A nonlinear mixed integer multiperiod "rm model - Semantic Scholar

Int. J. Production Economics 67 (2000) 183}199

A nonlinear mixed integer multiperiod "rm model Ralf OG stermark *, Hans Skrifvars, Tapio Westerlund Department of Business Administration, As bo Akademi University, Henriksgatan 7, 20500 Turku, Finland Department of Chemical Engineering, As bo 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 SoK derlund and OG stermark [1] to mixed integer problems. We will subsequently maximize the dis* Corresponding author. Tel.: 00358-2-215-4669; fax: 00358-2215-4806. E-mail address: ralf.ostermark@abo." (R.OG stermark)

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 statements, but there is no "nance theory to support them. In this paper we are concerned with optimizing some main economic variables of the "rm over a multi-year planning horizon, corresponding to critical "nancing, operating and investment decisions. The key property of our model is the simplicity in generating optimal "rm strategies. The key contribution of our study is in the derivation of a managerially convenient planning

0925-5273/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 5 2 7 3 ( 0 0 ) 0 0 0 1 9 - 0

R. O$ stermark et al. / Int. J. Production Economics 67 (2000) 183}199

184

constraint

Table 1 The objective functions in the di!erent test settings

f (x, y)!k)0 Solution space Decision variables

Linear

Nonlinear

Continuous Discrete

Net income (LP) Net income (MILP)

ROI (NLP) 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 machinery investment decisions. The features of this model have in#uenced various details of our design.

and the corresponding variable k is minimized. A sequence of points +(xI, yI), k"0, 1, 2, K, 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 +c2 x #c2y ,, V I W I +x , y ,3X , k"0, 1, 2, K, (2.7) I I I where X is de"ned by I X "¸5+x, y"l (x, y))0, k"0, 1,2, K!1,, ) I (2.8) x y I I

where

l (x, y)"f (xI, yI)#a ) I I I

The MINLP problem used in the method may be formulated as follows: min +c2 x#c2y,, +x, y,3N, V W VW where N"+x, y"u(x, y))0,.

(2.1) (2.2)

c and c are vectors with constants, x is a vector V W with continuous variables, y is a vector with integer variables and u(x, y) is a vector with nonlinear differentiable functions, all de"ned on a set ¸"X6>,

(2.3)

where X is an n-dimensional compact polyhedral convex set, X"+x"Ax)a, x31L,

(2.4)

*f I *y

*f I *x

(x!xI)

xI yI

(y!yI) (2.9) and f (x, y) is the function g (x, y) corresponding to I G max +g (xI, yI),. G G From the de"nition of the X set it follows from I Westerlund et al. [10] that the optimal solution +xH, yH, of (2.1) is a subset of X for convex and I quasi-convex problems, and #

2. The MINLP-algorithm

(2.6)

xI yI

X LX L2LX L2LX . (2.10) ) )\ I From Eq. (2.10) it also follows that the solutions Z "min+c2x #c2y , form a monotonically inI V I W I creasing sequence, Z *Z *2*Z *2*Z . (2.11) ) )\ I The convergence of the sequence, +Z , k"0, 1,2, K,, to the optimal solution in N is I shown for convex problems in [9], and for quasiconvex problems in [10].

and > is a "nite discrete set de"ned by >"+y"By)b, y3ZK,.

(2.5)

In the case of a nonlinear objective function f (x, y), the objective function can be written as a nonlinear

3. Model speci5cation Our "rm model encompasses a planning horizon as speci"ed by the management. In strategic "rm


planning, where we are dealing with aggregate decisions, a planning perspective of "ve years or longer is usually desirable [[11], p. 713]. The set of variables 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 depreciation } both connected to investments } in#uence net income. Like the interest expenses, the depreciations 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 investments, new loans, new issues, loan amortization, dividend payments, depreciation, sales volume and

185

production volume. Since sales and production are unsynchronized, inventory accumulation is possible. Inventory valuation is a crucial issue involving tax problems and matching 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 fundamental 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

Table 2 The objective functions, restrictions, decision variables and parameters of the "rm model Discounted objectives

Restrictions

O1 Net income R1

R2 R3

R4 R5 R6 R7 R8 R9

Sales } upper bound is a function of production capacity # inventory Amortization } equals a proportion of long-term debt New issues } upper bound is a proportion of stockholders' equity Dividends } upper bound is unrestricted shareholders equity Dividends }lower bound is a fraction of capital stock Depreciation } lower bound is a proportion of "xed assets Equity/Debt } lower bound Nonnegativity of cash Nonnegativity of debt

R10 Nonnegativity of "xed assets

Variables

Parameters

SALES R

Sales

mindep Minimal depreciation

PROD R

Production

tax

Taxes

cost

Operating costs/Turnover

NEWDEBT New debt R AMO R

Amortization sr

INV R

Investments

NEWISS R

New issue

DIVID Dividends R DEP Depreciation R Deviation variables AMODIFF in R2 EQUITYDIFF in R3 MAXDIVDIFF in R4 MINDIVDIFF in R5 DEDIFF in R7

Sales receivable/Turnover

mxiss

Maximal new issue/Stockholders equity minequ Stockholders' equity/Liabilities r cl o"

Interest rate on long-term debt Current liabilities/Operating costs Other "nancial items/Other "nancial assets d Discount factor X Machine cost p Unit sales price at time t R Factor Production capacity factor rep Amortization/Long-term debt mindiv Minimal dividend/Capital stock


186 Table 3 Balance sheet of the "rm model Assets

Symbols

Equations

Liabilities

Symbols

Equations

Fixed assets

FIXASS R

FIXASS #INV !DEP R\ R R

Capital stock

EQUITY R

EQUITY #NEWISS R\ R

Sales receivable

SALESREC R

sr TURNO R "SALES *SPRICE R R

Other unrestricted equity

OUEQUITY R

OUEQUITY R\ #NETINCOME R\ !DIVID R

Cash and bank deposits

CASH R

CASH #NETINCOME Net income R\ R !DEP #SALESREC R R !SALESREC #OTFINASS R\ R !OTFINASS !INV R\ R #NEWISS !DIVID R R !CURRLIAB #CURRLIAB R R\ !AMO R

NETINCOME R

See statement of income in Table 4

Other "nancial assets

OTFINASS R

OTFINASS R\ #OTFINASS R

Shareholders' equity

TOTEQUITY R

Current liabilities Long-term debt Total liabilities

CURRLIAB R DEBT R LIAB R

EQUITY R\ #OUEQUITY R #NETINCOME R cl cost TURNO R DEBT #NEWDEBT R\ R CURRLIAB #DEBT R R

Table 4 Statement of income Item

Symbols/Equations

Item

Equations

#Sales !Production costs ! Operating costs

SALES R PROD R cost

!Interest payments #Other "nancial income !Taxes

!Depreciation

DEP R

NETINCOME R

r(DEBT #NEWDEBT ) R\ R o";OTFINASS R tax;[TURNO (1!cost)!DEP R R !r;(DEBT #NEWDEBT )#o";OTFINASS ] R\ RG R [1!tax];[TURNO (1!cost)!DEP ! R R r;(DEBT #NEWDEBT )#o";OTFINASS ] R\ RG R

sanctioned deviation variables in the equations. The optimization problem can be formulated as follows: MAX y"f (x ) R VR s.t. A x )b , R R R x c 31L, R R

t"1,2, h, b 31K, A 31K"L, R R

(3.1)

where h is the planning horizon. In the test below we use h"5.

For each period, x consists of the following 13 R variables: x "(sales volume, production volume, new R debt, amortization, investments, new issue, dividends, depreciation, amortization di!erence, equity di!erence, maximal dividend di!erence, minimal dividend di!erence, depreciation di!erence) . R 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].


187

3.1. The objective functions

R (1!tax)[(1!cost)TURNO !r R [NEWDEBT !AMO ]!DEP ] G H GH GH G NETINCOME: (1#d)G G

R !PENALTY* Dev , GH G H R NETINCOME (1/(1#d)G) G G ROI: R (INV !DEP )/(1#d)G#FIXASS G G G R !PENALTY* Dev . GH G H

(3.3)

PENALTY is a positive value su$ciently large to make deviations undesirable. Dev refers to the GH 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 capacity 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) P /unit "Factor;FIXASS R , R X/unit

(3.4)

where the sales price, P , can vary over time. The R 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 "1000), the sales value of production is R (1000/0.2)P "5000P . The production costs are R R assumed to be constant. Only the sales price per unit (P ) varies and it is assumed to be known. The R

(3.2) capacity constraint is formulated as follows:

P SALES (Factor; R FIXASS R X

R # [INV !DEP ] . G G G

(3.5)

3.2.2. R2: Loan repayment The second restriction concerns the level of repayment. The amortization amount equals, as far as possible, a fraction of long-term debt, i.e.

R AMO "rep; DEBT # NEWDEBT G R G R\ (3.6) ! AMO #AMODIFF . R G G 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 constant 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 outstanding. 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


188

as follows: NEWISS )mxiss;EQUITY R #EQUITYDIFF . (3.7) R 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 complicated in comparison to the expected utility. 3.2.4. R4, R5: Upper/lower bounds on dividends Dividends are limited by upper and lower bounds. The upper bound is de"ned as free unrestricted equity, i.e. retained earnings: R DIVID (OUEQUITY # NETINCOME R G G R\ ! DIVID #MAXDIVDIFF . (3.8) G R G This is in order to protect creditors from excessive dividend payments. A minimum level of payments 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:

R DIVID ' mindiv EQUITY # NEWISS R G G !MINDIVDIFF . (3.9) R The deviation variable MAXDIVDIFF was included to guarantee feasibility in cases where unrestricted 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 mathematical expression is

R DEP ' mindep FIXASS # INV R G G R\ ! DEP . (3.10) G G 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 introduced, 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 intangible assets [21,22]). On the other hand, the pecking order theory [23] explains why some pro"table "rms borrow less, as they do not need outside money. Kjellman and HanseH n [24] found that most Finnish "nancial managers seek to maintain a constant 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:

R EQUITY # [NEWISS #NETINCOME G G G !DIVID ]#DEDIFF G R

'minequ DEBT #CURRLIAB R

R # [NEWDEBT !AMO ] . (3.11) G G G Equity is made up of new issues#retained earnings!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 balance sheet.


189

3.2.7. R8: Nonnegativity of cash The cash #ow is de"ned as CASH #[SALESREC !SALESREC ] R\ R *0. #[OTFINASS !OTFINASS ]![CURRLIAB !CURRLIAB ] R\ R R\ R # R [NETINCOME #DEP !INV #NEWDEBT #NEWISS !DIVID ] G G G G G G G

3.2.8. R9: Nonnegativity of debt Nonnegativity of long-term debt is de"ned as DEBT "DEBT R R # [NEWDEBT !AMO ]'0. G G G (3.13) Normally, debt will be positive while the amortization rate is less than unity. 3.2.9. R10: Nonnegativity of xxed assets The "nal constraint concerns nonnegativity of "xed assets: R FIXASS "FIXASS # [INV !DEP ]'0. R G G G (3.14) If selling of "xed assets is allowed, this must be included in the restriction. Otherwise it is redundant, since depreciation is always a nonnegative fraction of total assets. 4. A numerical experiment Our model has been tested on data from some Finnish listed companies. A hypothetical "rm is

(3.12)

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 determines 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 economy 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 capital 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 judgement of future development. The historical statements along with the optimal projected statements and the optimal programs are given in Tables 6}9.

Table 5 Parameters based on historical development Symbols

Values

Symbols

mindep tax cost sr d mxiss minequ r cl

0.06 0.25 1.00 0.15 0.15 0.02 1.00 0.08 0.15

o" t X P R Factor rep mindiv

Values

t"5 1.05

t"6 1.10

0.67 5 1.00 t"7 1.30 0.80 0.08 0.01

t"8 1.20

t"9 1.25

Symbols

Values

FIXASS SALESREC CASH OTFINASS EQUITY NETINCOME OUEQUITY DEBT CURRLIAB

1078.00 95.00 12.00 30.00 630.00 85.00 0.00 600.00 0.00


190

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: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13:

733.25 810.66 0.00 48.00 0.00 14.30 6.44 64.68 0.00 0.00 0.00 0.00 0.00

787.92 776.43 0.00 44.16 19.16 14.30 6.59 61.95 0.00 0.00 0.00 0.00 0.00

1307.51 1141.61 145.44 52.26 547.56 14.30 6.73 91.09 0.00 0.00 0.00 0.00 0.00

1290.09 1290.09 165.50 61.32 288.54 14.30 6.87 102.93 0.00 0.00 0.00 0.00 0.00

1490.94 1490.94 172.04 70.18 370.01 14.30 7.02 118.96 0.00 0.00 0.00 0.00 0.00

1 8

2 8

3 8

7 0.00

8 0.00

Sales Production New debt Amortization Investments New issues Dividends Depreciation Dividend deviation (Divdi!) Equity deviation (EKdi!) Debt-Equity deviation (D/E}di!) Repayment deviation (REPdi!) Max dividend deviation (MAXdivdf)

Historical period: Inventory volume

0 17

Planning period: Inventory volume Financial statements

5 177.40

6 165.91

Historical accounts 0

1

4 100 9 0.00

Forecasted accounts

2

3

4

5

6

7

8

9

Assets Fixed assets Valuation items Inventory Sales receivables Cash Other "nancial assets Financial assets Assets

859.00 914.00 0.00 0.00 10.00 5.00 150.00 180.00 10.00 15.00 30.00 30.00 190.00 225.00

1016.00 1070.00 1078.00 1013.32 0.00 0.00 0.00 0.00 5.00 5.00 100.00 177.40 170.00 190.00 95.00 115.49 15.00 15.00 12.00 0.00 30.00 30.00 30.00 30.00 215.00 235.00 137.00 145.49

970.53 1427.01 1612.62 1863.67 0.00 0.00 0.00 0.00 165.91 0.00 0.00 0.00 130.01 254.97 232.22 279.55 0.00 0.00 0.00 0.00 30.00 30.00 30.00 30.00 160.01 284.97 262.22 309.55

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 Other restricted equity Other unrestricted equity Net income for the year

500.00 500.00 0.00 0.00 0.00 0.00 109.00 84.00

550.00 0.00 0.00 71.00

550.00 0.00 0.00 85.00

630.00 644.30 658.60 0.00 0.00 0.00 0.00 78.56 17.84 85.00 !54.13 !17.84

Shareholders' equity 609.00 584.00 Accumulated depreciation di!erence 0.00 0.00 Reserves 0.00 0.00 Valuation items 0.00 0.00 Current liabilities 50.00 60.00 Long-term debt 400.00 500.00 Liabilities 450.00 560.00

621.00 0.00 0.00 0.00 65.00 550.00 615.00

635.00 0.00 0.00 0.00 75.00 600.00 675.00

715.00 0.00 0.00 0.00 0.00 600.00 600.00

Liabilities and shareholders+ equity

668.72 0.00 0.00 0.00 115.49 552.00 667.49

1059.00 1144.00 1236.00 1310.00 1315.00 1336.21

658.60 0.00 0.00 0.00 130.01 507.84 637.85

672.90 0.00 !6.73 189.82 855.99 0.00 0.00 0.00 254.97 601.02 855.99

687.20 0.00 176.21 74.00

701.50 0.00 243.20 141.91

937.42 1086.61 0.00 0.00 0.00 0.00 0.00 0.00 232.22 279.55 705.20 807.06 937.42 1086.61

1296.45 1711.97 1874.83 2173.22


191

Table 6 (continued) Financial statements

Historical accounts 0

1

Forecasted accounts

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

52.26 547.56

61.32 288.54

70.18 370.01

Other information Amortization Investments New issues Dividends

30

40 110

50 162

50 119

55 78

48.00 0.00

44.16 19.16

0 109

50 84

71

0 85

80 0

14.3 6.443

14.3 6.586

14.3 6.729

14.3 6.872

Table 7 Maxmizing net income in the linear mixed-integer case (MLP) Example

Optimization


5

6

7

8

9

1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13:

734.00 811.00 0.00 48.00 0.46 14.30 6.44 64.71 0.00 0.00 0.00 0.00 0.00

782.00 774.00 0.00 44.16 15.51 14.30 6.59 61.76 0.00 0.00 0.00 0.00 0.00

1311.00 1142.00 145.05 52.23 551.12 14.30 6.73 91.12 0.00 0.00 0.00 0.00 0.00

1291.00 1291.00 166.09 61.34 289.26 14.30 6.87 103.01 0.00 0.00 0.00 0.00 0.00

1493.00 1493.00 172.58 70.24 371.62 14.30 7.02 119.12 0.00 0.00 0.00 0.00 0.00

1 8

2 8

3 8

7 0.00

8 0.00


Historical period: Inventory volume Planning period: Inventory volume

0 17 5 177.00

6 169.00

4 100 9 0.00

14.3 7.015

192



Historical accounts

Forecasted accounts

0

1

2

3

4

5

6

859.00 0.00 10.00 150.00 10.00 30.00 190.00

914.00 1016.00 1070.00 1078.00 1013.75 0.00 0.00 0.00 0.00 0.00 5.00 5.00 5.00 100.00 177.00 180.00 170.00 190.00 95.00 115.61 15.00 15.00 15.00 12.00 !0.02 30.00 30.00 30.00 30.00 30.00 225.00 215.00 235.00 137.00 145.59

7

8

9


967.50 1427.50 1613.75 1866.25 0.00 0.00 0.00 0.00 169.00 0.00 0.00 0.00 129.03 255.65 232.38 279.94 !0.35 !0.37 !0.37 !1.11 30.00 30.00 30.00 30.00 158.68 285.28 262.01 308.82

1059.00 1144.00 1236.00 1310.00 1315.00 1336.34 1295.18 1712.78 1875.76 2175.07


500.00 0.00 0.00 109.00

500.00 0.00 0.00 84.00

550.00 0.00 0.00 71.00

550.00 0.00 0.00 85.00

630.00 644.30 658.60 0.00 0.00 0.00 0.00 78.56 17.85 85.00 !54.13 !18.14

Shareholders' equity Accumulated depreciation di!erence Reserves Valuation items Current liabilities Long-term debt Liabilities

609.00 0.00

584.00 0.00

621.00 0.00

635.00 0.00

715.00 0.00

668.73 0.00

658.31 0.00

856.48 0.00

937.98 1087.39 0.00 0.00

0.00 0.00 50.00 400.00 450.00

0.00 0.00 60.00 500.00 560.00

0.00 0.00 65.00 550.00 615.00

0.00 0.00 75.00 600.00 675.00

0.00 0.00 0.00 600.00 600.00

0.00 0.00 115.61 552.00 667.61

0.00 0.00 129.03 507.84 636.87

0.00 0.00 255.65 600.66 856.30

0.00 0.00 0.00 0.00 232.38 279.94 705.41 807.74 937.79 1087.68


672.90 0.00 !7.02 190.60

687.20 0.00 176.21 74.07

701.50 0.00 243.76 142.13

1059.00 1144.00 1236.00 1310.00 1315.00 1336.34 1295.18 1712.78 1875.76 2175.07

Statement of income Turnover Operating costs Operating income Depreciation Operating income after depreciation Interest expenses Other "nancial income Extraordinary income and expenses Allocations Taxes Net income

500.00 300.00 200.00 50.00 150.00

510.00 321.00 189.00 55.00 134.00

510.00 324.00 186.00 60.00 126.00

550.00 340.00 210.00 65.00 145.00

600.00 770.70 380.00 734.00 220.00 36.70 70.00 64.71 150.00 !28.01

860.20 1704.30 1549.20 1866.25 782.00 1311.00 1291.00 1493.00 78.20 393.30 258.20 373.25 61.76 91.12 103.01 119.12 16.44 302.18 155.19 254.13

!40.00 !50.00 !55.00 !58.00 !62.00 !44.16 !40.63 !48.05 !56.43 !64.62 20.00 20.00 20.00 20.00 20.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

21.00

20.00

20.00

22.00

0.00 23.00 !18.04

0.00 !6.05

0.00 63.53

0.00 24.69

0.00 47.38

109.00

84.00

71.00

85.00

85.00 !54.13 !18.14

190.60

74.07

142.13


30

109

40 110 0 84

50 162 50 71

50 119 0 85

55 78 80 0

48.00 0.46 14.3 6.443

44.16 52.23 61.34 70.24 15.51 551.12 289.26 371.62 14.3 14.3 14.3 14.3 6.586 6.729 6.872 7.015


193

Table 8 Maxmizing ROI in the nonlinear continuous case (NLP) Example

Optimization


5

6

7

8

9

1: Sales 2: Production 3: New debt 4: Amortization 5: Investments 6: New issues 7: Dividends 8: Depreciation 9: Dividend deviation (Divdi!) 10: Equity deviation (EKdi!) 11: Debt-Equity deviation (D/E}di!) 12: Repayment deviation (REPdi!) 13: Max dividend deviation (MAXdivdf)

731.68 810.00 1.46 48.12 0.00 14.30 6.44 64.68 0.00 0.00 0.00 0.00 0.00

784.60 774.80 0.00 44.27 17.00 14.30 6.59 61.82 0.00 0.00 0.00 0.00 0.00

1311.52 1142.35 144.29 52.27 550.58 14.30 6.73 91.14 0.00 0.00 0.00 0.00 0.00

833.73 1073.81 193.62 63.58 0.00 14.30 6.87 85.68 0.00 0.00 0.00 0.00 0.00

1249.45 1009.38 118.01 67.93 0.00 14.30 7.02 80.54 0.00 0.00 0.00 0.00 0.00

1 8

2 8

3 8

Historical period: Inventory volume

0 17

Planning period: Inventory volume Financial statements

5 178.98

6 169.18

7 0.01

Historical accounts

4 100

8 240.09

9 0.02

Forecasted accounts

0

1

2

3

4

5

6

7

8

9

Fixed assets Valuation items Inventory Sales receivables Cash Other "nancial assets Financial assets

859.00 0.00 10.00 150.00 10.00 30.00 190.00

914.00 0.00 5.00 180.00 15.00 30.00 225.00

1016.00 1070.00 1078.00 1013.32 0.00 0.00 0.00 0.00 5.00 5.00 100.00 178.98 170.00 190.00 95.00 115.24 15.00 15.00 12.00 !0.37 30.00 30.00 30.00 30.00 215.00 235.00 137.00 144.87

968.50 1427.93 1342.26 1261.72 0.00 0.00 0.00 0.00 169.18 0.01 240.09 0.02 129.46 255.75 150.07 234.27 !0.37 !0.37 !0.38 504.59 30.00 30.00 30.00 30.00 159.09 285.37 179.69 768.86

Assets

1059.00 1144.00 1236.00 1310.00 1315.00 1337.17

1296.77 1713.31 1762.04 2030.60

Assets

Shares equity and liabilities Capital stock Other restricted equity Other unrestricted equity

500.00 0.00 0.00

Net income for the year

109.00

84.00

71.00

85.00


609.00 0.00

584.00 0.00

621.00 0.00

635.00 0.00


0.00 0.00 50.00 400.00 450.00

500.00 0.00 0.00

0.00 0.00 60.00 500.00 560.00

550.00 0.00 0.00

0.00 0.00 65.00 550.00 615.00

550.00 0.00 0.00

0.00 0.00 75.00 600.00 675.00

630.00 0.00 0.00

658.60 0.00 17.70

672.90 0.00 !7.09

85.00 !54.27 !18.06

190.67

715.00 0.00 0.00 0.00 0.00 600.00 600.00

644.30 0.00 78.56

687.20 0.00 176.70

701.50 0.00 186.62

16.93

127.00

668.58 0.00

658.23 0.00

856.47 0.00

880.84 1015.12 0.00 0.00

0.00 0.00 115.24 553.34 668.58

0.00 0.00 129.46 509.08 638.54

0.00 0.00 255.75 601.09 856.84

0.00 0.00 0.00 0.00 150.07 234.27 731.13 781.21 881.20 1015.48

1059.00 1144.00 1236.00 1310.00 1315.00 1337.17 1296.77 1713.31 1762.04 2030.60


194 Table 8 (continued) Financial statements

Historical accounts

Forecasted accounts

0

1

2

3

4

5

500.00 300.00 200.00 50.00 150.00

510.00 321.00 189.00 55.00 134.00

510.00 324.00 186.00 60.00 126.00

550.00 340.00 210.00 65.00 145.00

600.00 768.27 380.00 731.68 220.00 36.58 70.00 64.68 150.00 !28.10

6

7

8

9


863.06 1704.98 1000.47 1561.81 784.60 1311.52 833.73 1249.45 78.46 393.46 166.75 312.36 61.82 91.14 85.68 80.54 16.64 302.31 81.07 231.83

!40.00 !50.00 !55.00 !58.00 !62.00 !44.27 !40.73 !48.09 !58.49 !62.50 20.00 20.00 20.00 20.00 20.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

21.00

20.00

20.00

22.00

0.00 23.00 !18.09

0.00 !6.02

0.00 63.56

0.00 5.64

0.00 42.33

109.00

84.00

71.00

85.00

85.00 !54.27 !18.06

190.67

16.93

127.00

Other information Amortization Investments New Issues Dividends

30

109

40 110 0 84

50 162 50 71

50 119 0 85

55 78 80 0

48.12 0.00 14.3 6.443

44.27 52.27 17.00 550.58 14.3 14.3 6.586 6.729

63.58 0.00 14.3 6.872

Table 9 Maxmizing ROI in the nonlinear mixed-integer case (MINLP) Example

Optimization


5

6

7

8

9

1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13:

731.00 810.00 1.55 48.12 0.00 14.30 6.44 64.68 0.00 0.00 0.00 0.00 0.00

785.00 774.00 0.78 44.34 15.94 14.30 6.59 61.76 0.00 0.00 0.00 0.00 0.00

1310.00 1142.00 143.47 52.27 551.12 14.30 6.73 91.12 0.00 0.00 0.00 0.00 0.00

834.00 1073.00 193.32 63.55 0.00 14.30 6.87 85.65 0.00 0.00 0.00 0.00 0.00

1249.00 1010.00 118.02 67.91 1.24 14.30 7.02 80.59 0.00 0.00 0.00 0.00 0.00

1 8

2 8

3 8


Historical period: Inventory volume Planning period: Inventory volume

0 17 5 179.00

6 168.00

7 0.00

8 239.00

4 100 9 0.00

67.93 0.00 14.3 7.015


195


Historical accounts

Forecasted accounts

0

1

2

3

4

5

6

859.00 0.00 10.00 150.00 10.00 30.00 190.00

914.00 1016.00 1070.00 1078.00 1013.32 0.00 0.00 0.00 0.00 0.00 5.00 5.00 5.00 100.00 179.00 180.00 170.00 190.00 95.00 115.13 15.00 15.00 15.00 12.00 !0.34 30.00 30.00 30.00 30.00 30.00 225.00 215.00 235.00 137.00 144.79

7

8

9


967.50 1427.50 1341.85 1262.50 0.00 0.00 0.00 0.00 168.00 0.00 239.00 0.00 129.53 255.45 150.12 234.19 2.60 !0.28 0.58 503.22 30.00 30.00 30.00 30.00 162.13 285.17 180.70 767.41

1059.00 1144.00 1236.00 1310.00 1315.00 1337.11 1297.63 1712.67 1761.55 2029.91


500.00 0.00 0.00 109.00

500.00 0.00 0.00 84.00

550.00 0.00 0.00 71.00

550.00 0.00 0.00 85.00

630.00 644.30 658.60 0.00 0.00 0.00 0.00 78.56 17.67 85.00 !54.30 !18.03


609.00 0.00

584.00 0.00

621.00 0.00

635.00 0.00

715.00 0.00

668.55 0.00

658.23 0.00

856.15 0.00

880.59 1014.77 0.00 0.00

0.00 0.00 50.00 400.00 450.00

0.00 0.00 60.00 500.00 560.00

0.00 0.00 65.00 550.00 615.00

0.00 0.00 75.00 600.00 675.00

0.00 0.00 0.00 600.00 600.00

0.00 0.00 115.13 553.42 668.55

0.00 0.00 129.53 509.87 639.40

0.00 0.00 255.45 601.07 856.52

0.00 0.00 0.00 0.00 150.12 234.19 730.84 780.95 880.96 1015.14


672.90 0.00 !7.09 190.35

687.20 0.00 176.38 17.01

701.50 0.00 186.38 126.89

1059.00 1144.00 1236.00 1310.00 1315.00 1337.11 1297.63 1712.67 1761.55 2029.91


500.00 300.00 200.00 50.00 150.00

510.00 321.00 189.00 55.00 134.00

510.00 324.00 186.00 60.00 126.00

550.00 340.00 210.00 65.00 145.00

600.00 767.55 380.00 731.00 220.00 36.55 70.00 64.68 150.00 !28.13

863.50 1703.00 1000.80 1561.25 785.00 1310.00 834.00 1249.00 78.50 393.00 166.80 312.25 61.76 91.12 85.65 80.59 16.74 301.88 81.15 231.66

!40.00 !50.00 !55.00 !58.00 !62.00 !44.27 !40.79 !48.09 !58.47 !62.48 20.00 20.00 20.00 20.00 20.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

21.00

20.00

20.00

22.00

0.00 23.00 !18.10

0.00 !6.01

0.00 63.45

0.00 5.67

0.00 42.30

109.00

84.00

71.00

85.00

85.00 !54.30 !18.03

190.35

17.01

126.89


30

109

40 110 0 84

50 162 50 71

50 119 0 85

55 78 80 0

48.12 0.00 14.3 6.443

44.34 52.27 15.94 551.12 14.3 14.3 6.586 6.729

63.55 0.00 14.3 6.872

67.91 1.24 14.3 7.015

196


The optimal trajectories for the critical success factors 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.

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 imperfections, we may constrain the sales of the "rm through a demand constraint. A particular demand relation is given by the well-known constant elasticity of demand (CED) function q"(t)"A(t)p(t)C",

5. Conclusion

(5.1)

where A(t) is a time-varying parameter estimated from historic data of the "rm,

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-

*q"(t) p(t) e"" *p(t) q"(t)

Fig. 1. Maximizing net income (NI) in the continous linear case (cf. Table 6).

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

(5.2)


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).

Fig. 5. ROI trajectory when maximizing net income (NI) in the continous linear case (cf. Table 6).

197

198


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).

Fig. 8. NI tracetory when maximizing ROI in the mixed-integer nonlinear case (cf. Table 9).


is the (constant) price elasticity of demand. Its timevarying counterpart with nonconstant price elasticity e"R 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 quantities 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 MonteCarlo 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 simultaneous 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 position deserves further attention in future research. Acknowledgements Financial support from the Academy of Finland is gratefully acknowledged. References [1] K. SoK derlund, R. OG stermark, A multiperiod "rm model for strategic decision support, Working paper, As bo Akademi University, 1995. [2] E. Kasanen, R. OG stermark, The managerial viewpoint in interactive programming with multiple objectives, Kybernetes 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 priorities, European Journal of Operational Research 30 (1987) 13}23.

199

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