Multi-criteria Analyses

Abstract

An introduction to various approaches for modeling systems where different policy makers and/or stakeholders have different and often conflicting goals regarding desired system performance.

16.1 Introduction

Rarely do people and organizations have just one goal they are trying to satisfy. Furthermore, some of those goals or objectives may conflict with others. There may be no plan or policy that everyone will agree is the best. Just what combination of objective values is considered best will differ depending on who is being asked and sometimes when they are asked. Deciding what to do or policy to implement takes place in a political process. The role of modelers is to inform the debates that take place in these political decision-making processes. Modelers can help identify and evaluate the alternative plans or policies available and define the tradeoffs among conflicting stakeholder goals and other measures of system performance (Fig. 16.1).

Fig. 16.1

Modeling assisting stakeholders who want different policies, programs, and outcomes

Given multiple performance criteria measured in different ways, there are a variety of modeling approaches that can be used to identify their tradeoffs, if any. In this chapter, some ways of including multiple objectives in models are reviewed. Multi-criteria or multi-objective analyses are not designed to identify the best solution in cases of conflict among these objectives, but only to provide information on the tradeoffs between given sets of quantitative performance criteria. Political decisions are likely to be based on qualitative judgements in addition to any quantitative information derived from models. They will not be determined by a computer or mathematical model, but the political debates that take place prior to decision-making can often be guided by the information resulting from analysts and their models.

For example, consider the resource allocation problem introduced and solved in previous chapters. Each allocation resulted in net benefits. The objective was to find the allocations that maximized the total net benefits obtained from all allocations. A second objective may be to distribute these maximum net benefits in an equitable way. Both objectives are measured in monetary units. Even assuming everyone may agree to maximize total net benefits, subject to any environmental, ecological, legal, and social constraints, not everyone is likely to agree as to how those net benefits should be allocated among all the stakeholders. This could lead to a decision that does not maximize net economic benefits but a decision that seems more acceptable and fairer to all who are impacted by the allocation decision.

A general multi-objective optimization problem can be viewed as having a vector of objectives. Let the vector X represent the set of unknown decision-variable values that are to be determined, and Zj(X) be a performance criterion or objective function whose value is determined by the values of X. Each possible vector of feasible values of the decision variables X represents a plan. Each performance criterion or objective j is an indicator of the impact of that plan. If all n objectives Zj(X) are to be maximized, the model can be written

$$\begin{gathered} {\text{Maximize }}[Z1\left( {\varvec{X}} \right),Z2\left( {\varvec{X}} \right), \ldots ,Zj\left( {\varvec{X}} \right), \ldots ,Zn\left( {\varvec{X}} \right)]. \hfill \\ {\text{Subject to all the constraints that must be satisfied}}{.} \hfill \\ \end{gathered}$$

The objective is a vector consisting of n separate objectives. The constraints define the feasible region of decision variable values.

A vector optimization representation of a multi-objective problem may be a concise way of defining a model, but it is not very useful in solving it. Multiple objective models can be solved only if their multiple objectives can be reduced to a single-objective. Thus, the multi-objective planning problem defined above cannot, in general, be solved without additional information and some modeling modifications. There are many ways to do this. This chapter introduces some of them.

16.2 Efficiency Concept

One of the goals of multi-objective planning is to identify technologically efficient tradeoffs among mutually exclusive feasible plans. These are plans that are on the tradeoff frontier (e.g., ‘b’ in Fig. 16.2). Feasible plans that are not on this frontier (e.g., ‘a’) are inferior in the sense that it is always possible to identity alternatives that will improve one or more objective values without making others worse. The goal of multi-objective modelling is the generation of a set of technologically feasible and efficient values of all unknown decision variables and objective functions. An efficient plan is one in which any objective value cannot be improved without causing a less desirable value of one or more other objectives.

Fig. 16.2

Feasible tradeoff frontier among two maximization objectives showing the maximum value of one objective given a value of another objective

16.3 Dominance

A plan i having multiple decision variable values, X_i dominates all others if it results in an equal or superior value for all objectives j, Zj(Xi), and at least one objective value is strictly superior to those of each other plan. In symbols, assuming that all objectives j are to be maximized, plan alternative i, denoted as Xi, dominates if all objectives j Zj(Xi) ≥ Zj(Xk) for all plans k and at least one objective j* is better for some plan k: Zj*(Xi) > Zj*(Xk).

It is seldom that one plan i, X_i, dominates all others. If it does, choose it! It is more often the case that different plans will dominate all the other plans based on different objectives. Plan h may be best based on one objective, while plan k may be best based on another objective. However, if there exists two plans k and h such that Zj(Xk) ≥ Zj(Xh) for all objectives j and for some objective j*, Zj*(Xk) > Zj_(Xh), then plan k dominates plan h and plan Xh can be dropped from further consideration, at least with respect to the objectives being considered.

Referring to Fig. 16.3, plan A dominates plan C and hence C can be dropped from consideration, at least based on the two objectives shown. While plans A and B are both dominant plans, plan C may be considered best based on other considerations or objectives not included in the analysis. If some objectives are not, or cannot be, included in the analysis, inferior plans with respect to the objectives that are included in the analysis should not be rejected from eventual consideration. Dominance analysis can only deal with the objectives being explicitly considered.

Fig. 16.3

Plot showing three discrete mutually exclusive plans and their two objective values

16.4 Satisficing

Defining which plans are dominant does not help us decide which among those dominant plans may be better than others. Satisficing, illustrated in Fig. 16.4, is one approach for selecting the best plan among those being considered.

Fig. 16.4

Plot showing the objective values of multiple plans illustrating the satisficing approach for selecting the best plan

Assume all objectives are to be maximized. Satisficing requires that the participants in the decision-making process specify a minimum acceptable value for each objective. Those alternatives whose objective values do not meet these minimum acceptable values are eliminated from further consideration. If only one alternative meets these minimum requirements, select it as the best. If no alternatives have objective values that meet these minimum requirements, either reduce these requirements until an alternative meets one or more of them or enlarge the options being considered, i.e., enlarge the system. If multiple alternatives remain, those that remain can again be screened by increasing one or more of the minimum requirements until only one alternative remains, such as shown in Fig. 16.4. When used in an iterative fashion, satisficing can identify what the participants consider best of multiple alternative plans or policies.

Of course, sometimes the participants in the decision-making process will be unwilling or unable to narrow down the set of available non-inferior plans sufficiently with the iterative satisficing method. Then it may be necessary to examine in more detail the possible tradeoffs among the competing alternatives.

16.5 Lexicography

Another simple approach for determining the best alternative is called lexicography. To use this approach, the participants in the decision-making process must rank the objectives in order of priority. The plan that is the best with respect to the highest priority objective will be the one selected as superior. If there is more than one plan that has the same value of the highest priority objective, then among this set of preferred plans the one that achieves the highest value of the second priority objective is selected. If here too there are multiple such plans, the process can continue until there is a unique plan selected.

Referring to Fig. 16.3, if Z1 is the most important objective, then the best decision is B as shown in that figure. If Z2 is the most important objective, then the best decision is alternative A.

This method assumes such a ranking of the objectives is possible. Often the relative importance of various objectives may change over time or depend on other factors affecting stakeholder or decision-maker opinions. Consider, for example, the problem of purchasing an apple or an orange. Assuming you like both apples and oranges, which should you buy if you can only buy one? If you know you already have lots of apples, but no oranges, perhaps you would buy an orange, and vice versa. Hence, the ranking of objectives can depend on the current state and needs of those who will benefit from the plan and these states or needs can change.

16.6 Indifference Analysis

Another method of selecting the best plan is called indifference analysis. To illustrate the possible application of indifference analysis to plan selection, consider a simple situation in which there are only two alternative plans (A and B) and two planning objectives (1 and 2) being considered. Let Z1A and Z2A be the values of the two respective objectives for plan A and Z1B and Z2B be the values of the two respective objectives for plan B. This situation can be plotted such as shown in Fig. 16.2 where plan C is not being considered. Comparing both plans A and B when one objective is better than another for each plan can be difficult. Indifference analysis can reduce the problem to one of comparing the values of only one objective.

Indifference analysis first requires the selection of an arbitrary value for one of the objectives, say Z2*, for objective 2. It is usually a value within the range of the values Z2A and Z2B, or in a more general case between the maximum and minimum of all objective 2 values. Next, a value of objective 1, say Z1, must be selected such that the participants involved are indifferent between the hypothetical plan that would have as its objective values (Z1, Z2^*) and plan A that has as its objective values (Z1A, Z2A). In other words, Z1 must be determined such that (Z1, Z2^*) is as desirable as or equivalent to (Z1A, Z2A):

$$(Z1,Z2^{*} ) \cong (Z1A,Z2A).$$

Next, another value of the first objective, say Z1^*, must be selected such that the participants are indifferent between a hypothetical plan (Z1^*, Z2^*) and the objective values (Z1B, Z2B) of plan B:

$$(Z1^{*} ,Z2^{*} ) \cong (Z1B,Z2B).$$

These comparisons yield hypothetical but equally desirable plans for each actual plan. These hypothetical plans differ only in the value of objective 1 and, hence, they are easily compared. If both objectives are to be maximized and Z1 is larger than Z1^*, then the first hypothetical plan yielding Z1 is preferred to the second hypothetical plan yielding Z1^*. Since the two hypothetical plans are equivalent to plans A and B, respectively, plan A must be preferred to plan B. Conversely, if Z1^* is larger than Z1, then plan B is preferred to plan A.

This process is illustrated in Fig. 16.5.

Fig. 16.5

Indifference analysis involving two maximization objectives. Steps of the process are shown in red

This process can be extended to a larger number of objectives and plans, all of which may be ranked by a single common objective. For example, assume that there are three objectives Z1i, Z2i, Z3i, and n alternative plans i. Consider any plan i. A reference value Z3^* for objective 3 can be chosen and a value Z1 estimated such that one is indifferent between (Zl, Z2, Z3^*) and (Zl, Z2, Z3).

The second objective value remains the same as in the actual alternative and in the hypothetical alternative. Thus, the focus is on the tradeoff between the values of objectives 1 and 3.

Next, a new hypothetical plan containing a reference value Z2^* together with Z3^* can be created and compared with hypothetical alternative (Zl, Z2, Z3^*). The focus is on the tradeoff between the values of objectives 1 and 2 since the third objective values are the same. A value of Z1^* must be selected along with the value of Z1 such that the participants are indifferent between (Z1^*, Z2, Z3^*) and (Z1, Z2^*, Z3^*).

$$(Z1^{*} ,Z2,Z3^{*} ) \cong (Z1,Z2^{*} ,Z3^{*} ).$$

Hence, the participants are indifferent between two hypothetical plans that are both equivalent to the actual one. The last hypothetical plans, (Zl, Z2^*, Z3^*), differ only by the value of the first objective. The plan that has the largest value for objective 1 will be the best plan. This was identified using only pair-wise comparisons among multiple objective values.

All n plans can be ranked just based on the value of this single-objective.

All the methods presented so far deal with discrete mutually exclusive plans, each defined by known discrete values of their decision variables. The remaining methods assume these values are unknown but will depend on the relative importance of each objective. Objective values are allowed to vary continuously over all possible feasible values. The purpose of these methods is to identify efficient combinations of objective values, along with their corresponding decision variable values, and the tradeoffs among them.

Two common approaches for identifying non-dominated plans that together identify the efficient tradeoffs among all the objectives Zj(X) are the weighting and constraint methods. Both methods require numerous solutions of a single-objective optimization model to generate points on the objective functions’ efficiency frontier.

16.7 The Weighting Method

The weighting approach involves assigning a relative weight to each objective and adding them together. This converts the objective vector to a scalar. This scalar is the weighted sum of the separate objective functions. The multi-objective model becomes

$$\begin{gathered} {\text{Maximize}}\;Z = [w1Z1\left( {\varvec{X}} \right) + w2Z2\left( {\varvec{X}} \right) \ldots + wjZj\left( {\varvec{X}} \right) \, \ldots + wJZJ\left( {\varvec{X}} \right)]. \hfill \\ {\text{Subject to all the relevant constraints}}{.} \hfill \\ \end{gathered}$$

The non-negative weights, wj, are constants specified by the modeler.

The values of these weights, wj, can be varied systematically, and the model solved for each combination of weight values, to generate a set of technically efficient (non-inferior) plans.

The foremost attribute of the weighting approach is that the tradeoffs or marginal rate of substitution of one objective for another at each identified point on the objective function’s efficiency frontier is dependent on the relative weights. The marginal rate of substitution between any two objectives Zj and Zk, at specified values of the decision variables X, is

$$[dZj/dZk] = wk/wj.$$

This applies when each of the objectives is continuously differentiable at the point X.

The relative weights can be varied over reasonable ranges to generate a wide range of plans that reflect different priorities among the objectives. Alternatively, specific values of the weights can be selected to reflect preconceived ideas of the relative importance of each objective. The prior selection of weights requires value judgments. As analysts, we are not asking decision-makers to give us their preferred relative weights or ranking of objectives. It seems unlikely any decision-maker would want to do this for a variety of reasons. We as analysts are picking various combinations of weights to identify the efficiency frontier among conflicting objectives. It is then up to the decision-makers to decide what point on this frontier represents the best combination of objective values, and hence the best decision variable values.

If each objective value is ‘normalized’ by dividing by its maximum feasible value, then the weights can range from 0 to 1 and sum to 1, to reflect the relative importance given to each objective. Otherwise, if the values of one objective are very large compared to the values of another objective, the weight on the lower value objective must be much larger than the weight on the higher value objective to get any change in the two objective values.

Fortunately, here we are not concerned with finding the best set of weights, but merely using these weights to identify the efficient tradeoffs among the conflicting objectives. After a decision is made, the weights that produced that solution might be considered the best, at least under the circumstances and at the time when the decision is made. They will probably not be the weight values that will apply in other places in other circumstances at other times.

A principal disadvantage of the weighting approach is that it cannot generate the complete set of efficient plans unless the efficiency frontier is strictly concave (decreasing slopes) for maximization objectives. If the objective value frontier, or any portion of it, is convex, as shown in Fig. 16.6, then only the endpoints of the convex portions of the efficiency frontier will be identified using the weighting method when maximizing. Similarly, for minimizing concave portions of efficiency frontiers. These limitations are overcome by using the constraint method.

Fig. 16.6

Using different values of the weights, w, to identify different locations on the efficiency frontier of two conflicting maximization objectives

16.8 The Constraint Method

The constraint method for multi-objective planning can produce the entire set of efficient plans for any shape of efficiency frontier assuming there are tradeoffs among the objectives. In this method, one objective, say Z_k, is maximized subject to lower limits, Lj, on the other objectives, j ≠ k. The solution of the model, corresponding to any set of feasible lower limits Lj, produces an efficient alternative if the lower bounds on the other objective values are binding (Fig. 16.7).

Fig. 16.7

The constraint method for finding values on the efficiency frontier of two maximization objectives. Z1 is being maximized subject to lower bounds on Z2.  

In its general form, the constraint model is

$$\begin{gathered} {\text{Maximize}}\;Zk\left( X \right). \hfill \\ {\text{Subject to, in addition to the other constraints in the model,}} \hfill \\ Zj\left( X \right) \ge Lj\quad \quad \forall j \ne k. \hfill \\ \end{gathered}$$

Note that the dual variables associated with the right-hand-side values Lj are the marginal rates of substitution or rate or change of Zk(X) per unit change in Lj (or Zj(X) if binding).

An efficiency frontier identifying the tradeoffs among conflicting objectives can be defined by solving the model for many values of the lower bounds. Just as with the weighting method, this can be a tedious job if there are many objectives. If there are more than three objectives, all the tradeoffs cannot be plotted. Pair-wise tradeoffs that can easily be plotted do not always clearly identify non-dominated alternatives.

The number of solutions to a weighting or constraint method model can be reduced considerably if the participants in the decision-making process can identify the acceptable ranges of the values of weights or lower limits. However, this is not the language of decision-makers. Decision-makers who count on the support of each interest group represented by each objective are not likely to specify weights that imply the relative importance of those various stakeholder interests. In addition, decision-makers should not be expected to know what they may want until they know what they can get, and at what cost (often politically as much as economically). However, there are ways of modifying the weighting or constraint methods to reduce the amount of effort in identifying these tradeoffs as well as the amount of information generated that is of no interest to those making decisions. This can be done using interactive methods that are discussed shortly.

The weighting and constraint methods are among many methods available for generating efficient or non-inferior solutions. The use of methods that generate many solutions, even just efficient ones, assumes that once all the non-inferior alternatives have been identified, the participants in the decision-making process will be able to select the best compromise alternative from among them. In some situations, this has worked. However, in many multi-objective planning situations, they are not sufficient in themselves. This is because the number of feasible non-inferior alternatives is simply too large. Participants in the decision-making process will not have the time or patience to examine and evaluate each alternative efficient plan. Participants may also need help in identifying which alternatives they prefer, and some may prefer ones that are not on any efficiency frontier, as previously discussed.

There are a few methods available for assisting decision-makers in selecting their most desirable non-dominated plan. Some of the more common ones are described next.

16.9 Goal Attainment

The goal attainment method combines some of the advantages of both the weighting and constraint plan generation methods already discussed. The participants in the planning and management process specify a set of goals or targets Tj for each objective j and, if applicable, a weight, wj, that reflects the relative importance of meeting that goal compared to meeting other goals. If the participants are unable to specify these weights, the analyst must select them and then later change them based on their reactions to the generated plans (Fig. 16.8).

Fig. 16.8

Determining efficient values of the two maximization objective functions using a goal attainment approach

The goal attainment method identifies the plans that minimize the maximum weighted deviation of any objective value, Zj(X), from its specified target, Tj. The problem is to find the values of the decision variables X and objective function values that

$$\begin{gathered} {\text{Minimize }}D \hfill \\ {\text{Subject to, in addition to the other constraints in the model,}} \hfill \\ wj[Tj{-}Zj\left( {\varvec{X}} \right)] \, \le D\quad j = 1,2, \ldots ,J. \hfill \\ \end{gathered}$$

This method of multi-objective analysis can generate efficient or non-inferior plans by adjusting the weights and targets. If some targets Tj are less than Zj(X), some plans generated from goal attainment may be inferior with respect to the objective functions being maximized. Again, this model assumes all objectives are being maximized. If not, change the terms wj[Tj − Zj(X)] in the constraints to wj [Zj(X) –Tj].

16.10 Goal-Programming

Goal-programming methods also require specified target objective values, along with relative losses or penalties associated with deviations from these target values. The objective is to find the plan that minimizes the sum of all such losses or penalties. Assuming for this illustration that all such losses can be expressed as linear functions of deviations from target values, and assuming each objective is to be maximized, the general goal-programming problem is to

$$\begin{gathered} {\text{Minimize }}\Sigma_{j} [vjDj + wjEj]. \hfill \\ {\text{Subject to, in addition to the other constraints in the model,}} \hfill \\ Zj\left( X \right) = Tj{-}Dj + Ej {\text{ for each objective }}j. \hfill \\ \end{gathered}$$

The parameters vj and wj are the penalties (weights) assigned to objective value deficits or excesses, as appropriate. The weights and the target values, Tj, can be changed to get alternative solutions, or tradeoffs, among the different objectives.

16.11 Interactive Methods

Interactive methods allow participants in the decision-making process to explore the range of possible decisions without having to generate all of them, especially those of little interest to anyone (Fig. 16.9).

Fig. 16.9

Iterative method starting at an obviously inferior plan and progressively improving the two maximization objective values until an acceptable plan is reached

Some iterative methods begin with an obviously inferior solution. Based on a series of questions concerning how much more important it is to obtain various improvements of each objective, the methods proceed incrementally from that inferior solution to more improved solutions. The result is either a solution everyone agrees is best, or an efficient one where no more improvements can be made in one objective without decreasing the value of another.

16.12 Plan Simulation Performance Measures

The methods outlined above provide a brief introduction to some of the simpler approaches available for plan identification and selection. Details on these and other potentially useful techniques can be found in many books, some of which are devoted solely to this subject of multi-objective planning. Most have been described in an optimization framework to focus on those alternatives that are considered dominant and efficient.

This section describes ways of evaluating alternative plans or policies based on performance criteria values derived from simulation models. Simulation models of systems yield sets of output variable values. These are values of multiple system performance criteria, each possibly pertaining to a specific interest and measured in its appropriate units.

There are numerous ways of summarizing sets of output data that might result from simulation analyses. Calculating arithmetic or geometric mean values and their standard deviations are two ways of summarizing multiple data. Other indications of system performance include reliability, resilience, and vulnerability measures.

Reliability

The notion of reliability requires defining ranges of values of each performance criterion or objective that are considered satisfactory and the ranges of values that are considered unsatisfactory. The number of simulated values of a performance measure in the satisfactory range divided by the total number of simulated values is a measure of its reliability.

$${\text{Reliability }} = {\text{ number of satisfactory values}}/{\text{total number of values}}.$$

Reliability values associated with any objective or performance criterion range from 0 to 1.

Is a system, or model of it, that produces more reliable output over time (e.g., the red time series in Fig. 16.10) better than a less reliable (e.g., the green time series) system? Reliability measures tell one nothing about how quickly a system that produces an unsatisfactory output value recovers and returns to producing satisfactory values, nor does it indicate how bad an unsatisfactory value might be should one occur. It may well be that a system that fails relatively often, but by insignificant amounts and for short durations, will be preferable to one whose reliability is much higher but when a failure does occur, it is likely to be much more severe and take longer to return to a satisfactory state.

Fig. 16.10

Time series of two simulation model outputs, divided into satisfactory and unsatisfactory values

Resilience and vulnerability measures can quantify these vulnerability and resilience system characteristics.

Resilience

Resilience can be defined as the probability that if a system output value is unsatisfactory, the next value will be satisfactory. It is the probability of having a satisfactory value in period t + 1, given an unsatisfactory value in any period t. It can be calculated as

$$\begin{aligned} {\text{Resilience}} & = \left[ {\text{number of times a satisfactory value follows an unsatisfactory value}} \right]/ \\ & \quad \left[ {\text{number of times an unsatisfactory value occurred}} \right]. \\ \end{aligned}$$

Resilience ranges from 0 to 1 and is not defined if no unsatisfactory values occur in a particular time series.

Vulnerability

Vulnerability is a measure of the extent of the differences between the threshold value, T, that divides values into satisfactory and unsatisfactory ones, and the unsatisfactory values. Clearly, this is a probabilistic measure since such deviations from the threshold value will differ. Some analysts use expected values, some use maximum observed values, and others may quantify vulnerability in terms of a probability of exceedance distribution.

Assuming an expected value measure of vulnerability is to be used:

$$\begin{aligned} {\text{Vulnerability}}\left[ {{\varvec{deviation}}} \right] & = \left[ {{\text{sum of unsatisfactory deviations from threshold }}T} \right]/ \\ & \quad \left[ {\text{number of times an unsatisfactory value occurred}} \right], \\ \end{aligned}$$

$$\begin{aligned} {\text{Vulnerability}}\left[ {{\varvec{duration}}} \right] & = \left[ {\text{sum of failure durations}} \right]/ \\ & \quad \left[ {\text{number of failure events}} \right]. \\ \end{aligned}$$

An Example:

For an example consider the two hypothetical time series of values of a performance measure shown in Fig. 16.11. They have the same mean, 4.6, and the same variance, 7.66. One is just the 180-degree rotation of the other about the mean. Hence if the objective being maximized was the mean, or if the objective being minimized was the variance, both series would give identical values of those objectives. However, their reliability, resilience, and vulnerability measures differ. There are tradeoffs among them.

Fig. 16.11

Two time series of values of a particular performance measure

Just looking at Fig. 16.11, we can see that the reliability of the red series is 70%. The blue series reliability is 90%.

The resilience of the red series is 33%. The blue series resilience is 100%. If vulnerability is based on maximum failure, that of the blue series is greater than that of the red series. If vulnerability is based on maximum duration that of the red series is greater than that of the blue series.

Exercises

• 1. Determining efficiency frontiers by weighting and constraining multiple objectives:

  • (a) Express the following model in a form used for defining the efficiency frontier (tradeoff between the two objectives) using the weighting method and the constraint method.

    $$\begin{gathered} {\text{Maximize }}Z_{1} = \, 4X_{1} {-} \, X_{2} \hfill \\ {\text{Maximize Z}}_{{2}} = \, - {\text{2X}}_{{1}} + {\text{ 6X}}_{{2}} \hfill \\ {\text{Subject to}}: \hfill \\ \quad \quad \quad \quad \quad X_{1} \le \, 4 \hfill \\ \quad \quad \quad \quad \quad X_{1} + \, X_{2} \le \, 6 \hfill \\ \quad \quad \quad \quad \quad X_{1} \ge \, 0 \hfill \\ \quad \quad \quad \quad \quad X_{2} \ge \, 0. \hfill \\ \end{gathered}$$

  • (b) Plot the efficiency frontiers in decision (x₁ vs. x₂) and objective (z₁ vs. z₂) spaces.

• 2. Resource allocation

Consider again the resource allocation problem where three users obtain benefits B(X) from the resources X they get allocated to them. The functions B(X) and their maximum values are shown below.

$\begin{array}{*{20}c} {{\text{Function}}} & {{\text{Optimal }}X} & {\text{Optimal value of function}} \\ {B_{1} \left( {X_{1} } \right) \, = \, 6X_{1} - X_{1}^{2} } & {X_{1} = \, 3} & {B_{1} \left( 3 \right) \, = \, 9} \\ {B_{2} \left( {X_{2} } \right) \, = \, 7X_{2} - 1.5X_{2}^{2} } & {X_{2} = \, 7/3} & {B_{2} \left( {7/3} \right) \, = \, 147/18} \\ {B_{3} \left( {X_{3} } \right) \, = \, 8X_{3} - 0.5X_{3}^{2} } & {X_{3} = \, 8} & {B_{3} \left( 8 \right) \, = \, 32} \\ \end{array}$

Instead of finding the values of each allocation that maximizes the total benefits, assuming only 6 resources are available, each user wants to maximize their own benefits. This is now a multi-objective problem. Show how to find the tradeoffs among each user using the weighting, constraint, goal attainment and goal-programming methods.

• 3. Reliability, resilience, and vulnerability performance measures:

Generate a time series of random variable values from a probability distribution you select and for a specified threshold value separating satisfactory values from unsatisfactory values determine values of reliability, resilience, and vulnerability.

• 4. A multiple objective optimization problem:

Show how you could use the weighting and constraint and goal attainment methods to identify the tradeoff among various maximum values of Z1 and Z2.

$$\begin{gathered} {\text{Maximize }}Z1 \hfill \\ {\text{Maximize }}Z2 \hfill \\ Z1 = 2X. \hfill \\ Z2 = 3Y. \hfill \\ X^{2} + Y^{2} \le 16. \hfill \\ \end{gathered}$$