Lee Merkhofer Consulting Priority Systems

Technical Terms Used in Project Portfolio Management (Continued)

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Term
Explanation

M

MACBETH

Stands for Measuring Attractiveness by a Categorical Based Evaluation TecHnique, a multicriteria analysis (MCA), decision-making approach based on an additive value model. The unique feature of MACBETH is that the model is constructed and applied solely by asking the decision maker to supply qualitative ("categorical") judgments of relative attractiveness in paired comparisons.

Software tools for applying MACBETH have been available since the early 1990s. The tools interactively guide users (individual decision makers or groups) through a six-step process:

Step 1 is to structure the decision problem. Like the analytic hierarchy process (AHP), which MACBETH resembles, MACBETH encourages thinking in terms of a hierarchical structure that identifies and connects a decision goal, with criteria, and then with alternatives.


MACBETH decision structure

MACBETH decision structure for problem with 4 criteria and 3 alternatives


Step 2 is to obtain judgments regarding the relative attractiveness of the alternatives with respect to each criterion. The tool chooses two of the alternatives and asks the decision maker to select from seven candidate phrases the phrase that best describes his or her judged difference in preference between the more- and less-preferred alternatives. The phrases are: "no difference", "very weak", "weak", "moderate", "strong", "very strong", and "extreme".

Step 3 is accomplished by the software. As the judgments are entered, a comparison matrix is filled out and the consistency of responses is checked. For example, if the difference between alternatives A and B and between B and C are "moderate," then the difference between A and C must be at least "moderate." If an inconsistency is detected, a warning appears ("Inconsistent judgment!"), and the user must revise judgments. The process continues until consistent preferences have been obtained for every pair of alternatives with respect to every criterion.

Step 4, also accomplished by the software, is to transform the ordinal ranking of alternatives into values expressed on a cardinal scale. The the least attractive option is assigned a value of 0 and the most attractive option is assigned a value of 100. The software then calculates numerical values for the other alternatives compatible with the qualitative comparisons. The cardinal values are assigned via the solutions to a series of linear programming (LP) problems. The objective functions for the linear programs is minimization of the value score assigned to the more preferred alternative. The constraints are those imposed by the qualitative rankings. Within the intervals between the alternatives, the value function is assumed to be linear and specified by two parameters, a slope and a constant.


MACBETH comparison matrix

Sample comparison matrix and transformed cardinal values


Step 5is to derive weights for the additive value function using the swing weight method. First, "good" and "neutral" reference levels are defined for each criterion. The decision maker is then asked to compare the desirability of improvements from neutral to good on two criteria at a time, with the differences in preferences specified using the same seven qualitative phrases specified above. The same LP approach is then used to translate the qualitative rankings into a set of quantitative weights. The swing weights are scaled to sum to 100.

Step 6 is the determination of weighted, summed values for each alternative, taking into account the values computed for the alternatives against each criterion and the criteria weights. The alternatives are ranked and, naturally, the one with the highest overall score is recommended for selection.

As noted above, MACBETH and AHP are similar. Both rely on qualitative, pairwise comparisons. The calculation of priorities and weights in the original form of AHP is achieved using an eigenvalue method, whereas MACBETH uses a set of linear programs to gain these results. The inherent precision of either approach is limited by the precision attained from qualitative-based judgments. However, MACBETH uses an interval scale while AHP adopts a ratio scale. Thus, AHP would likely be less accurate and more difficult to use than AHP on problems where the performance of alternatives against objectives varies by orders of magnitude.

MACBETH is based on utility theory, and, therefore, possesses the advantages and limitations of that theory. However, because MACBETH is limited to using an additive value model for the utility function, it is subject to serious errors if the criteria defined aren't additive independent. The decision hierarchies of the AHP and MACBETH differ in that AHP encourages using a hierarchy of criteria and then limits comparisons to the sets of criteria under each higher-level criterion. This means that MACBETH requires more comparisons, the number of which can be a problem for all approaches that derive values from pairwise comparison.

marginal probability distribution

Relevant when a joint probability distribution describes the collective, uncertain behavior of two or more random variables. The marginal probability distribution is simply the probability distribution that describes uncertainty in one of the variables alone, without reference to the values of the other variables. See the explanation of the joint probability distribution for an illustration.

mandatory project

Also called a mandated project, a project deemed a "must do." A common example is a project necessary to comply with regulatory requirements (for instance, the requirement to conduct an environmental impact analysis prior to doing construction on federal land). A project may or may not be considered mandatory based on the opinions of the organization's senior executives (depending on whether such opinions are sufficient to determine the organization's project choices). The criteria for labeling a project mandatory should be carefully designed, as labeling too many projects mandatory reduces flexibility and may result in insufficient resources for undertaking some high-value, but discretionary projects.. Mandatory projects may or may not be formally evaluated within a Project portfolio management (ppm) process. Regardless, project portfolio management tools typically provide capability to force mandatory projects into project portfolios.

marginal probability distribution

Relevant when a joint probability distribution describes the collective, uncertain behavior of two or more random variables. The marginal probability distribution is simply the probability distribution that describes uncertainty in one of the variables alone, without reference to the values of the other variables. See the explanation of the joint probability distribution for an illustration.

market price

The price at which a good or service is exchanged for another good or service or for money.

market risk

Also called systemic risk, the risk that a project or other investment will decline in value due to factors that are external to the investment and tend to impact the market or business as a whole. Common examples of market risk factors for financial investments (stocks, bonds, etc.) include interest rates, foreign exchange rates, and commodity prices. Such factors are also often important for project investments, but there are many other external risks (e.g., a labor strike, severe weather, breakdown in corporate governance) that could simultaneously and adversely impact many unrelated projects and would, therefore, be categorized as market or systemic risks. Because market risks tend to affect many projects, they can have a major impact on the risk of project portfolios. To adequately quantify project portfolio risks, it is frequently necessary to estimate and quantify the impact of market risks on portfolio value.

mean

The average value of a set of numbers. In the case of a random variable, the mean value is the expected value.

measurable

Capable of being measured and assigned a number.

measurable value function

See value function.

methodology

A collection of related methods, concepts, and procedures relevant to some field or technical topic, such as the valuation of project investments.

metric

A measure for quantifying some aspect of business or organizational performance, for example cost, percent sales returned, market share, and return on investment. Metrics may be used both to assess previous performance (e.g., What were our monthly sales over the past 6 months?) and as a vehicle for forecasting future performance (e.g., What is our projection of sales for next month?). To address uncertainty over future performance, rather than provide a point estimate, a range or probability distribution might be assigned to a metric.

Metrics for forecasting play a critical role in project prioritization because knowing which projects would have the greatest positive impact on organizational performance is key to deciding the projects to conduct. Although there is not universal agreement over terminology, the term performance measure is used here and elsewhere to describe a metric with characteristics that make it well-suited for use in a project selection decision model.

Microsoft Project (MP)

Also referred to as Microsoft Office Project (MOP), a popular PC-based software application sold by Microsoft for supporting project management. MP is designed to help project managers develop project plans, create schedules, assign resources to tasks, track progress, manage the budget, analyze workloads, and create reports. The user can create Gannt charts, PERT charts, and project network diagrams; identify critical paths; and perform earned value analysis. The software has been distributed in various editions corresponding to different years of release and in Standard and Professional versions. A wizard is provided that walks users through the process of project creation, from assigning tasks and resources to reporting results. Project files are in a proprietary file format with extension .mpp. MP is the dominant project management software application in the PC space. Many project portfolio management tools allow project data to be transferred to and from MP.

mission statement

A (typically) brief, formal statement, that explains what the organization does. For example, the mission statement for the Nissan automotive company is, "Nissan provides unique and innovative automotive products and services that deliver superior, measurable values to all stakeholders in alliance with Renault." If an organization has developed a mission statement, it can typically be found on its website.

A mission statement communicates to employees and other stakeholders the agreed upon aim and purpose of the organization. As such, it can provide a guide for decision making. An organization's mission statement is often a good source of fundamental objectives for creating an objectives hierarchy, a useful step for developing performance measures for evaluating and prioritizing projects.

mixed integer programming

See integer programming.

model

A graphical, mathematical, or verbal representation of a decision, event, phenomenon, system, or some other aspect of the real world. A model is a simplified version of reality that eliminates considerations that are of lesser importance to the purpose of the model. A model can be useful for 1) facilitating understanding, (2) aiding decision making by providing a platform for simulating 'what if' scenarios, and (3) explaining, controlling, and predicting events on the basis of past observations and model relationships.

A mathematical model is a model expressed in the language of mathematics. Mathematical models are often used for prediction, which allows predicted outcomes to be compared with reality. The comparison frequently provides data useful for improving the mathematical model.

modeling

The process of building a model. Modeling requires creative skills as well as understanding of the thing being modeled.

modern portfolio theory (MPT)

A method developed by Nobel Prize winner Harry Markowitz for finding "efficient portfolios," portfolios that have the minimum possible risk for a given expected return. Also called portfolio management theory (or, more simply, portfolio theory), MPT provides a relationship between the market price of an investment, the investment's expected return, and the risk of that investment relative to the market as a whole. The relationship demonstrates that diversification—including within the portfolio different types of investments— often reduces risk. Although questions have been raised about some assumptions underlying MPT, the theory is often used by financial investment managers to help make investment allocation decisions.

From the standpoint of project prioritization, MPT has a major limitation—The theory is designed to be used for optimizing portfolios of financial securities, such as stocks and bonds, not projects. There are important differences between financial and project investments. With a stock portfolio, for example, an investor can choose any level of investment in each security. Project portfolios, on the other hand, typically require the organization to choose to do, or not to do, each project. Tool providers that claim their tools are based on MPT are likely being disingenuous—Any techniques provided for optimizing project portfolios probably have nothing to do with MPT. However, there are some situations where MPT can appropriately be applied to value projects and optimize project portfolios (see below).

In MPT, the uncertain return from an investment is represented as a random variable characterized by its mean (average or expected value) and standard deviation (a measure of variability about the mean). If the investment is a financial security (e.g., a stock) these statistics can be estimated from historical data. The standard deviation is interpreted as a measure of the risk associated with the investment. The theory assumes that, among those portfolios with a given expected return, the most attractive is the one having the least risk.

MPT shows that a key determinant of the risk of a portfolio is the degree of correlation among the individual investments; that is, the extent to which their prices (or returns) tend to move together. For example, an up tick in oil prices is often good for oil companies, but bad for airlines. Thus, oil stocks and airline stocks tend to be negatively correlated. Diversifying by including negatively correlated (or even uncorrelated) investments in a portfolio tends to decrease portfolio risk.

The risk that cannot be avoided, no matter how much you diversify, is referred to as systematic or market risk. Market risk stems from correlations among securities that arise because there are economy-wide perils that impact all businesses. MPT shows that the contribution of an investment to the risk of a well-diversified portfolio is determined not by its riskiness in isolation, but rather by its market risk (measured using a coefficient called beta). A related result is that, on average, the market provides higher expected returns for investments with higher market risks. Note that the way that the market increases the return on riskier assets is for the asset to trade at a lower price than does a similar but lower risk asset.

One situation in which MPT could be applied to projects is the following. Suppose a project would create a factory for producing some commodity traded in the marketplace, such as corn or gasoline. Commodities are similar in some respects to financial instruments; their prices are determined by large numbers of buyers and sellers, and historical data is available for estimating the statistics needed for MPT. The value of the proposed factory will largely be determined by the prices for the commodity that prevail during the lifetime of the factory. MPT could be used to understand how market risks impact the value and risk of the project and project portfolios that include the project. As another potential application, MPT can sometimes be used to compute a risk-adjusted discount rate for net present value (NPV) analysis or other techniques, such as economic value added (EVA), that account for the cost of capital.

monetize

To express a benefit (e.g., from a project) in equivalent monetary units.

monontonic function

Any function that is entirely nondecreasing (or nonincreasing) across its range.

monotonic transformation

Any transformation that changes one set of numbers into another set of numbers that preserves the order of the numbers. For example, transforming any variable using a monotonic (strictly increasing) function, such as that shown below, will preserve order and therefore qualify as a monotonic transformation.


Example monotonic transformation

Monte Carlo analysis

Also called Monte Carlo simulation, a risk analysis method for quantifying uncertainty over some quantity or quantities of interest. The technique requires a mathematical model, such as one constructed with Excel, that calculates as its outputs the quantities of interest. Uncertainties over the model's input variables are described by probability distributions. Rather than choose one value for each uncertain input, Monte Carlo analysis randomly selects values from the corresponding probability distributions. The model is used to calculate the output variables over and over, each time using a different set of random values from the probability distributions. Depending upon the number of uncertainties and the ranges specified for them, a Monte Carlo simulation could involve thousands or tens of thousands (or more) of recalculations. By plotting the output variables as histograms or frequency charts, the simulations produce probability distributions for the model's output variables.

Monte Carlo analysis

Monte Carlo analysis was invented by scientists working on the atomic bomb in the 1940s, who named it for the city in Monaco famed for its casinos and games of chance. Some project portfolio management tools use Monte Carlo analysis to quantify project or portfolio risk.

motivation biases

A category of related judgmental biases with the common characteristic that they tend to promote behavior consistent with one's incentives and motivations. Motivation biases include the tendency of people to behave differently when they think they are being observed, to unconsciously distort judgment to "look good" and "get ahead", and to remember their decisions as being better than they were. Many comfort zone biases are also categorized as motivation biases, since behaving in ways consistent with one's motivations and incentives typically feels comfortable.

multi-attribute decision making (MADM)

A term used by some (though not on this website) to distinguish between multi-objective decision problems wherein the decision variables are continuous versus discrete. Under this alternative terminology, multi-objective decision making (MODM) problems are those with choices that are continuous variales (for which there are an infinite number of possibilities are termed problems for multi-attribute decision. In contrast, discrete decision problems are those that have a limited number of alternatives are termed problems for multi-attribute decision making (MADM).

multi-attribute utility analysis (MUA)

Also called multi-attribute utility theory (MAUT), multi-attribute decision analysis (MADA), multi-objective decision analysis (MODA) and multi-criteria decision analysis (MCDA), MUA is a sub-field of decision analysis focused on aiding decisions with multiple objectives. The MUA consists of methods and processes for creating a model that quantifies the value of things (e.g., projects) based on a multi-attribute utility function. MUA is useful for project portfolio management because it provides a highly defensible way to quantify project value, including non-financial (or "intangible") components of value.

The term MUA derives as follows. In economics, "utility" is a measure of the value or satisfaction derived from something. Sometimes, utility depends on a single attribute. For example, the utility to me of the cash in my wallet depends on one attribute; namely, the number of dollars. The more dollars I have, the greater the utility of those dollars. Mostly, though, the utility of something depends on more than one attribute. For example, if I travel to Europe, the utility of the cash in my wallet would depend on two attributes—the number of dollars and whether those dollars are US or Euros. "Multi-attribute utility" is a measure of value that depends on or is determined by more than one attribute of the thing being valued.

More precisely, MUA is an approach for deriving a "utility function" (a decision model) that, according to decision theory, quantifies a decision maker's preferences over the available alternatives to a decision. The utility function, U, is such that the best alternative is the one that maximizes U. Thus, if we could determine the function U, we could calculate which of the alternatives to a decision is the most desirable. MUA is a step-by-step process for determining U, a process that is efficient in the common case where multiple criteria (attributes) determine the desirability of alternatives.

Utility functions for evaluating projects are typically multi-attribute because the desirability of a project depends not just on the dollars the project returns (one attribute), but also on various additional criteria that capture other types of benefits that may accrue from the project (e.g., improved safety, increased knowledge, etc.). The concept of MUA is that the correct, multi-attribute function U will be much easier to find if it can be written as some simple combination of single-attribute functions (e.g., as a sum of a utility function describing the relative desirability of various levels of financial performance and another utility function describing the desirability of various levels of safety). If the utility function separates in this way, then, deriving the function is easier because each single-attribute function can be assessed without reference to the other attributes.

MUA provides a step-by-step process for deriving a multi-attribute utility function as a combination of single-attribute functions. The first step is to structure the objectives of the decision into a hierarchy in such a way as to meet certain criteria, including preferential independence, that ensure that the utility function will separate into single attribute functions. The next step is to define attributes or performance measures that quantify the achievement of each objective and to develop a single-attribute utility function (often called a scaling function) for each measure. Tests are performed on the measures to determine how the single-attribute utility functions should be combined (e.g., added, multiplied, etc.). Since the objectives are not necessarily equally important, weights must be assigned, and techniques (e.g., the swing weight method) are provided to correctly set the weights based on the decision maker's answers to questions indicating willingness to trade off various levels of performance on the different measures.

MUA can be easily integrated with more traditional financial measures of project value. For example, within MUA, net present value (NPV) can be used as a performance measure for the project's direct financial value. Likewise, MUA can be used with expected net present value (ENPV) to explicitly address project uncertainties.

MUA has been applied to many different types of decision problems and there is a vast literature on the topic. The major benefit of the approach is that it produces a single number, expressible in equivalent dollars, that measures the overall value (utility) of a project. Since this number is derived through a step-by-step process beginning with the specification of objectives, the logic is open and explicit, can be reviewed, and may be changed if any assumption made at any step is judged to be inappropriate. Scores and weights are also explicit and are developed according to established techniques and can often be cross-referenced to other sources of information on relative values, such as results from contingent valuations and dollar tradeoffs used in government cost benefit analyses. Because MUA provides a detailed "audit trail" of assumptions for reviewers, it has been used by government agencies to help make controversial public policy decisions.

A key characteristic of MUA is its explicit and extensive reliance on judgment. The necessary judgments include value judgments provided by policy makers (e.g., Which do we want more: a project that would generate $2 million or a project that would reduce the average number of annual worker injuries by 25%?) and technical judgments provided by specialists (e.g., How do we estimate the impact of a project on the annual number of worker injuries?). This reliance on subjective judgment is sometimes interpreted as a weakness, as applications may appear overly subjective. Judgments, however, are required for virtually all important decisions. The fact that MUA makes those judgments explicit is an advantage. Since the judgments and assumptions are represented as inputs to a decision model, interested parties can explore via sensitivity analysis whether changes would alter conclusions.

A limitation of MUA is that it is not easy to apply correctly. Meeting the technical requirements necessary to satisfy the assumptions of the approach requires skill, and applications generally must be guided by experienced specialists in the field. Furthermore, like all other decision tools, the decision model produced by MUA will necessarily involve simplifications that may introduce errors into recommendations.

multi-criteria analysis (MCA)

Also called multi-criteria method, an umbrella term used to describe formal decision-making methods that involve evaluating the performance of alternatives with respect to more than one criterion. Most decisions require considering multiple criteria because decision maker's have multiple objectives, and choices impact more than one of those objectives. For example, when people choose which products to purchase they typically want high quality and low price. But, high quality typically means paying a higher price. Therefore, deciding which products to purchase requires applying more than one criterion to evaluate and compare alternatives.

Researchers have devised numerous multi-criteria methods, including:

MCA methods can be categorized as being either compensatory or non-compensatory. With a compensatory method, poor performance against one or more criteria can be compensated for by good performance relative to other criteria. With a non-compensatory method, poor performance against a criterion can disqualify an alternative regardless of how well it might perform relative to other criteria.

Most of the methods listed above are compensatory methods. Outranking methods, such as ELECTRE are mostly non-compensatory, but some versions include comparisons across criteria. The dominance method is an example of a purely non-compensatory MCA method.

,

The application of a compensatory methods typically involves four steps:

  1. Specifying criteria for evaluating alternatives
  2. Assigning weights to indicate the importance of the criteria
  3. Judging the performance of each alternative with respect to each criterion
  4. Applying an aggregation equation or mathematical algorithm to combine the weights and performance judgments to compute a "desirability" measure (number) for ranking the alternatives

Though most compensatory methods proceed following the same four steps, they differ from one another in many ways, including the assumptions required for their application and especially in Step 4, the the equation or algorithm by which the measures of performance relative to the criteria and the weights are combined to produce the overall measure of desirability that is used to rank the alternatives. For example, the weighted sum model (WSM) multiplies the estimates of performance against each criterion by a criterion weight and adds the results. The weighted product model (WPM), and as the name suggests, uses an aggregation equation that involves multiplication rather than summation. With most of the others, including the analytic hierarchy process (AHP), TOPSIS, and ELECTRE, the mathematics for obtaining the ranking metrics are more complex.

The main strength of MCA methods is their capability to integrate a diversity of criteria in a multidimensional way. As seen by the numerous versions of the methods, they can be adapted for a wide range of contexts, including the prioritization of projects. Nearly all project portfolio management tools with the capability to prioritize projects do so using some sort of MCA method. Because every MCA method has its own strengths and limitations, there is debate over which is best suited for project prioritization. However, it should be noted that many MCA methods provide ordinal rather than cardinal measures for project ranking. Ordinal MCA methods are unable to provide a quantitative measure of project value.

multi-objective decision analysis (MODA)

See multi-attribute utility analysis (MUA).

multi-objective linear programming

A variation of the linear programming problem formulation wherein more than one linear objective function is specified. For example, choose values for the 3 variables x1, x2, and x3 that maximize

y = ax1 + bx2 + cx3

and

z = cx1 + dx2 + ex3

subject to one or more linear constraints. As illustration, x1, x2, and x3 might be the investment levels for 3 proposed new product projects, y might be total profit, and z might be total sales. If N is the available budget, the constraint would be expressed by an equation of the form:


x1 + x2 + x3N .

With a multiobjective linear program, no prioritization of the multiple objectives is expressed or implied. In the example, it is merely assumed that more profit is preferred over less profit and more sales is preferred over fewer sales. Multiobjective linear programs may be solved using a multiobjective analog of the simplex method for solving ordinary linear programs.

A key issue for multiobjective linear programs is that there is typically no solution that simultaneously optimizes all objectives. In the case of the example, there may be no combination of funding levels for the 3 projects that produces the greatest possible profit and sales (for example, more funding for a new, low-cost product might increase sales but reduce profits). Accordingly, multiobjective linear programming seeks a list of "non-dominated solutions." A non-dominated solution is one from which it is impossible to improve performance on one objective without some sacrifice in at least one other objective. Thus, in the example, there might be several combinations of project funding choices which would produce different levels of profit and sales. The approach is to list each funding combination wherein it is impossible to increase one (e.g., profit) without producing a decrease in the other (sales).

In principle, multiobjective linear programming appears to have a major advantage relative to other multiobjective methods, such as multi-attribute utility analysis, that require as input difficult judgments regarding decision-maker willingness to make tradeoffs. The concept with multiobjective linear programming is to present to decision makers a variety of potential solutions, with no judgment about which is preferred. In practice, however, there is a problem. Oftentimes, there will be a number of fixed non-dominated solutions, plus an infinite number of additional solutions that represent linear combinations of the fixed solutions. If there are more than two objectives, it may not be possible to even plot the non-dominated solutions, making it very difficult for decision makers to comprehend the options let alone choose from among those options. (See goal programming for another multiobjective approach related to linear programming that avoids this problem.)

multiple account evaluation (MAE)

A form of multi-criteria analysis (MCA) wherein alternatives are evaluated separately with regard to different criteria (or categories of criteria), referred to as "accounts." For example, one evaluation might be relative to financial performance (financial account), another relative to impacts on the natural environment (environment account), another relative to socio-economic performance (social account), etc. The various evaluations are not integrated, which means that there is no need for the analysis to contain weights that imply explicit judgments regarding the relative value of improving performance for the various accounts. Basically, MAE produces a series of analyses that typically produce different priorities representing the different points of view represented by the different types of criteria. In order to make choices, decision makers can then implicitly judge the relative value of the accounts.

MAE was originally developed to support public policy analysis by the Province of British Columbia (BC) in the 1990s as an alternative to cost benefit analysis (CBA), which may be criticized as being incapable of expressing in dollar units all of the costs and benefits of policy alternatives as well as sometimes specifying misleading or controversial dollar values. In Canada especially, though elsewhere as well, MAE is a method used often by governments to prioritize public sector investments, such as transportation projects.

MAE has been described as involving three basic steps: (1) specification of the evaluation accounts, (2) assessment of advantages and disadvantages of each alternative within each account, and (3) presentation of results to decision makers. The specific methods for assessing advantages and disadvantages generally follow the principles of CBA. However, the philosophy of MAE is that it may not be constructive to assign explicit monetary values to some costs and benefits, even if techniques exist for accomplishing that task. For the evaluation within some accounts, simple scoring models may be used, and, to distinguish this from the more rigorous CBA that may be used within other accounts, the scoring may be referred to as "qualitative analysis" (even though numbers are assigned).

A criticism of MAE is that failing to make explicit the value judgments to which policy choices are sensitive suggests a less transparent decision making process. In contrast, other MCA methods often advocate assigning explicit weights for integrating the evaluations with respect to different criteria, and then varying those weights in order to explore the sensitivity of priorities to the differing weights that different stakeholders would assign. In the absence of explicit methods for combining the account evaluations, decision makers may tend to weight the accounts more or less equally. According to one critic, this can lead to the "billion-dollar cost of the project as being no more or less important than any of the other accounts, however trivial." The main advantage to MAE is that by avoiding weight assignment entirely, it provides a less controversial form of analysis that may be acceptable in situations where other forms of analysis would not be.

multiplicative utility function

A generalization of the additive utility function that allows for preferences that depend on interactions among the attributes. If there are N attributes denoted xi, the equation for the multiplicative utility function has the form:


Multiplicative utility function

(The second line of the equation uses the shorthand π symbol to indicate that the terms following the symbol and inside the brackets are multiplied.)

The Ui are single attribute utility functions and the ki (see below) are weights. The equation is termed multiplicative because it is obtained by multiplying component factors, [1+kkiUi(xi)], each of which depends on only a single attribute. The scaling parameter k is the solution to the equation:


Equation for k

Because the multiplicative function is composed of single-attribute utility functions, it, like the additive utility function, is easy to assess. The Ui may be assessed from the decision maker using the certain equivalents method. The single attribute functions are scaled so that a utility of zero is assigned if the corresponding attribute is set to the least desirable level and equal to one if the attribute is at its most desirable level. Each ki weight equals the utility when all the attributes but the i'th are at their worst level while the i'th attribute is at its best level. Thus, the ki can likewise be assessed via the certain equivalents method. Once the ki are determined, the scaling parameter k, which depends on all of the ki, may be found by finding the non-zero solution to the above equation:

If the factors in the multiplicative utility function are multiplied out, the result looks something like this:

Equation for k

The two-variable version is:


Equation for k

As can be seen from the above two forms of the function, the multiplicative utility function is an additive utility function to which additional terms have been added. These additional terms are multiplications of the single attribute functions with the parameter k and its various powers controlling the strength of these interactions. If the interaction factor is zero, all of the higher order terms in the above equation drop out, and the utility function reduces to the additive form.

Whether k is greater or less than zero determines whether the attributes behave in a complementary or supplementary fashion. Considering the two-variable case, if k is greater than zero, obtaining high performance with respect to one attribute is more highly valued if a high degree of performance is attained with respect to the other attribute. In this sense, the attributes may be described as being complementary. If, on the other hand, k is less than zero, obtaining a high performance with respect to one attribute is less important if a high degree of performance is attained with respect to the other attribute. In this case, the attributes behave in a supplementary fashion. It can also be shown that if k is less than zero, the decision maker's preference for lotteries will show risk aversion, while a value greater than zero reflects a preference for taking risks.

Though not as restrictive as the additive independence requirement for justifying an additive utility function, the multiplicative utility function requires a fairly strong independence assumption known as mutual utility independence. Attribute X is utility independent of attribute Y if conditional preferences for lotteries over X given a fixed value for Y do not depend on the particular value of Y. Note that utility independence (in contrast to additive independence) is not symmetric: it is possible that attribute X is utility-independent of attribute Y but not vice versa. To obtain mutual utility independence, it should be possible to partition the attributes into two subsets in any way, and then consider lotteries in one subset, holding the attributes in the other subset fixed. As long as preferences for the lotteries do not depend on the level of the remaining attributes, mutual utility independence holds and the multiplicative utility function should provide a good model of the decision maker’s preferences.

multiplicative value function

A value function that can be decomposed to a mathematical form that is a product of factors each of which is a function of a single-attribute value function. A multiplicative value function is analogous to a multiplicative utility function except the individual, single-attribute functions in the factors that are multiplied are value functions rather than utility functions.

multi-project management

The simultaneous planning, monitoring, and management of multiple projects interconnected by their demand for common resources. Multi-project management is more difficult than the management of a single project due to the need for integration and coordination. Tools for multi-project management generally provide a system for standardized project workflows and reporting. However, in contrast with tools for project portfolio management, tools for multi-project management do not typically provide capabilities for project prioritization or portfolio optimization.

multi-tenancy

Term used to describe a computing architecture wherein there is a single running instance of an application that simultaneous serves multiple users, user groups or organizations ("tenants"). Project portfolio management tools delivered over the internet as a service (SaaS) use this approach. Each tenant, can customize the software's preferences and settings to the extent that such settings are provided within the common software instance. Although the data from different customers may reside on a same server, security controls are used to prevent the data from passing from one customer to another.

The main benefit of multi-tenant software is ease of software management. With traditional, single-tenant architecture, each organization gets its own database and instance of the software application. When the software needs to be changed, each instance of the software must be updated. But in a multi-tenant environment, just one update to the one instance is required. This saves the software provider time and money.

multivariate

Having or involving a two or more independent mathematical or statistical variables. For example, a joint probability distribution is multivariate.

multiple account evaluation (MAE)

A form of multi-criteria analysis (MCA) wherein alternatives are evaluated separately with regard to different criteria (or categories of criteria), referred to as "accounts." For example, one evaluation might be relative to financial performance (financial account), another relative to impacts on the natural environment (environment account), another relative to socio-economic performance (social account), etc. The various evaluations are not integrated, which means that there is no need for the analysis to contain weights that imply explicit judgments regarding the relative value of improving performance for the various accounts. Basically, MAE produces a series of analyses that typically produce different priorities representing the different points of view represented by the different types of criteria. In order to make choices, decision makers can then implicitly judge the relative value of the accounts.

MAE was originally developed to support public policy analysis by the Province of British Columbia (BC) in the 1990s as an alternative to cost benefit analysis (CBA), which may be criticized as being incapable of expressing in dollar units all of the costs and benefits of policy alternatives as well as sometimes specifying misleading or controversial dollar values. In Canada especially, though elsewhere as well, MAE is a method used often by governments to prioritize public sector investments, such as transportation projects.

MAE has been described as involving three basic steps: (1) specification of the evaluation accounts, (2) assessment of advantages and disadvantages of each alternative within each account, and (3) presentation of results to decision makers. The specific methods for assessing advantages and disadvantages generally follow the principles of CBA. However, the philosophy of MAE is that it may not be constructive to assign explicit monetary values to some costs and benefits, even if techniques exist for accomplishing that task. For the evaluation within some accounts, simple scoring models may be used, and, to distinguish this from the more rigorous CBA that may be used within other accounts, the scoring may be referred to as "qualitative analysis" (even though numbers are assigned).

A criticism of MAE is that failing to make explicit the value judgments to which policy choices are sensitive suggests a less transparent decision making process. In contrast, other MCA methods often advocate assigning explicit weights for integrating the evaluations with respect to different criteria, and then varying those weights in order to explore the sensitivity of priorities to the differing weights that different stakeholders would assign. In the absence of explicit methods for combining the account evaluations, decision makers may tend to weight the accounts more or less equally. According to one critic, this can lead to the "billion-dollar cost of the project as being no more or less important than any of the other accounts, however trivial." The main advantage to MAE is that by avoiding weight assignment entirely, it provides a less controversial form of analysis that may be acceptable in situations where other forms of analysis would not be.

multi-project management

The simultaneous planning, monitoring, and management of multiple projects interconnected by their demand for common resources. Multi-project management is more difficult than the management of a single project due to the need for integration and coordination. Tools for multi-project management generally provide a system for standardized project workflows and reporting. However, in contrast with tools for project portfolio management, tools for multi-project management do not typically provide capabilities for project prioritization or portfolio optimization.

multi-tenancy

Term used to describe a computing architecture wherein there is a single running instance of an application that simultaneous serves multiple users, user groups or organizations ("tenants"). Project portfolio management tools delivered over the internet as a service (SaaS) use this approach. Each tenant, can customize the software's preferences and settings to the extent that such settings are provided within the common software instance. Although the data from different customers may reside on a same server, security controls are used to prevent the data from passing from one customer to another.

The main benefit of multi-tenant software is ease of software management. With traditional, single-tenant architecture, each organization gets its own database and instance of the software application. When the software needs to be changed, each instance of the software must be updated. But in a multi-tenant environment, just one update to the one instance is required. This saves the software provider time and money.

multivariate

Having or involving a two or more independent mathematical or statistical variables. For example, a joint probability distribution is multivariate.

murder board

Term used to describe the role of a panel established and used by some organizations to facilitate the evaluation and prioritization of projects. A project proponent must present the arguments for conducting each project to the board. The panel, composed individuals selected from different parts of the organization, raises questions about the project and present arguments against conducting it. The proponent attempts to counter the concerns. The exchange helps uncover problems with project proposals that might otherwise be overlooked. The term originated from the U.S. military.

mutual preferential independence

An independence condition relevant to a multi-attribute utility function. Mutual preferential independence is the condition where each attribute and each combination of attributes selected to measure performance against the decision maker's objectives is preferentially independent of all other attributes. Mutual preferential independence is important because it is a necessary condition for obtaining an additive value function.