A Goal Programming Model for Capital Rationing with a Linear Cash Fluctuations Measure

Transcription

1 Joural of Admiistrative Scieces Ad Ecoomics Vol A Goal Programmig Model for Capital Ratioig with a Liear Cash Fluctuatios Measure Dr. M. Asaad Elidai Busiess Admiistratio Departmet College of Maagemet ad Techology Arab Academy for Sciece ad Techology Ad Maritime Trasport 17

2 Joural of Admiistrative Scieces Ad Ecoomics Vol A GOAL PROGRAMMING MODEL FOR CAPITAL RATIONING WITH A LINEAR CASH FLUCTUATIONS 1- Itroductio MEASURE Capital ratioig arise i situatios where the total available resources (capital, labor, materials, etc.) is less tha the resource requiremets for all ivestmet opportuities beig cosidered by maagemet [1]. Therefore firms eed to devise procedures for ratioig i order to select the optimal group(ll of ivestmets uder the restrictio of scarce resources. Several capital ratioig techiques for the rakig of alteratives is preseted i the literature [1-3]. Most of these approaches will provide good meas of rakig the alteratives. Put this way, maagemet the selects from the list util either the list or the available resources is exhausted. Optimality is ot guarateed through this procedure. Cosider the followig example. Example 1 Cosider Table (1) below: Proposal Capital NPV ,000 12,000 4,000 2,500 Table (1) Example 1 data 3 9,000 2,200 Note that ivestmet proposals are raked accordig to the values of the Net Preset Value (NPV). If the capital available is (l) Optimality is cosidered with respect to the objective fuctio beig evaluated. Differet objectives i most cases will yield differet optimal groups. 19

3 A Goal Programmig Model for Capital Ratioig Dr. M. Asaad Elidai 25,000 the oly the first alterative will be selected. I this case a total NPV of 4,000 is realized. If o the other had, ivestmet opportuities 2 ad 3 are selected (total capital expediture would be 21,000) the a total NPV of 4,700 is realized. Other techiques as liear programmig may be used. I this case the objective fuctio is to maximize the total NPV realized. I may cases the eed arises for achievig more tha oe objective. Goal programmig provided a way of realizig these objectives simultaeously. The cocept of goal programmig was iitially developed i the early sixties by Chares ad Cooper [4]. I 1965, Ijiri [5] itroduced additioal defiitios ad refiemets of the techique. Several applicatios ad extesios were later give by Igizio [6] ad Lee [7]. The cocept was applied to capital ratioig to allow for multiple objective ratioig [3, 8-15]. I this paper a goal programmig model is developed that selects a group of ivestmet opportuities which maximized the aual cash flow fluctuatios. Both objective fuctios are liear mixed iteger fuctios. Several measures of deviatio are itroduced, compared, ad aalyzed via computer rus usig RISK [16]. Test problem were radomly geerated by BUDGEN [17]. Implemetatio of the model, test examples, ad cocludig remarks are preseted. Special costraits ad cases of the capital ratioig problem are preseted i Appedix A. 2- Notatio The followig otatio is used throughout the paper: I.=.the set of all ivestmets i; 0 ~ i ~ m (where i=o represet ivestmets). 20

4 Joural of Admiistrative Scieces Ad Ecoomics Vol x1 l 0. otherwise j fij = cash flow of ivestmet i i year j; l ~ j ~ ( 1. if ivestmet i is selecte~ 2 l Fj = aual combied cash flows of curret ad selected ivestmets i year j. Fj = L/iJXi iel ; 1, j, (1) CJ -;;- stadard deviatio of the combied aual cash flows of Pi curret ad selected ivestmets. -;;- et preset value of i cash flows geerated by ivestmet opportuity i. CL -;;- total capital available i local currecy. CF -;;- total capital available i foreig currecy. C -;;- total capital available, irrespectful of currecy. eli -;;- capital required for proposal i i local currecy. cf; -;;- capital required for proposal i i foreig currecy. 3- a. (2) ci -;;- capital required for ivestmet opportuity i, irrespectful Assumptios of currecy. The proposed model relies o the followig set of assumptios: Idivisible ivestmet opportuities: proposed ivestmets ca The variable x 0 represets the curret ivestmets ad will always equal to 1. This meas, it is a costat ot a variable. Yet, to simplify the otatio ad the calculatio of the cash fluctuatio fuctio, it will be cosidered a variable i the otatio ad will be treated as a costat i the developmet of the model as will be show later. 21

5 A Goal Programmig Model for Capital Ratioig Dr. M. Asaad Elllidalli ot be broke up ito parts; either it is udertake as a whole or it is ot udertake at all. b. Sigle period budgetig; oly projects requirig capital expeditures at the preset are cosidered. c. Equal lives: all proposed ivestmet opportuities ad all curret ivestmets are assumed to have equal lives. 4. Model Developmet we will first develop the simplified sigle objective models, the maximizatio ad miimizatio models. Both models will have the same set of costraits, ad differ oly i the objective fuctio. The first model, the Capital Ratioig Maximizatio model (CRMAX), maximized the NPV of the firm. The secod model, the Capital Ratioig Miimizatio model (CRMIN) miimizes the fluctuatios of aual combied cash flows. The goal programmig model ( CRGP) is the preseted ad tested usig LINDO [30]. 4.1 The Capital Ratioig Maximizatio Problem I this model, the maximizatio of the total NPV of cash flows of curret ad selected ivestmets is cosidered. This presetatio would ot be complete without a ote o the selectio of a Miimum Acceptable Rate of Retur (MARR), that will be used i the calculatio of the NPV. The literature preseted may alteratives for determiig MARR [1-3, 18-27]. It is beyod the scope of this work to discuss the cotroversy related to the selectio of MARR. A reasoable discout factor that 22

6 Joural of Admiistrative Scieces Ad Ecoomics Vol will be used is the cost of capital iquiry. Hece, it is assumed that the discout factors for all ivestmet opportuities are equal. The mathematical model of the Capital Ratioig Maximizatio problem (CRMAX) is give by: (CRMAX) Maximize L P;x; subject to: iej iel ;'ViEl,i=tO 4.2 The Capital Ratioig Miimizatio Problem Risk aalysis allows the otio of the degree of deviatio of possible outcomes of a fiacial elemet from their calculated mea. Risk cosidered i this paper, measures the fluctuatios i aual cash flows of a certai group of ivestmet opportuities. Thus, the optimum group of ivestmets, uder the miimizatio model, is the oe that yields cash flows with the least possible fluctuatios from year to year, subject to the costrait of limited resources. To evaluate the rate of fluctuatio i a give set of cash flows, two measures are applied: the Stadard Deviatio, ad the Average Absolute deviatio. Let F be the mea of aual cash flows of a selected group of ivestmet opportuities, the: L~ F=f! Hece, the stadard deviatio ( cr) may be calculated as follows: 23 (2)

7 A Goal Programmig Model for Capital Ratioig Dr. M. Asaad Elidai (3) ad the average absolute deviatio (A) may be calculated as follows: fifj-pi A = (4) -'-i=_i--- Extedig equatios (2), (3) ad (4) to represet their elemetary compoets, X; adfij, results i the followig: L:L:J;jxi (5) p = J=l iel a= 1 L (L/;Jxi- J=I iei J=I ;EI (6) A =,!_ Measurig fluctuatios i cash flows, usig (6) results i a oliear objective fuctio: (7) (8) The average absolute deviatio measure, as give by (7), provides a mea of measurig the desired fluctuatios via liear relatios, therefore, a 0-1 iteger programmig model may be costructed. This should be easier to solve tha the oliear model. 24

8 Joural of Admiistrative Scieces Ad Ecoomics Vol Therefore, the objective fuctio will take the form: 11 L. L..f]x, - j=l iel ;=I iei Miimize A = ---' '- The miimum value of A subject to the costraits of the problem is equal to the miimum value of *A subject to the same set of costraits, therefore, (9) may be reduced to the followig: Miimize A= L Lfij- -'--j=- j=l iel LLfijxi (9) (10) Compariso betwee the two Measures of Risk It will be show that the average absolute deviatio model yields results that are very close to those provided by the stadard deviatio model. Because either measure simply determies how differet alteratives compare i terms of cash fluctuatios, close results are cosidered satisfactory for risk measuremet. It should be oted that the stadard deviatio does provide a more accurate measure of dispersio (the degree to which a set of values vary about their mea) tha the average absolute deviatio o the same sets of data [28]. Therefore, it is valid to say that selectios made by the two measures, may be differet. We eed to aswer two questios: how much do they differ? Ad, how ofte? The program BUDGEN [17] was used to radomly geerate hudreds of test cases. The program RISK [16] was the used to test 25

9 A Goal Programmig Model for Capital Ratioig Dr. M. Asaad Elidai ad evaluate these cases. Each case cosists of a predefied umber of alteratives ad their aual cash flows. The followig measures were computed: (A) The Cosecutive Absolute Differece: ~J L:l~ -~+11 j=l (11) (B) The Relative Absolute Differece: ~I F F L: j j+l J=l (12) (C) The Average Absolute Deviatio of cash flows: (D) ~J L:I~-PI (13) j=l The Stadard Deviatio of cash flows: L(~ -F)2 (14) \ i=i A group of oe hudred sets of problems (cases) were geerated. Each set is composed of four ivestmet opportuities. For each alterative, cash flows for eight years were radomly geerated. The four measures were computed. The total umber of times where each measure selects the same alterative as the oe selected by the variace is registered. The results are summarized i Table (2) below. 26

10 Joural of Admiistrative Scieces Ad Ecoomics Vol Number of times measure selects the same proposal as the oe selected by the Measure of Risk stadard deviatio A Cosecutive Absolute Differece 35 B Relative Absolute Differece 22 c Average Absolute Differece 86 Table (2) The umber of times each measure selects the same alterative as the oe selected by the stadard deviatio It is clear from Table (2) that the average absolute deviatio measure was much closer tha the absolute ad relative differeces i the selectio of alteratives. The absolute differece missed 65% of the cases ad was oly correct 35% of the times. The Relative differece missed 78% of the cases ad was correct 22% of the times. Although such results may ot be geeralized, they are good eough to determie that the average absolute deviatio measure is superior to the absolute ad relative differeces i measurig risk. This triggers the ext test. I order to get a better feel of how close the average absolute deviatio is i selectig ivestmet proposals as compared to the stadard deviatio, a program RISKAUTO [29] was used. This program uses data geerated by BUDGEN [17], tests them usig RISK [16] ad risk measure. A success is registered whe a risk measure selects the same alterative as the oe selected by the stadard deviatio. RISKAUTO provides three values: MINDIFF (the miimum differece), MAXDIFF (the maximum differece), ad AVGDIFF (the average differece). 27

11 A Goal Programmig Model for Capital Ratioig Dr. M. Asaad Elidai To illustrate, cosider this example. The results of oe of the test sets with 6 ivestmet alteratives. A alterative may cosist of more tha oe ivestmet opportuity. A group of ivestmet opportuities that do ot violate ay of the problem costraits may hece be called a ivestmet alterative 3 are show i Table (3). The Stadard ad average absolute deviatio of the aual cash flows were calculated for each alterative. Ivestmet Alteratives Year ,000 1,000 6,000 1, , ,600 1,500 4,500 1,000 1,400 3, , ,500 1, , ,800 1,500 2,800 1,800 1,500 2, ,500 1, , ,800 2,000 1,500 1,200 1,200 2, , ,000 1, , , ,000 2, , ,000 2,000 1,000 1, , ,500 1, , ,000 Table (3) Cash flows, Stadard Deviatio ad Average Absolute Deviatio of 6 ivestmet alteratives Accordig to Table (3), the best alterative accordig to the stadard deviatio is the fourth Ccr =371), ad the best alterative accordig to the average absolute 'deviatio is the fifth (A =290). (3) A alterative may cosist of more tha oe ivestmet oppmtuity. A group o ivestmet oppmtuities that do ot violate ay of the problem costraits may hece be called a ivestmet alterative. 28

12 Joural of Admiistrative Scieces Ad Ecoomics Vol Such a case will be registered as a failure for the average absolute deviatio for selectig a alterative other tha that selected by the stadard deviatio. I this case, we calculate the differece betwee the stadard deviatio of the fourth ad fifth alteratives ad fid it to be 2 (0.54%). This differece is the saved ad compared to other differeces for other test sets. From these differeces the three values MINDIFF, MAXDIFF ad A VGDIFF are registered. Three groups of problems were tested based o the above test procedure. The first cosisted of ivestmet opportuities (alteratives) with lives equal to 10 years, the secod 15 years ad the third 20 years. I each group, differet problem sizes were tested, ragig from sets with 3 alteratives each, through sets with 20 alteratives each. O each problem size, 100 sets of radomly geerated problems were tested from which the three values, MIND IFF, MAXDIFF ad A VGDIFF were calculated. I additio, the total umber of successes were reported as the umber of matches. A total of 5,400 test cases were geerated, tested ad reported. Table (4) below summarizes the results of these tests. 29

13 A Goal Programmig Model for Capital Ratioig Dr. M. Aslllld Elidai Number ot Aual cash Bows No. of Cash flows= 10 Cash flows= 15 Cash flows = 20 iv. No. ot IJJtterece No. ot IJttterece No. ot IJttterece alt. mate he~ mi max avg matches mi max avg matches mi max avg j YO :1.:115 ),()4 Y3 l.'j/. LU o.oo O.II O.II mi max avg * Zeros i the table idicate very small positive umbers. Table (4) Test cases showig the relatio betwee the average absolute deviatio ad the stadard deviatio as measures for risk Accordig to Table (4), the average absolute deviatio successfully selected the same alterative as the stadard deviatio i 83% of the cases (o the average) i the 10 year category. The rage of success was from a low 74% (18 alteratives) to a high 91% (4 ad 5 alterative). The average success was 85% for the 15 year category, with a low of 75% ad a high of 93%. The average success was 87% for the 20 year category, with a low 30

14 Joural of Admiistrative Scieces Ad Ecoo.mics Vol of 78% ad a high of 94%. The MINDIFF, MAXDIFF, ad A VGDIFF values are show i the secod, third ad forth colums respectively of each category. The MINDIFF value for the 10 year category scored a average of 0.55% ad the MAXDIFF for the same group scored a average of 18.44% with a overall average of 6.6%. For the secod group, the MINDIFF was of 0.33% o the average, ad the MAXDIFF was 12.71%, with a overall average of 4.77%. Fially the third category values were 0.44%, 11.69% ad a average 4.34%. The differece betwee the stadard deviatio of the alterative selected by the average absolute deviatio measure rages betwee a low 0.01% ad a high of 28.18% (both from the secod category). The overall average of differeces betwee the two stadard deviatios is 5.24% (computed by takig the average of the averages). Accordig to these results, it is safe to coclude that the average absolute deviatio is a reasoable approximatio of the stadard deviatio as a risk measuremet as tested o actual problems. Ad sice the average absolute deviatio measure could be trasformed to liear relatios it will be adapted to measure the cash fluctuatios. The stadard deviatio although a more accurate measure of dispersio will ot be used for it is based o oliear relatios Simplifyig the Objective Fuctio of the Miimizatio problem Two objectives i the multi-objective model are to be satisfied simultaeously, the maximizatio of the total et preset values of selected group of ivestmets, ad the miimizatio of the overall risk measured i terms of the fluctuatios i aual cash flow. 31

15 A Goal Programmig Model for Capital Ratioig Dr. M. Asaad Elidai Before the multi-objective model ca be preseted, the objective fuctio of the miimizatio model as give by (10) must be put i a liear form. The mea of cash flows of all selected ivestmets, as defied i (5), may be preseted as follows: LLf1 Lf;Ixi + Lf2x; + + Lfxi p = j=l iel _ iej iel iej fo1xo + f11x1 + f21x2 + + fmlxm + fo2xo + f12x1 + f22x2 + + fm2xm + xo(joj + fo2 + + fo )+xlf + h2 + + h )) ( _ + +xm(fml + fm2 + + fm) _ (Jol + f fo) (JII + f f1) - Xa +XI +... /ml + fm2 + + fm) +xm ( (15) The values betwee the brackets are the meas of the cash flows of ivestmet opportuities i=o, I,...,m, let that mea be F; (15) the becomes: F = F 0 x 0 + F;x F,xm, that is, F = :L"P;x; iej (16) 32

16 Joural of Admiistrative Scieces Ad Ecoomics Vol Hece, becomes: A= L LfJxi- L:F;x; j=l iel iel = f foj_xo + fi_}_xl + + ~ljxm J=l -F 0 x 0 -~x }mxm = L:lxoCfoj- Fa )+xl~j- F; ) xm(fmj- F;, )I j=l To ease the otatio, lets..= a_ - F), ad sice x =1, the A IJ Vi) I 0 r A= L (LS; 1 X; )+S (17) 01 j=l z=l hece, the objective fuctiq becomes: r Miimize A= L CL:S, 1 x; + S (18) ) 01 j=l i=l Let Yi, be defied as follows: m Yi=CL:SiJxi )+ SOJ i=j Because Yi may take positive ad egative values, it is redefied i terms of two o-egative variables y+ ady as follows: J J (19) The mathematical model of the capital ratioig miimizatio problem (CRMIN) may ow be preseted. (CRMIN) Miimize A= L (~+ + ~-) (20) J=l 33

17 A Goal Programmig Model for Capital Ratioig Dr. M. Asood Elidai subject to: m y+ - y- - "s - s f ; L...,;x, - o; i=l ;1S:jS: (21) iel x 0 = 1 X;= 0,1 y+ > 0 y~ > 0 J - ' J - ;ViEl,i-:t-0 ; 1 S:j S: (22) It is possible ot to cosider the multi-objective model which is preseted i the ext sectio The Capital Ratioig Goal Programmig Model ( CRG P) The goal programmig model CRGP may be writte as follows: (CRGP) subject to: i=l 17 L 0"7 + ~~ ) - z2 ::;; 0 J=l m ~+- ~~- Lsifxi =So] i=l ; 1 S:j S: (23) (24) iel Xo = 1 X;= 0,1 ~+,~-,z 1, z 2 ;::: 0 ; 1 S:i S:m ; 0 S:j S: Where W" W 2 are weights that may be used to emphasize the importace of oe objective over the other< 4 >. Costrait (23) results (4) A value of 1 may be used for both weights if o special preferece is desired. 34

18 ]ouriull of Admiistrative Scieces Ad Ecoomics Vol i the maximizatio of the total et preset value of the curret ad selected ivestmet opportuities, resulted from the positive coefficiet of z 1 i the objective fuctio Costrait (24) results i the miimizatio of the cash fluctuatios, due to the egative coefficiet of z 2 i the objective fuctio. The followig two simple examples illustrate how the model performs i selectig the group of ivestmet opportuities such that the total NPV is maximized ad yearly cash fluctuatios are miimized. Example 2 Cosider the followig problem with two ivestmet opportuities each with a life of two years. The problem data is summarized i Table (5) below. Total capital available is 800. A discout rate of 12% was used i the calculatio of the et preset values. Ivestmet Opportuities Curret First Secod Cost (500) (632) (708) Year 1 1, ,000 sij = ~j- Fj SOi slj s2j -1, ,000 Year2 3,000 1,000 3,000 1, ,000 Mea 2,000 1,850 2,000 NPV 2,486 2,300 2,300 Table (5) Example 2 data the form: The mathematical model of the above problem may the take Maximize z 1 - z 2 35

19 A Goal Programmig Model for Capital Ratioig Dr. M. Asaad Elidai subject to: Y Y 2- + Y Y 2- - Z2 ~ X X 2 ~ 800 ZpZz ~ 0 This problem was solved usig LINDO [30] as the mixed iteger problem optimizatio program. Data was fed to the program via two matrices: G, H, ad the vector S 0 See Appedix B for a defiitio of each matrix ad the costrait matrix of the problem. The solutio is give i Table (6). Variable Value x1 1 x2 0 z z2 300 y+ 1 0 y y y 2 0 Table (6) Variable values of Example 2 36

20 Joural of Admiistrative Scieces Ad Ecoomics Vol Accordig to Table (6), the first alterative that yields the miimum risk desired, is selected. Example 3 I example 2 both ivestmet opportuities had the same NPV. But the selectio of the first ivestmet resulted i a lower risk caused by lower fluctuatio of the aual cash flows of curret ad selected ivestmets. I this example, the NPV of both ivestmet opportuities are also equal but the risk is higher i the first alterative. Therefore, the model should select the secod ivestmet. Total capital available is 800. A discout rate of 12% was used i the calculatio of the et preset values. The problem data is show i Table (7). Ivestmet Opportuities Curret First Secod sij = J;j- Fj Cost (500) (638) (709) SOi s,j Szj Year 1 1, ,000-1,000-1,200-1,088 Year2 3,000 3, ,000 1,200-1,088 Mea 2,000 2,000 1,913 NPV 2,486 2,346 2,346 Table (7) Example 3 Data The mathematical model of the above problem is give by: 37

21 A Goal Programmig Model for Capital Ratioig Dr. M. Asaad Elidai subject to: 2,346 X 1 + 2,346 x 2 - z, ~ X X 2 $. 800 Y Y ,200 x 1 + 1,088 x 2 = 1,000 The solutio is give i Table (8) Variable Value XI 0 x2 1 Z1 2,346 z2 176 y+ I 88 y I 0 Y/ 0 y2 88 Table (8) Variable values of E-wmple 3 38

22 Joural of Admiistrative Scieces Ad Ecoomics Vol Computatioal Results The CRGP was tested o data from the Egyptia Govermet idustrial sector [31]. Results were compared by the actual decisios cocerig capital ivestmets ad foud to be similar. Detailed discussio o the use of goal programmig ad CRGP may be foud i [31]. The size of CRGB, is give by the umber of variables ad the umber of costraits for ay give ad m. Tfiere are three types of variables: x, Y ad z. The x variables (m, biary) represet the ivestmet opportuities, the Y variables (2 for each year, cotiuous) are used for the trasformatio of the absolute value relatio i the costrait set, ad the z variables (two, cotiuous) are used to attai the two defied goals of the model. Thus, the total umber of variables i the problem is equal to 2+m+2. There is a total of + 3 costraits. 39

23 A Goal Programmig Model for Capital Ratioig Dr. M. Asaad Elidai APPENDIX A A.1 Capital Requiremets It is possible i may cases that a ivestmet opportuity requires fuds both i local ad foreig currecies. Thus, a costrait for each type of capital is required as follows: Lcl;x;::::; CL A-1 IE] L:cfx; ::::;CF A-2 zei A. 2 Miimum Number of Selected Ivestmets This costrait states that a miimum umber of ivestmets are to be udertake. It is assumed however, that this predetermied miimum value does ot violate the resource availability costraits, otherwise, the problem would have o feasible solutios. zei' ;I'r:;;;,I A-3 A.3 Maximum Number of Selected Ivestmets If "o" is the maximum umber of ivestmets to be selected i ay give year, the: ; 1" r:;;;, I A-4 A. 4 Mutually Exclusive Ivestmet Opportuities Two ivestmet opportuities "s" ad "t" are said to be mutually exclusive if the selectio of oe prevets the selectio of the other. The associatig costrait will the take the form: 40

24 Joural of Admiistrative Scieces Ad Ecoomics Vol X 5 +X 1 ~ 1 A-5 This costrait is to be repeated for each patr (group) of mutually exclusive proposals. A. 5 Cotiget Ivestmet Opportuities Ivestmet opportuity "v" is said to be cotiget upo the selectio of ivestmet opportuity "w" if it ca ot be udertake uless ivestmet opportuity "w" is selected. This costrait will the take the form: A-6 A. 6 Limited Resources Limited resources does ot come oly i the form of capital. Several authors [ 1-3] had idicated the ature ad form of such costraits. If B is the limited resource (labor, materials, etc..), ad b; is the amout of that resource required whe ivestmet opportuity i is selected the: A-7 A. 7 Added Value Two or more ivestmet opportuities may yield a extra added value whe selected together. This ca be easily icorporated i the objective fuctio via a oliear relatio (o additioal costraits are required). The followig may be added to the objective fuctio of the maximizatio problem: 41 A-8

25 A Goal Programmig Model for Capital Ratioig Dr. M. Asaad Elidai Whe I"' C I is the set cosistig of ivestmet opportuities that would geerate the added value u. There is a alterative liear formulatio to this case. If xk ad x; geerate a added value u, if selected together, the a ew variable x' may be added to the set of ivestmet opportuities /. The objective fuctio coefficiet of x' would equal to Pk + P;+ u Two costraits eed be added to the costraits set to prevet xk ad X; from beig selected with x'. x' + xk < 1 x' +XI< 1 Note that sice Pk + P 1 < Pk + PI + u, xk ad xi will ot be selected together, because the maximizatio problem would force the selectio of x' istead or simply oly oe of the two variables is selected, i which case the x' = 0. 42

26 Joural of Admiistrative Scieces Ad Ecoomics Vol APPENDIX B The costrait matrix of CRGP takes the followig form: XI Xz X y+ y y + y... y + y zl Zz RHS m I I 2 2 Il Il PI p2 pm ci Cz em c -Stt -Szt -S. D SOl -Stz -Szz -Sm Soz -Stu -Sz... Smm Data is supplied to the program i two matrices, G, H, ad the vector S 0 The matrix G is 2 by m ad the matrix His m+ 1 by. To defie G, let be the value of the etry i row k ad colum i of G, the: gli = pi ad g2i = ci Let hij be the value of the etry i row i ad columj of H, the: 43

27 A Goal Programmig Model for Capital Ratioig Dr. M. Asaad Elidai Refereces [1] Arago, George A., Fiacial Maagemet. Massachusetts: Ally ad Baco, [2] Brigham, Eugee F., Fudametals of Fiacial Maagemet, fifth editio. The Dryde Press, Florida, [3] Caada, R. Joh, ad White, A. Joh, Capital Ivestmet Decisio Aalysis for Maagemet ad Egieerig: Pretice-Hall, Ic., New Jersey, [4] Chares, A., ad Cooper, W. W., Maagemet Models ad Idustrial Applicatios of Liear Programmig. Joh Wiley & Sos, Ic., New York, [5] Ijiri, Y., Maagemet Goals ad Accoutig for Cotrol. Rad McNally College Publishig Compay, Illiois, [6] Igizio, J. P., Goal Programmig ad Extesios. D. C. Health & Compay, Massachusetts, [7] Lee, S. M., Goal Programmig for Decisio Aalysis. Auerbach Publishers, Ic., Pesylvaia, (8] de Acharya, P. K. D., ad Sahu, K.C., "A Chace-Costraied Goal Programmig Model for Capital Budgetig". Joural of the Operatioal Research Society, Vol. 33, No.7, July 1982 ( ). [9] Igizio, J. P., "Capital Budgetig Via Iteractive Goal Programmig". AilE Trasactios, Fall ( ). 44

28 Joural of Admiistrative Scieces Ad Ecoomics Vol [10] Keow, A. J. ad Marti, J. D., "A Iteger Goal Programmig Model for Capital Budgetig i Hospitals". Fiacial Maagemet, Vol. 5, No. 3, Autum (28-35). [11] Keow, A. J., ad Taylor, A. W. III, "A Chace-Costraied Iteger Goal Programmig Model for Capital Budgetig i the Productio Area". Joural of Operatioal research Society, Vol. 31, No.7, July ( ). [12] Kumar, P. C., Philippatos, G. C., ad Ezzell, J. R., "Goal Programmig ad the Selectio of Portfolios by Dual-Purpose Fuds". Joural of Fiace, Vol. 33, No.1, March ( ). [13] Lawrece, K. D., ad Reeves, G. R., "A Zero-oe Goal Programmig Model for Capital Budgetig i a Property ad Liability Isurace Compay". Computers & Operatios Research, Vol. 9, No. 4, ( ). [14] Taylor, B. W. III, ad Keow, A. J., "A Goal Programmig Appliocatio of Capital Project Selectio i the Productio Area". AilE Trasactios, Vol. 10, No. 1, March (52-57). [15] Hiller, Frederick S., ad Lieberma, Gerald J., Itroductio to Operatios Research, third editio Hode Day, Ic., Califoria, [16] Elidai, M. Asaad, RISK. Busiess Admiistratio Departmet, Faculty of Admiistrative Scieces ad Ecoomics, The Uiversity of Qatar, Doha, Qatar, P.O. Box

29 A Goal Programmig Model for Capital Ratioig Dr. M. AslUld Elidai [17] Elidai, M. Assad, BUDGEN. Busiess Admiistratio Departmet, Faculty of Admiistrative Scieces ad Ecoomics, The Uiversity of Qatar, Doha, Qatar, P.O. Box [18] Atkis, D. R. ad Ashto, D. J., "Discout Rates i Capital Budgetig: A Reexamiatio of the Bauol & Quadt Paradox". The Egieerig Ecoomist, Vol. 21, No.3, ( ). [19] Baumol, W. J., "A Expected Gai-Cofidece Limit Criterio for Portfolio Selectio". Maagemet Sciece, Vol. 10, No. 1, ( ). [20] Berhard, R. H., "Some Problems i the use of a Discout Rate for Costraied Capital Budgetig". AilE Trasactios, Vol. 3, ( ). [21] Bradely, S. P., Frak, R.S., ad Frey, S.C., "Determiig the Appropriate Discout Rates i Pure Capital Ratioig". Decisio Scieces, Vol. 9, No. 3, July ( ). [22] Durad, D., "The Cost of Capital i a Imperfect Market: A Reply to Modigiliai ad Miller". America Ecoomic Review, September ( ). [23] Ederigto, L. H., ad Hery, W. R., "O Costs of Capital i Programmig Approaches to Capital Budgetig". Joural of Fiacial & Quatitative Aalysis, Vol. 14, No. 5, December ( ). 46

30 Joural of Admiistrative Scieces Ad Ecoomics Vol [24] Hoe, James C. Va, Fiacial Maagemet ad Policy. Pretice Hall, Ic., New Jersey, [25] Hays, J. W., "Discout Rates i Liear Programmig Formulatios of the Capital Budgetig Problem". Egieerig Ecoomist Joural, Vol. 29, No. 2, Witer ( ). [26] Lusztig P., ad Schwab, B., "A Note o the Applicatio of Liear Programmig to Capital Budgetig". Joural of Fiacial & Quatitative Aalysis, Vol. 3, No.4, ( ). [27] Solomo, Ezra, The Maagemet of Corporate Capital. The Free Press, New York, [28] Daiel, Waye W., ad Terrell, James C., Busiess Statistics: Basic Cocepts ad Methodology. Houghto Miffli Compay, Bosto, [29] Elidai, M. Asaad, RISKAUTO. Busiess Admiistratio Departmet, Faculty of Admiistrative Scieces ad Ecoomics, The Uiversity of Qatar, Doha, Qatar, P.O. Box [30] Lius Schrage, LINDO/PC, Califoria, USA, [31] Hidy, Mueir 1., ad Elidai, M. Asaad, "A Proposed Approach for Capital Ratioig". The Joural of the Faculty of Commerce, Uiversity oftata, Vol. 1-12,