System analyses involve modeling. The only way I know how to become good at model development and use is to practice. Opportunities to practice are given throughout the remaining chapters of this book starting in this chapter.

3.1 Let’s Model

To develop mathematical models, we need to use some notation for defining systems and their inputs, outputs, and various measures or indicators of performance. This chapter uses some simple examples to illustrate the modeling process and some common notations modelers use.

Many models consist of equations and inequalities that contain variables whose values are unknown and parameters whose values are assumed known. Together they define the system components and their interactions, and the system performance measures.

For example, consider creating a local community park having a specified area, A, that is to be surrounded by a fence. The perimeter of the park, P, i.e., the total length of fencing, is to be determined (Fig. 3.1).

Fig. 3.1
figure 1

A park area surrounded by a fence

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Depending on the area’s dimensions, which we don’t know, there are many possible values of P for a fixed area A.

Consider a rectangular area as illustrated in Fig. 3.2. If the area is rectangular with length L and width W, then the area A is LW and the total length of fencing P is 2L + 2W.

Fig. 3.2
figure 2

A rectangular area having length L and width W

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There are many combinations of L and W that can enclose a specified land area A. If we want to find the minimum value of P needed to enclose the specified area, A, and it is not already obvious, we can develop an optimization model of this system. Optimization models have an objective function that is to be maximized or minimized and various constraints that define the relationships among the system variables and parameters. In other words, they define the system. In this case, the objective is to find the minimum value of the length of fencing P that encloses the known area A. Hence, the model’s objective function is as follows:

\( \text{Minimize} \,\,\,\,\,\, P \):

the length of the fence needed

and expressions that define P in terms of the dimensions of A.

P ≥ 2(L+W) :

The length of the fence must at least surround the area.

A ≤ LW:

The area must be no less than some specified (known) A.

P ≥ 0:

The total length of the fence is non-negative.

\( L \ge 0, \,\,\, W \ge 0\):

The variables L and W are non-negative.

The objective function and all the inequality constraints just listed make up a model of this rectangular park. The variables P, L, and W are the unknown decision variables. The known area A is a parameter, along with the number 2.

If the area of the park is a circle, the radius, R, of the circle is unknown but is constrained by A (Fig. 3.3).

$$ A \le \pi R^2 $$
Fig. 3.3
figure 3

A circular area having radius R and circumference P

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The value of π is a known parameter, 22/7.

The needed fencing must at least surround the circular park.

$$ P \ge 2\pi R \; \text{and}\; R \ge 0. \, P \ge 0$$

The two unknown variables are non-negative.

Now obviously the solution to the circle problem is P = 2 π R and R = √(A/π) so we don’t need an optimization model to find the minimum value of P. But in the case of a rectangle, it may not be obvious what the values of the L and W are that minimize P given A. But even here a little thought will convince anyone that L will equal W and thus each will equal the square root of A, √A. But if the fence had to be of different types for the four different sides, each costing different amounts per unit length, and the objective was to minimize total cost, the solution would not be so obvious.

Before leaving this park problem, an equivalent modeling approach is to maximize the area, A, of the park given a fixed known length of fencing, P, available. Its solution will be the same as the solution to the previously defined models if the input parameter values are the same.

In the real world, this community park fencing problem may be a little more complex in that neither a rectangle nor a circle is desired or possible. Also of possible interest may be the gain in fencing that may be required for a unit gain in the area. One can determine these values by changing the parameter value A and resolving the model.

These simple examples serve to illustrate what modeling may look like and some of the notation used in defining models. Models consist of mathematical expressions that define the objectives or system performance measures as well as the constraints that specify conditions that have to be satisfied while minimizing or maximizing an objective function. The mathematical expressions contain decision variables whose values we seek and parameters whose values we assume we know.

The models just developed involving areas and their perimeters can be extended to consider three dimensions, i.e., volumes, rather than just two. Referring to the tanks shown in Fig. 3.4, there are many possible combinations of values of their dimensions that will satisfy any specified required volume, V. The best combination of values for the dimensions will depend on the design objective. One possible objective might be to minimize the area of material used for the tank’s sides, its base, and top. Another may be to minimize the total cost of the tank’s surfaces, where the costs per unit area of each surface can differ. Models can be developed that when solved will identify the values of l, w, and h of a rectangular tank, or R and H of a circular tank, as shown in Fig. 3.4, that achieve some objective, while meeting a volume V constraint.

Fig. 3.4
figure 4

Dimensions of a circular and rectangular tank

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There are many ways one can model this design problem. Different people may create different models, all of which when solved will yield the same solution if the assumed objective and parameter values are the same. Modeling is an art, and different artists rarely paint the same scene in the same way. But all models consist of equations and inequalities and each term within each equation or inequality has the same units of measure.

Assume the goal of a community public works department is to increase the reliability of the community water supply. They can do this by building a water storage tank. The greater the tank capacity, the greater will be the water supply reliability. But the greater the tank’s capacity, the greater its cost. Assume the community doesn’t want to spend more money than it has to but it has not decided what that amount should be. To help them make such a decision, they would like to know the relationship between cost and tank volume. Obviously, for a specified volume, there could be many costs depending on the tank’s dimensions. Hence, what is desired is the function defining the minimum cost associated with any specified tank volume. In other words, it wants to know the tradeoff between tank volume capacity and its minimum total cost. This tradeoff can be defined using an optimization model.

What costs money are the surfaces of the tank. For a rectangular tank, these surface areas are defined by the tank’s length l, width w, and height h. The cost per unit surface area may depend on the particular surface area, whether it is the tank’s bottom, sides, or top.

The rectangular tank’s capacity or volume, V, is the product of its length, width, and height, lwh.

To minimize the tank’s total cost, we are minimizing the cost of the sides having a total area of 2(wh + lh) and the top and bottom each having a total area of lw. Multiplying the unit cost (cost per unit area) of each surface area (CSIDE, CTOP, and CBASE) times the area defines the total cost of that surface area. Adding these total surface costs gives us the total tank cost.

The minimum cost model can be written as follows:

$$ \begin{array}{*{20}l} { {\text{Minimize Total}}\_{\text{Cost}} = {\text{C}}_{{\text{SIDE}}} 2h\left( {l + w} \right) \, + \, \left( {{\text{C}}_{{\text{TOP}}} + {\text{ C}}_{{\text{BASE}}} } \right) \, \left( {lw} \right)} \hfill \\ { {\text{Subject to:}}\,\,lwh \ge V.} \hfill \\ \end{array} $$

Solving this model for various values of the volume, V, will define the minimum cost function for storage volume, as illustrated in Fig. 3.5. Knowing the minimum (and marginal) cost associated with any particular volume should be useful information to those having to decide what the tank’s capacity should be.

Fig. 3.5
figure 5

Minimum cost function derived from the solution of the minimum cost model for various values of volume V

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In this example as with the others, there are many possible feasible solutions, i.e., solutions that satisfy the constraints. We identify and use an objective to determine the best value of all the unknown decision variables (in this case l, w, and h) associated with that objective. Different objectives may result in different ‘best’ solutions for various volumes V.

Before leaving this example problem, it is worth mentioning that there is often more than one way to view an optimization problem. For example, this problem could be viewed as finding the maximum volume V that can be obtained given a budget constraint, i.e., the money available to spend on the surfaces of the tank. The variable ‘Total_Cost’ in the above model is now known, and the objective becomes Maximize V. Nothing else changes. Again, for various values of Total_Cost, the model solution will identify the maximum volume that can be obtained and its associated dimensions l, w, and h.

Clearly, the values of the cost per unit area parameters, CSIDE, CTOP, and CBASE, will influence the resulting values of l, w, and h. If these unit costs are all the same, then we are finding the minimum total surface area associated with any volume V. In this case, the tank becomes a cube where l = w = h = \(\sqrt[3]{V.}\)

3.2 Types of Models

The examples just discussed involve finding the ‘optimal’ values of all the unknown decision variables of a particular ‘system’. Optimization models are used to find those decision variable values that maximize or minimize some function that represents some system performance goal or objective. Examples are the maximization of net economic benefits; the minimization of costs; the maximization of equity; the minimization of risks of various types; the maximization of measures of ecological, environmental, or human health; and so on. There are many different types of optimization models. The following chapters introduce some of them. They all have their advantages and limitations, and there is no one optimization method that is best for all optimization problems. But all optimization models focus on addressing ‘what should be’ the values of all the unknown decision variables given all the assumed parameter values, constraints, and system performance goals.

As opposed to optimization, simulation models focus on addressing ‘what if’. What will be the performance of the system given assumed values of all parameters and decision variables? In these models, the values of all decision variables are specified, and the model output indicates how the system performs given the various inputs and decision variable values.

The difference between optimization and simulation is illustrated in Fig. 3.6.

Fig. 3.6
figure 6

a Schematic of optimization modeling where the optimal decision variable values of a system are determined based on an assumed performance goal. b Schematic of simulation modeling where decision variable values of a system are specified, and the performance of the system is to be evaluated

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3.3 Why Model?

The reason we develop and solve models of systems is to better understand how to improve their performance and to estimate the impacts of doing so.

In both public and private sectors, there are often certain systems that may not be functioning as well as expected or desired, or there may be opportunities for modifying existing systems or building new ones that would increase social welfare or economic benefits or environmental quality or better satisfy some other system performance objective or goal. When there are many possible decisions or actions that could be taken and the best set of decisions or actions is not obvious, it often makes sense to use models to identify what decisions may have better outcomes than others. Solving models is one way of estimating the various impacts resulting from various decisions. We build and solve models to get useful information. We use models to aid us in identifying and evaluating alternative decisions in our search for the best.

Public policy modeling involves the use of tools taken from the disciplines of economics, planning, political science, operations research, statistics and probability theory, and applied mathematics. When applicable and depending on the issue or system being analyzed, it will also draw on agriculture, ecology, environmental management and policy, transportation engineering, law, and other disciplines as applicable and needed.

We often deal with systems that are so complex as to be beyond the limits of our intuitive understanding. If it is not obvious what decisions to take that will maximize system performance, then by definition, the system is complex. In these cases, we can construct models to help us study that which we seek to understand better.

Whether a model is right or wrong or too simplistic or too complex is simply a value judgment. Whether it is correct or incorrect, or a good model or bad model, depends on how well it serves its purpose, given the information needed and the time and data available. The most important question to ask is how well it promotes our understanding of how to improve the design and/or management or operation of a system and the resulting impacts. The extent to which a model aids in the development of our understanding is the basis for deciding how good the model is. Many find that just the process of building models gives them a greater understanding of the system they are modeling even before attempting to solve them.

When developing models there is always a tradeoff between reality and simplicity. A model is inevitably a simplification of reality. The question is always what to include and what to exclude. If relevant components are excluded, there is a chance that the model will be too simple to be useful. On the other hand, if too much detail is included, the model may become so complicated that, again, it fails to promote the stakeholder trust needed to fully accept its output. A recommended approach to model building is to start simple and add detail only as needed and after successfully solving the simpler model.

3.3.1 Some Cravats

Our job as modelers is to construct models with sufficient detail to provide decision-makers with the understanding and precision they need or want about the system or process of interest and for which decisions will be made. They may want to know the following:

  • What to do.

  • Where to do it.

  • How much to do—to what extent.

  • When to do it.

  • Why—what are the economic, social and environmental, or other impacts?

These questions should be answered at the level of detail, and in terms, appropriate for the level of decision-making and issues being addressed.

Modeling can help address these questions but will be based on a given set of assumptions. What are the best assumptions? Models can be helpful in determining the best decisions given the assumptions, and the objective(s), but not on identifying what assumptions are best, or correct, or true. Modeling can, therefore, help focus the political debate on just what assumptions are best rather than spending time determining what decisions are best given any assumptions.

This suggests that a modeler’s job is not over until a ‘sensitivity analysis’ is performed. In a sensitivity analysis, the assumptions should be varied over their likely values to determine just how sensitive the model’s decision variable values are to changes in the assumptions. If, as one hopes, the changes in those decision variable values are not significant, there may be less need to spend a lot of time debating the assumptions. Otherwise, there may be a greater need to find a robust set of decision values that will ensure satisfactory system performance no matter what assumptions turn out to be true.

3.3.2 Limitations and Common Sins

  • Models cannot help us invent new ideas or creative alternatives that are not considered in our models. For example, a model for determining the most economical dimensions of a rectangular tank will not suggest a circular tank may be better.

  • Modeling can be seductive—the danger of modelers or users of models believing the model is the real world.

  • Incorrect, ambiguity, or errors in model inputs result in errors in model outputs. For example, what does 8/2(2 + 2) equal? One or sixteen? Different calculators may give different answers.

  • Difficulty in verifying uncertain (future) data and assumptions.

  • Insufficient attention to the sensitivity of assumptions and uncertainty analyses.

  • Temptation to shape model results to what the client (or teacher?) wants to hear.

3.3.3 A Word of Caution

For anyone learning how to develop and solve various types of mathematical models to address various problems and issues, it is easy to become enamored with the potential power of this methodology for identifying and evaluating alternatives, and indeed for finding mathematically optimal solutions. This especially applies to those who enjoy the subject and enjoy solving puzzles. They tend to trust their models. But when a computer program says an optimal solution is found, one should look at it and ask, does the solution make sense? Are the results surprising? If so, there may be a good chance that there is an error in the model or its input. If you cannot find one, then maybe you should do all the tests and sensitivity analyses you can think of to be sure you have actually created some new knowledge or understanding. If that is the case, then brag about it! But more to the point, we all have mental models of what may be the best decision, at least generally if not in its details. These mental models may be influenced by factors not included in the mathematical ones. Hence, do not ignore your mental models and others who have them, including those as illustrated in Fig. 3.7.

Fig. 3.7
figure 7

We all have mental models, and we should not ignore them when evaluating our mathematical ones

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3.3.4 Subscripted Variables

When constructing models, it is often convenient to use subscripts or superscripts to distinguish among different variables. For example, consider allocating a resource to n different activities. Let the subscript i represent a particular activity. Then Xi can represent the allocation to the ith activity. If R is the total amount of resources available, then an obvious constraint on all the allocations is that their sum cannot exceed R.

$$ X_1 + X_2 + \cdot \cdot \cdot + X_i + \cdot \cdot \cdot + X_n \le R. $$

This can also be written using the summation sign ∑.

$$ \sum_{i = 1,n} {X_i \le R \, \text{or} \, \sum_{i = 1}^n {X_i \le R} } . $$

If for some reason you wanted to know the product of all the Xi variables, it could be written using the product sign ∏.

$$ X_1 X_2 X_3 \cdot \cdot \cdot X_i \cdot \cdot \cdot X_n = \prod\limits_{i = 1}^{n} X_i . $$

Assuming each of these allocations must be non-negative, then

$$ X_i \ge 0, \, i = 1,2, \cdot \cdot \cdot ,n $$

or if n is understood you can use the ‘for all’ sign \( \forall . \)

$$ X_i \ge 0, \, \forall i $$

It doesn’t matter what letters are used for subscripts or superscripts as long as what they signify are defined.

For example, if the subscript i denotes a location and the subscript j a particular product, and if Xij is the number of products of type j sent to location i, then

$$ \sum_j {X_{ij} } = {\text{total number of all products}}\,\,j\,\,{\text{sent to location}}\,\,i, $$
$$ \sum_i {X_{ij} } = {\text{total number of product}}\,\,j\,\,{\text{ sent to all locations}}\,\,i, $$
$$ \sum_i {\sum_i {X_{ij} } } = {\text{total number of all products sent to all locations}}. $$

where it is assumed understood how many locations i and how many different products j exist and each sum includes all the values of the associated subscript.

There will be other symbols we will be using, some of which are shown in Table 3.1. We will define others when we need them.

Table 3.1 Some modeling operations and notations (The use of the constant e will be discussed later.)
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Exercises

  1. 1.

    If \(\sum_{i = 2,4} {A\left( i \right) = A\left( 2 \right) + } A\left( 3 \right) + A\left( 4 \right)\), write out the sum: \(\sum_{i = 1,3} {\sum_{0 < j \le i} {\left( {X_{ij} } \right)} }\).

  2. 2.

    Given that \(\sum_{1}^{n}.\) represents a sum and \(\prod_{1}^{n}\) represents a product of n terms, what is the value of \({{\sum_{i = 1}^3 {\prod_{j = 1}^4 {\left( {i + j} \right)} } } / {\sum_{k = 2}^6 k }}\) = ?

  3. 3.

    Construct a conceptual model (a picture or a node-link network) of a multiple component system. Then identify what decisions are to be made and potential objectives or measures of performance.

  4. 4.

    Define the ‘modeling process’ in your own words.

  5. 5.

    What are the possible sources of uncertainty in any planning or management model and how can one deal with them?

  6. 6.

    Distinguish between simulation and optimization.

  7. 7.

    Identify some pitfalls of modeling.

  8. 8.

    Consider the following five alternative plans for providing for more security and better road maintenance. Whatever the units of performance are, they differ. Assume the alternative plans are all feasible, i.e., can be implemented but only one is to be selected.

    Alternative

    Security benefits

    Road maintenance costs

    A

    25

    30

    B

    10

    32

    C

    20

    35

    D

    15

    21

    E

    5

    25

    Which alternative would be the best in your opinion and why? Why might a decision-maker select alternative E even realizing other alternatives exist that can give more security and road maintenance?

  9. 9.

    Define a mathematical model for finding the dimensions of a cylindrical tank that minimizes the total cost of storing a specified volume of liquid. What are the unknown decision variables? What are the model parameters? How would you solve this model?