Linear Programming

1
Linear Programming

• Optimization problems
• Linear programs
• Simplex algorithm
• Refinery blending operation

2
Introduction

• Optimization problems in chemical engineering
• Process equipment design
• Model parameter estimation
• Data reconciliation
• Optimal steady-state operation
• Optimal control
• Scheduling planning
• Formulation of optimization problems
• Select an objective function
• Develop the process model constraints
• Identify the decision variables
• Simplify the objective function process model
• Compute the optimal solution
• Perform sensitivity analysis

3
Optimization Problems

• Objective function f(x)
• Scalar function to be minimized or maximized
• Raw material costs, energy costs, total profit,
  least-squares fit to data
• Decision variables x
• Variables that can be adjusted to change
  objective function
• Flow rates, temperature, pressure,
  concentrations, etc.
• Multivariable optimization problem
• Maximization equivalent to minimizing negative of
  objective function
• Unconstrained problem decision vector x can be
  adjusted freely
• Constrained optimization decision vector x must
  satisfy certain constraints (variable bounds,
  model equations)
• Degrees of freedom difference between the number
  of variables the number of equality
  constraints must be greater than zero

4
Linear Programs

• Characteristics
• Linear objective function
• Linear equality inequality constraints
• Examples of linear constraints
• Production raw material limitations
• Product specifications
• Safety requirements
• Material energy balances
• Standard form of linear program (LP)
• n-m degrees of freedom available for optimization

5
Flux Balance Analysis

• Stoichiometric model Av b
• v n-dimensional vector of unknown fluxes
• A (mxn)-dimensional matrix of stoichiometric
  coefficients
• b m-dimensional vector of measured transport
  rates fluxes
• Underdetermined system m lt n
• More unknowns than equations
• Infinite number of solutions exist
• Least-squares solution v AT(AAT)-1b
• Not biologically plausible
• Linear program formulation
• Maximal growth objective f(x) cTv
• Determine fluxes that maximize growth

6
Feasibility and Optimality

• Linear program (LP)
• Feasible point
• Any point x that satisfies the constraints
• Generally not optimal
• Optimal solution
• Yields largest possible f(x) among all feasible
  points
• Possibly more than one optimal solution
• Basic feasible solution
• Feasible solution in which at least n-m variables
  are zero
• At least one optimal solution is also a basic
  feasible solution

7
Solution of Linear Programs

• Fundamental property
• Optimal solution always lies on domain boundary
  at the intersection of active constraints
  (vertices)
• These vertices are basic feasible solutions
• Only a finite number of vertices must be checked
• Potentially very large number of vertices

8
Simplex Algorithm

• Background
• Most widely used LP solution method
• Provides efficient solution of very large LPs
• Iterative algorithm
• Moves between basic feasible solutions while
  always increasing the objective function value
• Example

9
Simplex Table and Basic Feasible Solution

• Simplex table
• Basic variables columns with only one non-zero
  entry (x3, x4)
• Non-basic variables columns with multiple
  non-zero entry (x1, x2)
• Basic feasible solution
• Obtained by setting non-basic variables to zero
• Algorithm iteratively generates better solutions

10
First Iteration

• Pivot selection
• Use first column with negative entry in first row
  (column 2)
• Divide RHS by corresponding entry of selected
  column. Use row with smallest quotient (row 3)
• Eliminate entries above and below the pivot
  (entry in column 2 and row 3) using Gauss-Jordan
  elimination
• Basic feasible solution

11
Second Iteration

• Pivot selection
• Use first column with negative entry in first row
  (column 3)
• Divide RHS by corresponding entry of selected
  column. Use row with smallest quotient (row 2)
• Eliminate entries above and below the pivot
  (entry in column 3 and row 2) using Gauss-Jordan
  elimination
• Basic feasible solution
• Optimal solution achieved since T2 has no
  negative entries in first row

12
Refinery Blending Operation

• Optimization problem
• Determine the production rates of E F that
  maximize operating profit
• Reactions
• Process 1 AB ? E
• Process 2 A2B ? F
• Variables
• x1 lb/day A consumed
• x2 lb/day B consumed
• x3 lb/day E produced
• x4 lb/day F produced

13
Economic and Operating Data
14
Problem Formulation and Solution

• Operating profit (/day)
• Sales income 0.4x30.33x4
• Feed cost 0.15x10.2x2
• Operating cost 0.15x30.05x4350200
• Total profit P 0.25x30.28x4-0.15x1-0.2x2-550
• Material balances constraints
• x1 0.667x30.5x4
• x2 0.333x30.5x4
• Variable constraints
• 0 lt x1 lt 40,000
• 0 lt x2 lt 30,000
• 0 lt x3 lt 30,000
• 0 lt x4 lt 30,000
• Solution obtained with LP solver
• P 5100 /day, x1 35000 lb/day, x2 25000
  lb/day,
• x3 30000 lb/day, x4 3000 lb/day