Multiple Objective Linear Programming in ...

August 26, 1999

Multiple Objective Linear Programming in Supporting Forest Management Pekka Korhonen (December1998) International Institute for Applied Systems Analysis A-2361 Laxenburg, AUSTRIA and Helsinki School of Economics and Business Administration Runeberginkatu 14-16, 00100 Helsinki, FINLAND Tel. +358-9-431 38525 Fax. +358-9-431 38535 E-mail: [email protected]

The research was supported, in part, by grants from the Foundation of the Helsinki School of Economics. All rights reserved. This study may not be reproduced in whole or in part without the author’s permission. (Forthcoming in Proceedings of 2nd Berkeley-Kvl Conference on Natural Resource Management and Workshop, August 6-12, 1998, Copenhagen.)

D:\Projects\Kvl\kvl41.doc

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Multiple Objective Linear Programming in Supporting Forest Management Pekka Korhonen, Helsinki School of Economics, Runeberginkatu 14-16, 00100 Helsinki, FINLAND, E-mail: [email protected], Tel: +358-9-431 38525, Fax: +358-9-431 38535, Home-page: http://www.hkkk.fi/methods/korhonen.html

Abstract. This paper gives a brief introduction into multiple objective linear programming and discusses its possibilities in the area of forest management. We overview basic concepts, formulations, and principles of solving multiple objective linear programming problems. As such problems s always require the intervention of a decision-maker behavioral aspectsare also considered. We use an application by Chang and Buongiorno [1981] to illustrate how a reference direction approach developed for multiple objective linear programming can be used to deal with a forest management problem. Keywords: Multiple Criteria, Multiple Objective Linear Programming, Interactive Approach, Forest Management.

1. Introduction Before taking a closer look at Multiple Objective Linear Programming (MOLP), it may be useful to first define the general concept Multiple Criteria Decision Making (MCDM). Many different definitions exist about what MCDM actually is, but most researchers working in the field might accept the following generalone: Multiple Criteria Decision Making (MCDM) refers to the solving of decision and planning problems involving multiple (generally conflicting) criteria. "Solving" means that a decision-maker (DM) will choose one "reasonable" alternative from among a set of available ones. It is also meaningful to define that the choice is irrevocable. For an MCDM-problem it is typical that no unique solution for the problem exists. Therefore to find a solution for MCDM-problems requires the intervention of a DM. In MCDM, the word "reasonable" is replaced by the words "efficient/nondominated". They will be defined later on. Actually the above definition is a strongly simplified description of the whole (multiple criteria) decision making process. In practice, MCDM-problems are not often so wellstructured that they can be considered just as a choice problem. Before a decision problem is ready to be "solved", the following questions require a lot of preliminary work: How to structure the problem? How to find essential criteria? How to handle uncertainty? These questions are by no means outside the interest area of MCDMresearchers. The Outranking Method by Roy [1973] and the AHP (the Analytic Hierarchy Process) developed by Saaty [1980] are examples of MCDM-methods, in which a lot of effort is devoted to problem structuring. Both methods are well known and widely used in practice. In both methods, the presence of multiple criteria is an essential feature, but the structuring of a problem is an even more important part of the solution process.

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When the term "support" is used in connection with MCDM, we may adopt a broad perspective and refer with the term to all research associated with the relationship between the problem and the DM. In this paper we take a narrower perspective and focus on a very essential supporting problem in Multiple Criteria Decision Making: How to assist a DM to find the “best” solution from among a set of available "reasonable" alternatives, when the alternatives are evaluated by using several criteria? Available alternatives are assumed to be defined implicitly by means of a mathematical model. This kind of problem is referred to as a Multiple Criteria Design Problem or a Continuous Multiple Criteria Problem and the term Multiple Objective Programming (MOP) is used to refer to a corresponding mathematical model. In addition, when all constraints and objective functions are linear, the model is a Multiple Objective Linear Programming (MOLP) model. The paper consists of 7 sections. In Section 2, we give a brief introduction to some foundations of multiple objective linear programming. How to generate potential "reasonable" solutions for a DM's evaluation is considered in Section 3, and in Section 4, we review general principles to solve a multiple objective programming problem. In Section 5, a reference direction approach and its dynamic version Pareto Race is introduced. A public forest management problem is solved in Section 6. Concluding remarks are given in Section 7.

2. Multiple Objective Linear Programming The MOLP problem can be written in the general form as follows: “max”

q = Cx

s.t.

(2.1) x ∈ X = {x Ax ≥ b, x ≥ 0}

where x ∈ ℜn, b ∈ ℜm, the constraint matrix A ∈ ℜm×n is of full rank m, and the objective function matrix C ∈ ℜk×n. Set X ⊂ ℜn is a feasible set and the space ℜn is called a variable space. The functions qi, i = 1, 2, …, k are objective functions. The set Q = {q  q = f(x), x ∈ X} ⊂ ℜk is called a feasible region, and it is of special interest in MOLP. The space ℜk is called a criterion space. The MOLP-problem seldom has a unique solution, i.e. an optimal solution that simultaneously maximizes all objectives. Conceptually the multiple objective linear programming problem may be regarded as a value (utility) function maximization program: max s.t.

v(q) q ∈ Q,

(2.2)

where v is a real-valued function, which is strictly increasing in the criterion space and defined at least in the feasible region Q. It maps the feasible region into a onedimensional value space ℜ. Function v specifies the DM’s preference structure over the

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feasible region. However, the key assumption in multiple objective linear programming is that v is unknown. Solutions for MOLP-problems are all those alternatives which can be the solutions of some value function v: Q → ℜ. Those solutions are called efficient or nondominated depending on the space where the alternatives are considered. The term nondominated is used in the criterion space and efficient in the variable space. Any choice from among the set of efficient (nondominated) solutions is an acceptable and “reasonable” solution, unless we have no additional information about the DM's preference structure. Definition 1. In (2.1), x* ∈ X is efficient iff there does not exist another x ∈ X such that Cx ≥ Cx* and Cx ≠ Cx*. Definition 2. In (2.1), x* ∈ X is weakly efficient iff there does not exist another x ∈ X such that Cx > Cx*. Vectors q ∈ Q corresponding to efficient solutions are called nondominated criterion vectors and vectors q ∈ Q corresponding to weakly efficient solutions are called weakly nondominated criterion vectors. The set of all efficient solutions is called the efficient set, and the set of all nondominated criterion vectors is called the nondominated set. Figure 1 can be used to illustrate the definitions. A shaded area denotes a feasible region in the two-dimensional criterion space. A nondominated set is line [B,C]. All other points are dominated, but a set of weakly nondominated points also consists of lines [A, B) and (C, D] - in addition to line [B,C]. For other points we can always find a better point on both criteria. For instance, for g1 point q1 is clearly a dominating point. The final (“best”) solution q ∈ Q of the problem (2.1) is called the Most Preferred Solution. It is a solution preferred by the DM to all other solutions. At the conceptual level, we may think it is the solution maximizing an (unknown) value function (2.2). How to find this solution is a key problem in MCDM, in general. Unfortunately, the above characterization of the most preferred solution is not very operational, because no system allows the DM to simultaneously compare the final solution to all other solutions with an aim to check if it is really the most preferred or not. It is also just as difficult to maximize a function we do not know. For example, some properties for a good system arethat it convinces the DM that the final solution is the most preferred one, does not require too much time from the DM to find the final solutionin order to give reliable enough information about alternatives, etc. Even if it is impossible to say which system provides the best support for a DM for his multiple objective linear programming problem, all proper systems have to be able to recognize, generate and operate with nondominated solutions.

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3. Generating Nondominated Solutions Despite many variations among different methods of generating nondominated solutions, the ultimate principle is the same in all methods: a single objective optimization problem is solved to generate a new solution or solutions. The objective function of this single objective problem may be called a scalarizing function according to Wierzbicki [1980]. It typically has the original objectives and a set of parameters as its arguments. The form of the scalarizing function, as well as what parameters are used, depends on the assumptions made concerning the DM’s preference structure and behavior. Two classes of parameters are widely used in multiple objective optimization: 1) weighting coefficients for objective functions and 2) reference/ aspiration/ reservation levels for objective function values. Based on those parameters, several ways exist to specify a scalarizing function. An important requirement is that this function completely characterizes the set of nondominated solutions: "for each parameter value, all solution vectors are nondominated, and for each nondominated criterion vector, there is at least one parameter value, which produces that specific criterion vector as a solution" (see, for theoretical considerations, e.g. Wierzbicki [1986]).

3.1.

A Linear Scalarizing Function

A classic method to generate nondominated solutions is to use the weighted-sums of objective functions, i.e. to use the following linear scalarizing function: max {λ’Cx  x ∈ X}.

(3.1)

Using the parameter set Λ = {λ λ > 0} in the weighted-sums linear program we can completely characterize the efficient set. However, Λ is an open set, which causes difficulties in a mathematical optimization problem. If we use cl(Λ) = {λ λ ≥ 0} instead, the efficiency of solution x cannot be guaranteed anymore. It is surely weaklyefficient, but not necessarily efficient. (See, e.g. Steuer [1986, p. 215 and 221].) When the weighted-sums are used to specify a scalarizing function in multiple objective linear programming (MOLP) problems, the optimal solution corresponding to nonextreme points of X is never unique. The set of optimal solutions always consists of at least one extreme point, or the solution is unbounded. In early methods, a common feature was to use λ’Cx as a scalarizing function, limiting considerations to efficient extreme points (see, e.g., Zionts and Wallenius [1976]).

3.2.

A Chebyshev-type Scalarizing Function

Currently, most solution methods are based on the use of a so-called Chebyshev-type scalarizing function first proposed by Wierzbicki [1980]. The same function was also used by Steuer and Choo [1983], but in a somewhat different form. We will refer to this function by the term achievement (scalarizing) function. The achievement (scalarizing) function projects any given (feasible or infeasible) point g ∈ ℜk onto the set of nondominated solutions. Point g is called a reference point, and its components

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represent the desired values of the objective functions. These values are aspiration levels.

called

The simplest form of achievement function is: gk - qk s(g, q, w) = max [ w , k ∈ K] k

(3.2)

where w > 0 ∈ ℜk is a (given) vector of weights, g ∈ ℜk, and q ∈ Q. The vector w is used to scale the objective functions (roughly) onto the same scale and/or to control the direction of projection. By minimizing s(g, q, w) subject to q ∈ Q, we find a weakly nondominated solution vector q* (see, e.g. Wierzbicki [1980], [1986]). However, if the solution is unique for the problem, then q* is nondominated. If g ∈ Q is feasible, then q* ≥ g. To guarantee that only nondominated (instead of weakly nondominated) solutions are generated, more complicated forms for the achievement function have to be used, for example: k gk - qk s(g, q, w,ρ) = max [ w ] + ρ ∑ (gi - qi), (3.3) k k∈K i=1 where ρ > 0 is small. In practice, we cannot operate with a definition "any positive value". We have to use a pre-specified value for ρ or to use a lexicographic formulation (see, e.g. Steuer [1986] or Korhonen and Halme [1996]. To apply the scalarizing function (3.3) is easy, because given g ∈ ℜk, the minimum of s(g, v, w, ρ) is found by solving the following LP-problem:

min

k ε + ρ ∑ (gi - qi) i=1

s.t.

(3.4) q ∈Q ε ≥ (gi - qi) / wi , i =1, 2, . . . , k ,

The problem (3.4) can be further written as: k

min

ε + ρ ∑ (gi - qi) i=1

s.t.

(3.5) q ∈Q q + εw - z = g z ≥ 0.

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To illustrate the use of the achievement scalarizing function, consider a two-criteria problem with a feasible region having five extreme points {(0,0), (0,3), (2,3), (7,0.5), (7,0)}, as shown in Figure 1.

q2 4

A

B g2

q1 2

q2 g1 0

C 4

D

8

q1

Figure 1. Illustrating the projection of a feasible and an infeasible aspiration level point onto the nondominated surface. In Fig.1, the thick solid lines describe the indifference curves when ρ = 0 in the achievement scalarizing function. The thin dotted lines stand for the case ρ > 0. Note that the line from (2,3) to (8,0) is nondominated and the lines from (0,3) to (2,3) (except point (2,3)) and (7,0.5) to (7,0) (except point (7,0.5)) are only weaklynondominated, but dominated. Let us assume that the DM first specifies a feasible aspiration level point g1 = (2,1). Using a weight vector w = (2,1), the minimum value of the achievement scalarizing function (-1) is reached at a point v1 = (4,2) (cf. Figure 2). Correspondingly, if an aspiration level point is infeasible, say g2 = (8,2), then the minimum of the achievement scalarizing function (+1) is reached at point v2 = (6,1). When a feasible point dominates an aspiration level point, then the value of the achievement scalarizing function is always negative; otherwise it is nonnegative. It is zero, if an aspiration level point is weakly-nondominated.

4. Solving Multiple Objective Problems Currently, the systems developed for solving multiple objective (linear) programming problems are interactive. The specifics of these procedures vary, but they have several common characteristics. For example, at each iteration a solution, or a set of solutions are generated for a DM’s examination. As a result of the examination, the DM inputs information in the form of tradeoffs, pairwise comparisons, aspiration levels, etc. The responses are used to generate a presumably improved solution. The ultimate goal is to find the most preferred solution of the DM. Which search technique and termination rule is used heavilydepends on the underlying assumptions postulated about the behavior

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of the DM and the way in which these assumptions are implemented. In MCDMresearch there is a growing interest in the behavioral realism of such assumptions. Based on the role that the value function (2.3) is supposed to play in the analysis, we can classify the assumptions into three categories: 1.

Assume the existence of a value function v, and assess it explicitly. (Actually, this approach is not usually classified under the MCDM-category.)

2.

Assume the existence of a stable value function v, but do not attempt to assess it explicitly. Make assumptions of the general functional form of the value function. (see, e.g. Geoffrion, Dyer, and Feinberg [1972]; Zionts and Wallenius [1976]).

3.

Do not assume the existence of a stable value function v, either explicit, or implicit (see, e.g. (Wierzbicki [1980]; Steuer and Choo [1983]).

For an excellent review of several interactive multiple criteria procedures, see Steuer [1986]. See also for surveys Hwang and Masud [1979], Shin and Ravindran [1991], and White [1990]). Other well-known books which provide a deeper background and additional references especially in the field of Multiple Objective Optimization include Cohon [1978], Haimes et al. [1990], Ignizio [1976], Sawaragi et al. [1985], Yu [1985], and Zeleny [1982].

5. A Reference Direction Approach Our purpose is to describe an interactive method called a Reference Direction Approach developed by Korhonen and Laakso [1986]. The Achievement Scalarizing Function is the main theoretical basis on which the method lies. By parametrizing the function, it is possible to project the whole vector onto the nondominated frontier as originally proposed by Korhonen and Laakso [1986]. The vector to be projected is called a Reference Direction Vector and the method is called Reference Direction Approach, correspondingly. When a direction is projected onto the nondominated frontier, a curve traversing across the nondominated frontier is obtained. Then an interactive line search is performed along this curve. The idea enables the DM to make a continuous search on the nondominated frontier. The corresponding mathematical model is a simple modification from the original model (3.5) developed for projecting a single point:

min

k ε + ρ ∑ (gi - qi) i=1

s.t.

(5.1) x ∈X q + εw - z = g + tr z ≥ 0,

where t : 0 → ∞ and r ∈ ℜk is a reference direction. In the original approach, a reference direction was specified as a vector starting from the current solution and

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passing through the aspiration levels. The DM was asked to give aspiration levels for the criteria. The basic idea of the original method is illustrated in Figure 2.

Projection of the Reference Direction A Reference Direction

g

i

q

i+1

Nondominated Set

q

i

Feasible Region

Figure 2. Illustration of the Reference Direction Approach Korhonen and Wallenius [1988] improved upon the original procedure by making the specification of a reference direction dynamic. The dynamic version was called Pareto Race. In Pareto Race, the DM can freely move in any direction on the nondominated frontier he/she likes, and no restrictive assumptions concerning the DM’s behavior are made. Furthermore, the objectives and constraints are presented in a uniform manner. Thus, their role can also be changed during the search process. The method and its implementation is called Pareto Race. The whole software package consisting of Pareto Race is called VIG (Korhonen [1987]). In Pareto Race, a reference direction r is determined by the system on the basis of preference information received from the DM. By pressing number keys corresponding to the ordinal numbers of the objectives, the DM expresses which objectives he/she would like to improve and how strongly. In this way he/she implicitly specifies a reference direction. Figure 3 shows the Pareto Race interface for the search. Thus Pareto Race is a visual, dynamic, search procedure for exploring the nondominated frontier of a multiple objective linear programming problem. The user sees the objective function values on a display in numeric form and as bar graphs, as he/she travels along the nondominated frontier. The keyboard controls include an accelerator, gears, brakes, and a steering mechanism. The search on the nondominated frontier is like driving a car. The DM can, e.g., increase/decrease the speed, make a turn at any moment he/she likes. To implement those features, Pareto Race uses certain control mechanisms, which are controlled by the following keys:

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(SPACE) BAR: An “Accelerator” Proceed in the current direction at constant speed. F1: ”Gears (Backward)” Increase speed in the backward direction. F2: ”Gears (Forward)” Increase speed in the forward direction. F3: “Fix” Use the current value of objective i as the worst acceptable value. F4: “Relax” Relax the “bound” determined with key F3. F5: ”Brakes” Reduce speed. F10: “Exit” num: ”Turn” Change the direction of motion by increasing the component of the reference direction corresponding to the goal's ordinal number i ∈ [1, k] pressed by DM. Pareto Race does not specify restrictive behavioral assumptions for a DM. He/she is free to make a search on the nondominated surface, until he/she believes that the solution found is his/her most preferred one.

6. A Case: Model for Multiple Use Forestry Chang and Buongiorno [1981] studied a resource allocation problem with the aim of developing a methodology for the problem of multiple use planning in public forests. They developed a mathematical model for this problem by combining goal programming and input-output analysis.1 A public forest provides society with several products and services. The management problem is to decide how much of each product and service should be provided. Since many management activities use the output of other management activities as their input, input-output tables are a convenient way to describe the relationships between various activities. First, the authors introduced an optimization model based on inputoutput tables, where the weighted sum of the final demand vector was maximized. The prices for outputs were proposed to be used as weights. Because there is not realistic to assume the availability of such prices, the authors proposed the use of the goal programming model, in which a desired goal was set to each component of the final demand vector. The authors were not fully satisfied with their model because of 1

Description of a public forest management problem is adopted from Chang and Buongiorno [1981]

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the lack of generality. To set goals, priorities, and weights is tentative in the beginning. That’s why to use the model requires several trial and error iterations. Chang and Buongiorno [1981] did not report their original model in their paper, but only a small hypothetical example. In the following, we use their example to illustrate how an interactive multiple objective linear programming method (a reference direction approach) provides a flexible tool to deal with this problem. The problem consists of eight management activities (G1 … G8) which are listed in Table 1 together with their management goals. Table 1. Management Activities and Goals for Hypothetical Multiple Use Problem # Activity Management Goal G1 Timber Harvest Supervise 5350 acres of timber harvest G2 Timber Sale Preparation Prepare 6200 acres of timber sale G3 Merchantable Stand Maintain 125 000 acres of merchantable Management growing stock at the end of the year G4 Non- Merchantable Maintain 104 000 acres of non-merchantable Stand Management stand G5 Camping Management Provide 258 000 visitor days of camping opportunities G6 Snowmobiling Management Provide 90 000 visitor days of snowmobiling opportunities G7 Road Maintenance Maintain 24 miles of roads for general uses G8 Road Construction Construct 1.5 miles of roads for general uses To produce outputs also requires resources. Seven resources are assumed to be available to achieve the various goals are listed in Table 2. Table 2. Available Resources for Hypothetical Multiple Use Forest Management Problem # Type Amount R1 Merchantable Growing Stock 127 955 acres R2 Non-Merchantable Growing Stock 111 657 acres R3 Camp Ground 582 tent pads R4 Snowmobile Trails 580 miles R5 Roads 340 miles R6 Manpower 7225 man-days R7 Budget 430 000 $ The structure of the model is given in Table 3. The coefficients on each column indicate the inputs required to produce one unit of output for that particular management activity. For instance, column 5 (x5) in Table 3 shows that to produce one camping day requires 0.93 acres of merchantable stands (G3), 12.17/1000 miles of road maintenance (G7), 1.74 tent pads (R3), 7.54 man-days (R6), and $440 (R7). All goals (G1 … G8) are formulated as the objective functions to be maximized. In the original model, the right-hand column corresponding to those rows consisted of

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goal values (Table 1), and to each row a positive and negative deviation variable was attached. Negative deviations were minimized. Table 3. The Multiple Objective Linear Programming Multiple Use Management Problem x1 x2 x3 x4 x5 x6 1 G1 1 G2 -1 1 -0.93 -1.26 G3 -.05 1 -1.04 G4 1 G5 1 G6 G7* -1.21 -0.67 -.004 -.005 -12.17 -11.62 G8* -4.36 0.95 R1 1 R2 1.74 R3 2.30 R4 R5 .10 .31 .002 .002 7.54 6.35 R6 5.73 17.59 0.11 0.11 440 476 R7 ) * The row values are multiplied by 1000 **) The coefficients are given for 1/1000 miles

Model for the Hypothetical x7**

x8**

1 1 .002

.004 .02 1.21

.045 2.55

To solve the problem we use Pareto Race as illustrated in Figure 3. Pareto Race Goal 1 (max ): Timber Har ==> ■■■■■■■■■■■■■■■■■ 5350.54 Goal 2 (max ): Timber Sal ==> ■■■■■■■■■■■■■■■■■ 6200.74 Goal 3 (max ): MerchSMan ■■■■■■■■■■ 69.873 Goal 7 (max ): Road Maint