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# 6.2 Structure of Mathematical Models for Decision Support

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The components of Decision support Mathematical Models

All quantitative models are typically made up of four basic components (see Figure 6.1): result (or outcome) variables, decision variables, uncontrollable variables (and/or param-eters), and intermediate result variables. Mathematical relationships link these components together.

In nonquantitative models, the relationships are symbolic or qualitative. The results of decisions are determined based on the decision made (i.e., the values of the decision variables), the factors that cannot be controlled by the decision maker (in the environment), and the relationships among the variables. The modeling process involves identifying the variables and relationships among them. Solving a model determines the values of these and the result variable(s).

Result (Outcome) Variables Result (outcome) variables reflect the level of effectiveness of a system; that is, they indicate how well the system performs or attains its goal(s). These variables are outputs. Examples of result variables are shown in Table 6.2. Result variables are considered dependent variables. Intermediate result variables are sometimes used in modeling to identify intermediate outcomes. In the case of a depend-ent variable, another event must occur first before the event described by the variable can occur. Result variables depend on the occurrence of the decision variables and the uncontrollable variables.

Decision Variables Decision variables describe alternative courses of action. The decision maker controls the decision variables. For example, for an investment problem, the amount to invest in bonds is a decision variable. In a scheduling problem, the decision variables are people, times, and schedules. Other examples are listed in Table 6.2.

Uncontrollable Variables, or parameters In any decision-making situation, there are factors that affect the result variables but are not under the control of the decision maker. Either these factors can be fixed, in which case they are called uncontrollable variables, or parameters, or they can vary, in which case they are called variables. Examples of factors are the prime interest rate, a city’s building code, tax regulations, and utilities costs. Most of these factors are uncontrollable because they are in and determined by elements of the system environment in which the decision maker works. Some of these variables limit the decision maker and therefore form what are called constraints of the problem.

Intermediate result Variables Intermediate result variables reflect intermediate outcomes in mathematical models. For example, in determining machine scheduling, spoilage is an intermediate result variable, and total profit is the result variable (i.e., spoilage is one determinant of total profit). Another example is employee salaries. This constitutes a decision variable for management: It determines employee satisfaction (i.e., intermediate outcome), which, in turn, determines the productivity level (i.e., final result).

The structure of Mathematical Models

The components of a quantitative model are linked by mathematical (algebraic) expressions— equations or inequalities.

A very simple financial model is P = R – C

where P = profit, R = revenue, and C = cost. This equation describes the relationship among the variables. Another well-known financial model is the simple present-value cash flow model, where P

= present value, F = a future single payment in dollars, i = inter-est rate (percentage), and n = number of years. With this model, it is possible to determine the present value of a payment of $100,000 to be made 5 years from today, at a 10% (0.1) interest rate, as follows:

P = 100,000/ (1 + 0.1)5 = 62,092

**Structure of Mathematical Models for Decision Support**

- Non-Quantitative Models (Qualitative)
- Quantitative Models: Mathematically links decision variables, uncontrollable variables, and result variables

**Table 6.2 **Examples of Components of Models

Area | Decision Variables | Result Variables | Uncontrollable Variables and Parameters |

Financial investment | Investment alternatives and amounts | Total profit, risk Rate of return on investment (ROI) Earnings per share Liquidity level | Inflation rate Prime rate Competition |

Marketing | Advertising budget Where to advertise | Market share Customer satisfaction | Customer’s income Competitor’s actions |

Manufacturing | What and how much to produce Inventory levels Compensation programs | Total cost Quality level Employee satisfaction | Machine capacity Technology Materials prices |

Accounting | Use of computers Audit schedule | Data processing cost Error rate | Computer technology Tax rates Legal requirements |

Transportation | Shipmentsschedule Use of smart cards | Total transport cost Payment float time | Delivery distance Regulations |

Services | Staffing levels | Customer satisfaction | Demand for services |