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

$$\begin{array}{rlrl}\underset{x}{\min }\varphi =c^{\mathbf{T}}x& & \leftarrow & \text{Objective function}\\ \text{s.t}& & \leftarrow & \text{Constraints}\\ A_{h}x=b_h& & \leftarrow & \text{Equality constraints}\\ A_{g}x\le bg\le 0& & \leftarrow & \text{Inequality constraints}\\ x_{lb}\le x\le x_{ub}& & \leftarrow & \text{Variable Bounds}\end{array}$$

where:

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

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 ${\sum }_{\text{each coefficient}}(\frac{\text{Proposed change}}{\text{Allowable change}})\le 100\%$ 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