## Dynamic Control with Discrete Variables

One often encounters problems in which manipulated variables or adjustable parameters must be selected from among a set of discrete values. Examples of discrete variables include ON/OFF state, selection of different feed lines, and other variables that are naturally integers. Many dynamic optimization problems are discrete in nature.

At first glance it might seem solving a discrete variable problem would be easier than a continuous problem. After all, for a variable within a given range, a set of discrete values within the range is finite whereas the number of continuous values is infinite. When searching for an optimum, it seems it would be easier to search from a finite set rather than from an infinite set.

This is not the case, however. Solving discrete problems is harder than continuous problems. This is because of the combinatorial explosion that occurs in all but the smallest problems. For example if we have two variables which can each take 10 values, we have 10*10 = 100 possibilities. If we have 10 variables that can each take 10 values, we have 10^10 possibilities. Even with the fastest computer, it would take a long time to evaluate all of these.

Discrete decision variables are those that have only certain levels or quantities that are acceptable at an optimal solution. Examples of discrete variables are binary (e.g. off/on or 0/1), integer (e.g. 4,5,6,7), or general discrete values that are not integer (e.g. 1/4 cm, 1/2 cm, 1 cm).

#### Exercise 1

**Objective:** Solve a discrete optimization problem with the branch and bound technique. *Estimated time: 1 hour.*

#### Solution

#### Exercise 2

Design a thermostat for a home that has a furnace with On/Off control. The home is initially unheated in 40 degF weather. When the heater is turned on, the temperature in the home reaches 63% of the steady state value in 120 minutes, starting from 40 degF. If left on indefinitely, the home would eventually reach 100 degF. Optimize the heater response for the first 2 1/2 hours starting from a cold home at 40 degF. The target range for the home is between 68 and 70 degF.