Linear Optimization in Economics Analysis

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

Linear Optimization in Economics Analysis

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Sensitivity reports

Decision variables

Constraints

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