An analytic model for strategic control of corporate development

OMEG.4 The Int. JI of Mgmt Sci., Vol. 12, No. I, pp. ~3-51. 1984 Printed in Great Britain. All nghts reserved

0305-0483 84 5300+0.00 Copyright (2" 1984 Pergamon Press Ltd

An Analytic Model for Strategic Control of Corporate Development JACQUES

SARRAZIN

Ecole Polytechnique Superieur, Paris, France TIMOTHY

RUEFLI

University of Texas at Austin, USA (Received January 1983; in revised f o r m June 1983)

The concept of strategic control as des~eribed by the authors 18] partitions the decision horizon of the firm into two segments. In the long term, information is not precise, goals are vague and constraints are unclear. The problem for the firm is, in spite of this ambiguity, to establish a development window at the beginning of this period that reflects long-run factors and can serve as a guide for short-term behaviour. In the short run, information is precise enough for conventional mathematical programming models to be used to guide attainment of the development window. This paper presents an analytic model that solves the long-run problem defined by the strategic control framework and generates the required development window. A numerical example is presented for illustrative purposes.

INTRODUCTION

As Fig. 1 illustrates, the problem in the near term is to guide corporate development through a relatively certain environment to a state, g (r), (called the corporate development window) where it will be in a position most likely to enable it to reach a set of long-range goals in a more uncertain environment. The decision problem in the short run can be represented by extant goal programming formulations and, hence, will not be treated further here. Rather, this paper will be concerned with developing a mathematical model that solves the problem of generating the corporate development window, g(r), given long-range fuzzy goals and constraints. The parameters of this window will then serve as the primary goal inputs to the aforementioned goal programming problem for the short run. The general long-range problem for a firm with development horizon, T, and intermediate horizon, r, is to: Find g(r) such that

IN AN EARLIER paper [8], we presented in general terms, a new model of corporate development under ambiguous circumstances. This model partitioned the development space of the corporation into two parts and postulated a decision problem for each subspace. Analytically, the problem for the firm in the nearterm period, [o, r], for which relatively certain information is available is to: Min subject

II h,[x,(t),x2(t)

..... x.(t)l-g(r)

II

to:

f[,[xt(t ), x2(t ) . . . . . x,(t)]

_>d ; ( t )

for i~l,

t = 1. . . . . r

xj(t)>O

for all j, t,

(1)

where h,[] = describes the performance of the firm at time r, xj(t) = the j t h decision variable at time t, f[,[] > d~(t) = the ith constraint for period

g(r)+h[xt(r

+ 1) . . . . . x . ( r + 1"). . . . .

x.(T)]=g(T)(2)

subject to

t, fi.,[x,(t) . . . . .

g(r)

= a vector of near-term goals, I = the set of constraint indices.

x.(t)l
xl(t ) . . . . . x . ( t ) > 0 43

for t = r + 1. . . . .

is[

T

44

Sarrazin. Ruefli--Strategic Control of Corporate Development

Fuzzy

~r(c)

constraints

I

.3 : i

I

Near term

Long

term

F hme

m

ization of a non-linear relationship is not as critical a problem as it might be in the near-term case. Secondly, the fuzzy set approach which will be employed allows for variations in the slope and intercept of the approximating linear functions, in effect, making the set of constraints representing the fuzzy constraint non-linear. Finally, the same property permits the representation of known non-linearities by linear approximations. This leads to a more compact formulation of the initial long-range model: Find g(r) such that

F i g . I. Near-term and long-term spaces o f corporate d e v e l opment in relation to strategic control. ( a ) = development of

AX =g(T)-g(r)

the firm in the period [t0, r]. g(r) = intermediate non-fuzzy goals for time "r'.

BX < D

(3)

where g(T), g(r) are vectors representative of objectives

where g(T)=the vector of long-range fuzzy goals, xj(t) = the j t h decision variable at time t, hr[] = describes organizational performance, f.,[] < di(t) = the ith (fuzzy) constraint, g (r) = the corporate development window, I = the set of constraint indices. A modified form of this problem can be derived by approximating h (x) by a linear function such that: h[x,(r + 1) . . . . .

x>0

x.(T)]

_~ aj(r + l ) x t ( r + 1) + . . .

+

a.(T)x.(T),

where al(r + 1) . . . . .

a.(T)

might have fuzzy values. In the same manner, the fuzzy long-term constraints f., can be rewritten such that:

X = [ x , ( r + t) . . . . .

:%(T)]'

is a vector of the decisions made during the period [r, T]. A, B, D, g ( T ) are matrices or vectors whose elements might be fuzzy. S O L U T I O N PROCESS The solution process described here is a satisricing approach, based on the distinction between the future development of the organization in the short run, when reliable information can be collected, and its evolution in the long run, when the hypothesis is that information is fuzzy. Accordingly, the application of the mathematical model will differ in the short run and in the long run, leading to two successive and complementary processes comprising strategic control (see Fig. 2).

Strategic control system

bij(t)xj(t ) < d , ( t ) J

t > r

1

i~l

where bi4(t ) and di(t) might have fuzzy values. The linearization of the problem provides significant computational advantages and can be justified on several grounds. First, the problem being linearized has been defined to occur in a future state in which information reliability is not high, thus possibly inappropriate linear-

[

I

J B.or, ron e oraoess t~o

(21

t~r

Forwards (carried out second}

FIG. 2.

Schematic

rao e t~f

11)

rooe,, t~7-

Backwards ( c a r r i e d out f i r s t ]

of the two stages of the solution process.

Omega,

Vol. t2, No.

Applied to corporate development in the long run. i.e. after time "r'. the model will operate ~backwards'. That is, given the fuzzy long-term objectives g (T) and constraints between time "r" and time 'T', it will give a minimal level of organizational performance g(r), or an associated range of values, at time r most likely to be needed for the achievement of the long-term objectives. The use of the term 'most likely' here means that the determination o f g ( r ) might not be fully reliable, but reflect only a satisficing level, to be chosen by the enterprise in relation to its attitude towards risk. It must be stressed that in this process, the determination of the set of objectives g(r) results from the interactions of both long-term constraints and objectives and not from the knowledge of the set of long-term objectives alone. The long run problem can be formulated according to several different approaches and solved by the appropriate solution procedures. The approach to be followed here is based on Zadeh's concept of fuzzy set theory [9] and the fuzzy decision-making model of Bellman and Zadeh [1]. Other possible approaches include chance-constrained programming [4], or the goal-interval formulation of Charnes and Collomb [3]. A future paper will compare the three approaches. The proposed procedure for determining g (r) will be based on four steps: (I) Determination of the fuzzy set of polyhedral convex sets of long-term constraints BX <_D, f >_O. (2) Representation of the fuzzy set of 'functionals' AX = K, K being a parameter. (3) Determination of the fuzzy intersection, (k7,/~), of both preceding fuzzy sets. (4) Selection of [ g ( T ) - g(r)] as the maximal space (or interval) interior to all possible spaces (k,k) having a grade of membership greater than a specified value. Lastly, g(r) is deduced from g(T). v

A

SOLUTION PROCEDURE These four steps will now be detailed, and illustrated with the case of a single goal, two constraint and two variable model. Problem (3) can then be stated as: This approach was suggested by an a n o n y m o u s referee.

45

I

Find g(r) such that al.,q z- azxz = g ( T ) - g ( r ) b l . t x ~ + h l z . v z <_d I

(C!)

bz.~x I + bz.zx: < d,

(C:)

x t . x: >_ 0

(-t)

The ith fuzzy linear constraint (C~) above is completely described by the three terms di, bi.~. and bi.2. Given the fuzzy sets D,, &.~, and B~.: in R, it might then be represented by a fuzzy set of constraints in R 3. The conventional approach in fuzzy set theory is to define the membership grade of the intersection of two fuzzy sets to be the minimum of the membership grade of those sets, i.e. ,u(A f3 B) = min{# (A), I~(B)}.

As Kickert [7] reports, Zimmerman et al. [10] have pointed out this is a very conservative approach and one that does not represent managers' concepts of intersection. A more realistic measure would be to define: ~(a riB) = [~,(A).~,(B)]!. This is the approach we will follow, t This fuzzy set can be represented graphically in the space of variables xj. Figure 3 shows a case of two variables and two constraints.

Representation of the fuzz)' set of jimctionals AX = K, K being a parameter We will use here the concepts of null space,

N(A), and range space, R(A+), of a transformation [2, 6]. N(A) and R(A +) are characterized by a null-space basis and a range-space basis, respectively. One can show that such bases can be obtained by starting with the matrix A and then diagonalizing the matrix by elementary row operations. If there are any fuzzy elements in A, their fuzziness is reflected directly by the fuzziness of the spaces N(A) and R ( A ' ) via their bases. That is, fuzzy elements in A will yield bases that have membership grades reflective or the membership grades of the elements of A. The interest in the concepts of null space and range space for this model comes from the result that N(A) and R(A +) are orthogonal. As the sum of their dimensions is equal to the dimension of the space of X, it follows that X can be expressed as a unique sum of a vector in N(A) and a vector in R(A +). Furthermore, the transformation A is fully characterized by the knowl-

46

Sarra-in. Ruefli--Strategic Control of Corporate Development

xa ~

I 1

~

@ . ~

)

(CI)

NI ¥ I N I

,

Fuzzy set of constromts

l(Cz~

\

\

0

-:('~

FIG. 3. Constraint space. On this diagram, the f u z z y set associated with the constraint (C t ) has three elements, represented here by solid lines. On the other hand, the f u z z y set related to (C.,) has two elements, which are iUustrated b)' the two dotted lines. When the two non-negativity constraints are taken into account, we come up with a fitzzy set o f sLr polyhedral sets o f constraints, each made up o f four constraints. edge of N(A) and R(A +). This property will now be exploited in the second step of the solution procedure of the long-range problem. Let us consider the case of two variables x~ and x_, and a functional AX reduced to the following linear function (atxt + azx,_). If the coefficients a~ and a., are fuzzy, the transformation A = (a~, a2) is also fuzzy, as are the null and range spaces N(A) and R(A +). The fuzzy sets associated with N(A ) and R (A +) are illustrated in Fig. 4, in the case where the fuzzy sets of a~ and a,_ have only two elements each. In this example of two variables x~ and x2 and a single functional, N(A) is a one-dimensional space, as is R(A +). A particular solution to the equation AX = k, k being an arbitrary value, is represented by the point Xo on R(A +), such that the metric distance between Xo and the origin 0 is k. The complete solution space to AX = k is

the line parallel to N(A) and going through X,,, since N(A) is the space of vectors X such that AX = 0 and since N(A) is orthogonal to R(A +) ([6], Appendix A).

Determination of the filzzy intersection [[(, ~c] of the set of constraints and of the functional The procedure in this third step will first be described in the case of the simple two variable model (3); generalization will then be made. We have seen in Fig. 4 that the solution space AX = K is a line parallel to N(A) and going through the point Xo on R(A +), such that the distance from 0 to Xo is K. It follows that if K varies, the solution space will be translated parallel to N(A). Let us now introduce the convex set of constraints (C~) and (C:) and assume at this stage of the discussion that A, (C~) and (C.,) are non-fuzzy. It is then

x2

~

(oI o21

"4 O X1

0

FIG. 4.

Omega, Vol. I2. No. I

47

Y 2

"\

"~

\

/

\

k

(Cz)

X.

Fio. 5.

clear that the solution space will be non-empty only if k stays inside the range generated by the projection of the convex set of constraints into R(A +) parallel to N(A). This range [/~,/~] is illustrated graphically in Fig. 5. In our model, K is equal to [g(T)-g(r)], More generally, given a specific set of linear constraints and a specific linear transformation A, the projection of the set of constraints on the range space of A +, R (A +), parallel to the null space of A, N(A), will give the maximum space inside which [ g ( T ) - g ( r ) ] can vary while preserving the existence of a solution X. If now the set of constraints and the matrix A are fuzzy, the combination of all the elements of the fuzzy set of polyhedral sets of constraints with all the elements of the fuzzy set of range spaces R(A +) two by two will result in a fuzzy set of intervals (#i,v k,)^ and the grade of membership for each (ki, ki) will be defined as the product of the membership grades of the corresponding polyhedral set of constraints and range space R (A +) [7].

Determination of g(r) [g(T)-g(r)] will be defined as the intersection of all possible ranges (/~,/~), whose membership grades exceed an arbitrary given value. Such a selection procedure combines two major advantages: (1) it results in a limited range for [g(T) - g (r)]; (2)it takes into account the most 'probable' values of the fuzzy coefficients and thus maximizes the reliability of the determination of g(r). For instance, the procedure will come up with: ~,<_g(T)-g(r)<_l~,,.

If no intersection exists for a given mere-

OME

12 I ~ D

bership value, one increases the value until an intersection is found. Increasing the required membership grade for the range (/(,/,:). indeed, will reduce the actual set of ranges to be considered. It may be the case that there is no membership grade that yields a non-empty intersection. In this case, the fuzzy constraints must be relaxed and the solution procedure reinstated to attempt to find a non-empty solution space. The value of g(r) is then deduced from the value of g(T) and the knowledge of the intersection [g(T)-g(r)]. As the long-term goals g(T) are themselves fuzzy, one can either express g(r) as a fuzzy set, the membership function of which is directly translated from the membership function of g(T), or arbitrarily reduce this fuzzy set to a range of equally acceptable values, or select the value with the highest acceptable membership grade as the value of g(r). But if g(r) has to be non-fuzzy, only the two latter selection procedures can be adopted. It may be that given (I,~,,~,) that a desired g(T) does not yield a feasible g(r). In this case, g(T) must be redefined at a reduced level until g(r) is feasible. The procedure for the solution of the longrange strategic control model (3) leads to the determination of an intermediate objective, g(r), which takes into account both the longterm objectives and the long-term constraints of the firm in an explicit way. This point must be emphasized because it is a response to an endless traditional debate on whether objectives or constraints are defined first and on what kinds of links exist between them. Another major point of interest in the solution procedure is in the simple relation between g(T) and g(r), that is, between long-term and

Sarrazin. Ruefli--Strategic Control of Corporate Development

48

intermediate objectives. The output of the procedure, indeed, is the difference [ g ( T ) - g ( r ) ] . the formulation of which depends only on the long-term organizational objectives. Changes in g(T) will result in a straightforward modification of the intermediate objectives g (r) and reciprocally.

where each value is immediately followed by its membership grade in parens. The fuzzy set related to the first constraint (Ct) is then composed of four elements: with associated membership grades:

NUMERICAL EXAMPLE This section will develop an example of application of the strategic control process investigated in this paper. More precisely, we will assume that a firm has as a long run goal to be in the upper half of its industry with respect to sales volume. Management has estimated that this will require sales of eight million dollars with membership grade 0.70 and six million dollars with membership grade 0.40. Sales growth is constrained by market growth and by government regulation. The problem for the firm is to: Find g such that ax~+x,=G-g bx t+3x 2
(Ct)

cx t < 10 (C,) x t, x2_>0

(4a)

G being the fuzzy long-term objectives, g being the intermediate (non-fuzzy) objectives, a, b, c, d being fuzzy parameters. To be able to represent the steps of the procedure graphically, we have deliberately reduced the set of variables to two elements, which can be considered as either two activities and there is then a time-lag of one period between long-term and intermediate objectives, or one activity over two periods of time. The fuzzy sets associated with a, b, c and d will be exhibited as they are required by the procedure. In this example, we will follow the four steps of the procedure discussed earlier:

(I) Determination of the polyhedral convex sets of constraints Let B, C and D be the fuzzy sets associated with the parameters b, c and d, defined as: B = { I ( 0 . 8 ) , 3(0.5)}

x ~ + 3 . v : < 12

,z = 0 . 6 9

.vl+3.v_,< 15

~L = 0 . 4 0

3x~+3x.,<12

,tL=0.55

3,v1 + 3 x : < 1 5

It=0.32.

In the same manner, the fuzzy set associated with the second constraint (C:) is formed by two elements: 2x t < 10

#=0.4

3x t_< 10

,u = 0 . 3 .

The constraints (C~) and (C,) joined to the non-negativity constraints can then be represented by a fuzzy set of polyhedral sets S of constraints, each polyhedral set being composed of two elements of the fuzzy sets related to (Ct) and (C2), respectively and by the axes "x~ = O, x, = 0". In other words, we come up with a fuzzy set of eight polyhedral sets, defined in Table 1.

(2) Representation of the fuzzy set of functionals We will now consider the functional: ax I "Jr-x, = k where a is a fuzzy parameter with an associated fuzzy set A, given by: A = {3(0.8), 2(0.4)}.

As noted the null space N(A ) and range space R (A +) are then unidimensional. N ( A ) is given by the direction ( - 1 , a) and R(A +) by the direction (a, 1). The fuzzy set A being composed of two elements, the fuzzy set

TABLE 1 D

30

S:,

,u = 0.35

,u = 0.45

(33) Ss'~'=0"35S>I'=0"4720 (33) S~''=0"30S~*'=03030

C = {2(0.4), 3(0.3)} D = {12(0.6), 15(0.2)}

&,

x~, x., > 0 in all cases.

Omega. I.'ol. 12. 3,o

1

49

I

X2

X

@\

×

X

S! •

\

\

{

-

o\

~=

*

"

....1t-<11.,, ~ 3

x\

FiG.

a 5

/

=

6.

T~BLE 2

Polyhedral set oF constraints SI

Extremal point

(10)

E I 5.~-

Projection on R L(A ") ,~,~ = 0

Membership grade

Projection on R,I'A ~ ) k", = 0

Membership grade

k:'ll = 5.80

0.57

/'~_'L= 5 9 6

0.40

S:

E< \(1o 3q ,T' 9-]

/$,: = 4.39

0.53

k',__,= 4.72

0.37

S3

E3 5. ~

/ ~ = 5.48

0.64

k'_,3 = 5 5 2

045

Sa

Ea~7, if)

k u = 4.08

0.60

/$> = 4,27

0.42

SS

Es(5.01

'~5 = 4.74

0.53

/$:s = 4.47

0.37

/10 26\

/lo 5\

S~

E ~ - . 3)

/~ls = 3.69

0.49

L:> = 3.73

0.35

$7

E7(4.0)

/~7 = 3.79

0.61

/~:7 = 3 5 8

0.43

S,

/i0 2\ E , /\ ~ - . 51/

k,~ = 3.37

0.49

/(z~ = 3.28

0.35

50

Sarra_-in, Ruefli--Strategic Control of Corporate Development

of range spaces will also have two elements, R~(A -) and R:(A -), with the same membership grades as the corresponding values of the parameter a and defined as: R~(A-)=(3.1)

# =0.8

R:(.4")=(2. I)

# =0.4.

and

(3) Determination of the Jitzzy intersection (k, k) of both preceding fi~zzy sets The set of polyhedral sets S of constraints as well as the set of range spaces are represented graphically in Fig. 6. The projection of each of the eight polyhedral sets on each of the two possible range spaces results in 16 possible intervals (k",/S), where/~ is always the origin and L: is the orthogonal projection of the extremal point of each polyhedral set in the direction of R(A +) on the two possible range spaces R~(A +) and R2(A *). In other words, we come up with eight projections /~ on R~(A+) and eight on R,_(A +), defined in Table 2. The grade of membership of each range (/~,/~) being determined as the grade of the intersection of the corresponding polyhedral set of constraints and range space R (A +).

(4) Determination of g [G - g ] will be defined as the intersection of all the ranges (/~,/S) above, whose membership grades exceed an arbitrary given value, say, 0.55. In this example, as ,( is the origin in every range (/~,/~), such an intersection exists. It is the interval common to the four ranges (/S,/~), (/(,/~t3), (k,/~t4), (/(,/~7), that is, the range (/(,,,/~o), such that: I~,,= 3.79.

It follows that: 0 ___
(4b)

the top half of its industry with membership grade 0.65 at T then at time r. they should achieve sales of at least 54.21 million. The short-run goal-programming formulation would then attempt to minimize deviations of performance below $4.21 million in sales. If management was willing to settle for the lower membership grade of 0.49 for being in the upper half of their industry, then the goal at time r would be $2.21 million. Notice that in this example the development window was defined in terms of its 'sill', in other cases an upper limit could also be generated. The process of generating the development window involved mathematical techniques no more complicated than diagonalizing a matrix, finding intersections of linear functions and finding orthogonal projections. The sheer number of these simple calculations can, however, become cumbersome for large problems because of the combinatorial nature of the formulation. The number of intersections and the number of projections required goes up exponentially with the number of fuzzy variables, and the range of values they can assume. Mitigating this limitation is the small number of dimensions usually involved in strategic decision making in a long-term problem. For a problem with 10 equations in 20 variables, with an average of 3 fuzzy coefficients per relation each evaluated at 2 values, an upper bound of approx. 25,000 can be calculated for the number of intersections and orthogonal projections that might be required. Since many of the intersections (and their resultant projections) would be ruled out by the logic of an applied problem, problems of this magnitude are feasible on a large mainframe computer. For larger problems, the alternative techniques of chance-constrained or goalinterval programming mentioned earlier in this paper would be necessary.

Since G is itself fuzzy and defined by the fuzzy set:

CONCLUSION

G = [8(0.7), 6(0.4)]

The model presented in this paper represents an analytic statement of the problem of strategic control. Two models are required to state the corporate development problem over the development horizon because of the differences in the informational environment in the near term and the long term. The key to modelling the strategic control approach lies in being able to formally generate the corporate development win-

g can also be identified as a fuzzy set, in this case: g = {4.21 (0.65), 2.21 (0.49)I..

(4c)

Given our previous interpretation of a sales goal, (4c) says that if the top management of the firm wishes to have a reasonable chance to be in

Omega, Vol. I..'. ,%0. I

dow. The approach followed here, relying on fuzzy set theory', defined a development window in terms of a set of parameters that would serve as the input to the short-term development problem. This latter problem was defined in terms of well-known, existing models. While alternative models and solution processes, both for the long-term and short-term problems, can be adapted to address the requirements of strategic control, those presented here have the virtue of being both fairly robust and relatively straightforward.

REFERENCES 1. Bellman R and Zadeh L (1970) Decision-making in a fuzzy environment. Mgmt Sci. 17(4), BI41-BI64. 2. Ben-Israel A and Charnes A (1963) Contributions to the theory of generalized inverses. S I A M Jl 11(3).

5t

3. Charnes A and Collomb B 11972) Optimal economic stabilization polic',: Linear goal-interval programming models. Socio-Econ. Plann. Sci. 6. 4. Charnes A and Cooper WW 11959) Chance-constrained programming. M U m Sci. 6(1), 73-79. 5. Charnes A and Cooper WW (1961) 5Ianagement 3,Iodels and In&~strial Applications o1" Linear Programming. Vol. 2, Chapter VIII, John Wiley. New York. 6. [jiri Y (1965) Management Goals and Accounting ~or Control. North-Holland, Amsterdam. 7. Kickert WJM (1978) Fazzv Theories on Decisionmaking. Martinus Nijhoff. Leiden. 8. Sarrazin J and Ruefli TW (1981) Strategic control of corporate development under ambiguous circumstances. Mgmt Sci. 27(10), 1158-1170. 9. Zadeh L. (1965) Fuzzy sets. Inf Control 8. 10. Zimmerman HJ et al. (1977) Results of empirical studies in fuzzy set theory. Working paper, Int. Con. on Applied General Systems Research, Binghampton. New Jersey. Profi, ssor T W Ruefli, The [nstitute fi)r Constructive Capitalism, The University o/" Texas at Austin, 2815 San Gabriel, ,4ustin, TX 78705, USA.

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