Regional Science and Urban Economics 13 (1983) 31-53. North-Holland

A MULTIOBJECTIVE INTEGER PROGRAMMING MODEL FOR THE LAND ACQUISITION PROBLEM

Jeffrey WRIGHT

Purdue University, West Lafayette, IN 47907, USA

Charles ReVELLE and Jared COHON

The Johns Hopkins University, BaEtimore, MD 2I218, USA

The purpose of this paper is to present a multiobjective integer programming formulation for the analysis of the Iand acquisition problem. There are two important contributions of this paper. First, the model incorporates the discrete and multiobjective nature of land acquisition. Second, we present an effkient, specialized algorithm for finding non-inferior solutions of a multiobjective integer program, a problem <for which a general-purpose algorithm, applicable to moderately sized problems, does not exist. c

1. Introduction

The basic problem addressed in the paper exists in both the public and private sectors. The problem is general, in that land must be acquired, whenever it is not already owned, for the construction of virtually any structure. In some instances, as in the formulation of public parks, the acquired land is itself the facility. Whatever the setting, the decision-maker - the person or organization who needs to acquire land - is faced with the problem of assembling enough parcels to meet his or her needs. This is not an easy problem. As we will show later, there are many factors, some in conflict with each other, to be considered.

There has been little quantitative work devoted to the land acquisition problem. Facility location models have been formulated for related problems. McClellan and Medrich (1969) used a m icroeconomic approach to identify factors that should be included in an outdoor recreation planning model. More quantitative work has incorporated these and other factors into sophisticated public lands planning models [see, e.g., ReVelle and Swain (X970), Rojeski and ReVelle (1970), Cicchetti (1979), Schuler and Meadows (1975), Weintraub and Navon (1976), and Webber (1980)]. While these and other efforts have resulted in the development of powerful planning tools, they deal almost exclusively with the allocation of existing land-based resources rather than with the acquisition of new ones.

The work by Meier (1968) is a notable exception. He presents a linear

0166-0462/83/0000-0000/$03.00 0 1983 North-Holland

32 J. Wight et al., A multiobjective integer programming model

programming formulation for the purpose of the acquisition of recreational lands with the single objective of maximizing recreational value. Constraints include restrictions on the supply of and demand for different land ‘types’ as well as budget lim itations. The model includes decision variabIes for the quantity of land type i to be acquired in region j to meet demand in region k. Following a detailed description of the basic formulation, the model is expanded to include changes (per planning period) in supply, demand and cost. Other factors that could potentially affect land acquisition are listed but not discussed.

The methodology offered by Meier has two main shortcomings. First, the continuous nature of the decision variables does not coincide with the discrete nature of land acquisition. Suggesting to a decision-maker that an optimal policy would be to acquire N acres of land type i in region j may be optimal in terms of the model structure, but infeasible in real terms. Land ownership is inherently discrete and one is unlikely to sell a fraction of one’s holdings simply because they display specific characteristics. Second, the formulation presented by Meier is unable to handle the multiobjective nature of the land acquisition problem. It cannot, for example, address the issues of contiguity or compactness in an explicit manner. Both of these deficiencies are addressed in the present research.

In the next section, the several factors which bear on land acquisition are discussed. A multiobjective integer programming model is presented in section 3, followed by a description of the solution algorithm, sample results and computational experience.

2, Multiple objectives in land acquisition

The acquisition of land is inherently multiobjective in nature. In the simple case of an individual wishing to invest in real estate, one’s objectives m ight be to maximize the ‘value’ of the purchase while trying to hold down total cost. Value in this instance m ight be represented by area or development potential or estimated return on investment. If the purpose of the acquisition is to build an airport, we m ight be less concerned with cost- than with being able to buy a sufficient contiguous area. On a larger scale, if we are purchasing land for the purpose of assembling a national park or a wilderness area, compactness or accessibility m ight be the overriding concern. In each case, however, we are interested in identifying the tradeoff(s) between objectives.

Whatever the motivation, any particular acquisition of land is also likely to be influenced by a variety of physical and/or economic factors over which the decision-maker may or may not have any control. For example, the current pattern of land ownership in a region m ight preclude certain action or raise the cost of others. The decision-maker m ight be forced to purchase a

J. Wright et al., A multiobjective integer programming model 33

relatively large and expensive parcel of land to insure contiguity. For purposes of policy analysis, these physical and economic influences manifest themselves either as system constraints or as goals and objectives.

The principal features of the land acquisition problem can be represented in matrix form as shown in table 1. Each column represents a physical or economic constraint imposed on the system either by the problem environment or by the decision-maker. Each row is a representative set of objectives for land acquisition. A problem scenario can be specified by making entries in the columns associated with those constraints that apply.

TabIe 1 Matrix representation of the land acquisition problem.

STATEMENT OF OBJECTIVES

MAXI?.l IZE COMPACTNESS l4AXIMIZE AREA

MAXIMIZE COMPACTNESS MAXIMIZE AREA MINIMIZE COST

MAXIMIZE COMPACTNESS MAXIMIZE AREA

MAXIMIZE COMPACTNESS MAXIMIZE VALUE

MAXIMIZE COMPACTNESS MINIM12 COST

MAXIMIZE COMPACTNESS MINIMIZE COST

-

34 J. Wkight et al., A multiobjective integer programming model

The ‘scenarios provided as examples in the table are among those addressed here and in Wright (1982).

The representation in table 1 is not intended to be exhaustive, but rather is included to show that, while the physical and economic influences on land acquisition are relatively few in number, the kinds of different problem specifications are not. As will be shown, the nuances of the problem have important implications for the kinds of analytical tools that can be applied to its solution.

For any specified problem scenario, there exist physical and economic factors which influence not only the range of choice in land acquisition but the way in which that choice is made-as well. The physical factors of area, compactness, contiguity, and configuration are of particular importance, and they are discussed more fully below, as are economic factors in land acquisition-

2.1. Lund area

In the most general sense, land area must represent a constraint on any land acquisition problem since there must be a finite area under consideration. Furthermore, the boundaries of this area are usually fully known, and the availability of land within these boundaries fully understood. Additional lim itations on choice m ight be imposed, for example, by requiring certain parcels to be purchased or requiring total area to be larger than some pre-specified m inimum. It is also conceivable that a ceiling m ight be imposed on total area, say, if the use intended for the land being acquired has some negative effects such as waste disposal or the development of a facility which could potentially have adverse societal implications.

Besides constraining the range of choice, land area m ight also represent the motivation behind that choice. The decision-maker m ight be interested in ‘maximizing’ the total area acquired within the lim itations of the problem setting. Increased land area becomes something that is desired rather than required.

Whether land area is perceived as an objective or as a constraint, it behaves rather well as an attribute that can be included in an analytical procedure. That is, the solution space can be represented by a finite set of parcels, each having a distinctive size (area) and shape. By acquiring a specific parcel of land, the total land area increases regardless of the size, shape or proximity of the other parcels.

2.2. Compactness

Unlike land area, compactness is a much less predictable factor in the land acquisition problem. Several observations can be made in this regard. First,

J. Wright et al., A multiobjective integer programming model 35

any particular parcel of land has no convenient quantitative measure of compactness. One measure that has been suggested is the ratio of border length to total area. Consider the two shapes presented in figs. l(A) and l(B). Since the area enclosed by these two shapes is precisely the same, and since the lengths of their borders are identical, a ratio of border to area provides no information about compactness.

Second, the contribution to total compactness by any individual parcel depends on the configuration of the other parcels in the solution. This is shown graphically in fig. l(C), in which the shaded area represents land which is owned or has been designated for purchase. Faced with the decision of whether to purchase cell 1 or cell 2, one would like to acquire that cell which increased compactness the most. Clearly, since the two cells are identical in size and shape, we would choose cell 2 over cell 1. The value of

(A) (B)

(Cl

Fig. 1. Compactness as a factor in land acquisition.

36 J. Wright et al., A multiobjective integer programming model

cell 2 in increasing compactness is dictated by the configuration of -the shaded area.

2.3. Contiguity

In most applications, the issue of contiguity is a central one. It is viewed as an absolute constraint of a binary nature; either the resulting land acquisition has a continuous border or it has not. The desire to ‘maximize’ contiguity has, in most cases, no direct physical meaning. Consequently, contiguity iS treated as a constraint imposed on the system rather than as an objective.

An explicit statement of contiguity in the land acquisition problem. is relatively straightforward. It is possible to classify any border of any particular parcel or cell of land as being exactly one of three types:

Internal border =A border separating two parcels of land, both of which are acquired.

External border = A border separating two parcels of land, exactly one of which is acquired.

Open border =A border separating two parcels of land, neither of which is acquired.

Contiguity in land acquisition simply requires that for each parcel of land acquired, at least one of its borders must be internal (except, of course, for the uninteresting case of only one parcel being acquired).

As a constraint on land acquisition, contiguity not only has a direct and major influence on the problem solution, but also places constraints on the techniques that can be applied to problem resolution as well.

2.4. Conflguvation

As with contiguity, land configuration may influence the kinds of analytical techniques that can be applied to the land acquisition problem. The issue of land configuration concerns primarily the pattern of border designation, which represents a possible constraint on the land acquisition problem and is an important factor in developing a methodology for approaching the problem. Figs. 2(A) and (B) depict the two types of border designations that are possible in any particular feasible area. The ‘regular’ grid configuration presented in fig. 2(A) has the characteristics that each parcel has the same area and shape and exactly the same border structure. In contrast, fig. 2(B) shows the ‘irregular’ grid configuration in which the size

J. Wright et al., A multiobjective integer programming model

(C)

37

Fig. 2. Configuration as a factor in land acquisition.

and shape of each cell may be unique. Furthermore, the border structure is completely different in the irregular case; both the length and number of borders vary from cell to cell.

Though the n-regular grid configuration itself precludes generalizations about border structure, it is possible to overlay a regular grid of higher resolution over an irregular grid as shown in fig. 2(C).

2.5. Economic factors

In addition to the physical factors discussed above several economic influences may be involved in the formulation of the land acquisition problem. Furthermore, the effects of these influences may take the form of either constraints or objectives.

3x J. FWight et al., A multiobjective integer programming model

A budget limitation represents an economic constraint on the acquisition of land. In general, this constraint requires total expenditure for land to be bounded (usually from above) by sorue amount, with each parceI of land having a pre-defined and explicit dollar cost. Alternatively, or in addition, a ‘value’ coefficient could be established for each parcel based on cost-per-unit- area, for example, and used as a lower bound constraint.

In other instances, cost or value might be more properly viewed not as firm constraints, but as quantities to be optimized. The minimizing of costs and maximizing of value are two conventional ways of incorporating economic influences into an analytical process. A decision-maker might be more interested in the concept of profit- potential than with cost or value. In the public sector, land may be acquired for its ecological, aesthetic or environmental values for a game or forest preserve, park or other recreational purposes. Such examples present interesting problems of evaluation, a subject beyond the scope of this research. Whatever the case, economic parameters are coefficients attached to each ce31 which reflect their relative attractiveness in the land acquisition problem.

3. A multiobjective land acquisition model

In light of the previous discussion, the challenge is to formulate a solvable model that captures the salient features of the land acquisition problem in a fashion that is meaningful and useful for decision makers. A basic formulation, corresponding to entries in the bottom two rows of table 1, is presented in this section. The model is a multiobjective integer program; a solution technique is presented in the next section.

In the formulation presented in table 2, the three factors of land costs, acquired area and compactness are captured. The issue of contiguity is not modelled explicitly, but it is addressed in the sample results presented in

-section 5. The basic model formulation, presented in table 2, is a three-objective

integer program for maximizing acquired area and compactness and minimizing cost. The main decision variables are Xi, i = 1, . . . , vl, which are 0,l integer variables representing the acquisition of parcel i. The area and cost objectives are straightforward. Compactness is represented by the minimization of external border, a representation discussed below.

A given area is maximally compact if the border which encloses that area is a circle. Furthermore, any border of such an area which is not circular has a total length which is greater than its spherical counterpart. Consequently, one might argue that the most compact arrangement of a collection of cells is that which has the smallest total perimeter. In terms of our previous border classification scheme, we seek to minimize total external border; i.e., borders separating cells which are acquired from those which are not.

J. might et al., A multiobjective integer programming model 39

Table 2 Basic model formulation.

Min - WI 2 AiXi+ W, t C,X,+ W, 2 c SiI.Pij+N,), i=l i=l i=L jeT*

(1)

subject to

Xi-Xj-FP,+Nij=O Vi, jE q, (2)

Xi, F,, Nij=O, 1 Vi, j E Ti, (3) where

Ai = the area of cell i, Sij = the length of the border between cells i and j, Ci = the cost of acquiring cell i, i = the index for land parcels, n = the number of cells in the feasible area, W, = the weight on the area objective, W, = the weight on the cost objective, W, = the weight on the compactness objective, z =the set of cells adjacent to cell i,= Xi = 1 if cell i is acquired,

= 0, otherwise, F, = 1 if Xi= I and Xi=O,

= 0, otherwise, IV,=1 ifXj=l andXi=O,

= 0, otherwise.

‘L

“T is the set of adjacent cells, but constraint (2) need not be written for all cells adjacent to all cells. That is if, say, for cell 6, cell 8 E T6, then constraint 2 need not be written for both i= 6 and j=S and i=8 and j=6.

This measure of compactness is captured by the third term of the objective function [eq. (l)] and constraint (2) which requires that exactly one of Pij or N, will have a value of one whenever exactly one of Xi or Xj has a value of one. For example, suppose that at optimality cell i is acquired (Xi = I) and cell j is not (Xj= 0). Then eq. (2) would be satisfied if P, = 1 and IV, = 0. Since external border is being minimized, there is downward pressure on P, and Nij such that their values at optimality will be as low as possible while still satisfying eq. (2); 1 and 0, respectively in this case.

Now suppose that both Xi and Xj are acquired at optimality; i.e., have a value of one. The smallest values assignable to P, and Nij which would satisfy eq. (2) would now be Pij =0 and N,= 0. Thus, the border separating cell i and cell j would not be counted in the objective function since it is not an external. border. The argument for the case of an open border (neither parcel acquired) is the same as for the internal border.

The weights in the objective function (W,, W,, WC in 1) are parameters for the weighting method of multiobjective programming [Cohen (197g)], used in the solution algorithm u of the next section. The weights, which must be non-negative, are varied to generate non-inferior land acquisition plans.

40 J+ FWight et al., A multiobjective integer programming model

The formuIation of table 2 is a general model that can take -on vari.ous alternative forms, depending on solution approach, the region’s configuration and other problem features. W right (1982) presents many variations on this basic problem structure. One’ variation of particular interest, which is used in subsequent sections, is for a regular grid configuration [see fig. 2(A)]. In this. case, all cells have the same area, i.e., Aj = A, Vi. The basic formulation becomes

subject to

~ Xi=~, i=l

(5)

Xi-Xj-PP,j+Nij=O Vi, jE z,

Xi, Pij, N,=O, 1 Vi,jEz

(6)

(7)

in which the area objective has become a constraint on parcel acquisition. By varying M , the total number of parcels (or area) to be acquired, a particularly meaningful set of two-objective tradeoff curves can be generated.

Note that for the ‘uniform’ (square and regular) grid, S, = a,~ Vi, j which are not corner cells, and S,= 2a for corner cells where a is the length of the side of a cell. However, any value of a can be used without affecting the outcome; a= 1 in the computational results presented below.

4. Solution algorithm

The feasible regions of integer programming problems are, by definition, _ non-convex sets. This presents a particularly challenging problem in multiobjective integer programs since ‘duality gaps’ are likely to exist. A duality gap means that there exist non-inferior (or non-dominated or efficient) solutions which cannot be found with weighting techniques. Fig. 3 demonstrates the problem. Points A and B are non-inferior integer solutions for a problem with two objectives (2, and 2,). These points can be found via a weighting technique with appropriate values for the weights W I and IV, on 2, and 2,. Point C, on the other hand, cannot be found with any set of weights; yet, it is non-inferior. We .will call such a point a ‘gap point’. Bitran (1979) calls gap points, ‘weakly efficient’ solutions.

Generating all non-inferior integer solutions for large-scale problems is an unsolved problem, Bitran (1979) and others have presented general-purpose codes, but these techniques are restricted to small problems. We have developed a technique for finding all non-inferior integer solutions for the

J. WPight et al., A multiobjective integer programming model 41

B

<-P, WI >P

Fig. 3. Duality gaps in multiobjective integer programs. [Figure 10-2 from Cohon (1978).j

problem in (4j-(7). It is a specialized algorithm, created for this problem alone.

The use of the uniform grid configuration provides a means for being able to specify the precise set of non-inferior integer solutions for the land acquisition problem in (4) through (7). Because of its unique structure, the uniform grid provides a solution space that is both finite and discrete. One is thus able to determine in advance exactly where non-inferior solutions will be found. This, coupled with a tight stopping rule results in a powerful solution methodology that allows the generation of the complete tradeoff surface for the land acquisition problem.

Two specific observations can be made to support this claim. First, for any specified number of cells to be acquired, the number of possible external borders falls within some finite range. Second, within this finite range of solutions, the number of different external border configurations is finite and

‘discrete. We consider each of these claims in turn more carefully. Suppose we are interested in the tradeoff between cost and compactness

(as measured by external border) for the purchase of a specific number of cells, say M . Suppose further that the number of candidate cells is at least 2M. Then, the maximum number of external borders possible for such an acquisition would be 4M, the case where no acquired cell is adjacent to any other acquired cell. This number represents a theoretical upper bound on the number of external borders that are possible for a purchase of M cells. The practical upper bound, call it Bg=, is likely to be much smaller and can be determined by simply finding the number of external borders that would result from the purchase of the M least expensive cells. If any of these M cells is adjacent to one or more of these other M cells, BzaX will be less than 4M. In any case, it becomes unnecessary to examine solutions which have more than Bga, external borders, for these solutions must be inferior.

42 J. VWight et al., A multiobjective integer programming model

The lower lim it on the number of possible external borders (B~in) is less intuitive, but can be computed from the following formula:

Bzi, = (<a>) + 2t, where 09

t=O if M-(<&G>)2=0,

= 1 if Mc(<m>)2 +(<a>)+ 1,

=2 if M>(<.J’%>)” +(<J%>) -I- 1, and

<,,&?> = the integer part of J%?.

As with the upper lim it on the solution space, it is unnecessary to examine solutions which have less than &$,, external borders. Consequently, meaningful integer solutions for the land acquisition problem on a regular grid are possible only for values of total external border which arc between BFi, and BzaX, inclusive.

The second proposition is equally useful. It suggests that within the range specified by B~i, and Bza,, only certain values for external border are possible. This results from the special adjacency features of the regular grid configuration. Consider the scenario posed by fig, 4. Suppose the shaded area represents land which we have already decided to acquire. We are interested in identifying all possible ways in which the addition of a cell m ight affect compactness as measured by external border. There are only five such possibilities as indicated by the candidate cells A through E. If cell A is added to the solution, all of its borders (which were formerly classified as external) become internal. The net change in number of external borders is ~4, an improvement in compactness. The addition of either of the cells labeled B’ also results in an improvement in compactness since only one external border is added while three are removed. By similar reasoning, cell C does not influence the present measure of compactness while the addition of cells I3 or E would increase total external border.

The important point is that regardless of what the final pattern of land acquisition becomes, the total number of ekternal borders at optimality will be a positive even integer. The implications of these two propositions for solving the land acquisition problem can be summarized in the form of the following proposition:

Proposition. Non-inferior solutions for the multiobjective land acquisition problem on a uniform grid can only occur for positive, even and integer values of total external border between Bmin and B,,, inclusive.

J. Wight et al., A multiobjective integer programming model 43

EXTERNAL EXTERNAL i3ORDER BORDER NET

CELL ADDED DELETED CHANGE

A 0 4 -4 B 1 3 -2 C D %

2 0 1 2

E 4 a 4

Fig. 4. The effect of land acquisition on compactness for the case of the regular grid.

The availability of this greatly reduced solution space provides the basis for a solution algorithm which can generate the tradeoff surface for the land acquisition problem to whatever level of accuracy one desires.

The algorithm presented -below applies to the following equivalent form of problem (4H7):

min t CiXi, subject to i=l

Xi-xj-PP,+N,=O Vi,jET,

ifil & sij(pij + Nijl = Lo ‘

Xi, Pij, Nij = 0, 1 Vi,jE z,

(9

(11)

(13)

44 J. Wright et al., A multiobjective integer programming model

where L is a required value of compactness to be varied over all allowable values of compactness.

4.1. A solution algorithm for land acquisition on a uniform grid

The solution procedure for generating tradeoff information can be thought of as having two phases. In phase I the solution space is defined and the precise locations in objective space of possible non-inferior integer solutions (in terms of values of external border) are identified. This is accomplished simply by using the proposition of the last section. Phase II involves solving an integer, programming problem one- or more times at each potential solution location either to fined a non-inferior solution at that iocation or to show that one- does not exist there. Once both of these phases have been completed, the precise tradeoff curve for cost and compactness for a given acquired area will have been identified.

The model used in phase II is the formulation in (9) through (13). The changes in the structure of the formulation have been made solely to facilitate the solution process, and are possible only because of the knowledge of a reduced solution space. Notice the weighting method is no longer necessary. Instead, a cut constraint has been added in the form of eq. (12) which specifies the value of total external border in the current solution. By ranging the value of L from Bmin to B,,, in increments of 2, all non- inferior integer solutions can be found. This procedure is similar to the constraint method [Cohen (1978)] with the added feature that we know in advance the feasible values of L. The specific solution algorithm operates as follows.

Step 1.

step 2. Step 3.

Step 4.

Step 5.

Step 6.

Determine the value of B,,, by summing the external borders of the M cells with the lowest cost. If there are ties for the last cell to be included in the least cost configuration, select that configuration with the lowest sum of external borders. [Note: B,,, could be obtained by solving (4H7) with W, =0, but this would entail an unnecessary use of linear, perhaps integer, programming.] Compute Bmin using formula (8) from the previous section. Set the right-hand side of eq. (12) equal to Bmin and solve the problem. If the solution is integer, go to Step 5; if not, continue. Using a branch and bound procedure, find the optimal integer solution at L=E&,; this will be a non-inferior integer solution for this value of compactness. Add 2 to the right-hand side of eq. (12) and solve. If the solution is integer, go to Step 7; if not, continue. Compute C* = mink,, C,, where 1= the set of all previously found non-inferior solutions with compactness CL. Using a branch and

J. Wright et al., A multiobjective integer programming mode. 45

bound procedure with an upper cost of C*, find the non-inferior solution at the present value of L if one exists. Go on to Step 7.

Step 7. If L= B,,,, STOP; the solution algorithm is complete for the current value of M . If L is less than B,,,, go to Step 5 and continue.

Upon termination of this procedure, the complete tradeoff curve for cost and compactness for the acquisition of A4 parcels will have been fully specified. Those solutions found without resorting to branch and bound techniques lie on the convex hull and as such, would have been found using Formulation (4)-(7) for some combination of weights on cost and compactness. Those which required branch and bound are likely to be gap points or weakly efficient solutions which would have been m issed using the weighting method for problem solution.

A branch and bound algorithm employs a bound on the maximum value (when m inimizing) of the objective function at which an optimal integer solution may exist. The lower this bound, the more efficient is the algorithm. In Step 6 the concept of dominance is used to tighten the bound in the standard branch and bound algorithm. The upper bound on cost becomes the m inimum cost of all those previously found integer solutions with a value of external border lower than the current value of L. Once the cost of any particular solution, whether integer or not, exceeds C*, that particular branch can be terminated since an integer solution with cost greater than C* must be inferior. If this is the case for all such active nodes, a non-inferior solution for the value of L does not exist.

5. Sample results

To understand more fully the value of the algorithm, it is useful to step through a sample problem. Using a sample uniform grid configuration of thirty cells (six cells’by five cells), the tradeoff curve for cost and compactness as measured by external border was generated using the algorithm described in the previous section. A summary showing relevant information for each non-inferior integer sohition is presented in table 3. A graphical depiction of the tradeoff curve is shown as fig. 5. A detailed description of the use of the solution algorithm follows.

Step 1 of the algorithm called for defining BE=, which in this example is called B&tX. By inspecting the problem data, one can identify and label the 10 ieast expensive parcels of land. Should there be alternate optima present, the solution having the lowest number of external borders should be used. For this sample problem, the solution corresponding to B&z, is shown as point A in fig. 5. By counting the number of external borders in this solution, we determine BkzX to be 30.

Table 3 Summary of results for M= 10.

Borders Sollution cost Compactness constrained at C* Comments

A B B’

C D E

E H

74 30 Unconstrained 97 14 14 98 16 16

X6 84 80 77 76 75

18 20 22

$2 28

E 97

18 97 20 86 22 84 24 80 26 77 28 76

Hull point X -1 z2- Dominated by B Hull point X -0 16,z.S - X -1 16-- Hull point Alt. optima Alt. optima

T 110

loo- I B. B'*

60-

@ HULL POINTS

EXTERNAL BORDER Fig. 5. Tradeoff curve (M = IO).

J. Wright et al., A multiobjective integer programming model 47

Next, we determine &!$,, using formula (8):

= 4(3) + 2t,

where: t=l since 10~(<~>)‘+(<~>)+1 or lOt9+3+1 so: B,!$,,= 4(3) + 2(l) = 14.

We have thus identified the solution space for the sample problem; non- inferior integer solutions will be found only for even integer values of total external border between 14 and 30, inclusive. The non-inferior set for the acquisition of 10 parcels can consist of no more than 9 solutions. We can now proceed to find those solutions.

Proceeding to Step 3 we set L= Bk:,, = 14 in eq. (12) and solve formulation (9H13). In this case, the problem terminates fractional so we proceed to Step 4. Using a conventional branch and bound algorithm we are able to locate an integer non-inferior solution for L= 14 which we label as point B on fig. 5. ,

In Step 5 we increment the constraint on total external border by 2 (L= 16) and resolve the integer program. Again, the problem terminates fractional. This time, however, the branch and bound procedure specified in Step 6 with C* = 97 (the cost at point B) does not locate a non-inferior solution. In normal applications we would have simply terminated the branch and bound algorithm without finding an integer solution, since there existed no feasible solution with cost less than 97. For demonstration purposes, we continued to branch and bound to find solution B’ which, of course is dominated by solution B as can be seen in table 3 and in fig. 5, confirming that there does not exist a non-inferior integer solution for this problem having a total external border of 16.

Moving on to Step 7, we find that since we have not examined all possible solution locations, we must continue by returning to Step 5. Setting L= 18 and resolving (9H13) results in an integer solution without resorting to branch and bound. This, solution is a hull point and, unlike point B, would have been found using formulation (4)-(7) for some combination of weights W, and W,.

The algorithm continues in this fashion until all possible solution locations have been explored. In the sample problem, non-inferior integer solutions were found at 8 out of the possible 9 locations. These solutions are labeled A, B, C, D, E, F, G, and H in table 3 and on fig. 5, It should be noted that the tradeoff ‘surface’ in objective space is, in fact, a set of unconnected points. As discussed above, there exist no feasible solutions at values of compactness other than those explored in the algorithm. The lines connecting solutions in fig. 5 were included only to show the relationships among solutions. We have

48 J. Wright et al., A multiobjectiue integer programming model

used and will continue to use the words “curve’ and ‘surface’ below with these qualifications in mind.

In the same manner, the entire tradeoff surface in three dimensions can be generated using this solution algorithm. The process is simply repeated for other values of M, a value of the area maximization objective. Depending on the level of detail required by the problem application, one can uncover the tradeoff surface for any desired degree of precision. The results for several values of M are shown in fig. 6.

4

3oc

25x:

XX

s 15(:

s -101:

5c

‘-----.. ‘\.- ---. M=25

.

\ ‘--.

----. \ ---.

-------.-, ..*-. ml=20

‘-Y\ ‘--. ‘1

‘\.\ *A

----. , M=15

‘----A. .-.‘. .---.P._* M = lo

I I I 10 20 30

BORDER Fig. 6. Tradeoff surface for the sample problem.

J. Wright et al., A multiobjective integer programming model 49

Each curve in fig. 6 is comprised of a set of non-inferior integer solutions for a specified value of M . These tradeoff curves represent the precise and complete set of solutions to the land acquisition problem on the regular sample grid for these specified areas. Together they describe a portion of a three-dimensional tradeoff surface for cost, compactness and area. Had curves been generated for all values of M from I to 30, fig. 6 would present the entire non-inferior set of this problem.

A very interesting feature of the land acquisition problem can be seen by comparing the solution ranges for each value of M in fig. 6. Notice that as M increases from 5 to 10 to J5, the range of possible values of external border increases while both of the lim its I$!$;,, and Bga, increase. This simply means that as M increases, BEax increases faster than BEin. However, notice that as we move from M = 20 to M = 25, Bf$,, actually decreases such that the range of the solution space is diminished. This counter-intuitive result is due to the fact that the feasible set of parcels is finite for this problem. Crowding results as higher values of M are investigated such that external borders occur less frequently. We can say, therefore, that the feasible area has an inherent ‘carrying capacity’ which serves to indu.ce compactness for large acquisitions.

Table 4 presents a summary of the results shown in fig. 6. Notice that for the five tradeoff curves only three of the 45 non-inferior solution locations were found to be ‘empty’; i.e., they did not contain a non-inferior integer solution.

Table 4 Summary of results: Optimization procedure (cost as a

function of border and number of cells).

Number of cells

Border 5 10 15 20 25

10 40 12 37 14 36 97 16 33 a 159 18 32 86 153 230 20 84 149 215 289 22 80 146 212 282 24 77 142 209 278 26 76 137 203 273 28 75 134 201 270 30 74 130 198 267 32 127 195 265 34 a 193 262 36 a 191 38 126 190

“No solution found for this value of external border.

50 J. Wiight et al., A multiobjective integer programming model

On contiguity

Contiguity was not exphcitly addressed in the formulations presented in this paper, but contiguity can be examined with this model. As expected, all of the solutions represented by the tradeoff surface presented in fig. 6 are not contiguous. However, each curve contains at least one contiguous non- inferior solution. Recall that for every specified number of parcels (value of A4) a value for 13zn exists and can be computed. Any solution having total external border length equal to @& is necessarily contiguous. Consequently, the leftmost solution on each curve of fig. 6 is a non-inferior contiguous solution. Fig. 7 presents a single tradeoff -curve containing these points. Specifically, this curve represents the non-inferior solution set when the constraint on contiguity is imposed.

6. Computational

5 10 15 20 25

AREA Fig. 7. Tradeoff curve for contiguous solutions.

experience

Linear programming solutions were obtained with the XMP code [Marsten (1979)] on a Digital Equipment Corporation DECsystem 10 computer. Branch and bound was performed with a FORTRAN program using XMP as a subroutine on the same computer. The 30 cell problem had

J. Wiight et al,, A multiobjective integer programming model 51

146 0,l integer variables and 69 constraints, a size that is well beyond the capabilities of any general-purpose multiobjective integer programming algorithm known to the authors.

The computational effort required to generate the solutions presented in fig. 6 has been summarized in table 5. The amount of computer time (CPU

Table 5 Summary of computational effort: Optimization routine (CPU seconds).”

Number of cells

Border 5 10 15 20 25 Total

10

12

14

16

1X

20

22

24

26

28

30

32

34

36

38

Total

Average

Hull points

12.37 (2) 4.97

(0) 88.30 30.76

(14) (8) 6.13 6.40

(0) (1) 6.08 6.12

(0) (0) 36.51 (6) 13.70 c-4 5.97

(0) 6.13

(1) 7.11

(1) 6.73

(01

117.85 119.43 (16) (19) 23.57 13.27 (3.2) (1.8)

3 3

55.90 (8) 59.30 (8)

21.60 (2)

25.60 (4)

28.60 (4) 6.27

(1) 81.40 (8) 7.08

(0) 7.97

(0) 7.55

(0) 7.24

(0) 7.45

(0)

78.20 (10)

7.83 (0)

75.71 (10) 171.94 (18)

8.64 (0)

24.03 (3)

26.14 (41

32.17 (6)

91.72 (10) 87.42 (6) 6.72

(0)

36.10 (61

27.40 (4)

205.40 (26) 41.00 (6) 30.10 (4) 5.97

0 13.73 (2) 5.97

(0)

315.96 528.52 365.67 (35) (67) (48) 26.33 48.05 45.70 (2.8) (6.1) (6)

3 3 1

12.37 (2) 4.97

(0) 119.06 (22) 68.43 (9)

149.70 (18) 102.04 (14) 142.41 cw

424.31 (481 62.04 (8)

142.64 (16) 45.92 (41 53.87 (8)

105.24 (10) 94.66 (6) 14.70 (0)

“Numbers in parentheses indicate number of branchings required.

52 J- Wkight et al., A multiobjective integer programming model

seconds) required to generate each solution is presented. In addition, the number of branchihgs of the branch and bound procedure at each location is included in parentheses just below each entry. Zero branchings implies a hull point solution which was found upon initial problem solution. One branching simply means that the initial solution terminated fractional but the first branching resulted in an integer alternative optimum to that value.

By inspecting the column of total computer time for a solution having a specific external border, one cannot generalize about which possible locations require the most computation to explore. It is interesting, however, to note that solutions restricting external border to 26 were strikingly more efficient than others. Similarly, the amount of computation required to generate any particular tradeoff curve follows no discernible pattern. One would expect that the level of computation required would be directly related to the number of possible solution locations. This does not seem to be the case, however, as can be seen by comparing the numbers for M = 5 and M = 10, and again for M = 15 and M = 25.

7. _ Conclusions

The land acquisition problem is of general interest, and its multiobjective and discrete nature presents an analytical challenge. The model presented here captures many of the factors that bear on the problem, but there are other factors which require further research. Though contiguity can be addressed with our model, research on the explicit representation of this factor would be very useful. There are likely to be other considerations of importance that will be identified only when analytical procedures ai-e applied to actual land acquisition problems.

The algorithm presented here is a powerful solution technique for a specialized multiobjective integer program. Further research is certainly needed on generalized algorithms and the development of special algorithms for other problems of this nature.

References

Bitran, Gabriel P., 1979, Theory and algorithms for linear multiple objective programs with zero-one variables, MathematicaI Programming 17, 362-390.

Cicchetti, C.J., 1979, Some economic issues in planning urban recreation facilities, Land Economics 1 I, 3 11-3 15.

Cohon, Jared L., 1978, MuItiobjective programming and planning (Academic Press, New York). Marsten, Roy E., 1979, The design of the XMP linear Programming Library, Management

Information Systems technical report no. X0-2 (University of Arizona, Tucson, AZ). McClellan, Keith and Elliot A. Medrich, 1969, Outdoor recreation: Economic consideration for

optimal site selection and development, Land Economics 45, no. 2, 175-182. Meier, Robert C., -1968, Programming of recreational land acquisition, Socio-Economic Planning

Sciences 2, 15-24. ReVelle, C. and R. Swain, 1970, Central facilities location, Geographical Analysis 2.

J. Dwight et ‘&l., A multiobjective integer programming model c 53

Rojeski, Peter and Charles ReVelle, 1970, Central facilities location under an investment constraint, Geographical Analysis 8, no. 4, 342-360.

Schuler, A. and J.C. Meadows, 1975, Planning resource use on national forests to achieve multiple objectives, Journal of Environmental Management 3, 351-366.

Webber, M.J., 1980, A theoretical analysis of aggregation in spatial interaction models, Geographical Analysis 12, 129-141.

Weintraub, Andres and Daniel Navon, 1976, A forest management planning model integrating silivicultural and transportation activities, Management Sciences 22, no. 1.

Wright, Jeffrey, 1982, A multiobjective approach to land acquisition, PbD. thesis (Department of Geography and Environmental Engineering, The Johns Hookins University, Baltimore, MD).