Linear Optimization in Economics Analysis

See also slides, optimization

Linearization around first order Taylor series expansions

Usage:

• Resource allocation
• Project selection
• Scheduling and Capital budgeting
• Energy network optimization

Criteria for optimization models

• comprised of only continuous variables
• linear objective function
• either only linear constraints or inequality constraints

$$\begin{align*} \min_{x} \phi = c^\mathbf{T} \mathcal{x} & &\leftarrow &\space \text{Objective function} \\\ \text{s.t} & &\leftarrow &\space \text{Constraints} \\\ A_h \mathcal{x} = \mathcal{b}_h & &\leftarrow &\space \text{Equality constraints} \\\ A_g \mathcal{x} \leq \mathcal{b}g \leq 0 & &\leftarrow &\space \text{Inequality constraints} \\\ \mathcal{x}_{lb} \leq \mathcal{x} \leq \mathcal{x}_{ub} & &\leftarrow &\space \text{Variable Bounds} \end{align*}$$

where:

• $\mathcal{x} \rightarrow j^{\text{th}}$: decision variables
• $c \rightarrow j^{\text{th}}$: cost coefficients of the $j^{\text{th}}$ decision variable
• $a_{i, j}$: constraint coefficient for variable $j$ in constraint $i$
• $b_i \rightarrow \text{RHS}$: coefficient for constraint $i$
• $(A_k \mid k = \lbrace \mathcal{h}, \mathcal{g} \rbrace)$: matrix of size $\lbrack m_k \times n \rbrack$

Sensitivity reports

Decision variables

Reduced cost: the amount of objective function will change if variable bounds are tighten

Allowable increase/decrease: how much objective coefficient must change before optimal solution changes.

100% Rule

If there are simultaneous changes to objective coefficients, and $\sum_{\text{each coefficient}}(\frac{\text{Proposed change}}{\text{Allowable change}}) \leq 100 \%$ then the optimal solution would not change.

Constraints

Final value: the value of constraints at the optimal solution

Shadow price: of a constraint is the marginal improvement of the objective function value if the RHS is increased by 1 unit.

Allowable increase/decrease: how much the constraint can change before the shadow prices changes.

See lemon_orange.py