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Integer goal programming model for the implementation of multiple corporate objectives
Integer Goal Programming Model for the Implementation of Multiple Corporate Objectives Arthur J. Keown,
Bernard
W. Taylor III
Virginia Polytechnic Institute and State University
The problem of long-range planning and investment project evaluation is complicated by the presence of multiple conflicting goals measured in incommensurable units, indivisibility of alternative projects, and the desire of management to consider mutually exclusive marketing-pricing strategies. The
model proposed in this study attempts to allow for these complexities through the use of integer goal programming. It is the hope of the authors that this model will provide management with an additional decision-making tool for implementation of multiple corporate objectives.
As a firm attempts to evaluate alternative investment projects related to the development of new products or the expansion of existing products, it is frequently faced with multiple, conflicting objectives. However, the appropriate method for bridging the gap between recognition of these conflicting objectives and reflecting them in the decision making/planning process is often unclear. This article proposes an integer goal programming model as a planning tool for the incorporation of multiple, conflicting corporate objectives and strategies within a product expansion and investment framework. The advantage of this approach is that it allows for a multidimensional objective function with goals measured in incommensurable units to be solved in both a sequential and/or simultaneous manner. The model thus provides a framework for management to include their own individual goals and decision environment in the planning function. As a basis for the formulation of this model, a framework combining corporate objectives and strategies developed by Ramsey [ 131 is employed. MODEL METHODOLOGY The existence of multiple, conflicting goals in corporate decision making has been given considerable attention in both behavioral Address correspondence to: Arthur J. Keown, Department of Business Administration, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061. JOURNAL
OF BUSINESS
0 Elsevier North-Holland,
RESEARCH Inc., 1978
6 (1978)
221-235 0148-2963/78/0006-0221$01.75
221
222
Arthur J. Keown and Bernard W. Taylor III
and quantitative decision-making settings. In the behavioral area, Ansoff [ 11, Hill and Granger [ 61, Paine and Naumes [ 121, and Ramsey [ 131 all acknowledge the existence of multiple corporate goals and strategies. In the quantitative decision-making area, Cooper [3], Encarcion [ 51, Lee [7], Lintner [9], Osteryoung [ 101, Shubik [ 141, Solomon [ 16, Chapter 11 I, and Weston [ 18, p. 841 have expressed the view that the firm does in fact have multiple goals that are to be satisfied in a lexicographic manner. Osteryoung [ 10, p. 2061, in citing his own survey of the capital budgeting decision process of the Fortune “500” companies [ 1 I], states, “If 95 percent of these firms considered more than a single goal, it would seem not only relevant but important to incorporate a multiple goal criteria in the asset selection decision model.” The existence of multiple goals rather than the single classical goal of profit maximization when evaluating projects related to product development occurs for several reasons. First, because of uncertainty and subsequent estimation problems, the determination of long-term profit becomes an unmanageable task. For this reason, various surrogates such as sales, market share, and market growth rate of the products are used in lieu of long-term profit. In addition, the separation of ownership and management in modem business has led to the incorporation of alternative goals, reinforcing the security of the management’s position into the firm’s objective function. This separation has also helped produce a satisficing rather than maximizing behavior on the part of some firms [ 151. To provide for multiple and conflicting goals in a manner congruent with empirical observation, Charnes and Cooper [21 first proposed the use of goal programming. This methodology avoided the traditional problems of translation of incommensurable goal measurements into common profit or utility measurements. This was accomplished through the introduction of slack variables, which were then minimized from the desired goal levels. Not only did this approach yield itself well to the problem of incommensurable goals, but it also allowed the goal deviations to be ordinally ranked and/or scalar weighted. This enabled goals to be treated in a sequential and/or simultaneous manner, thus avoiding the problem of infeasible solutions. In the corporate decision-making framework, any attempt to convert goals into common measurement units would be futile. Ramsey, for example, cited corporate goals of market share by segment, actual dollars overall, overall profit growth rate, and
Interger Goal Programming of Multiple Objectives
223
number of market segments. An attempt to convert these goals into a common measurement would involve the arbitrary assignment of weights and tradeoffs to the goals and thus produce a meaningless objective function. For this reason, conversion of these goals into a unidimensional objective function becomes an unworkable solution leading to a goal programming formulation. Moreover, goal programming allows for goals to be treated in a lexicographic manner, which appears to be consistent with empirical observation. The goal programming approach has been modified somewhat in order to allow for indivisibility of inputs [8]. As pointed out by Danzig [4] and Weingartner [ 171, in the case of indivisibility of projects, the use of noninteger programming will many times lead to suboptimal results. For this case, the model is formulated using zero-one integer goal programming, which forces all projects to be completely accepted or rejected. In the product planning/ capital budgeting area, this obviously is a necessity, as fractional portions of capital projects are infeasible. Moreover, the introduction of integer programming allows for the incorporation of mutually exclusive and contingent projects in addition to allowing for the consideration of multiple marketing-pricing strategies for each investment proposal. MODEL FORMULATION A general model based on the Ramsey framework will first be presented, followed by a hypothetical example. The purpose of this model is not to provide the only possible goal formulation but to provide a general framework that is flexible enough to reflect multiple and conflicting corporate goals. The Ramsey framework identifies five areas in which corporate performance objectives can be identified: (1) product market identification, (2) economic concerns, (3) desired growth, (4) stability considerations, and (5) public image considerations. Within each of these areas various attributes are identified by Ramsey as well as examples of measurement yardsticks and possible goal values. The goal programming model presented here will include the same five general areas proposed by Ramsey but will not include all possible attributes within each area (for the sake of brevity).i However, the reader should be aware that the 1 While the five general areas of Ramsey’s framework are followed closely in the model formulation, the attributes are often altered, especially in the recommended means of quantification.
Arthur J. Keown and Bernard W. Taylor III
224
model can be expanded or contracted to reflect individual decision environments. Model Variables The following variables will be employed in the goal programming model: Xi = investment project i (i = 1, 2, .**, N); xi = 0 if project i is rejected; Xi = 1 if project i is accepted. Since a firm often desires to evaluate the various projects it is considering under different marketing-pricing strategies, the variables can further be defined as: Xij = investment project i under strategy j (i = 1, 2, .**,N) and Cj = 1, 2, ***,M).
MODEL GOAL CONSTRAINTS Strict Constraints The indivisibility requirements for project proposals require that each alternative take on a zero or one value. In addition, any project which is by definition mutually exclusive or contingent would require the inclusion of a strict constraint representing this condition. For example, if projects X1 1, X2 1, and X3 1 were mutually exclusive, this could be expressed as
If the acceptance of project X, 1 were dependent on the acceptance of either project X, 1 or project X3 1, this could be written as
x
11
-x21 -x31
GO.0
(2)
Moreover, the inclusion of alternative marketing-pricing strategies for each project would result in the following set of equations:
x,,
+x22
+“-+X,j<
1.0 (3)
Xi,
+ Xi, + ‘. ‘+ Xij ~ 1.0
Corporate Objective Constraints The remaining the goal programming model reflects the corporate specified in the Ramsey model.
formulation of objective areas
Interger
Goal Programming
of Multiple
Objectives
22s
A. Product Market Identification I. Market share by segment This goal constraint reflects management’s desire to achieve control of a desired percentage of a product market segment from the acceptance of a project (i.e., a new product or product expansion) :
izzl
PMijAXij ~ DPMA
(A = 1, 2, .. ..A*)
(4)
j=l
where PMijA =
2.
the percentage of product market A associated with the acceptance of project i under marketing-pricing strategy j; DPMA = the desired total market percentage of market segment A. Product introduction position Management’s desire to maintain their position as an innovative leader in their field may necessitate the acceptance of at least two of the following projects: X1 1, X3 1, X4 1, X6 1, X7 r . This objective is formulated as
C1
Xi1 22
(5)
i=
i+2.5
B.
Economic Concerns 1. Actual dollars overall This goal constraint reflects management’s desire to achieve an actual increase in accounting profit over a specified time period resulting from the acceptance of new projects. This goal may be the result of management’s desire to provide a pattern of earnings growth sufficient to satisfy current shareholders and maintain future control. 2 izl
2
APtjTXij Z DAPT
(T = 1, 2, . . . . T”)
(6)
j=l
where APijT =
the accounting profit in period T resulting from the acceptance of project i under marketingpricing strategy j;
Arthur J. Keown and Bernard W. Taylor IZZ
226
DAPT = 2.
the desired increase in accounting profit period T from the acceptance of new projects.
in
Net present value
Provided all management goals and objectives can be attained, the firm may wish to maximize net present value of those projects accepted.
i=l j=l
NPVijXij ~ K
(7)
where NPVi =
the net present value associated
with project
i;
K = a large unattainable number. C Desired Growth Overall sales growth rate This constraint I.
reflects management’s desire to achieve specified levels of increases in sales by dollar volume over their current sales volume.
SVijTXij > DSVT i=l
(T= 1, 2, ..e, T”)
(8)
j=l
where SVijT =
DSV= =
the increase in sales by dollar volume in time period T above the current level associated with the acceptance of profit i under marketing-pricing strategy j; the desired increase in sales over the current level by dollar volume in period T from the acceptance of new projects.
D. Stability Considerations I. Degree of reliability on general economy
In order to reduce the variability of sales due to swings in the general economy, management may wish to accept at least two projects that are considered to have inelastic demand. If projects X,i , X6 r , X7 i , Xs i , and X a i are considered to be inelastic with respect to demand, this goal can be written as
~ Xii ~ 2.0 i= 5
(9)
Interger
2.
Goal Programming
of Multiple
Objectives
221
Limitation of risky ventures Management may wish to limit either the number of or dollar amount invested in risky projects in an attempt to keep the overall risk of the firm at an acceptable level. Management may state that of the projects it accepts, some maximum number can be classified as risky. N
M
R ijXij ~ MNP
(10)
where Rij =
MNP = 3.
risky venture index given a value of 1 if Xii is classified as a risky project and 0 if Xij is classified as a safe investment; the desired maximum number of new risky projects to be accepted.
Resource limitations Resource limitations in the form of budget constraints or manpower limitations can be easily incorporated into the model. While capital rationing is not theoretically justifiable, its existence is commonplace; hence, it has been included in this model formulation. For example, a capital rationing situation could be expressed as ~ i=l
~
CijXij ~ BRL
(11)
j=l
where Cij =
BRL = 4.
the cash outflow associated with project i under marketing-pricing strategy j. For the sake of simplicity, all cash flows are assumed to take place initially; budget resource limit.
Management depth This goal constraint reflects the degree to which management must go outside its own firm for management resources for each of the new projects. ~ MDijXij f DMD i=
1
(12)
Arthur J. Keown and Bernard W. Taylor 111
228
where MD,, =
the number of new management personnel that are estimated to be hired if project Xii is accepted; DMD = the maximum desired number of new management personnel to be hired for all accepted projects. E. Public Image Considerations 1. Public service image desired This objective reflects management’s desire to maintain a good public image. As such, management may state that of the projects it accepts, some maximum number can be classified as bad public image risks. ~ i=l
~ j=l
PSijXij
~ DNS
(13)
where PSij =
DNS =
public image index given a value of 1 if Xij is considered a bad public image risk and 0 if Xij is considered a good public image risk; the desired maximum number of new projects classified as bad public image risks that can be accepted.
DEMONSTRATING
THE MODEL
Within the set of corporate objectives defined above, the firm will attempt to minimize the deviations from the desired goal levels, which are ordinally ranked and/or cardinally weighted. This allows management to incorporate the important factors of their individual decision environment into the model. In order to demonstrate the establishment of priorities within an actual objective function and the implementation of the goal programming model, a hypothetical example is presented. This example involves the evaluation of 11 projects where projects Xii and Xsj are classified as highly price elastic, projects Xsj, X*j, and X~j are classified as moderately price elastic, and projects Xsj through Xr lj are classified as price inelastic. Three marketingpricing strategies are then evaluated for projects classified as highly
Interger
Goal Programming
of Multiple
Objectives
229
price elastic, two for projects classified as moderately price elastic, and one for price inelastic projects. It should be noted that the corporate performance objectives and goal structure illustrated here are just one of many possible alternative structures. The appropriate goal structure and objectives would of course be dependent on management’s particular utility function. The relevant input information concerning the alternative investment proposals is given in Table 1. The indivisibility requirement on all investment proposals is implicitly taken care of by the zero-one goal programming algorithm. The hypothetical corporation’s performance objectives in order of importance are assumed to be as follows: P, : The first corpoate objective is to allow for the acceptance of a maximum of one marketing-pricing strategy for each investment alternative. In addition, the mutual exclusiveness of projects Xdj and Xrej and the dependence of project Xdij on the prior acceptance of project Xsj are included as strict constraints. P, : The budget ceiling on total expenditures is $600,000. Pa : The management depth goal limiting maximum number of new management personnel to be hired for all newly accepted projects is set at 15. P,: In order to stabilize the firm’s cyclical sales trend, at least two projects must be selected from the set of projects X,j, X,j, X,j, Xaj, and Xgj. In addition, management wishes to limit the number of projects classified as risky to three or less. The firm’s management is indifferent between the achievement of these two goals; thus, they are weighted equally at the same priority level. P, : In order to maintain their position as an innovative leader, they must accept at least two projects out of the set X,j, Xaj, Xdj, and X,j. In addition, management wishes to maintain a good public image and as such wishes to limit the number of projects classified as bad public image risks, Xaj, X,j, and Xloj, to two or less. Again, these goals are weighted equally at the same priority level. P, : The sixth priority goal is to provide 8.0% and 7.0% control of product market I and II, respectively, from the acceptance of new projects. P, : The seventh firm goal is to achieve profit of $70,000, $110,000, and $150,000 from the acceptance of new
Ci (dollars)
100,000 110,000 120,000 180,000 190,000 200,000 130,000 140,000 270,000 300,000 60,000 65,000 90,000 140,000 210,000 120,000 290,000 340,000
X11
x12 x13 x21 x22 x23 x31 x32 x41 x42 x51 x52 x61 x71 x81 x91 x10,1 x11,1
(X,)
Project
Table 1: Summary
2.0 2.4 2.8 0.0 0.0 0.0 1.7 2.0 5.5 6.8 0.0 0.0 1.9 0.0 4.8 0.0 6.8 0.0 3.8 4.3 4.8 0 0 0 0 1.0 1A 0 3.2 0 2.3 0 8.1
0
0 0
PM!'
(perkent)
PM{ 15,000 13,000 10,000 35,000 30,000 25,000 25,000 15,000 30,000 25,000 15,000 10,000 10,000 35,000 20,000 25,000 40,000 35,000
(dollars)
Apil
Projects
(percent)
of Investment Api
20,000 16,000 14,000 42,000 35,000 29,000 35,000 20,000 40,000 30,000 20,000 14,000 14,000 35,000 40,000 27,000 50,000 55,000
(dollars)
APi
25,000 18,000 16,000 50,000 40,000 33,000 45,000 25,000 50,000 35,000 25,000 18,000 20,000 35,000 60,000 30,000 60,000 80,000
(dollars)
SVil
30,000 35,000 50,000 60,000 70,000 85,000 40,000 70,000 80,000 100,000 20,000 30,000 30,000 30,000 65,000 50,000 90,000 130,000
(dollars)
SVi2
110,000 135,000 60,000 100,000 140,000 165,000 30,000 45,000 40,000 35,000 80,000 50,000 140,000 130,000
50,000 60,000 80,000 90,000
(dollars)
MDi
1 1 1 3 3 3 6 6 8 8 2 2 3 2 6 2 3 4
(people)
Ri
NPVi
110,000 70,000 25,000 20,000 30,000 60,000 80,000 50,000 150,000 180,000
40,000 32,000 25,000 70,000 55,000 40,000 50,000 35,000
(O-l index) (dollars)
%
2 f a g_ b 5 2
k 2 $ .; *
Interger
Goal Programming
of Multiple
Objectives
231
projects during each of the first three periods, respectively. P,: The eighth goal is to achieve a level of sales growth by dollar volume of $240,000 in period 1 and $350,000 in period 2 over current dollar sales volume. P, : The final goal is to maximize the net present value of the projects selected. RESULTS Given the input values listed in Table 1, the example problem is formulated in the Appendix. Of the 11 projects considered, the following solution was achieved: X 1,l = 1.0 x2,3
=
1.0
x,,, = 1.0 X8,1
=
1.0
All other Xii = 0 In this solution, the five first goal priorities were completely satisfied, while the final priorities P6 through P9 were not completely attained. The degree of goal attainment for those priorities was as follows: P, : While the desired goal of 8.0% control of product market I was completely attained, the goal of 7.0% control of product market II fell short by 2.2%. P, : While in the first period the desired profit goal of $70,000 from the acceptance of new products was exactly attained, the desired goals of $110,000 in period 2 and $150,000 in period 3 fell short by $27,000 and $17,000, respectively. P, : The level of sales growth goals in periods 1 and 2 were underachieved by $30,000 and $45,000, respectively. Pg : The net present value of the projects accepted was $190,000. In examining the final solution, it should be noted that the existence of more than one unsatisfied goal level demonstrates the fact that a linear programming formulation of this example in which the final goal was made the objective function and all other goals formulated as constraints would result in an infeasible solution. Moreover, the use of integer goal programming has allowed for the inclusion of mutually exclusive projects and multiple
232
Arthur
marketing-pricing tions.
strategies
J. Keown
in addition
and Bernard
to requiring
W. Taylor III
integer solu-
CONCLUSION The problem of investment project evaluation is complicated by the presence of multiple conflicting goals measured in incommensurable units, indivisibility of alternative projects, and the desire of management to consider mutually exclusive marketingpricing strategies. To ignore these complications would result in meaningless model building and arbitrary solutions. This study has been an attempt to provide planners with a model robust enough to operate within the framework of multiple, conflicting objectives and strategies such as those developed by Ramsey [ 131. It is hoped that this approach will provide planners with a useful tool in the pragmatically difficult task of decision making.
Appendix:
Integer Goal Programming
Model
A. Strict Constraints l.X,i 2.Xa1 3.X31 4.X,, 5.X,, 6. X,1 7.X*1
+x12 +xia +Xz2 +Xz3 +Xs2 +d,+Xd2 +d,+X5, +d,+X,, +XIO,l +x42 -X,1
B. Budget Expense
+d,+dz-da+ -d4+ -d,+ +d,+d,-
Goal
8. 100,000X11 + 110,OOOX~~ 19o,ooox2z + 200,000xza 270,000X41 + 300,000X42 90,000X6 1 + 140,000X, 1 290,000X, o, 1 + 340,000X1 C. Management 9.X11
- di+ = 1.0 = 1.0 = 1.0 = 1.0 = 1.0 -d,+ = 1.0 - d7+ = 1.0 -d2+
+ 12O,OOOX1a + 180,000Xz1 + 130,000xa 2 + 140,000xa 2 + 60,000X5 2 + 65,OOOX5a + 2 10,OOOXa i + 12o,ooox$l~ r ,
+
1 + d, - - d8 + = 600,000
Depth Goal
+xIz +x13 +3x,, +3X,, +3X,, +6X,1+6X,2 + 8Xd2 + 2X51 + 2X52 + 3X6,+ 2X7 1+ 6X,1 + 2x,, +3X,O,1 +4X,1,, +d,-d,+ = 15
8X,,
+ + +
+
Interger Goal Programming of Multiple Objectives
D. Limitation 10.Xr,r
of Degree of Reliability
+Xs2
E. Limitation
+X,,
+X,r
12.X,,
Innovation +Xrz
d IZ-
+Xar
+dlo-
Goal =2
-dlo+
Goal
11.x11 +x12 +x,, +x,r+x32 X 11,1 +d,,- dll+ = 3.0 F. Product
on General Economy
+X,,
of Risky Ventures
233
+x51
+X52
+X,1
+X91
+
+x61
+
Goal
+X13
+X31 +X,,
+X33
+x41
+X42.
- d12 + = 2.0
G. Public Service Image Goal 13.XB1 +Xs2
+Xsl
+X10,1
+d,a-
-dls+
=2.0
H. Market Share Goal 14. 2.0Xll 6.8X42 15. 3.8X2I 2.3X,r
+2.4X12 +2.8X1, + 1.9X6, + 4.8Xsr + 4.3X22 +4.8X23 + 8.1XIr,r +dls-
+ 1.7Xa1 +2.0X,, + 6.8Xle.r +dl,+ 1.0X~,r + 1.4X,z - d15+ = 7.0
+ 5.5X4, + - dr*+ = 8.0 + 3.2X7 r+
I. Profit Goal 16. 15,000XrI + 13,000X,2 + 1O,OOOXr3 + 35,000X21 30,OOOXaa + 25,000X,a + 25,OOOXar + 15,000x32 30,000X4 1 + 25,000X4 2 + 15,000x51 + 10,000x52 10,000X6 r + 35,000x, r + 20,000xs I+ 25,OOOXg 1 +d,,- d, 6 + = 70,000 40,000X, 0,l + 35,000Xrr,r 17. 20,000X, 1 + 16,000X1 2 + 14,000X1 3 + 42,000X,1 + 29,000X2 3 + 35,000XaI + 20,000X3 2 35,000X22 4o,ooox4~ + 3o,ooox4p, + 2o,ooox5~ + 14,OOOX5z 14,000xG 1 + 35,000X, r + 4O,OOOXsl+ 27,OOOXa I 5O,OOOX,o,l + 55,000Xll,l +dl,- dl,+ = 110,000 18. 25,000Xr1 + 18,000X12 + 16,000X,,+ 50,OOOXar 40,000X2 a + 33,000X2 a + 45,000X3 1 + 25,000X, s 50,000X,, + 35,OOOX42 + 25,000Xs1 + 18,OOOX5a 20,000X6 1 + 35,000X7 1 + 60,OOOXs 1 + 30,OOOXg z 60,000X1 o,l + 80,000X1 1 ,r + dl s- - d, a+ = 150,000
Arthur J. Keown and Bernard W. Taylor III
234
J. Overall Sales Growth
Goal
19. 30,000X1 1 + 35,000X12 + 50,000X13 + 6O,OOOXz1 70,000X2 2 + 85,000X2 3 + 40,000X3 I + 70,000X3 2 80,000X4 1 + 100,000X4 2 + 20,000X5 1 + 30,000X5 2 30,000Xs 1 + 30,000X, 1 + 65,000X, 1 + 50,000Xg 1 9O,OOOX,0,I + 13O,OOOXI1,1 +d19- d19+ = 240,000 20. 50,000X1 1 + 60,000X1 2 + 80,000X1 3 + 90,000X21 + 1 10,000Xz2 + 135,000X23 + 6O,OOOX31 + lOO,OOOX~,+ 140,000Xd1 + 165,000X42 + 30,000XS1 + 45,000X5,+ 40,000XG1 + 35,000X,1 + 80,OOOXs 1 + 50,OOOXg I+ 140,000xl~,l + 130,000x~~,1 +c&o- -c&()+ = 350,000 K. Maximize Net Present Value 21.
40,000X1 I 55,000x2 2 110,000X, 1 30,000X, 1 150,000X10,1
L. Objective
+ 32,000X1 2 + + 40,000x~ 3 + + 7O,OOOX42 + + 6O,OOOX,1 + + 18O,OOOX11,1
25,000X1 3 + 5o,ooox3~ + 25,000Xs1 + 80,000Xs 1 + +dzl- dzl+
7O,OOOXz1+ 35,000x32 + 2O,OOOX5z + 50,000X, 1 + = l,OOO,OOO
Function
Minimize Z = PI i
di+ + P,de + + Pa dg + + P4 (d, o-
+d,,+)+
i=l
P,(dlzd 17-
+d,s+)+P6(&4fdlg-1
+Ps(d,,-
+dls-l+P,(dls-+ +dzo-1
+Ps(d,,-1
References 1. 2.
Ansoff, H. I., Corporate Strategy, McGraw-Hill, New York, 1965. Charnes, A., and Cooper, W. W., Management Models and IndustrialApplications
of Linear Programming, Vols. 1 and 2, John Wiley & Sons, New York, 1961.
3.
Cooper, W. W., Theory of the Firm: Some Suggestions for Revision, Am. Econ. Rev. 34 (December, 1949): 1204-1222.
4.
Dantzig, G. B., On the Significance of Solving Linear Programming Problems with Some Integer Variables, Econometrica 28 (January, 1960): 30-44.
5.
Encarcion, J., Constraints and the Firm’s Utility Function, Rev. Econ. Studies 2 (April, 1964): 113-121.
6.
Hill, W. E., and &anger, C. H., Management Objectives and Bases for Evaluation, in Handbook for Industtil Research Management, C. Heyel, ed., Reinhold, New York. 1968.
lnterger Goal Programming of Multiple Objectives
235
1.
Lee, S. M., Goal Programming for Decision Analysis, Auerbach, Philadelphia, 1972.
8.
Lee, S. M., and Morris, R., Integer Goal Programming, Research Report, Department of Business Administration, Virginia Polytechnic Institute and State University, August 1975.
9.
Lintner, J., The Cost of Capital and Optimal Financing of Corporate Growth, J..Finance 18 (May, 1963): 292-310.
10.
Osteryoung, 3. S., Capital Budgeting: Long Term Asset Selection, Grid, Inc., Columbus, Ohio, 1974.
11.
Osteryoung, J. S., Multiple Goals in the Capital Budgeting Decision, in MuZtipZe Criterion Decision Making, James L. Cochrane and Milan Zehrey, eds., The University of South Carolina Press, Columbia, 1973, pp. 447-457.
12.
Paine, F. T., and Naumes, W., Strategy and Policy Formulation, W. B. Saunders Co., Philadelphia, 1974.
13.
Ramsey, J. E., A Framework for the Interaction of Corporate Value Objectives, Corporate Performance Objectives, and Corporate Strategy, J. Econ. Bus. 28 (Spring-Summer, 1976): 171-180.
14.
Shubik, M., Objective Functions and Models of Corporate Optimization, Econ. 75 (August, 1961): 345-375.
15.
Simon, H., A Behavioral Mode of Rational Choice, Q. J. Econ. 59 (February, 1955): 99-118.
16.
Solomon, E., The Theory of Financial Management, Columbia University Press, New York, 1963.
17.
Weingartner,
Q. Rev.
H. M., Mathematical fiogramming and the Analysis of Capital
BudgetingProblems,Prentice-Hall, Englewood Cliffs, N.J., 1963. 18.
Weston, J., The Scope and Methodology of Finance, Prentice-Hall, wood Cliffs, N.J., 1966.
Engle-
Bernard
W. Taylor III
Virginia Polytechnic Institute and State University
The problem of long-range planning and investment project evaluation is complicated by the presence of multiple conflicting goals measured in incommensurable units, indivisibility of alternative projects, and the desire of management to consider mutually exclusive marketing-pricing strategies. The
model proposed in this study attempts to allow for these complexities through the use of integer goal programming. It is the hope of the authors that this model will provide management with an additional decision-making tool for implementation of multiple corporate objectives.
As a firm attempts to evaluate alternative investment projects related to the development of new products or the expansion of existing products, it is frequently faced with multiple, conflicting objectives. However, the appropriate method for bridging the gap between recognition of these conflicting objectives and reflecting them in the decision making/planning process is often unclear. This article proposes an integer goal programming model as a planning tool for the incorporation of multiple, conflicting corporate objectives and strategies within a product expansion and investment framework. The advantage of this approach is that it allows for a multidimensional objective function with goals measured in incommensurable units to be solved in both a sequential and/or simultaneous manner. The model thus provides a framework for management to include their own individual goals and decision environment in the planning function. As a basis for the formulation of this model, a framework combining corporate objectives and strategies developed by Ramsey [ 131 is employed. MODEL METHODOLOGY The existence of multiple, conflicting goals in corporate decision making has been given considerable attention in both behavioral Address correspondence to: Arthur J. Keown, Department of Business Administration, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061. JOURNAL
OF BUSINESS
0 Elsevier North-Holland,
RESEARCH Inc., 1978
6 (1978)
221-235 0148-2963/78/0006-0221$01.75
221
222
Arthur J. Keown and Bernard W. Taylor III
and quantitative decision-making settings. In the behavioral area, Ansoff [ 11, Hill and Granger [ 61, Paine and Naumes [ 121, and Ramsey [ 131 all acknowledge the existence of multiple corporate goals and strategies. In the quantitative decision-making area, Cooper [3], Encarcion [ 51, Lee [7], Lintner [9], Osteryoung [ 101, Shubik [ 141, Solomon [ 16, Chapter 11 I, and Weston [ 18, p. 841 have expressed the view that the firm does in fact have multiple goals that are to be satisfied in a lexicographic manner. Osteryoung [ 10, p. 2061, in citing his own survey of the capital budgeting decision process of the Fortune “500” companies [ 1 I], states, “If 95 percent of these firms considered more than a single goal, it would seem not only relevant but important to incorporate a multiple goal criteria in the asset selection decision model.” The existence of multiple goals rather than the single classical goal of profit maximization when evaluating projects related to product development occurs for several reasons. First, because of uncertainty and subsequent estimation problems, the determination of long-term profit becomes an unmanageable task. For this reason, various surrogates such as sales, market share, and market growth rate of the products are used in lieu of long-term profit. In addition, the separation of ownership and management in modem business has led to the incorporation of alternative goals, reinforcing the security of the management’s position into the firm’s objective function. This separation has also helped produce a satisficing rather than maximizing behavior on the part of some firms [ 151. To provide for multiple and conflicting goals in a manner congruent with empirical observation, Charnes and Cooper [21 first proposed the use of goal programming. This methodology avoided the traditional problems of translation of incommensurable goal measurements into common profit or utility measurements. This was accomplished through the introduction of slack variables, which were then minimized from the desired goal levels. Not only did this approach yield itself well to the problem of incommensurable goals, but it also allowed the goal deviations to be ordinally ranked and/or scalar weighted. This enabled goals to be treated in a sequential and/or simultaneous manner, thus avoiding the problem of infeasible solutions. In the corporate decision-making framework, any attempt to convert goals into common measurement units would be futile. Ramsey, for example, cited corporate goals of market share by segment, actual dollars overall, overall profit growth rate, and
Interger Goal Programming of Multiple Objectives
223
number of market segments. An attempt to convert these goals into a common measurement would involve the arbitrary assignment of weights and tradeoffs to the goals and thus produce a meaningless objective function. For this reason, conversion of these goals into a unidimensional objective function becomes an unworkable solution leading to a goal programming formulation. Moreover, goal programming allows for goals to be treated in a lexicographic manner, which appears to be consistent with empirical observation. The goal programming approach has been modified somewhat in order to allow for indivisibility of inputs [8]. As pointed out by Danzig [4] and Weingartner [ 171, in the case of indivisibility of projects, the use of noninteger programming will many times lead to suboptimal results. For this case, the model is formulated using zero-one integer goal programming, which forces all projects to be completely accepted or rejected. In the product planning/ capital budgeting area, this obviously is a necessity, as fractional portions of capital projects are infeasible. Moreover, the introduction of integer programming allows for the incorporation of mutually exclusive and contingent projects in addition to allowing for the consideration of multiple marketing-pricing strategies for each investment proposal. MODEL FORMULATION A general model based on the Ramsey framework will first be presented, followed by a hypothetical example. The purpose of this model is not to provide the only possible goal formulation but to provide a general framework that is flexible enough to reflect multiple and conflicting corporate goals. The Ramsey framework identifies five areas in which corporate performance objectives can be identified: (1) product market identification, (2) economic concerns, (3) desired growth, (4) stability considerations, and (5) public image considerations. Within each of these areas various attributes are identified by Ramsey as well as examples of measurement yardsticks and possible goal values. The goal programming model presented here will include the same five general areas proposed by Ramsey but will not include all possible attributes within each area (for the sake of brevity).i However, the reader should be aware that the 1 While the five general areas of Ramsey’s framework are followed closely in the model formulation, the attributes are often altered, especially in the recommended means of quantification.
Arthur J. Keown and Bernard W. Taylor III
224
model can be expanded or contracted to reflect individual decision environments. Model Variables The following variables will be employed in the goal programming model: Xi = investment project i (i = 1, 2, .**, N); xi = 0 if project i is rejected; Xi = 1 if project i is accepted. Since a firm often desires to evaluate the various projects it is considering under different marketing-pricing strategies, the variables can further be defined as: Xij = investment project i under strategy j (i = 1, 2, .**,N) and Cj = 1, 2, ***,M).
MODEL GOAL CONSTRAINTS Strict Constraints The indivisibility requirements for project proposals require that each alternative take on a zero or one value. In addition, any project which is by definition mutually exclusive or contingent would require the inclusion of a strict constraint representing this condition. For example, if projects X1 1, X2 1, and X3 1 were mutually exclusive, this could be expressed as
If the acceptance of project X, 1 were dependent on the acceptance of either project X, 1 or project X3 1, this could be written as
x
11
-x21 -x31
GO.0
(2)
Moreover, the inclusion of alternative marketing-pricing strategies for each project would result in the following set of equations:
x,,
+x22
+“-+X,j<
1.0 (3)
Xi,
+ Xi, + ‘. ‘+ Xij ~ 1.0
Corporate Objective Constraints The remaining the goal programming model reflects the corporate specified in the Ramsey model.
formulation of objective areas
Interger
Goal Programming
of Multiple
Objectives
22s
A. Product Market Identification I. Market share by segment This goal constraint reflects management’s desire to achieve control of a desired percentage of a product market segment from the acceptance of a project (i.e., a new product or product expansion) :
izzl
PMijAXij ~ DPMA
(A = 1, 2, .. ..A*)
(4)
j=l
where PMijA =
2.
the percentage of product market A associated with the acceptance of project i under marketing-pricing strategy j; DPMA = the desired total market percentage of market segment A. Product introduction position Management’s desire to maintain their position as an innovative leader in their field may necessitate the acceptance of at least two of the following projects: X1 1, X3 1, X4 1, X6 1, X7 r . This objective is formulated as
C1
Xi1 22
(5)
i=
i+2.5
B.
Economic Concerns 1. Actual dollars overall This goal constraint reflects management’s desire to achieve an actual increase in accounting profit over a specified time period resulting from the acceptance of new projects. This goal may be the result of management’s desire to provide a pattern of earnings growth sufficient to satisfy current shareholders and maintain future control. 2 izl
2
APtjTXij Z DAPT
(T = 1, 2, . . . . T”)
(6)
j=l
where APijT =
the accounting profit in period T resulting from the acceptance of project i under marketingpricing strategy j;
Arthur J. Keown and Bernard W. Taylor IZZ
226
DAPT = 2.
the desired increase in accounting profit period T from the acceptance of new projects.
in
Net present value
Provided all management goals and objectives can be attained, the firm may wish to maximize net present value of those projects accepted.
i=l j=l
NPVijXij ~ K
(7)
where NPVi =
the net present value associated
with project
i;
K = a large unattainable number. C Desired Growth Overall sales growth rate This constraint I.
reflects management’s desire to achieve specified levels of increases in sales by dollar volume over their current sales volume.
SVijTXij > DSVT i=l
(T= 1, 2, ..e, T”)
(8)
j=l
where SVijT =
DSV= =
the increase in sales by dollar volume in time period T above the current level associated with the acceptance of profit i under marketing-pricing strategy j; the desired increase in sales over the current level by dollar volume in period T from the acceptance of new projects.
D. Stability Considerations I. Degree of reliability on general economy
In order to reduce the variability of sales due to swings in the general economy, management may wish to accept at least two projects that are considered to have inelastic demand. If projects X,i , X6 r , X7 i , Xs i , and X a i are considered to be inelastic with respect to demand, this goal can be written as
~ Xii ~ 2.0 i= 5
(9)
Interger
2.
Goal Programming
of Multiple
Objectives
221
Limitation of risky ventures Management may wish to limit either the number of or dollar amount invested in risky projects in an attempt to keep the overall risk of the firm at an acceptable level. Management may state that of the projects it accepts, some maximum number can be classified as risky. N
M
R ijXij ~ MNP
(10)
where Rij =
MNP = 3.
risky venture index given a value of 1 if Xii is classified as a risky project and 0 if Xij is classified as a safe investment; the desired maximum number of new risky projects to be accepted.
Resource limitations Resource limitations in the form of budget constraints or manpower limitations can be easily incorporated into the model. While capital rationing is not theoretically justifiable, its existence is commonplace; hence, it has been included in this model formulation. For example, a capital rationing situation could be expressed as ~ i=l
~
CijXij ~ BRL
(11)
j=l
where Cij =
BRL = 4.
the cash outflow associated with project i under marketing-pricing strategy j. For the sake of simplicity, all cash flows are assumed to take place initially; budget resource limit.
Management depth This goal constraint reflects the degree to which management must go outside its own firm for management resources for each of the new projects. ~ MDijXij f DMD i=
1
(12)
Arthur J. Keown and Bernard W. Taylor 111
228
where MD,, =
the number of new management personnel that are estimated to be hired if project Xii is accepted; DMD = the maximum desired number of new management personnel to be hired for all accepted projects. E. Public Image Considerations 1. Public service image desired This objective reflects management’s desire to maintain a good public image. As such, management may state that of the projects it accepts, some maximum number can be classified as bad public image risks. ~ i=l
~ j=l
PSijXij
~ DNS
(13)
where PSij =
DNS =
public image index given a value of 1 if Xij is considered a bad public image risk and 0 if Xij is considered a good public image risk; the desired maximum number of new projects classified as bad public image risks that can be accepted.
DEMONSTRATING
THE MODEL
Within the set of corporate objectives defined above, the firm will attempt to minimize the deviations from the desired goal levels, which are ordinally ranked and/or cardinally weighted. This allows management to incorporate the important factors of their individual decision environment into the model. In order to demonstrate the establishment of priorities within an actual objective function and the implementation of the goal programming model, a hypothetical example is presented. This example involves the evaluation of 11 projects where projects Xii and Xsj are classified as highly price elastic, projects Xsj, X*j, and X~j are classified as moderately price elastic, and projects Xsj through Xr lj are classified as price inelastic. Three marketingpricing strategies are then evaluated for projects classified as highly
Interger
Goal Programming
of Multiple
Objectives
229
price elastic, two for projects classified as moderately price elastic, and one for price inelastic projects. It should be noted that the corporate performance objectives and goal structure illustrated here are just one of many possible alternative structures. The appropriate goal structure and objectives would of course be dependent on management’s particular utility function. The relevant input information concerning the alternative investment proposals is given in Table 1. The indivisibility requirement on all investment proposals is implicitly taken care of by the zero-one goal programming algorithm. The hypothetical corporation’s performance objectives in order of importance are assumed to be as follows: P, : The first corpoate objective is to allow for the acceptance of a maximum of one marketing-pricing strategy for each investment alternative. In addition, the mutual exclusiveness of projects Xdj and Xrej and the dependence of project Xdij on the prior acceptance of project Xsj are included as strict constraints. P, : The budget ceiling on total expenditures is $600,000. Pa : The management depth goal limiting maximum number of new management personnel to be hired for all newly accepted projects is set at 15. P,: In order to stabilize the firm’s cyclical sales trend, at least two projects must be selected from the set of projects X,j, X,j, X,j, Xaj, and Xgj. In addition, management wishes to limit the number of projects classified as risky to three or less. The firm’s management is indifferent between the achievement of these two goals; thus, they are weighted equally at the same priority level. P, : In order to maintain their position as an innovative leader, they must accept at least two projects out of the set X,j, Xaj, Xdj, and X,j. In addition, management wishes to maintain a good public image and as such wishes to limit the number of projects classified as bad public image risks, Xaj, X,j, and Xloj, to two or less. Again, these goals are weighted equally at the same priority level. P, : The sixth priority goal is to provide 8.0% and 7.0% control of product market I and II, respectively, from the acceptance of new projects. P, : The seventh firm goal is to achieve profit of $70,000, $110,000, and $150,000 from the acceptance of new
Ci (dollars)
100,000 110,000 120,000 180,000 190,000 200,000 130,000 140,000 270,000 300,000 60,000 65,000 90,000 140,000 210,000 120,000 290,000 340,000
X11
x12 x13 x21 x22 x23 x31 x32 x41 x42 x51 x52 x61 x71 x81 x91 x10,1 x11,1
(X,)
Project
Table 1: Summary
2.0 2.4 2.8 0.0 0.0 0.0 1.7 2.0 5.5 6.8 0.0 0.0 1.9 0.0 4.8 0.0 6.8 0.0 3.8 4.3 4.8 0 0 0 0 1.0 1A 0 3.2 0 2.3 0 8.1
0
0 0
PM!'
(perkent)
PM{ 15,000 13,000 10,000 35,000 30,000 25,000 25,000 15,000 30,000 25,000 15,000 10,000 10,000 35,000 20,000 25,000 40,000 35,000
(dollars)
Apil
Projects
(percent)
of Investment Api
20,000 16,000 14,000 42,000 35,000 29,000 35,000 20,000 40,000 30,000 20,000 14,000 14,000 35,000 40,000 27,000 50,000 55,000
(dollars)
APi
25,000 18,000 16,000 50,000 40,000 33,000 45,000 25,000 50,000 35,000 25,000 18,000 20,000 35,000 60,000 30,000 60,000 80,000
(dollars)
SVil
30,000 35,000 50,000 60,000 70,000 85,000 40,000 70,000 80,000 100,000 20,000 30,000 30,000 30,000 65,000 50,000 90,000 130,000
(dollars)
SVi2
110,000 135,000 60,000 100,000 140,000 165,000 30,000 45,000 40,000 35,000 80,000 50,000 140,000 130,000
50,000 60,000 80,000 90,000
(dollars)
MDi
1 1 1 3 3 3 6 6 8 8 2 2 3 2 6 2 3 4
(people)
Ri
NPVi
110,000 70,000 25,000 20,000 30,000 60,000 80,000 50,000 150,000 180,000
40,000 32,000 25,000 70,000 55,000 40,000 50,000 35,000
(O-l index) (dollars)
%
2 f a g_ b 5 2
k 2 $ .; *
Interger
Goal Programming
of Multiple
Objectives
231
projects during each of the first three periods, respectively. P,: The eighth goal is to achieve a level of sales growth by dollar volume of $240,000 in period 1 and $350,000 in period 2 over current dollar sales volume. P, : The final goal is to maximize the net present value of the projects selected. RESULTS Given the input values listed in Table 1, the example problem is formulated in the Appendix. Of the 11 projects considered, the following solution was achieved: X 1,l = 1.0 x2,3
=
1.0
x,,, = 1.0 X8,1
=
1.0
All other Xii = 0 In this solution, the five first goal priorities were completely satisfied, while the final priorities P6 through P9 were not completely attained. The degree of goal attainment for those priorities was as follows: P, : While the desired goal of 8.0% control of product market I was completely attained, the goal of 7.0% control of product market II fell short by 2.2%. P, : While in the first period the desired profit goal of $70,000 from the acceptance of new products was exactly attained, the desired goals of $110,000 in period 2 and $150,000 in period 3 fell short by $27,000 and $17,000, respectively. P, : The level of sales growth goals in periods 1 and 2 were underachieved by $30,000 and $45,000, respectively. Pg : The net present value of the projects accepted was $190,000. In examining the final solution, it should be noted that the existence of more than one unsatisfied goal level demonstrates the fact that a linear programming formulation of this example in which the final goal was made the objective function and all other goals formulated as constraints would result in an infeasible solution. Moreover, the use of integer goal programming has allowed for the inclusion of mutually exclusive projects and multiple
232
Arthur
marketing-pricing tions.
strategies
J. Keown
in addition
and Bernard
to requiring
W. Taylor III
integer solu-
CONCLUSION The problem of investment project evaluation is complicated by the presence of multiple conflicting goals measured in incommensurable units, indivisibility of alternative projects, and the desire of management to consider mutually exclusive marketingpricing strategies. To ignore these complications would result in meaningless model building and arbitrary solutions. This study has been an attempt to provide planners with a model robust enough to operate within the framework of multiple, conflicting objectives and strategies such as those developed by Ramsey [ 131. It is hoped that this approach will provide planners with a useful tool in the pragmatically difficult task of decision making.
Appendix:
Integer Goal Programming
Model
A. Strict Constraints l.X,i 2.Xa1 3.X31 4.X,, 5.X,, 6. X,1 7.X*1
+x12 +xia +Xz2 +Xz3 +Xs2 +d,+Xd2 +d,+X5, +d,+X,, +XIO,l +x42 -X,1
B. Budget Expense
+d,+dz-da+ -d4+ -d,+ +d,+d,-
Goal
8. 100,000X11 + 110,OOOX~~ 19o,ooox2z + 200,000xza 270,000X41 + 300,000X42 90,000X6 1 + 140,000X, 1 290,000X, o, 1 + 340,000X1 C. Management 9.X11
- di+ = 1.0 = 1.0 = 1.0 = 1.0 = 1.0 -d,+ = 1.0 - d7+ = 1.0 -d2+
+ 12O,OOOX1a + 180,000Xz1 + 130,000xa 2 + 140,000xa 2 + 60,000X5 2 + 65,OOOX5a + 2 10,OOOXa i + 12o,ooox$l~ r ,
+
1 + d, - - d8 + = 600,000
Depth Goal
+xIz +x13 +3x,, +3X,, +3X,, +6X,1+6X,2 + 8Xd2 + 2X51 + 2X52 + 3X6,+ 2X7 1+ 6X,1 + 2x,, +3X,O,1 +4X,1,, +d,-d,+ = 15
8X,,
+ + +
+
Interger Goal Programming of Multiple Objectives
D. Limitation 10.Xr,r
of Degree of Reliability
+Xs2
E. Limitation
+X,,
+X,r
12.X,,
Innovation +Xrz
d IZ-
+Xar
+dlo-
Goal =2
-dlo+
Goal
11.x11 +x12 +x,, +x,r+x32 X 11,1 +d,,- dll+ = 3.0 F. Product
on General Economy
+X,,
of Risky Ventures
233
+x51
+X52
+X,1
+X91
+
+x61
+
Goal
+X13
+X31 +X,,
+X33
+x41
+X42.
- d12 + = 2.0
G. Public Service Image Goal 13.XB1 +Xs2
+Xsl
+X10,1
+d,a-
-dls+
=2.0
H. Market Share Goal 14. 2.0Xll 6.8X42 15. 3.8X2I 2.3X,r
+2.4X12 +2.8X1, + 1.9X6, + 4.8Xsr + 4.3X22 +4.8X23 + 8.1XIr,r +dls-
+ 1.7Xa1 +2.0X,, + 6.8Xle.r +dl,+ 1.0X~,r + 1.4X,z - d15+ = 7.0
+ 5.5X4, + - dr*+ = 8.0 + 3.2X7 r+
I. Profit Goal 16. 15,000XrI + 13,000X,2 + 1O,OOOXr3 + 35,000X21 30,OOOXaa + 25,000X,a + 25,OOOXar + 15,000x32 30,000X4 1 + 25,000X4 2 + 15,000x51 + 10,000x52 10,000X6 r + 35,000x, r + 20,000xs I+ 25,OOOXg 1 +d,,- d, 6 + = 70,000 40,000X, 0,l + 35,000Xrr,r 17. 20,000X, 1 + 16,000X1 2 + 14,000X1 3 + 42,000X,1 + 29,000X2 3 + 35,000XaI + 20,000X3 2 35,000X22 4o,ooox4~ + 3o,ooox4p, + 2o,ooox5~ + 14,OOOX5z 14,000xG 1 + 35,000X, r + 4O,OOOXsl+ 27,OOOXa I 5O,OOOX,o,l + 55,000Xll,l +dl,- dl,+ = 110,000 18. 25,000Xr1 + 18,000X12 + 16,000X,,+ 50,OOOXar 40,000X2 a + 33,000X2 a + 45,000X3 1 + 25,000X, s 50,000X,, + 35,OOOX42 + 25,000Xs1 + 18,OOOX5a 20,000X6 1 + 35,000X7 1 + 60,OOOXs 1 + 30,OOOXg z 60,000X1 o,l + 80,000X1 1 ,r + dl s- - d, a+ = 150,000
Arthur J. Keown and Bernard W. Taylor III
234
J. Overall Sales Growth
Goal
19. 30,000X1 1 + 35,000X12 + 50,000X13 + 6O,OOOXz1 70,000X2 2 + 85,000X2 3 + 40,000X3 I + 70,000X3 2 80,000X4 1 + 100,000X4 2 + 20,000X5 1 + 30,000X5 2 30,000Xs 1 + 30,000X, 1 + 65,000X, 1 + 50,000Xg 1 9O,OOOX,0,I + 13O,OOOXI1,1 +d19- d19+ = 240,000 20. 50,000X1 1 + 60,000X1 2 + 80,000X1 3 + 90,000X21 + 1 10,000Xz2 + 135,000X23 + 6O,OOOX31 + lOO,OOOX~,+ 140,000Xd1 + 165,000X42 + 30,000XS1 + 45,000X5,+ 40,000XG1 + 35,000X,1 + 80,OOOXs 1 + 50,OOOXg I+ 140,000xl~,l + 130,000x~~,1 +c&o- -c&()+ = 350,000 K. Maximize Net Present Value 21.
40,000X1 I 55,000x2 2 110,000X, 1 30,000X, 1 150,000X10,1
L. Objective
+ 32,000X1 2 + + 40,000x~ 3 + + 7O,OOOX42 + + 6O,OOOX,1 + + 18O,OOOX11,1
25,000X1 3 + 5o,ooox3~ + 25,000Xs1 + 80,000Xs 1 + +dzl- dzl+
7O,OOOXz1+ 35,000x32 + 2O,OOOX5z + 50,000X, 1 + = l,OOO,OOO
Function
Minimize Z = PI i
di+ + P,de + + Pa dg + + P4 (d, o-
+d,,+)+
i=l
P,(dlzd 17-
+d,s+)+P6(&4fdlg-1
+Ps(d,,-
+dls-l+P,(dls-+ +dzo-1
+Ps(d,,-1
References 1. 2.
Ansoff, H. I., Corporate Strategy, McGraw-Hill, New York, 1965. Charnes, A., and Cooper, W. W., Management Models and IndustrialApplications
of Linear Programming, Vols. 1 and 2, John Wiley & Sons, New York, 1961.
3.
Cooper, W. W., Theory of the Firm: Some Suggestions for Revision, Am. Econ. Rev. 34 (December, 1949): 1204-1222.
4.
Dantzig, G. B., On the Significance of Solving Linear Programming Problems with Some Integer Variables, Econometrica 28 (January, 1960): 30-44.
5.
Encarcion, J., Constraints and the Firm’s Utility Function, Rev. Econ. Studies 2 (April, 1964): 113-121.
6.
Hill, W. E., and &anger, C. H., Management Objectives and Bases for Evaluation, in Handbook for Industtil Research Management, C. Heyel, ed., Reinhold, New York. 1968.
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235
1.
Lee, S. M., Goal Programming for Decision Analysis, Auerbach, Philadelphia, 1972.
8.
Lee, S. M., and Morris, R., Integer Goal Programming, Research Report, Department of Business Administration, Virginia Polytechnic Institute and State University, August 1975.
9.
Lintner, J., The Cost of Capital and Optimal Financing of Corporate Growth, J..Finance 18 (May, 1963): 292-310.
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Osteryoung, 3. S., Capital Budgeting: Long Term Asset Selection, Grid, Inc., Columbus, Ohio, 1974.
11.
Osteryoung, J. S., Multiple Goals in the Capital Budgeting Decision, in MuZtipZe Criterion Decision Making, James L. Cochrane and Milan Zehrey, eds., The University of South Carolina Press, Columbia, 1973, pp. 447-457.
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Paine, F. T., and Naumes, W., Strategy and Policy Formulation, W. B. Saunders Co., Philadelphia, 1974.
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Ramsey, J. E., A Framework for the Interaction of Corporate Value Objectives, Corporate Performance Objectives, and Corporate Strategy, J. Econ. Bus. 28 (Spring-Summer, 1976): 171-180.
14.
Shubik, M., Objective Functions and Models of Corporate Optimization, Econ. 75 (August, 1961): 345-375.
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Simon, H., A Behavioral Mode of Rational Choice, Q. J. Econ. 59 (February, 1955): 99-118.
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Solomon, E., The Theory of Financial Management, Columbia University Press, New York, 1963.
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Weingartner,
Q. Rev.
H. M., Mathematical fiogramming and the Analysis of Capital
BudgetingProblems,Prentice-Hall, Englewood Cliffs, N.J., 1963. 18.
Weston, J., The Scope and Methodology of Finance, Prentice-Hall, wood Cliffs, N.J., 1966.
Engle-