected by

the cost of capital. The level of sales and

deprecia-tion

}

both connected to investments

}

in

#

uence

net income. Like the interest expenses, the

deprecia-tions operate as a tax shield. The amortization rate

is determined schematically as a fraction of

accountable debt (see Table 2).

The key

"

nancial decision variables are

invest-ments, new loans, new issues, loan amortization,

dividend payments, depreciation, sales volume and

production volume. Since sales and production are

unsynchronized, inventory accumulation is

pos-sible. Inventory valuation is a crucial issue

involv-ing tax problems and matchinvolv-ing of income and

expenses within accounting periods. Furthermore,

a set of deviation variables guaranteeing solvability

is speci

"

ed below. The structure of the

"

nancial

statements is described in Tables 3 and 4.

The

"

nancial constraints include some

funda-mental requirements such as nonnegativity of

assets and liabilities and some economic conditions

and aspiration levels. The restrictions are based

on accounting legislation (e.g. maximal dividend),

on accounting

}

technical logic (e.g. nonnegativity

of assets and liabilities), on requirements imposed

by the economic environment (e.g. debt

}

equity

ratio and other relations) and on restrictions on

the productive capacity. Some restrictions are

allowed to diverge if the solution is infeasible.

Mathematically, this is settled by introducing

(4)

Table 3

Balance sheet of the"rm model

Assets Symbols Equations Liabilities Symbols Equations

Fixed assets

FIXASS

t FIXASSt~1#INVt!DEPt Capital stock EQUITYt EQUITYt~1#NEWISSt Sales

receivable

SALESREC

t sr TURNOt

"SALES

t*SPRICE_t

Other unrestricted equity

OUEQUITY

t OUEQUITYt~2

#NETINCOME t~1 !DIVID t Cash and bank deposits CASH

t CASHt~1#NETINCOMEt

!DEP

t#SALESRECt

!SALESREC

t~1#OTFINASSt

!OTFINASS

t~1!INVt

#NEWISS

t!DIVIDt

!CURRLIAB

t#CURRLIABt~1

!AMO t

Net income NETINCOME

t See statement of income in Table 4

Other "nancial assets

OTFINASS

t OTFINASSt~1

#OTFINASS t

Shareholders' equity

TOTEQUITY

t EQUITYt~1

#OUEQUITY t

#NETINCOME t Current liabilities CURRLIAB

t cl cost TURNOt Long-term debt DEBT

t DEBTt~1#NEWDEBTt Total liabilities LIAB

t CURRLIABt#DEBTt Table 4

Statement of income

Item Symbols/Equations Item Equations

#Sales SALES

t !Interest payments r(DEBTt~1#NEWDEBTt)

!Production costs PROD

t #Other"nancial income o"]OTFINASSt

!Operating costs cost !Taxes tax][TURNO_t(1!cost)!DEP t

!r](DEBT_t~1#NEWDEBT

ti)#o"]OTFINASSt]

!Depreciation DEP

t NETINCOMEt _r[1!tax]][TURNOt(1!cost)!DEPt!

](DEBT_t~1#NEWDEBT

ti)#o"]OTFINASS_t]

sanctioned deviation variables in the equations.

The optimization problem can be formulated as

follows:

MAX

x

y

"

f

(

x

t

)

s.t.

A

t

x

t

)

b

t

,

t

"

1,

2 ,

h

,

x

t

'

c

t

3 Rn

,

b

t

3 Rm

,

A

t

3 Rm

C

n

,

(3.1)

where

h

is the planning horizon. In the test below

we use

h

"

5. For each period,

x

t

consists of the following 13

variables:

x@

t

"

(sales volume, production volume, new

debt, amortization, investments, new issue,

divi-dends, depreciation, amortization di

!

erence, equity

di

!

erence, maximal dividend di

!

erence, minimal

dividend di

!

erence, depreciation di

!

erence)

t

.

The objective functions are discounted to present

value by a discount factor (see Table 2 for a

description of the variables and parameters). The

variables are subject to 12 constraints as explained

below. For a multiobjective generalization of the

problem formulation, see, for example, [12,13].

(5)

3.1. The objecti

v

e functions

NETINCOME:

+

t

i

/1

C

(1

!

tax)[(1

!

cost)TURNO

i

!

r

+

t

j

/1

[NEWDEBT

ij

!

AMO

ij

]

!

DEP

i

]

(1

#

d

)

i

D

!

PENALTY

_*

+

t

i

/1

5 +

j

/1

Dev

ij

,

(3.2)

ROI:

+

t

i

/1

NETINCOME

i

(1/(1

#

d

)

i

)

+

_ti

_/1

(INV

i

!

DEP

i

)/(1

#

d

)

i

#

FIXASS

0 !

PENALTY

_*

+

t

i

/1

5 +

j

/1

Dev

ij

.

(3.3)

PENALTY is a positive value su

$

ciently large to

make deviations undesirable. Dev

ij

refers to the

deviation variables in the constants below.

3.2. The restrictions

The restrictions of our model are presented in

Table 2.

3.2.1. R1

:

The capacity constraint

Sales is de

"

ned as a function of production

capa-city of the

"

rm. Capacity, again, is related to

machinery as part of total

"

xed assets. The sales

value of production volume is as follows:

SALES VALUE OF PRODUCTION(

t

)

"

Factor

]

FIXASS

_t

P

t

/unit

X

/unit

,

(3.4)

where the sales price,

P

t

, can vary over time. The

symbol

X

stands for the machine cost per unit. To

illustrate: assume that we have acquired a machine

for 1000 money units. The estimated production

of the machine over its entire lifetime is 5.000 units

of a certain product. The machine cost per unit

is then,

_X

"

1000/5000

"

0.2. Assuming that this

machine is the only

"

xed asset (Factor

"

1;

FIXASS

t

"

1000), the sales value of production is

(1000/0.2)

P

t

"

5000

P

t

. The production costs are

assumed to be constant. Only the sales price per

unit (

P

t

) varies and it is assumed to be known. The

capacity constraint is formulated as follows:

SALES

t

(

Factor

]

P

t

X

C

FIXASS

0 #

+

t

i

/1

[INV

i

!

DEP

i

]

D

.

(3.5)

3.2.2. R2

:

Loan repayment

The second restriction concerns the level of

re-payment. The amortization amount equals, as far

as possible, a fraction of long-term debt, i.e.

AMO

t

"

rep

]

C

DEBT

0 #

t

+

i

/1

NEWDEBT

i

!

t

~1

+

i

/1

AMO

i

D

#

AMODIFF

t

.

(3.6)

The

"

rm is obliged to follow the plan of repayment,

but if necessary

}

for example, due to risk for

insolvency/bankruptcy

}

an exception is allowed.

In practice, the loan repayment schedule is of

course more complicated. Each loan has its own

amortization plan and repayments are not a

con-stant fraction of total debt. However, in the long

term, when the

"

rm approaches its equilibrium

level of operations, the total repayments will

amount to a fairly constant fraction of debts

out-standing. Before reaching this level, the repayments

could be modelled more exactly, for example, by

allowing the repayment fraction to vary over the

planning horizon.

3.2.3. R3

:

Upper bound on new issues

The upper bound on new shares is based on

rational economic arguments. The issue costs, the

possible negative reaction of the stock market (see,

for example, [14]) and the demand for new equity

limit the level of the issue. We restrict the new issues

to a fraction of stockholders

'

equity at the base year

(6)

as follows:

NEWISS

t

)

mxiss

]

EQUITY

0 #

EQUITYDIFF

t

.

(3.7)

The deviation variable allows new issues to diverge

if necessary. One may argue that there is also

a lower bound due to the

"

xed costs associated

with new issues. From a strategic planning point of

view, such costs are considered negligible, however.

In practice, the decision to issue new equity or to

prefer new debt, is governed by the target

equity/debt ratio or some other objective related to

controlling the

"

rm. Recognizing the

"

xed costs of

new issues (through appropriate binary-valued

variables) would make the model unduly

complic-ated in comparison to the expected utility.

3.2.4. R4

,

R5

:

Upper

/

lower bounds on di

v

idends

Dividends are limited by upper and lower

bounds. The upper bound is de

"

ned as free

un-restricted equity, i.e. retained earnings:

DIVID

t

(

OUEQUITY

0 #

t

+

i

/0

NETINCOME

i

!

t

~1

+

i

/1

DIVID

i

#

MAXDIVDIFF

t

. (3.8)

This is in order to protect creditors from

excess-ive dividend payments. A minimum level of

pay-ments is motivated by the shareholders

'

demand for

a stable dividend. A cutback of the dividend rate

would probably have a negative impact on the

market value of the

"

rm [15]. Especially minor

shareholders are guaranteed a fair dividend

through the lower bound:

DIVID

t

'

mindiv

C

EQUITY

0 #

t

+

i

/1

NEWISS

i

D

!

MINDIVDIFF

t

.

(3.9)

The deviation variable MAXDIVDIFF was

in-cluded to guarantee feasibility in cases where

unre-stricted equity is negative. In practice, the dividend

policy is much more complex than can be captured

by an upper and lower limit on dividends. Yet,

there is a signi

"

cant managerial interest in knowing

the

leeway

for dividend payment provided by the

optimal strategic scenario.

3.2.5. R6

:

Depreciations

The minimal depreciation level is governed by

economic and physical considerations. The

math-ematical expression is

DEP

t

'

mindep

C

FIXASS

0 #

t

+

i

/1

INV

i

!

t

~1

+

i

/1

DEP

i

D

.

(3.10)

3.2.6. R7

:

The equity

/

debt relation

This constraint controls the capital structure of

the

"

rm. According to Modigliani and Miller

'

s [16]

classical work on the theory of capital structure, the

mixture of

"

nancing investments does not a

!

ect the

value of the

"

rm in a world without taxes. When

taxes [17] and cost of bankruptcy [18] are

intro-duced, a trade-o

!

between these will lead to an

optimal capital structure (see also [19,20]). This

reasoning is partly supported by recent empirical

evidence, even though counterevidence does exist.

Firms with safe, tangible assets and plenty of

taxable income have higher debt-to-equity ratios

than an unpro

"

table and risky business with

intan-gible assets [21,22]). On the other hand, the

peck-ing order theory [23] explains why some pro

"

table

"

rms borrow less, as they do not need outside

money. Kjellman and Hanse

H

n [24] found that most

Finnish

"

nancial managers seek to maintain a

con-stant debt-to-equity ratio. A target debt ratio

is obviously a part of the

"

rms

' "

nancing policy.

In our model, the ratio is de

"

ned as the relation

between shareholders

'

equity and total liabilities:

EQUITY

0 #

t

+

i

/1

[NEWISS

i

#

NETINCOME

i

!

DIVID

i

]

#

DEDIFF

t

'

minequ

C

DEBT

₀

#

CURRLIAB

t

#

+

t

i

/1

[NEWDEBT

i

!

AMO

i

]

D

.

(3.11)

Equity is made up of new issues

#

retained

earn-ings

!

dividends.

The above seven restrictions describe the relation

between the

"

rm and its environment. The next set

de

"

nes the necessary nonnegativity relations for the

(7)

Table 5

Parameters based on historical development

Symbols Values Symbols Values Symbols Values

mindep 0.06 o" 0.67 FIXASS

4 1078.00

tax 0.25 t 5 SALESREC

4 95.00

cost 1.00 X 1.00 CASH

4 12.00

sr 0.15 t"5 t"6 t"7 t"8 t"9 OTFINASS

4 30.00

d 0.15 P

t 1.05 1.10 1.30 1.20 1.25 EQUITY4 630.00

mxiss 0.02 Factor 0.80 NETINCOME

4 85.00

minequ 1.00 rep 0.08 OUEQUITY

4 0.00

r 0.08 mindiv 0.01 DEBT

4 600.00

cl 0.15 CURRLIAB

4 0.00

3.2.7. R8

:

Nonnegati

v

ity of cash

The cash

#

ow is de

"

ned as

C

CASH

0 #

[SALESREC

t

~1

!

SALESREC

t

]

#

[OTFINASS

t

~1

!

OTFINASS

t

]

!

[CURRLIAB

t

~1

!

CURRLIAB

t

]

#

+

_ti

_/1

[NETINCOME

i

#

DEP

i

!

INV

i

#

NEWDEBT

i

#

NEWISS

i

!

DIVID

i

]

D

*

0. (3.12)

3.2.8. R9

:

Nonnegati

v

ity of debt

Nonnegativity of long-term debt is de

"

ned as

DEBT

t

"

DEBT

0 #

+

t

i

/1

[NEWDEBT

i

!

AMO

i

]

'

0. (3.13)

Normally, debt will be positive while the

amortiza-tion rate is less than unity.

3.2.9. R10

:

Nonnegati

v

ity of

x

xed assets

The

"

nal constraint concerns nonnegativity of

"

xed assets:

FIXASS

t

"

FIXASS

0 #

t

+

i

/1

[INV

i

!

DEP

i

]

'

0. (3.14)

If selling of

"

xed assets is allowed, this must be

included in the restriction. Otherwise it is

redund-ant, since depreciation is always a nonnegative

frac-tion of total assets.

4. A numerical experiment

Our model has been tested on data from some

Finnish listed companies. A hypothetical

"

rm is

studied below over a

"

ve year planning horizon.

The upper bound on sales volume is determined by

new investments, since the size of

"

xed assets

deter-mines the capacity of production. New investments,

again, are restricted by the

"

nancial structure of the

"

rm. The

"

nancing alternatives are internal or

external sources of funds, i.e. retained earnings, new

equity and new debt. Depreciation a

!

ects the

econ-omy of the

"

rm in three di

!

erent ways. Firstly, it

reduces the maximal allowed dividends. Secondly,

it decreases the value of

"

xed assets which a

!

ects

production capacity and reduces net income.

Thirdly, it provides a tax shield a

!

ecting cash

#

ows.

If new issues and internal funds do not, however,

su

$

ce to

"

nance new investment, the

"

rm is forced

to borrow. When maximizing net income the

capi-tal structure is particularly sensitive to the interest

rate and investments and new debts are negatively

correlated with the interest rate.

The base year (year 4)

"

nancial statements and

control parameters are presented in Table 5.

The parameters are speci

"

ed on the basis of

historical values (years 0

}

4) and a subjective

judge-ment of future developjudge-ment. The historical

state-ments along with the optimal projected statestate-ments

and the optimal programs are given in Tables 6

}

9.

(8)

Table 6

Maximizing net income in the linear continuous case (LP)

Example Optimization

Value of objective function 772.15

Decision variables 5 6 7 8 9

1: Sales 733.25 787.92 1307.51 1290.09 1490.94

2: Production 810.66 776.43 1141.61 1290.09 1490.94

3: New debt 0.00 0.00 145.44 165.50 172.04

4: Amortization 48.00 44.16 52.26 61.32 70.18

5: Investments 0.00 19.16 547.56 288.54 370.01

6: New issues 14.30 14.30 14.30 14.30 14.30

7: Dividends 6.44 6.59 6.73 6.87 7.02

8: Depreciation 64.68 61.95 91.09 102.93 118.96

9: Dividend deviation (Divdi!) 0.00 0.00 0.00 0.00 0.00

10: Equity deviation (EKdi!) 0.00 0.00 0.00 0.00 0.00

11: Debt-Equity deviation (D/E_}di!) 0.00 0.00 0.00 0.00 0.00

12: Repayment deviation (REPdi!) 0.00 0.00 0.00 0.00 0.00

13: Max dividend deviation (MAXdivdf)

0.00 0.00 0.00 0.00 0.00

Historical period: 0 1 2 3 4

Inventory volume 17 8 8 8 100

Planning period: 5 6 7 8 9

Inventory volume 177.40 165.91 0.00 0.00 0.00

Financial statements Historical accounts Forecasted accounts

0 1 2 3 4 5 6 7 8 9

Assets

Fixed assets 859.00 914.00 1016.00 1070.00 1078.00 1013.32 970.53 1427.01 1612.62 1863.67 Valuation items 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Inventory 10.00 5.00 5.00 5.00 100.00 177.40 165.91 0.00 0.00 0.00 Sales receivables 150.00 180.00 170.00 190.00 95.00 115.49 130.01 254.97 232.22 279.55

Cash 10.00 15.00 15.00 15.00 12.00 0.00 0.00 0.00 0.00 0.00

Other"nancial assets 30.00 30.00 30.00 30.00 30.00 30.00 30.00 30.00 30.00 30.00 Financial assets 190.00 225.00 215.00 235.00 137.00 145.49 160.01 284.97 262.22 309.55

Assets 1059.00 1144.00 1236.00 1310.00 1315.00 1336.21 1296.45 1711.97 1874.83 2173.22 Shares equity and liabilities

Capital stock 500.00 500.00 550.00 550.00 630.00 644.30 658.60 672.90 687.20 701.50 Other restricted equity 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Other unrestricted equity 0.00 0.00 0.00 0.00 0.00 78.56 17.84 !6.73 176.21 243.20 Net income for the year 109.00 84.00 71.00 85.00 85.00 !54.13 !17.84 189.82 74.00 141.91 Shareholders'equity 609.00 584.00 621.00 635.00 715.00 668.72 658.60 855.99 937.42 1086.61 Accumulated depreciation di!erence 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Reserves 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Valuation items 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Current liabilities 50.00 60.00 65.00 75.00 0.00 115.49 130.01 254.97 232.22 279.55 Long-term debt 400.00 500.00 550.00 600.00 600.00 552.00 507.84 601.02 705.20 807.06 Liabilities 450.00 560.00 615.00 675.00 600.00 667.49 637.85 855.99 937.42 1086.61

(9)

Table 6 (continued)

Financial statements Historical accounts Forecasted accounts

0 1 2 3 4 5 6 7 8 9

Statement of income

Turnover 500.00 510.00 510.00 550.00 600.00 769.91 866.71 1699.77 1548.11 1863.67 Operating costs 300.00 321.00 324.00 340.00 380.00 733.25 787.92 1307.51 1290.09 1490.94 Operating income 200.00 189.00 186.00 210.00 220.00 36.66 78.79 392.25 258.02 372.73 Depreciation 50.00 55.00 60.00 65.00 70.00 64.68 61.95 91.09 102.93 118.96 Operating income after depreciation 150.00 134.00 126.00 145.00 150.00 !28.02 16.84 301.17 155.09 253.78 Interest expenses !40.00 !50.00 !55.00 !58.00 !62.00 !44.16 !40.63 !48.08 !56.42 !64.56 Other"nancial income 20.00 20.00 20.00 20.00 20.00 0.00 0.00 0.00 0.00 0.00

Extraordinary income and expenses 0.00 0.00 0.00 0.00 0.00

Allocations 0.00 0.00 0.00 0.00 0.00

Taxes 21.00 20.00 20.00 22.00 23.00 !18.04 !5.95 63.27 24.67 47.30

Net income 109.00 84.00 71.00 85.00 85.00 !54.13 !17.84 189.82 74.00 141.91

Other information

Amortization 30 40 50 50 55 48.00 44.16 52.26 61.32 70.18

Investments 110 162 119 78 0.00 19.16 547.56 288.54 370.01

New issues 0 50 0 80 14.3 14.3 14.3 14.3 14.3

Dividends 109 84 71 85 0 6.443 6.586 6.729 6.872 7.015

Table 7

Maxmizing net income in the linear mixed-integer case (MLP)

Example Optimization

Value of objective function 772.60

Decision variables 5 6 7 8 9

1: Sales 734.00 782.00 1311.00 1291.00 1493.00

2: Production 811.00 774.00 1142.00 1291.00 1493.00

3: New debt 0.00 0.00 145.05 166.09 172.58

4: Amortization 48.00 44.16 52.23 61.34 70.24

5: Investments 0.46 15.51 551.12 289.26 371.62

6: New issues 14.30 14.30 14.30 14.30 14.30

7: Dividends 6.44 6.59 6.73 6.87 7.02

8: Depreciation 64.71 61.76 91.12 103.01 119.12

9: Dividend deviation (Divdi!) 0.00 0.00 0.00 0.00 0.00

10: Equity deviation (EKdi!) 0.00 0.00 0.00 0.00 0.00

11: Debt-Equity deviation (D/E}di!) 0.00 0.00 0.00 0.00 0.00

12: Repayment deviation (REPdi!) 0.00 0.00 0.00 0.00 0.00

13: Max dividend deviation (MAXdivdf)

0.00 0.00 0.00 0.00 0.00

Historical period: 0 1 2 3 4

Inventory volume 17 8 8 8 100

Planning period: 5 6 7 8 9

(10)

Table 7 (continued)

Financial statements Historical accounts Forecasted accounts

0 1 2 3 4 5 6 7 8 9

Assets

Fixed assets 859.00 914.00 1016.00 1070.00 1078.00 1013.75 967.50 1427.50 1613.75 1866.25

Valuation items 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Inventory 10.00 5.00 5.00 5.00 100.00 177.00 169.00 0.00 0.00 0.00 Sales receivables 150.00 180.00 170.00 190.00 95.00 115.61 129.03 255.65 232.38 279.94 Cash 10.00 15.00 15.00 15.00 12.00 !0.02 !0.35 !0.37 !0.37 !1.11 Other"nancial assets 30.00 30.00 30.00 30.00 30.00 30.00 30.00 30.00 30.00 30.00 Financial assets 190.00 225.00 215.00 235.00 137.00 145.59 158.68 285.28 262.01 308.82

Assets 1059.00 1144.00 1236.00 1310.00 1315.00 1336.34 1295.18 1712.78 1875.76 2175.07 Shares equity and liabilities

Capital stock 500.00 500.00 550.00 550.00 630.00 644.30 658.60 672.90 687.20 701.50 Other restricted equity 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Other unrestricted equity 0.00 0.00 0.00 0.00 0.00 78.56 17.85 !7.02 176.21 243.76 Net income for the year 109.00 84.00 71.00 85.00 85.00 !54.13 !18.14 190.60 74.07 142.13 Shareholders'equity 609.00 584.00 621.00 635.00 715.00 668.73 658.31 856.48 937.98 1087.39 Accumulated depreciation

di!erence

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Reserves 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Valuation items 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Current liabilities 50.00 60.00 65.00 75.00 0.00 115.61 129.03 255.65 232.38 279.94 Long-term debt 400.00 500.00 550.00 600.00 600.00 552.00 507.84 600.66 705.41 807.74 Liabilities 450.00 560.00 615.00 675.00 600.00 667.61 636.87 856.30 937.79 1087.68

Liabilities and shareholders+equity 1059.00 1144.00 1236.00 1310.00 1315.00 1336.34 1295.18 1712.78 1875.76 2175.07 Statement of income

Turnover 500.00 510.00 510.00 550.00 600.00 770.70 860.20 1704.30 1549.20 1866.25 Operating costs 300.00 321.00 324.00 340.00 380.00 734.00 782.00 1311.00 1291.00 1493.00 Operating income 200.00 189.00 186.00 210.00 220.00 36.70 78.20 393.30 258.20 373.25 Depreciation 50.00 55.00 60.00 65.00 70.00 64.71 61.76 91.12 103.01 119.12 Operating income after

depreciation

150.00 134.00 126.00 145.00 150.00 !28.01 16.44 302.18 155.19 254.13 Interest expenses !40.00 !50.00 !55.00 !58.00 !62.00 !44.16 !40.63 !48.05 !56.43 !64.62 Other"nancial income 20.00 20.00 20.00 20.00 20.00 0.00 0.00 0.00 0.00 0.00 Extraordinary income and

expenses

0.00 0.00 0.00 0.00 0.00

Allocations 0.00 0.00 0.00 0.00 0.00

Taxes 21.00 20.00 20.00 22.00 23.00 !18.04 !6.05 63.53 24.69 47.38

Net income 109.00 84.00 71.00 85.00 85.00 !54.13 !18.14 190.60 74.07 142.13

Other information

Amortization 30 40 50 50 55 48.00 44.16 52.23 61.34 70.24

Investments 110 162 119 78 0.46 15.51 551.12 289.26 371.62

New issues 0 50 0 80 14.3 14.3 14.3 14.3 14.3

(11)

Table 8

Maxmizing ROI in the nonlinear continuous case (NLP)

Example Optimization

Value of objective function 0.65

Decision variables 5 6 7 8 9

1: Sales 731.68 784.60 1311.52 833.73 1249.45

2: Production 810.00 774.80 1142.35 1073.81 1009.38

3: New debt 1.46 0.00 144.29 193.62 118.01

4: Amortization 48.12 44.27 52.27 63.58 67.93

5: Investments 0.00 17.00 550.58 0.00 0.00

6: New issues 14.30 14.30 14.30 14.30 14.30

7: Dividends 6.44 6.59 6.73 6.87 7.02

8: Depreciation 64.68 61.82 91.14 85.68 80.54

9: Dividend deviation (Divdi!) 0.00 0.00 0.00 0.00 0.00

10: Equity deviation (EKdi!) 0.00 0.00 0.00 0.00 0.00

11: Debt-Equity deviation (D/E_}di!) 0.00 0.00 0.00 0.00 0.00

12: Repayment deviation (REPdi!) 0.00 0.00 0.00 0.00 0.00

13: Max dividend deviation (MAXdivdf)

0.00 0.00 0.00 0.00 0.00

Historical period: 0 1 2 3 4

Inventory volume 17 8 8 8 100

Planning period: 5 6 7 8 9

Inventory volume 178.98 169.18 0.01 240.09 0.02

Financial statements Historical accounts Forecasted accounts

0 1 2 3 4 5 6 7 8 9

Assets

Fixed assets 859.00 914.00 1016.00 1070.00 1078.00 1013.32 968.50 1427.93 1342.26 1261.72

Valuation items 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Inventory 10.00 5.00 5.00 5.00 100.00 178.98 169.18 0.01 240.09 0.02 Sales receivables 150.00 180.00 170.00 190.00 95.00 115.24 129.46 255.75 150.07 234.27 Cash 10.00 15.00 15.00 15.00 12.00 !0.37 !0.37 !0.37 !0.38 504.59 Other"nancial assets 30.00 30.00 30.00 30.00 30.00 30.00 30.00 30.00 30.00 30.00 Financial assets 190.00 225.00 215.00 235.00 137.00 144.87 159.09 285.37 179.69 768.86

Assets 1059.00 1144.00 1236.00 1310.00 1315.00 1337.17 1296.77 1713.31 1762.04 2030.60 Shares equity and liabilities

Net income for the year 109.00 84.00 71.00 85.00 85.00 !54.27 !18.06 190.67 16.93 127.00 Shareholders'equity 609.00 584.00 621.00 635.00 715.00 668.58 658.23 856.47 880.84 1015.12 Accumulated depreciation

di!erence

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Reserves 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Valuation items 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Current liabilities 50.00 60.00 65.00 75.00 0.00 115.24 129.46 255.75 150.07 234.27 Long-term debt 400.00 500.00 550.00 600.00 600.00 553.34 509.08 601.09 731.13 781.21 Liabilities 450.00 560.00 615.00 675.00 600.00 668.58 638.54 856.84 881.20 1015.48

(12)

Table 8 (continued)

Financial statements Historical accounts Forecasted accounts

0 1 2 3 4 5 6 7 8 9

Statement of income

Turnover 500.00 510.00 510.00 550.00 600.00 768.27 863.06 1704.98 1000.47 1561.81 Operating costs 300.00 321.00 324.00 340.00 380.00 731.68 784.60 1311.52 833.73 1249.45 Operating income 200.00 189.00 186.00 210.00 220.00 36.58 78.46 393.46 166.75 312.36 Depreciation 50.00 55.00 60.00 65.00 70.00 64.68 61.82 91.14 85.68 80.54 Operating income after

depreciation

150.00 134.00 126.00 145.00 150.00 !28.10 16.64 302.31 81.07 231.83 Interest expenses !40.00 !50.00 !55.00 !58.00 !62.00 !44.27 !40.73 !48.09 !58.49 !62.50 Other"nancial income 20.00 20.00 20.00 20.00 20.00 0.00 0.00 0.00 0.00 0.00 Extraordinary income and

expenses

0.00 0.00 0.00 0.00 0.00

Allocations 0.00 0.00 0.00 0.00 0.00

Taxes 21.00 20.00 20.00 22.00 23.00 !18.09 !6.02 63.56 5.64 42.33

Net income 109.00 84.00 71.00 85.00 85.00 !54.27 !18.06 190.67 16.93 127.00

Other information

Amortization 30 40 50 50 55 48.12 44.27 52.27 63.58 67.93

Investments 110 162 119 78 0.00 17.00 550.58 0.00 0.00

New Issues 0 50 0 80 14.3 14.3 14.3 14.3 14.3

Dividends 109 84 71 85 0 6.443 6.586 6.729 6.872 7.015

Table 9

Maxmizing ROI in the nonlinear mixed-integer case (MINLP)

Example Optimization

Value of objective function 0.65

Decision variables 5 6 7 8 9

1: Sales 731.00 785.00 1310.00 834.00 1249.00

2: Production 810.00 774.00 1142.00 1073.00 1010.00

3: New debt 1.55 0.78 143.47 193.32 118.02

4: Amortization 48.12 44.34 52.27 63.55 67.91

5: Investments 0.00 15.94 551.12 0.00 1.24

6: New issues 14.30 14.30 14.30 14.30 14.30

7: Dividends 6.44 6.59 6.73 6.87 7.02

8: Depreciation 64.68 61.76 91.12 85.65 80.59

9: Dividend deviation (Divdi!) 0.00 0.00 0.00 0.00 0.00

10: Equity deviation (EKdi!) 0.00 0.00 0.00 0.00 0.00

11: Debt-Equity deviation (D/E}di!) 0.00 0.00 0.00 0.00 0.00

12: Repayment deviation (REPdi!) 0.00 0.00 0.00 0.00 0.00

13: Max dividend deviation (MAXdivdf)

0.00 0.00 0.00 0.00 0.00

Historical period: 0 1 2 3 4

Inventory volume 17 8 8 8 100

Planning period: 5 6 7 8 9

(13)

Table 9 (continued)

Financial statements Historical accounts Forecasted accounts

0 1 2 3 4 5 6 7 8 9

Assets

Fixed assets 859.00 914.00 1016.00 1070.00 1078.00 1013.32 967.50 1427.50 1341.85 1262.50

Valuation items 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Inventory 10.00 5.00 5.00 5.00 100.00 179.00 168.00 0.00 239.00 0.00 Sales receivables 150.00 180.00 170.00 190.00 95.00 115.13 129.53 255.45 150.12 234.19

Cash 10.00 15.00 15.00 15.00 12.00 !0.34 2.60 !0.28 0.58 503.22

Other"nancial assets 30.00 30.00 30.00 30.00 30.00 30.00 30.00 30.00 30.00 30.00 Financial assets 190.00 225.00 215.00 235.00 137.00 144.79 162.13 285.17 180.70 767.41

Assets 1059.00 1144.00 1236.00 1310.00 1315.00 1337.11 1297.63 1712.67 1761.55 2029.91 Shares equity and liabilities

Capital stock 500.00 500.00 550.00 550.00 630.00 644.30 658.60 672.90 687.20 701.50 Other restricted equity 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Other unrestricted equity 0.00 0.00 0.00 0.00 0.00 78.56 17.67 !7.09 176.38 186.38 Net income for the year 109.00 84.00 71.00 85.00 85.00 !54.30 !18.03 190.35 17.01 126.89 Shareholders'equity 609.00 584.00 621.00 635.00 715.00 668.55 658.23 856.15 880.59 1014.77 Accumulated depreciation

di!erence

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Reserves 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Valuation items 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Current liabilities 50.00 60.00 65.00 75.00 0.00 115.13 129.53 255.45 150.12 234.19 Long-term debt 400.00 500.00 550.00 600.00 600.00 553.42 509.87 601.07 730.84 780.95 Liabilities 450.00 560.00 615.00 675.00 600.00 668.55 639.40 856.52 880.96 1015.14

Liabilities and shareholders+equity 1059.00 1144.00 1236.00 1310.00 1315.00 1337.11 1297.63 1712.67 1761.55 2029.91 Statement of income

Turnover 500.00 510.00 510.00 550.00 600.00 767.55 863.50 1703.00 1000.80 1561.25 Operating costs 300.00 321.00 324.00 340.00 380.00 731.00 785.00 1310.00 834.00 1249.00 Operating income 200.00 189.00 186.00 210.00 220.00 36.55 78.50 393.00 166.80 312.25 Depreciation 50.00 55.00 60.00 65.00 70.00 64.68 61.76 91.12 85.65 80.59 Operating income after

depreciation

150.00 134.00 126.00 145.00 150.00 !28.13 16.74 301.88 81.15 231.66 Interest expenses !40.00 !50.00 !55.00 !58.00 !62.00 !44.27 !40.79 !48.09 !58.47 !62.48 Other"nancial income 20.00 20.00 20.00 20.00 20.00 0.00 0.00 0.00 0.00 0.00 Extraordinary income and

expenses

0.00 0.00 0.00 0.00 0.00

Allocations 0.00 0.00 0.00 0.00 0.00

Taxes 21.00 20.00 20.00 22.00 23.00 !18.10 !6.01 63.45 5.67 42.30

Net income 109.00 84.00 71.00 85.00 85.00 !54.30 !18.03 190.35 17.01 126.89

Other information

Amortization 30 40 50 50 55 48.12 44.34 52.27 63.55 67.91

Investments 110 162 119 78 0.00 15.94 551.12 0.00 1.24

New issues 0 50 0 80 14.3 14.3 14.3 14.3 14.3

(14)

Fig. 2. Maximizing net income (NI) in the mixed-integer linear case (cf. Table 7). Fig. 1. Maximizing net income (NI) in the continous linear case (cf. Table 6).

The optimal trajectories for the critical success

fac-tors are shown in Figs. 1

}

8 .

The results show that the discrete solutions are

slightly better than the continuous ones for the last

planning period, due to rounding errors in the cash

position. Some rounding errors are also observed

in the cash position for the nonlinear continuous

solution.

5. Conclusion

In the present study a fundamental assumption

has been that the sales versus production decisions

of the

"

rm do not a

!

ect market demand, in

consist-ence with perfect competition. Thus, the sales of the

"

rm are limited only by the internal conditions of

the

"

rm, in particular, productive capacity and

inventory volume. To allow for market

imperfec-tions, we may constrain the sales of the

"

rm

through a demand constraint. A particular demand

relation is given by the well-known constant

elas-ticity of demand (CED) function

q

D

(

t

)

"

A

(

t

)

p

(

t

)

e

,

(5.1)

where

A

(

t

) is a time-varying parameter estimated

from historic data of the

"

rm,

e

D

"

L

q

D

(

t

)

L

p

(

t

)

p

(

t

)

(15)

Fig. 3. Maximizing ROI in the nonlinear continous case (cf. Table 8).

Fig. 4. Maximizing ROI in the nonlinear mixed-integer case (cf. Table 9).

(16)

Fig. 6. ROI trajectory when maximizing net income (NI) in the mixed-integer linear case (cf. Table 7).

Fig. 7. NI trajectory when maximizing ROI in the continous linear case (cf. Table 8).

(17)

is the (constant) price elasticity of demand. Its

time-varying counterpart with nonconstant price

elastic-ity

e

D(

t

)

may also be used. Tenhunen [25] tested the

CED-function on Rautaruukki, a state-owned

Finnish steel manufacturing

"

rm, with yearly data

between 1990 and 1993. The estimated steel

quant-ities corresponded well with the realized

"

gures.

There are many possibilities to further re

"

ne and

develop our model. The riskiness of the business

environment can be recognized, e.g. by

Monte-Carlo simulations in the spirit of Kasanen et al. [13].

The impact of o

!

-balance sheet factors, such as

derivatives, is also relevant. Bessler and Booth [7]

have developed a bank model including derivative

securities. Another direction would be to concentrate

on techno-economic

"

rm planning, i.e. on

simulta-neous modelling of strategic decisions of the

"

rm and

calibration of its technical processes. Finally, our

"

rm-model could be extended to (multinational)

concerns, an important and worthwhile exercise.

Our results show that simple rounding of the

continuous solutions does not guarantee an optimal

mixed-integer solution in strategic

"

rm planning.

The problem with rounding errors in the cash

posi-tion deserves further attenposi-tion in future research.

Acknowledgements

Financial support from the Academy of Finland

is gratefully acknowledged.

References

[1] K. SoKderlund, R. OGstermark, A multiperiod"rm model for strategic decision support, Working paper, Asbo Akademi University, 1995.

[2] E. Kasanen, R. OGstermark, The managerial viewpoint in interactive programming with multiple objectives, Kyber-netes 16 (1987) 235}240.

[3] G.G. Booth, P.E. Kovesos, A programming model for bank hedging decisions, Journal of Financial Research 9 (1986) 271}279.

[4] H. Meyer zu Zelhausen, Commercial bank balance sheet optimization. A decision model approach, Journal of Banking and Finance 10 (1986) 119}142.

[5] A. Korhonen, A dynamic bank portfolio planning model with multiple scenarios, multiple goals and changing pri-orities, European Journal of Operational Research 30 (1987) 13}23.

[6] G.G. Booth, W. Bessler, W.G. Foote, Managing interest rate risk in banking institutions, European Journal of Operational Research 41 (1989) 302}313.

[7] W. Bessler, G.G. Booth, An interest rate risk management model for commercial banks, European Journal of Opera-tional Research 74 (1994) 243}256.

[8] D.W. Reid, G.L. Bradford, A"rm farm model of machin-ery investment decisions, American Journal of Agricul-tural Economics 69 (1987) 66}87.

[9] T. Westerlund, F. Pettersson, An extended cutting plane method for solving convex MINLP problems, Computers and Chemical Engineering 19 (Suppl.) (1995) S131}136. [10] T. Westerlund, H. Skrifvars, I. Harjunkoski, An extended

cutting plane method for a class of non-convex MINLP problems, Computers and Chemical Engineering 22 (1998) 357}365.

[11] R.A. Brealey, S.C. Myers, Principles of Corporate Finance, 4th Edition, McGraw-Hill, New York, 1991.

[12] R. OGstermark, Solving a linear multiperiod portfolio prob-lem by interior point methodology, Computer Science in Economics and Management 5 (1991) 283}302. [13] E. Kasanen, M. Zeleny, R. OGstermark, Gestalt system

of holistic graphics: New management support view of MCDM, in: A.G. Lockett, G. Islei (Eds.), Improving Deci-sion Making in Organizations, Lecture Notes in Econ-omics and Mathematical Systems, Springer, Berlin, 1989. [14] P. Asquith, D.W. Mullins, Equity issues and o!ering

dilu-tion, Journal of Financial Economics 15 (1986) 61}90. [15] P. Healey, K. Palepu, Earnings information conveyed by

dividend initiations and omissions, Journal of Financial Economics 21 (1988) 149}175.

[16] F. Modigliani, M.H. Miller, The cost of capital,

corpora-tion "nance and the theory of investment, American

Economic Review 48 (1958) 261}297.

[17] F. Modigliani, M.H. Miller, Corporate income taxes and cost of capital: A correction, American Economic Review 53 (1963) 433}443.

[18] J.B. Warner, Bankruptcy costs: Some evidence, Journal of Finance 32 (1977) 337}348.

[19] M.J. Gordon, Towards a theory of"nancial distress, Jour-nal of Finance 26 (1971) 337}348.

[20] M. Jensen, W. Meckling, Theory of the"rm: Managerial behaviour, agency costs, and ownership structure, Journal of Financial Economics 3 (1976) 305}360.

[21] J.B. Warner, Bankruptcy, absolute priority, and the pric-ing of risky debt claims, Journal of Financial Economics 4 (1977) 239}276.

[22] E. Altman, A further empirical investigation of the bank-ruptcy cost question, Journal of Finance 39 (1984) 1067}1089. [23] S.C. Myers, The capital structure puzzle, Journal of

Finance 39 (1984) 575}592.

[24] A. Kjellman, S. HanseHn, Determinants of capital structure: Theory vs practice, The Scandinavian Journal of Manage-ment 2 (1993) 91}102.

[25] L. Tenhunen, Supply functions and pricing behaviour in the theory of the"rm, University of Tampere. School of Busi-ness Administration, 1994, series A2. ISBN 951-44-3599-0.

(1)

Table 8 (continued)

Financial statements Historical accounts Forecasted accounts

0 1 2 3 4 5 6 7 8 9

Statement of income

Depreciation 50.00 55.00 60.00 65.00 70.00 64.68 61.82 91.14 85.68 80.54

Operating income after depreciation

150.00 134.00 126.00 145.00 150.00 !28.10 16.64 302.31 81.07 231.83 Interest expenses !40.00 !50.00 !55.00 !58.00 !62.00 !44.27 !40.73 !48.09 !58.49 !62.50

Other"nancial income 20.00 20.00 20.00 20.00 20.00 0.00 0.00 0.00 0.00 0.00

Extraordinary income and expenses

0.00 0.00 0.00 0.00 0.00

Allocations 0.00 0.00 0.00 0.00 0.00

Taxes 21.00 20.00 20.00 22.00 23.00 !18.09 !6.02 63.56 5.64 42.33

Net income 109.00 84.00 71.00 85.00 85.00 !54.27 !18.06 190.67 16.93 127.00

Other information

Amortization 30 40 50 50 55 48.12 44.27 52.27 63.58 67.93

Investments 110 162 119 78 0.00 17.00 550.58 0.00 0.00

New Issues 0 50 0 80 14.3 14.3 14.3 14.3 14.3

Dividends 109 84 71 85 0 6.443 6.586 6.729 6.872 7.015

Table 9

Maxmizing ROI in the nonlinear mixed-integer case (MINLP)

Example Optimization

Value of objective function 0.65

Decision variables 5 6 7 8 9

1: Sales 731.00 785.00 1310.00 834.00 1249.00

2: Production 810.00 774.00 1142.00 1073.00 1010.00

3: New debt 1.55 0.78 143.47 193.32 118.02

4: Amortization 48.12 44.34 52.27 63.55 67.91

5: Investments 0.00 15.94 551.12 0.00 1.24

6: New issues 14.30 14.30 14.30 14.30 14.30

7: Dividends 6.44 6.59 6.73 6.87 7.02

8: Depreciation 64.68 61.76 91.12 85.65 80.59

9: Dividend deviation (Divdi!) 0.00 0.00 0.00 0.00 0.00

10: Equity deviation (EKdi!) 0.00 0.00 0.00 0.00 0.00

11: Debt-Equity deviation (D/E}di!) 0.00 0.00 0.00 0.00 0.00

12: Repayment deviation (REPdi!) 0.00 0.00 0.00 0.00 0.00

13: Max dividend deviation (MAXdivdf)

0.00 0.00 0.00 0.00 0.00

Historical period: 0 1 2 3 4

Inventory volume 17 8 8 8 100

Planning period: 5 6 7 8 9

(2)

Table 9 (continued)

Financial statements Historical accounts Forecasted accounts

0 1 2 3 4 5 6 7 8 9

Assets

Fixed assets 859.00 914.00 1016.00 1070.00 1078.00 1013.32 967.50 1427.50 1341.85 1262.50

Valuation items 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Inventory 10.00 5.00 5.00 5.00 100.00 179.00 168.00 0.00 239.00 0.00

Sales receivables 150.00 180.00 170.00 190.00 95.00 115.13 129.53 255.45 150.12 234.19

Cash 10.00 15.00 15.00 15.00 12.00 !0.34 2.60 !0.28 0.58 503.22

Other"nancial assets 30.00 30.00 30.00 30.00 30.00 30.00 30.00 30.00 30.00 30.00 Financial assets 190.00 225.00 215.00 235.00 137.00 144.79 162.13 285.17 180.70 767.41

Assets 1059.00 1144.00 1236.00 1310.00 1315.00 1337.11 1297.63 1712.67 1761.55 2029.91

Shares equity and liabilities

Capital stock 500.00 500.00 550.00 550.00 630.00 644.30 658.60 672.90 687.20 701.50

Other restricted equity 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Other unrestricted equity 0.00 0.00 0.00 0.00 0.00 78.56 17.67 !7.09 176.38 186.38 Net income for the year 109.00 84.00 71.00 85.00 85.00 !54.30 !18.03 190.35 17.01 126.89 Shareholders'equity 609.00 584.00 621.00 635.00 715.00 668.55 658.23 856.15 880.59 1014.77 Accumulated depreciation

di!erence

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Reserves 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Valuation items 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Liabilities and shareholders+equity 1059.00 1144.00 1236.00 1310.00 1315.00 1337.11 1297.63 1712.67 1761.55 2029.91

Statement of income

Depreciation 50.00 55.00 60.00 65.00 70.00 64.68 61.76 91.12 85.65 80.59

Operating income after depreciation

150.00 134.00 126.00 145.00 150.00 !28.13 16.74 301.88 81.15 231.66 Interest expenses !40.00 !50.00 !55.00 !58.00 !62.00 !44.27 !40.79 !48.09 !58.47 !62.48

Other"nancial income 20.00 20.00 20.00 20.00 20.00 0.00 0.00 0.00 0.00 0.00

Extraordinary income and expenses

0.00 0.00 0.00 0.00 0.00

Allocations 0.00 0.00 0.00 0.00 0.00

Taxes 21.00 20.00 20.00 22.00 23.00 !18.10 !6.01 63.45 5.67 42.30

Net income 109.00 84.00 71.00 85.00 85.00 !54.30 !18.03 190.35 17.01 126.89

Other information

Amortization 30 40 50 50 55 48.12 44.34 52.27 63.55 67.91

Investments 110 162 119 78 0.00 15.94 551.12 0.00 1.24

New issues 0 50 0 80 14.3 14.3 14.3 14.3 14.3

(3)

Fig. 2. Maximizing net income (NI) in the mixed-integer linear case (cf. Table 7). Fig. 1. Maximizing net income (NI) in the continous linear case (cf. Table 6).

The optimal trajectories for the critical success

fac-tors are shown in Figs. 1

}

8 .

The results show that the discrete solutions are

slightly better than the continuous ones for the last

planning period, due to rounding errors in the cash

position. Some rounding errors are also observed

in the cash position for the nonlinear continuous

solution.

5. Conclusion

In the present study a fundamental assumption

has been that the sales versus production decisions

of the

"

rm do not a

!

ect market demand, in

consist-ence with perfect competition. Thus, the sales of the

"

rm are limited only by the internal conditions of

the

"

rm, in particular, productive capacity and

inventory volume. To allow for market

imperfec-tions, we may constrain the sales of the

"

rm

through a demand constraint. A particular demand

relation is given by the well-known constant

elas-ticity of demand (CED) function

q

D

(t)

"

A(t)p(t)

e

,

(5.1)

where

A(t) is a time-varying parameter estimated

from historic data of the

"

rm,

e

D

"

L

q

D

(t)

L

p(t)

(4)

Fig. 3. Maximizing ROI in the nonlinear continous case (cf. Table 8).

Fig. 4. Maximizing ROI in the nonlinear mixed-integer case (cf. Table 9).

(5)

Fig. 6. ROI trajectory when maximizing net income (NI) in the mixed-integer linear case (cf. Table 7).

Fig. 7. NI trajectory when maximizing ROI in the continous linear case (cf. Table 8).

(6)

is the (constant) price elasticity of demand. Its

time-varying counterpart with nonconstant price

elastic-ity

e

D(t)