Economic Optimization

See also slides

model-based

• conclusions from the model of the system

Components:

• decision variables
• constraints
• objectives
• functions: mathematical function that determines the objective as a function of decision variable

$$\begin{align*} \min_{x} \phi = f(x) & &\leftarrow &\space \text{Objective function} \\\ \text{s.t} & &\leftarrow &\space \text{Constraints} \\\ h(x) = 0 & &\leftarrow &\space \text{Equality constraints} \\\ g(x) \leq 0 & &\leftarrow &\space \text{Inequality constraints} \\\ x_{lb} \leq x \leq x_{ub} & &\leftarrow &\space \text{Bounds} \end{align*}$$

decision variables

discrete.

limited to a fixed or countable set of values

$$x_{\mathcal{D}} \mid a \in \mathcal{I} = \lbrace 1, 2, 3, 4, 5 \rbrace$$

continuous.

can take any value within a range

$$x_{\mathcal{C}} \subset \mathcal{R}$$

constraints

• physical limitations: cannot purchase negative raw materials

• model assumptions: assumptions about the system

domain of a definition

a decision upper and lower bounds ($x^{\mathcal{U}}$ and $x^{\mathcal{L}}$)

Properties

• Active/binding: $\exists \space x^{*} \mid g(x^{*}) = 0$
• Inactive: $\exists \space x^{*} \mid g(x^{*}) < 0$

graphing models

feasible set of an optimization model

The collection of decision variables that satisfy all constraints

$$\mathcal{S} \triangleq \lbrace x : g(x) \leq 0, h(x) = 0, x^L \leq x \leq x^U \rbrace$$

outcomes

optimal value

the optimal value $\phi^{*}$ is the value of the objective at the optimum(s)

$$\phi^{*} \triangleq \phi(x^{*})$$

Constraints satisfy, but it is not binding

Linear optimization problems

$$\begin{aligned} \underset{x_1,x_2}{\min} \space \phi &= 50x_1 + 37.5x_2 \\ &\text{s.t} \\\ 0.3x_1 + 0.4x_2 &\geq 2000 \\\ 0.4x_1 + 0.15x_2 &\geq 1500 \\\ 0.2x_1 + 0.35x_2 &\leq 1000, \\\ x_1 &\leq 9000 \\\ x_2 &\leq 6000 \\\ x_i &\geq 0 \end{aligned}$$

See also Linear Optimization