Investment planning for the development of a national resource —linear programming based approaches

Compul.

& Opm. Rer., Vol.

I, pp. 247-262.

Pergamon

Press, 1974. Printed in Great

Britain

INVESTMENT PLANNING FOR THE DEVELOPMENT OF A NATIONAL RESOURCE -LINEAR PROGRAMMING BASED APPROACHES WII_I_IAM W. SWART ‘* , CHARLES E. GEARING~~, TURGUT VAR~$ and GARY CANN~§ ‘Department of Management Science, University of Miami, Coral Gables, Fla. 33124, U.S.A. *The Tuck School of Business Administration, Dartmouth College, Hanover, N.H. 03755, U.S.A. ‘Department of Economics &Commerce, Simon Fraser University, Burnaby 2, B.C., Canada, and ‘Union Carbide Corporation, South Charleston, W. Va. 25303, U.S.A.

Scope and purpose -This paper develops a methodology which can aid a country in planning its tourism development. Specifically, the paper addresses itself to situations in which a capital budget must be allocated among a large group of competing touristic investment alternatives. The mathematical model which is developed to represent the problem situation described above is a zero-one integer programming model whose objective function explicitly includes the results of subjective evaluations concerning touristic attractiveness. The solution methodology presented is specifically developed for organizations who do not have the resources required to implement sophisticated non linear programming methodology. In particular, it is shown how linear programming concepts may be used to economically provide a wide range of solutions to the model. The computational aspects are illustrated through application to the tourism investment problem faced by Turkey for its third 5-year plan.

Abstract -This paper will review and generalize the work of Gearing, Swart and Var dealing with the development of a mathematical model to aid the government ofTurkey in determining the “best” allocation of the capital budget for tourism among a large group of competing proposals. One of the more unique aspects of this work was the development of a measure of benefit for particular allocation plans which allowed for subjective information to be an integral part of the investment planning model. The computational procedures developed heretofore to derive investment strategies from the model have been based on dynamic programming, integer programming or combined dynamic programming-integer programming approaches. This paper will show how various methods based on linear programming yield satisfactory answers to many questions regarding development policies. The primary advantage of these linear programming methods is that they can be implemented with standard linear programming computer packages which are generally available, and hence eliminate the high cost of developing special purpose computer codes.

‘* William W. Swat? is Associate Professor of Management Science at the University of Miami in Coral Gables. He received a B.S. in Industrial Engineering from Clemson University, an M.S. in Industrial and Systems Engineering and a Ph.D. in Operations Research from the Georgia Institute of Technology. He has been associated with International Paper Company, and E. I. duPont de Nemours, as well as the Organization for Economic Cooperation and Development(OECD). Professor Swart’s recent articles have appeared in Manugemenr Science, Journal of Travel Research. Tourist Review (The Journal of AlEST), and the METU Journal of Pure and Avulied Sciences.“Recently, with the collaboration GearingandVar, he has completed a forthcoming book on Tourism Planning. t Charles E. Gearing is a Visiting Associate Professor at the Amos Tuck School of Business Administration at Dartmouth College. He holds a B.E.E. in Electrical Engineering from the Georgia Institute of Technology and an M.S. and Ph.D. both in Industrial Administration from Purdue University. His research in Tourism is reported in a study of the Turkish Tourism Sector, sponsored and published by U.S.A.I.D.; and in collaboration with Swart and Var, two articles in Tourist Review.

of

241

248

WILLIAMW. SWAKT,CHARLESE. GEARING,TURCUTVAR and GARYCANN

$ Turgut Var is a Visiting Associate Professor at Simon Fraser University in the Department of Economics and Commerce. Professor Var holds a B.A. in Business Administration from Claremont Men’s College, and M.B.A. in Finance from the University of Chicago, and a Ph.D. in Accounting from the University of Ankara. Professor Var has been responsible for translating several English books and articles into Turkish and has published a bibliography of Management Science articles in Turkish. F Garv Cann is a Programmer Analvst with Union Carbide Corporation in South Charleston, West Virsinia. Mr. Cat& holds a B.S. inchemical Engineering from Yale University and an M.S. in Industrial Engineering-from West Virginia University.

INTRODUCTION

All nations are endowed, to varying degrees, with distinguishing natural, social, and historical characteristics. These characteristics, in some instances, have been or could be supplemented by, for example, the development of recreational and shopping facilities, the construction of plush resort hotels, and the introduction of gambling casinos and other forms of night-time entertainment. When such developments are introduced to primarily attract foreign tourists, then they can be regarded as a national resource geared to generate foreign exchange eamings. These earnings will have a direct contribution toward enhancing the country’s buying power in the community of nations. When the development of a tourism industry has been decided upon as a strategy for foreign exchange earnings generation, then its implementation frequently involves a high level of direction from the central government. When such governmental involvement extends to include the direct investment of public funds in touristic facilities, then the central authority is faced with the problem of determining the most appropriate program for allocating the capital investments. Gearing, Swart and Var[l] have developed a mathematical model for selecting the optimal tourism investment policy for a developing country. One of the salient features of this model is that it allows for direct incorporation of subjective evaluations regarding touristic attractiveness. The model was used to aid Turkey in planning its tourism investment policy for the third 5-year plan. Originally, a specialized algorithm was developed to solve the model which provided a complete sensitivity analysis for various model parameters. Although very efficient, the solution algorithm did require a level of mathematical and programming sophistication for its implementation which is not. in general, present in most tourism planning organizations. It is the purpose of this paper to re-examine the Gearing-Swart-Var model from a computational point of view. In particular, an alternate representation of the model will be developed which significantly reduces the computational efforts required for its solution. Furthermore, it will be shown how various linear programming based approaches yield satisfactory answers to the model, making it more available to tourism planning organizations not having access to a high level of mathematica1 and/or computer sophistication. To achieve its purpose the next section of the paper will develop the Gearing-Swart-Var model as presented in [I], followed by a section describing and justifying three distinct linear programming based approaches. The computational results obtained by applying these approaches to the tourism development problem for Turkey will then be presented, followed by the conclusion.

Investment planning for the development of a national resource-linear MODEL

programming based approaches

249

DEVELOPMENT

For the purposes of developing the model, consider that the country, or geographical area, under consideration is subdivided into N particular touristic locations, or “touristic areas” (t.a.) and that, at any t.a., say the ith, there exist Ki specific proposed projects which may be undertaken. These projects, then, represent competing investment proposals, and they cover a wide range of possible investments. Some examples are: excavation and/or restoration of ruins, construction or improvement of roads, hotels or motels, sports and recreational facilities, etc. In all cases, it is expressly assumed that each t.a. has included as the first two proposed projects the following: (1) A planning project, i.e., a proposal for a detailed development plan of the touristic area, and (2) A project which is designed to bring the infrastructure and food and lodging facilities of a given t.a. up to minimally sufficient level, which was designated “minimal touristic quality” (m.t.q.). At each La., the proposed projects, if undertaken, exhibit certain dependencies in the form of precedence relations derived from factors such as physical necessity, logical preference, and functional interdependence. These precedence relationships are independent between t.a.‘s but, at each, the following standard convention was adopted: (i) If a t.a. does not have a formal plan of development, the planning project precedes all others, and (ii) If a t.a. does not have infrastructure and food and lodging up to m.t.q. standards, the necessary improvements are considered as a single project to precede all others except the planning project. Associated with every proposed project j at touristic area i is an estimated cost of completion cij. If the assumption is made that, considering all N t.a.‘s, the total cost of project development is equal to the amount of touristic investment, then the total cost cannot exceed the amount, b, budgeted for capital expenditures in the tourism sector. From the previous discussion, it can now be established that any plan, in order to be considered, must satisfy the following restrictions:

ii1j$lcijxij sb xi,l_ 2 Xi,

i =

1, N; V(L, M)EP~.

(1) (2)

Where : xij =

1 If project j is to be developed at touristic area i, 0 Otherwise.

Pi = Index set of ordered pairs at touristic area i. Each element (L, M) of the set indicated that project L immediately precedes project M. Having defined the feasible region, the next task is to define a numerical measure of benefit (or “utility”) associated with any program satisfying (1) and (2). In order to assess the relative merits of alternative investment programs it is necessary to have some means of measuring the benefit associated with a particular project, and this measure must have the property of being summable across all the projects in a particular allocation plan so as to define a measure of the benefit of the entire plan. Since the thrust of the work is to provide a planning model, we have taken a long range perspective wherein, as opposed to measuring the impact of short range earnings, a measure was developed which assumes a long range perspective in which the country is trying to develop its total potential for tourism earnings. This potential is inherently linked to the total touristic attractiveness

250

WILLIAM W. SWART, CHARLES E. GEARING, TURGUT VAR and GARY CANN

of the country which is expressable, through our measure, as the summation of the “touristic attractiveness” of the country’s t.a.‘s when subject to specific tourism development plans. The specific approach taken involved the selection of seventeen criteria which the authors deemed constituted the essential ingredients of “touristic attractiveness”, and they were selected so as to approximate as nearly as possible the property of being “independent”, as far as contributing toward “touristic attractiveness” is concerned [2]. The criteria were grouped into five categories, and they are presented in Table 1, along with the types of considerations involved in making valuations on each of them. The determinate of the importance of a particular criterion is the extent to which a tourist is responsive to the property characterized by that criterion. To clearly establish the relative importance of the seventeen criteria, a system of measurement or weights was established. The numerical values assigned to the weights were obtained by applying a modification of a procedure due to Churchman and Ackoff [3]. The subjective judgements used were solicited in a careful and consistent manner from a group of twenty-six tourism “experts” who, in Table Group A.

Natural

heading factors

Consideration

Criterion (1) Natural

beauty

(2) Climate B.

C.

D.

Social factors

Historical

factors

Recreational and shopping facilities

(I) Artistic and architectural (2) Festivals

(3) Historical prominence (1) Sports facilities facilities

(3) Facilities conducive to health, rest and tranquility (4) Night-time recreation (5) Shopping facilities

E.

Infrastructure and food and shelter

General topography; flora and fauna; proximity to lakes, rivers, sea, islands and islets: hot and mineral water springs; caverns; waterfalls. Amount of sunshine; temperature: winds, precipitation discomfort index. Local architecture: mosques, monuments; art museums.

features

(3) Distinctive local features (4) Fairs and exhibits (5) Attitudes towards tourists (1) Ancient ruins (2) Religious significance

(2) Educational

1

(1) Infrastructure above “minimal touristic quality” (2) Food and lodging facilities above “minimal touristic quality”

Music and dance festivals; sports events and competitions. Folk dress; folk music and dances (not organized); local cuisine; folk handicrafts; specialized products. Normally of a commercial nature. Local congeniality and treatment of tourists. The existence, condition, and accessibility of ancient ruins. The religious importance, in terms of present religious interests, observances and practices. The extent to which a site may be well-known due to important historical events and/or legends. Hunting, fishing, swimming; skiing: sailing, golfing: horseback riding. Archeological and ethnographic museums; zoos; botanical gardens; aquarium. Mineral water spas: hot water spas, hiking trails, picnic grounds. Gambling casinos; discoteques; theatres; cinemas. Souvenir and gift shops; handicraft shops: auto service facilities (beyond gasoline dispensing stations); groceries and necessities. Highway and roads ; water, electricity, and gas : safety services; health services; communications; public transportation facilities. Hotels; restaurants; vacation villages; bungalows, motels; camping facilities.

Investment

planning

for the development

Table

2.

of a national

Relative

resource-linear

weights and rank order of seventeen “touristic attractiveness”

Criterion 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

programming

Natural beauty Climate Artistic and architectural features Folk festivals Distinctive local features Fairs and exhibits Attitudes towards tourists Ancient ruins Religious significance Historical prominence Sports facilities Educational facilities Resting and tranquility Night-time recreation Shopping facilities Infrastructure above m.t.q. Food and lodging above m.t.q.

based approaches

criteria

Weight

Rank

0.132 0099 0.05 1 0.029 @026 0.011 @054 0.057 0053 0.065 0.046 0.015 0.032 @045 0036 0.131 0.125

1 4 9 14 15 17 7 6 8 5 10 16 13 11 12 2 3

251

of

the writer’s opinion, were well qualified to represent the attitudes of tourists. The actual results obtained are summarized in Table 2. A detailed discussion and description of the procedures together with the complete results of the interviews with the “experts” can be found in [4]. With the quantitative expression of the relative importance given to the various criteria of touristic attractiveness, the next step was the evaluation of the various t.a.‘s as to the extent to which they possessed characteristics which satisfied the various criteria. To quantitatively express the extent with which a particular t.a. was endowed, the judgements of an “evaluation” team were sought. This “evaluation” team, ideally, would be composed of persons representing a number of disciplines such as tourism, architecture and planning, sociology, archeology, transportation, etc. The selected team was asked to assign, at each t.a., a valuation between zero and one on each of the seventeen criteria. On a particular criterion, the value of zero would represent the complete absence of the property described, whereas the value of one would represent the highest possible satisfaction of that property. The evaluation team was charged with providing valuations on each criteria at each t.a. when the corresponding t.a. is assumed to be in any state j. A particular t.a. is considered in state j when project j has been, or is assumed to have been, completed, including all other projects which must precede it. Consider the column vector or valuations made at touristic area i when it is assumed in state j as: AI”

for i = 1,2,. . . ,N andj = 1,2 ,..., Ki.

Then, the “impact” of project M being undertaken at touristic area i is given by: TIM) = Ai“” - AIL’ for M 2 3 and (L, M) E Pi

(3)

T!z’ = A!2’ I

(4)

T!” = 0.

(5)

I

252

WILLIAM W. SWART, CHARLESE. GEARING, TURGUT VAR and GARY CANN

Relations (4) and (5) are special cases since it should be recalled that project 1 is a planning project which does not have any impact on the attractiveness of a particular site. At the same time, project 2 is one designed to make a non-touristic site into one suitable for tourists by providing for minimal touristic comforts, hence the particular impact associated with project 2 is the full benefit of going from “nothing” (touristically, that is) to the valuation of the site on all criteria as found in its initial state. Denoting the relative weights of the tourist attractiveness criteria by the row vector W, the measure of benefit associated with project j at site i is given by dij = WTI”

for i = 1,2, . . . , N andj= 1,2 ,..., Ki.

It should be noted that di, = 0 for i = 1,2,. . . , N which is consistent with the fact that planning does not, by itself, alter the “touristic attractiveness” of an area. With the measure of benefit developed as shown, the mathematical expression of the decision problem can be given by: Maximize

~ i=l

~ dijXij

(6)

j=l.

(7) Xi, - Xi, 2 0

for i = 1,2,. . . , N and (L, M) E pi

(8)

Xii = 0 or 1

for i = 1,2,. . . , N andj= 1,2 ,..., Ki.

(9)

Alternate model representation In order to have a computational alternative to the above model, it is possible to obtain its “multi-compartment’ knapsack equivalent [5]. Specifically, recapping the results discussed in [5], the problem represented by (6)-(9) can be reformulated as a multiple choice, or multi-compartment knapsack problem, of the form Maximize

(10)

Subject to

(11) i = 1,2,...,N

(12)

for i = 1,2,. . . , N andj= 1,2 ,..., ti.

(13)

j=l Xi; = 0,l

Where: X!‘) = [X$X$

Xi&] - the Ith feasible solution to constraint sets (8) and (9) for a specific i (there are a total of ti feasible solutions to that set of constraints).

Investment planning for the development of a national resource-linear

programming based approaches

253

cil = 5 c,,X{:) - total cost of the Ith feasible solution to constraint sets (8) and (9) for a j=l specified i. di, = 5 j=l

x;, =

dijX$)-total benefit of the Ith feasible solution to constraint sets (8) and (9) for a specific i.

1 -if i

0

feasible solution 1 to constraint sets (8) and (9) for a specific i is selected.

-Otherwise.

As can be noted, the essential step of the transformation process is the generation of all xi’), the feasible solutions to the precedence constraints. This step can be accomplished most expediently by realizing that any combination of zeros and ones represents a possible feasible solution to the precedence constraints. The total number of these is 2Kc=

Kt j=O 4

Ki j I'

and can be generated by looking at the binary representation of the integers from zero to 2Ki - 1, which are contained internally in the computer. This particularly simple way of obtaining candidate solution vectors coupled with constraint checks and implicit elimination of candidate solution vector subsets from explicit examination has proven to be a very effective manner in which to obtain all feasible solutions to the precedence constraint sets.

COMPUTATIONAL

APPROACHES

This section of the paper will discuss some computational approaches for solving both models which are available to any organization having access to a digital computer and a linear programming code. Although exact methods have been developed to solve both models, the effort associated with developing the specialized software is rather substantial, while, for certain uses, the additional information obtained by having access to this software does not warrant the effort required to develop it. The first part of this section will show how linear programming concepts may be used to generate solutions to the tourism investment problem. Although an optimal solution cannot be guaranteed through this approach, an optimal solution will be obtained for the problem with perhaps a somewhat different value of the budget amount, b, initially stipulated (this is identical to the concept underlying the use of generalized Lagrange multipliers [6]). This shortcoming is, for many purposes, not significant since the budget amount usually reflects a requested amount subject to probable change before it is granted at some later date. The second part of this section will briefly describe the performance of IBM’s Mathematical Programming System Extended (MPSX) Mixed Integer Programming (MIP) [7] when used to obtain the optimal solution for a specific value of the budget amount b. The basic rationale for including this approach in the study is that it is not uncommon for organizations using commercial linear programming codes to have access to some form of integer programming capability. The third and last part of this section will describe how the results of the first part may be combined with dynamic programming concepts to yield an approach which may be used to solve very large problems (relative to available computer facilities) of the type discussed herein.

254

WILLIAM W. SWART, CHARLESE. GEARING,

TURGUT VAR and GARY CANN

Linear programming Precedence constrained model. Upon examination, it becomes apparent that the precedence constrained formulation has a constraint set which is “almost” unimodular. In particular, constraint sets (8) and (9) may be rewritten as:

Xi, - Xi~ 2 0 0 5 Xi < 1

for i = 1,2,. . , N and (L, M) E Pi

(14)

for i = 1,2,. . . ,N andj= 1,2 ,..., Ki.

(15)

The constraint coefficient matrix to the above equations satisfy the conditions for unimodularity as given by Heller and Tompkins [8] (this may be trivially verified by considering the transpose of the constraint coefficient matrix). Consequently, the only non-integer extreme points introduced in the solution space of the precedence constrained model are those formed by the budget constraint (7). From knowledge about the unimodular property of (14) and (15), it follows that there must be adjacent extreme points (adjacent to the noninteger extreme points), which have integer coordinates. Several of these integer extreme points may be found by performing a parametric analysis on the right hand side of the budget constraint, b. Specifically, given a non-integer extreme point, the effect of varying the right hand side, b, will be to eventually indicate that a change of basis is required if the solution is to remain feasible (non-negative). If we specify that the parametric analysis should be for values less than or equal to the original b, then it will normally be true that a basis change will occur when one variable, originally in the solution at a fractional value, has been reduced to a value of zero. If it so happens that this shift in the budget constraint was sufficient to generate an all integer solution, then a feasible solution to the total precedence constrained model has been found which, furthermore, is optimal for that value of b which induced the change in basis.

I

I 300

I 900

I

600

Budget

Fig. 1. Budget vs. optimal

I

1

/

1200

1500

1600

amount

I 2100

x IO5

benefit relationship

for small study.

Investment planning for the development of a national resource-linear

programming based approaches

2.55

To empirically test the above ideas, a small problem consisting of 4 touristic areas with a total of 25 projects was solved. A brief description of projects, their cost and benefit measures, and the integer results obtained through parametric analysis, are given in Table 3. From results presented in this table (projects undertaken are marked by an x), it was possible to obtain the typical convex curve (Fig. 1) that depicts the optimal objective function value as a function of the right hand side value of the budget constraint. Comparing these results to all integer solutions obtained by an integer programming scheme (at greater cost), it was decided that linear programming with parametric analysis would provide sufficient information for initial macro level planning purposes. Based upon the encouraging results from the small study, the study was broadened to include the sixty-five geographical areas which had been designated as “touristic areas” by the Planning Department of the Turkish Ministry of Tourism. These had been established for the purpose of planning for the development of the tourism industry in that country. The various development projects that had been proposed for each touristic area in the past plus projects currently in process of being submitted were reviewed for suitability and completeness with respect to all relevant factors. The total number of projects that remained as viable candidates for funding after this preliminary analysis was 372. The preliminary budget figure that was proposed for funding tourism development projects for all of Turkey over the next 5-year plan was 7 x lo* Turkish Lira. Even though it was not expected that the full amount would be granted, it was not expected that less than 5 x lo8 T.L. would be devoted to touristic development under any circumstance. Based upon the above information, tourism planning strategies were developed via parametric linear programming. The range within which the budget amount, b, was allowed to vary was from 5 x 10’ T.L. to 7 x lo* T.L. The integer solutions obtained in terms of cost and maximum benefit are listed in Table 4. Although it is not practical to list all the corresponding solutions for this problem, as was done in Table 3 for the small scale problem, the nature of the solutions are very similar. Multiple choice model. Proceeding in a similar manner as with the precedence constrained model, consider constraint sets (12) and (13) fi

1 X,Cj= 1 for i=

1,2,...,N

(16)

j=l

X:j 2 0

for i = 1,2,...,N and j = 1,2,. . . , ti

(17)

whose constraint coefficient matrix can also be verified to satisfy the Heller-Tompkins unimodularity conditions [8]. Consequently, the same type of properties apply to the multiple choice model as to the precedence constrained model. The results obtained by subjecting the multiple choice equivalent to the sixty-five touristic area problem described earlier to the parametric analysis procedure were identical to the results shown in Table 4. In general, it cannot be expected that the results of the parametric analysis applied to both models will be identical. Since there appears to be no measure to indicate, a priori, which formulation will yield the “best” results, the principal determinate of the relative desirability of one formulation over another will be the computational performance of one over the other.

WILLIAM W. SWART, CHARLESE. GEARING, TURGUT VAR and GARY CANN

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Investment

planning

for the development

of a national

Table 4. Solutions

Integer

Budget amount

Optimal benefit for given budget ( x 103)

7000 6985 6857 6157 6651 6627 6597 6535 6505 6405 6365 6325 6055 6005 5905 5884 5834 5804 5704 5659 5489 5364 5314 5264 5224 5194 5164 5134 5084 5009 5000

38512.4925 38507.73 38453.43 38427.83 38391.91 38380.02 38367.97 38341.86 38329.14 38284.56 38266.33 38248.07 38119.11 38094.97 38046.45 3803623 38011.88 37996.84 37946.29 37923.43 37835.54 37170.64 37144.64 377 1854 37697.10 3768@30 37663.50 37646.70 37618.69 37576.34 37571.0975

resource-linear to complete

programming

development

based approaches

251

problem

Remarks Noninteger solution Integer solution Integer solution Integer solution Integer solution Integer solution Integer solution Integer solution Integer solution Integer solution Integer solution Integer solution Integer solution Integer solution Integer solution Integer solution Integer solution Integer solution Integer solution Integer solution Integer solution Integer solution Integer solution Integer solution Integer solution Integer solution Integer solution Integer solution Integer solution Integer solution Noninteger solution

at upper limit for budget

at lower limifs for budget

programming

As indicated before, the rationale for including this section in a paper dealing with linear programming based approaches is that it is not uncommon for organizations which have access to a linear programming code to also have access to an integer or mixed integer option associated with that code. In particular, this paper will only present the results that were obtained by using the MIP option available with IBM’s MPSX program product. This particular option utilizes a two stage method for optimizing a given mixed integer program. First, the problem is solved neglecting the integrality requirements. The second stage is a branch and bound procedure based on an algorithm developed by Dakin[9]. Although the program provides facilities whereby control can be exercised over the search process (see [7] for a detailed description of the system), the results to be presented herein wel’e obtained by using only the standard features of MPSX. In contrast to results from the parametric analysis approach, the integer programming approach, when successful, yields a unique optimum solution to the problem under consideration. If additional information is required, such as the sensitivity of the solution to changes in the budget amount, b, then the entire problem has to be solved again.

C.A.O.R.,

Vol. I, No. 2--G

WILLIAM W. SWART, CHARLESC. GEARING, TURGUT VAR and GARY CANN

258

For organizations having access to a computer with limited core storage, or for organizations faced with solving a very large problem of the type discussed herein, the concepts of dynamik programming applied to results obtained from a variation of the parametric analysis method yields a viable approach for obtaining near optimal solutions to the problem. Basically, the procedure consists of applying a coarse grid approach, as discussed in Nemhauser [lo], to a dynamic program. Each stage of the dynamic program consists of an arbitrary number of t.a.‘s. The state variable associated with each stage is the budget amount still left to be allocated while the return function and decision variable are obtained from a parametric analysis on the subproblem defined by that stage (the formulation can be either the precedence constraint model or the multiple choice model). For the precedence constrained model, the approach would consist of defining, for each stage s, the decision variable to be the amount of money allocated for development. The return, as a function of the decision variable is obtained by solving the parametric problem : I

Maximum Subject to < R,(R)

=

f

2 dijXij

i=cts j=i

Xi~--Xi~20 O
Where: 6, is a parameter

(18)

5 5 CijXij I 13, i=a, j= 1 for i=ol,,Cr,ll,...,/?, and (L, M) E P;: 1 for i=cr,,~~+ andj= 1,2 ,...,

l,...,& K;.

such that ~9,= API,,, Aif, + 6, A$, + 26, . . . , A:‘,, .

~1,- p, : Index range of touristic areas included in stage s.

g1,

- g;,

: region over which dynamic programming

6: dynamic programming

grid is to be applied at stage S.

grid size.

The results obtained from (18), similar to those shown in Fig. 1, for each stage may then be unified to provide a solution to the overall problem. The overall optimization can be achieved by using the recursion. f,(ZJ = Ma~~~um~~~(~~) + f,- ,(Z, - 6)) 5

with:

(19)

Z, - 6, 2 0 for all s I = (AminrAmin+ 6, Amin+ 26, * * -7 A!&} .fo(~o) = 0

and whereJJZ,) is the optimal return when there still are Z, units left to be allocated at stage s. It should be noted that when the function R,(B,) is tabulated, the above recursion is very simple to impfement. The stipulation of the grid size, 6, governs the amount of calculation required for the recursion.

Investment planning for the development of a national resource-linear

programming based approaches

259

Given, (18) and (19) the overall solution process may be described by the following steps: (1) Set A$, = 0; A$‘,, = b; 6 = grid size. (2) Find R,(&) by solving (18). (3) Apply the dynamic programming

recursion (19). Denote optimal allocations by 0:.

(4) If optimal solution is same as for larger value of 6, go to 7. (5) SetA~l,=8,*-6;A~~,=8,*+6;6=~.6

(whereO
(6) Go to 2. (7) STOP. The above outlined sequence of steps constitutes the coarse grid approach applied to the tourism development problem formulated earlier. The authors applied this approach to the 65 t.a. problem by breaking it up into three stages composed of 16 t.a.‘s each and the fourth consisting of 17 t.a.‘s. The initial grid size was set at 350 units, at 50 units during the second iteration, and 25 during the third and last iteration. The resulting dynamic programming problems were simple and small enough so that recursion (19) was applied manually. For the problem solved, the overall global optimum was obtained, although this can generally not be guaranteed. COMPUTATIONAL

RESULTS

The computational data accumulated as a result of implementing the approaches discussed in the previous section is summarized in Table 5. An explanation of the various numerical upper scripts follows: (1) This includes 372 rows specifying that each Xij must not exceed one. (2) By the exact method described in [l] it was found that there are a total of 95 integer solutions within this range. (3) These figures could be zero if the data was already available in MPSX format. (4) Assumes that multiple choice formulation is already available. CPU time to accomplish transformation is 1 second. (5) This was total time to generate 160 optimal solutions to define the R,(8,) functions. (6) This includes one additional man day over (3) due to manually performing dynamic programming calculations. As can be seen, the computational cost and efforts are minimal when considering that the model is a zero-one integer program with 372 variables and 309 constraints. The computational approaches should not be compared to each other on the basis of computer resource expenditure since they do yield different types of results. The parametric analysis approach, for example, is particularly suited to answer broad questions of marginal nature such as the incremental maximum return to be obtained by additional budget allocations, as well as the implication that additional budget allocations may have on specific project undertakings. The MIP option approach yields a specific answer to the question of what the development policy should be for a specific budget allocation, and is not amenable to marginal analysis. Finally, the dynamic programming approach is ideally suited to solve “very large” problems (in relation to available computer facilities) in that it essentially decomposes the original

analysis

*All computations

Dynamic programming based on parametric analysis

MIP option

Parametric

Approach

constrained

were performed

104 102 89 77

193 189 163 139 system.

502

312

681”’

67

502

372

681”’

61

LP Columns

dimensionality

data obtained

LP Rows

Problem

on an IBM 370/165 computer

1 2 3 4

choice

constrained

choice

Stage Stage Stage Stage

Multiple

Precedence

Multiple

Precedence

Formulation

Table 5. Computational

64’5’

109

171

26

43

CPU time fsec)

described

128

256

256

256

256

Core storage K-bytes

from approaches

MPSX

MPSX/MIP

MPSX/MIP

MPSX

MPSX

Programming system

in paper*

characteristics

Seauence of imorovine solutions. Finai solution not necessarily optimal.

b=7000.

Optimal solution for specific budget

Optimal solution for specific budget b=7000.

29”’ integer solutions in budget range 6500 <: b 5 7000.

29”’ integer solutions in budget range 6500 I h 5 7000.

Output

2’4’

2

2’4’

2

Computer programming man-days’“’

8

N

Investment planning for the development of a national resource-linear

programming based approaches

261

problem into a number of manageable stages, solves each stage as a separate parametric linear program, and only uses the results of the parametric linear programs to provide an answer to the entire problem. The computational data presented in Table 5 was derived from solving one particular real problem. The computational performance of the approaches to different problems (from a problem dimensionality standpoint) is directly related to the performance of the linear programming codes to problems of different dimensions. Since very large linear programming problems can be solved today, there is no reason why very large problems of the type presented in this paper cannot be solved. The one exception is, of course, the MIP approach since it is well known that general branch and bound schemes perform in a rather unpredictable fashion. CONCLUSIONS

Many organizations desiring to use a model such as the one developed in this paper do not have the specialized talent and resources required to implement rather sophisticated solution algorithms which are guaranteed to yield optimal solutions in all cases. In this paper explicit recognition was given to the idea that more than one approach can be used to obtain solutions to a specific model. In particular, it was hypothesized that it would not be unreasonable to assume that user organizations for the tourism investment model have access to a linear programming code. Consequently, several methods were developed and tested which can now be used, in conjunction with a linear programming code, to provide user organizations with an operational system by which to implement the model developed herein. Symbols used in the text cij

Estimated cost of developing project j at touristic area i.

b

Total amount budgeted for capital expenditures in the tourism sector.

xii =

1

If project j is to be developed at touristic area i.

0

Otherwise.

N

Total number of touristic areas under consideration.

Ki

Number of possible development

Index set of project pairs which directly precede one another at touristic area i.

pi

Column vector of valuations made at touristic area i when it is assumed in state j (i.e. when project j and all its predecessors are assumed to have been completed).

A!" .'

“Impact” vector of project A4 being undertaken at touristic area i.

T!M'

Row vector of touristic attractiveness

VG

criteria weights.

Benefit associated with developing project j at touristic area i.

dij

xi; = i

4,

projects existing at touristic area i.

1

If feasible solution I to constraint sets (8) and (9) for a specific i is selected.

0

Otherwise. Total cost of the Ith feasible solution to constraint specific i.

sets (8) and (9) for a

262

WILLIAMW. SWART,CHARLESE. GEARING,TURGUTVAR and GARY CANN

ti

Total number of feasible solutions to constraint sets (8) and (9) for a specific i.

0, A’S! _ A(S) Ill,” max

Parameter

w%)

Optimal objective function value of (18).

a, -

B,

such that f3, = A:{,, A$, + 6, A$, + 26,.

Region over which dynamic programming

. , + AEh.

grid is to be applied at stage s.

Index range of touristic areas included in stage s.

6

Dynamic programming

MS)

Optimal return of the s-stage dynamic programming are still 1, units of budget to be allocated at stage s.

I-

Discrete set of admissible values of the parameter 0,.

grid size. process when there

REFERENCES I. Charles E. Gearing, William W. Swart and Turgut Var. Determining the optimal investment policy for the tourism sector of a developing country, Management Sci. 20 (4) 487-497 (1973). 2. P. C. Fishburn, Utility theory, Management Sci. 14, 335-378 (1968). 3. C. W. Churchman and R. L. Ackoff, An approximate measure of value, Ops Res. 2, 1954. 4. Charles E. Gearing, Turgut Var and William W. Swat?, Establishing a measure of touristic attractiveness, J. Travel Res. 13, l-8 (1974). 5. William W. Swart, A multi-compartment knapsack problem, presented at the TIMS-ORSA-AIIE Joint National Meeting, Atlantic City, New Jersey, November (1972). 6. H. Everett III, Generalized Lagrange multiplier method for solving problems of optimum allocation of resources, Ops Res. 11, 339-417 (1963). 7. Mathematical Programming System Extended (MPSX) Mixed Integer Programming (MIP) Program Description, IBM Corporation, White Plains, New York (1971). 8. I. Heller and C. B. Tompkins, An extension of a theorem of Dantzig’s in H. W. Kuhn and A. W. Tucker (eds.), Linear Inequality and Related Systems, Princeton University Press, Princeton (1956). 9. R. J. Dakin, A free search algorithm for mixed integer programming problems, Comput. J. 8 (3), (1965). 10. G. L. Nemhauser, Introduction to Dynamic Programming, John Wiley, New York (1966). (Paper presented at Computers & Operations Research Symposium 20-21 August 1973).