II. SITUACIÓN DE LOS PUEBLOS INDÍGENAS EN ALC……………………………………………………….11-64
2.2. Prosperidad y Buen Vivir…………………………………………………………………………28-43
2.2.4. Salud Intercultural
ater resource planning is a complex and interdisciplinary problem in-volving many stakeholders—engineers, system analysts, economists, social scientists, environmentalists, and decision-makers, to name a few. Each of these groups has its own perceptions and ideas on what should be the objectives in water resources planning. Also, within the same group, people may differ in their ideas of what exactly these should be. Maximizing aggre-gated net monetary benefits to all the parties that are affected by the water resources project (positively or negatively) is an important objective perceived by many, but there are other objectives relating to environmental quality, re-gional development, resource utilization, social well-being, etc. that are also important. Therefore, it is possible that we may have to consider multiple objectives in water resources planning. Some of these objectives can be fied, some are very difficult to quantify, while some others cannot be quanti-fied at all.
We shall restrict ourselves, in this book, to only those multiple objectives that can be quantified, and avoid discussion on what should be the real objec-tives in water resources planning, as this is simply a matter of policy.
Some of the multiple objectives may conflict with one another. For example, in a reservoir project intended to satisfy both irrigation and power generation from a powerhouse located downstream of the dam, water withdrawn for irri-gation is not available for power, and releases for power are not available for irrigation. Irrigation and power production, in this case, are conflicting with each other. If the reservoir is meant to serve recreation also, then, among the three objectives, irrigation, hydropower generation and recreation, each objec-tive is in conflict with the other two. This is because all of them share the same resource, and storage used for any one purpose is not available for the other two (recreation needs storage).
4.1 NONINFERIOR SOLUTIONS
We shall introduce some basic terminology and mathematical basis for multiobjective planning before discussing the popular techniques used therein.
The concept of noninferior solutions is basic to the mathematical framework for multiobjective planning. A noninferior solution is one in which no increase in any objective is possible without a simultaneous decrease in at least one of the other objectives.
Consider a problem in which two objectives, z1 and z2, are to be maximized and let both be functions of the decision variable x (Fig. 4.1).
Fig. 4.1 Noninferior Set of Solutions (x1 £ x £ x2)
It is clearly seen that solutions in the range x < x1 and x > x2 can be eliminated from consideration, as the objective values in this range are worse than some in the range x1£ x £ x2. This range is therefore called a noninferior range and solutions in this range are termed noninferior solutions or pareto-admissible solutions. Each of these solutions is such that one objective value cannot be increased unless the other objective value is simultaneously decreased.
In some cases, the objectives cannot be measured in monetary terms and they may just be expressed in units of their physical output.
Each of the solutions in the noninferior range gives the maximum value of any one objective for a given value of the other. Multiobjective problems, in general, do not have an ‘optimal solution’ per se. Therefore, the noninferior solutions are important. Each of these solutions may be interpreted as an out-come of putting the resource to its maximum use, or operating the system at its maximum efficiency. The decision-maker should look at these ‘efficient’ solu-tions and pick one, which in some sense is the ‘best’ (by a definition). This is called the best solution, best compromise solution, or the preferred solution.
The product transformation curve, or the efficiency frontier referred to in Section 3.3.1 is the boundary of the objective space containing the noninferior set of solutions. In a general problem, there can be many objectives and many plans (each containing a set of decisions). While the same objective may have different values for different plans, the same plan may result in different values for different objectives.
Let X be a vector of decision variables, X = (x1, x2, x3, … , xn), and Zj(X), j = 1, 2, …, p denote p objectives, each of which is to be maximized. The multiobjective problem may be written as
Maximize [Z1(X ), Z2(X ), …, Zp(X )]
subject to gi(X ) = bi i = 1, 2, …, m
The objective function in the problem is a vector consisting of p separate objectives. The constraints impose technical feasibility. This objective function in the vector form can be maximized only if it can be reduced to a scalar function, using a value judgment analysis of the different components in it.
4.2 PLAN FORMULATION
There are two essential steps in multiobjective analysis: plan formulation and plan selection. Plan formulation is aimed at generating the noninferior set of solutions (or set of technologically efficient solutions), and plan selection is the process of selecting the best compromise solution. Two of the most common approaches in formulating a multiobjective planning problem are the weighting method and the constraint method.
4.2.1 Weighting Method
In the weighting method, the objective function (in the vector form) is con-verted to a scalar by expressing it as a weighted sum of the various objectives by associating a relative weight to each objective function. If wj is the relative weight assigned to the objective Zj, then the multiobjective model is written as
Maximize Z = w1Z1 + w2Z2 + … + wpZp subject to gi(X ) = bi i = 1, 2, …, m
The relative weights, wj, reflect the trade-off or the marginal rate of transfor-mation of pairs of objective functions. These weights are varied systematically, and solutions obtained for each set of values. The solution obtained for a given set of weights gives one generated set of noninferior or efficient solutions or plans. By varying the weights in each case, a wide range of plans are obtained for further analysis before the best one is selected.
Weights imply value judgments. The determination of the set of relative weights is a complex exercise and has to be done keeping the preferences of the decision-makers in mind who, in turn, are supposed to represent the interest and preferences of the beneficiaries. This requires a study of the economic, societal and developmental priorities. For a given set of weights, however, it is easy to infer the relative values of the various objectives considered in the analysis.
One major limitation of the weighting approach is that it cannot generate the complete set of efficient plans unless the efficiency frontier is strictly convex.
If a part of it is concave, then, only the end points (and none on the curve in between them) of this part can be obtained by the weighting technique.
4.2.2 Constraint Method
In this method, one objective is maximized, with lower bounds on all the others.
Maximize Zj(X )
subject to gi(X ) = bi i = 1, 2, …, m and Zk(X ) ³ Lk for all k not equal to j
Any set of feasible values of Lk resulting in a solution with binding con-straints gives an efficient alternative (solution). If the constrained method of formulation can be solved using linear programming, it is particularly useful to conduct sensitivity analysis to infer the implied tradeoffs for given right-hand side values of the binding constraints. The dual variables of the binding con-straints with Lk on the right-hand side are the marginal rates of transformation of the objectives Zj(X ) and Zk(X ).
Example 4.2.1 Basic Problem Statement, following Loucks et al. 1981) Let objective Z1(X ) = 5X1 – 4X2 and Z2(X ) = –2 X1 + 8X2. Both are to be maximized. Assume the constraints on the variables X1 and X2 are
–X1 + X2 £ 6
X1 £ 12
X1 + X2 £ 16
X2 £ 8
X1, X2 ³ 0
1. Plot a pareto chart for admissible or noninferior solutions in decision space.
2. Plot the efficient combinations of Z1 and Z2 in objective space.
3. Maximize Z1(X ) and Z2(X ) using the weighting method, given the weights associated with Z1 and Z2 are 1 and 2 respectively, and illustrate the method in decision and objective space.
4. Illustrate the constraint method of defining all efficient solutions in the decision space.
Solution:
1. The decision space is plotted in the following figure. By plotting the constraint boundary lines, we find that the boundary of the feasible region is OBCDEF. The lines of constant Z1 and Z2 will be parallel to the Z lines shown in the figure. However, the line of maximum Z1 passes through the point F (12,0), and that of maximum of Z2 through C (2,8). This may be verified by comparing the slopes of the objective lines with those of the constraint lines. For example, the slope of the objective function line, Z1, is 5/4 (dx2/dx1 = 5/4), and the point on the boundary of the feasible region, farthest from the origin, which a line with this slope meets, is F. Similarly, Z2 is a maximum at the point C.
Therefore, the noninferior set of solutions are represented by the line CDEF. Note that the segment BC does not contain noninferior solu-tions, as both objectives Z1 and Z2 can be increased along BC (Z1 can be increased from –24 to –22, and Z2 from 48 to 60, by moving from B to C).
2. Evaluating the values of Z1(X ) and Z2(X ) at C, D, E, and F, the line of efficient combinations of Z1 and Z2 in the objective space is plotted.
This is the line CDEF in the following figure.
Decision Space
3. Weighting method
Maximize Z = w1Z1 + w2Z2 with w1= 1 and w2 = 2
Z = Z1(X ) + 2Z2(X ) = (5X1 – 4X2) + 2 (–2X1 + 8X2)
= X1 + 12 X2
Objective is to maximize Z = X1 + 12X2, subject to the given con-straints.
Decision space: The Z line, Z = X1 + 12 X2, has a slope of –1/12, in the decision space and Z is found to have a maximum value equal to 104 on the boundary of the decision space at D (8,8). [The student may verify this.]
Objective space: The objective function line Z = Z1 + 2Z2, has a slope of –1/2. The value of Z will be maximum at D (8,48) in the objective space, and is equal to Z = 8 + 2(48) = 104.
4. Constraint method
The problem may be solved using LP.
Maximize Z1(X )
subject to Z2(X ) ³ L2, along with the other constraints.
Any optimal solution for an assumed value of L2 is a noninferior solution, if the constraint with L2 on the right-hand side is binding. By varying the value of L2, we get different noninferior solutions.
An easier way to identify different noninferior solutions is to solve the following problem using LP.
Maximize Z1= 5X1 – 4X2 subject to – X1 + 4X2= L2
– X1 + X2£ 6 –X1£ 12 X1 + X2£ 16 X2£ 8 X1, X2³ 0
Solve for different values of L2 for which the problem is feasible. Each set of the maximized Z1 value and the corresponding L2 (Z2 value) defines one noninferior solution. The entire set of noninferior solutions may be obtained by solving the problem for all feasible values of L2 (problems of this type can be easily solved on a PC using PC software for LP such as LINGO).
Problems
4.2.1 A reservoir is planned both for gravity and lift irrigation through withdrawals from its storage. The total storage available for both uses is limited to 5 units each year. It is decided to limit the gravity irrigation withdrawal in a year to 4 units. If X1 is the allocation of water to gravity irrigation and X2 the allocation for lift irrigation, two objectives are planned to be maximized and are expressed as
Z1= 3X1 – 2 X2 Z2= –X1 + 4X2
1. Formulate a multiobjective planning model using weighting approach with weights for gravity and lift irrigation withdrawals being w1 and w2 respectively. Plot the decision space and the objective space and determine the optimal share of withdrawals for gravity and lift irrigations, if
(i) w1 = 1 and w2 = 2 (ii) w1 = 2 and w2 = 1.
2. Formulate the problem using the Constraint method.
[Ans: (i) X1 = 0; X2 = 5. (ii) X1 = 4; X2 = 0 to 1]
4.2.2 Refer to Example 4.2.1 illustrated earlier. Assume that while maximizing the weighted objective function, Z = w1 Z1 + w2 Z2, the decision-maker chose the point E (X1 = 12, X2 = 4) in the decision space as the preferred solution. Determine the value or range of values of the marginal rate of substitution of the objective Z2 to Z1. Interpret what it means in the perception of the decision maker.
[Ans: –10/9 > w1/w2 > –2]
4.3 PLAN SELECTION
Plan selection depends essentially on the relative preferences of the decision maker to the various objectives to be maximized. In the weighting method, e.g., if the relative weights attached to each objective are specified then the particular solution corresponding to those specified weights will be the selected plan. However, determination of these relative weights itself is a problem in multiobjective planning. Necessarily, multiobjective planning is an interactive and iterative process involving decision makers and systems analysts. Quantifi-cation of intrinsic preferences to different objectives is necessary for multi-objective planning and this is where the systems analyst’s role comes into play.
There are a number of plan generation and selection techniques that warrant a more elaborate discussion than is possible in an elementary book like this. The publications mentioned at the end of the chapter may be consulted for further reading on this.
REFERENCE
1. Loucks, D.P., J.R. Stedinger, and D.A. Haith, Water Resource Systems Planning and Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J. 1981.
2. Haimes, Y.Y., Hierarchical Analysis of Water Resources: Modeling and Optimization of Large-Scale Systems, McGraw-Hill, Inc., New York, 1977.
3. Haimes, Y.Y., W.A. Hall, and H.T. Freedman, Multiobjective Optimiza-tion of Water Resources Systems: The Surrogate Worth Trade-off Method, Elsevier, Amsterdam, 1975.