Group of Process Optimization Institute for Automation and Systems Engineering

Technische Universität Ilmenau

Systems Optimization

Chapter 3: Mixed-Integer Linear Optimization

Pu Li

pu.li@tu-ilmenau.de www.tu-ilmenau.de/prozessoptimierung

2 Example: Production Planning

3

Definition:

4 Aim: Minimization of the total consts under the restrictions of the product specifications

A mixed-integer linear programming (MILP) problem

5 Example: Design of a reactor system

Two types of reactors:

Reaktor I

Reaktor II

F 1

F 2

6 Fragen: • Reactor I or Reactor II? • Or both? • How large?

Die Superstruktur des Systems:

Definition:

i. e.

Then Product amount:

If than für

Reaktor I

Reaktor II

F 1

F 2

F

7 Logic restriction: at least one reactor must be selected, i. e.

Formulation of the optimization problem:

There are 3 possibilities (i.e. enumeration):

The solution is trivial: reaktor I will be selected with

8 Example: The aim of design of a process is to produce a main product and a byproduct. There is a plug-flow reactor, a stirred-tank reactor and two distillation columns. The super-structure is as follows:

The maximal number of possibilities:

216 = 65536

9 Modeling of integer variables Relations between logic and algebraic expressions for integer variabls

For the product, both reactor A and reactor B must be used.

For the product, reactor A or reactor B (at least one) can be used.

When reactor A is used, column B must be used. (when reactor A is not used, column B can be used) i. e.

Then or

• Conjunctive „and“:

• Disjunctive „oder“:

• Implication:

10 • Equivalence:

When colonm A is used, colonm B must be used and when colonm B is used, colonm A muss be used.

It follows

• „only one“ Among reactor A and reactor B, only one must be used.

• „at least one“

• „at most one“

• „B when A“

11 Formulation of logic relations with linear equalities or inequalities

(1) Expression in logic form (2) Transformation to conjunctive form (3) Formulation of lineare equalities and inequalities

Satz from DeMorgan:

Example: Formulating the following logic relation with linear inequalities:

Transform to the implication form:

From DeMorgan

and

and

12 Then

i. e.

with

Introduce integer variables:

It follows

and

for

From the expression we have:

or

or

13 Example:

• When , then , this is a feasible solution.

• When , then , there is no feasible solution.

There may be combinations of the integer variables which are infeasible. Therefore, we should search for the optimal solution in the feasible region. However, we do not know the feasible combinations a priori.

14 Mixed-Integer-Linear Programming (MILP)

Problem formulation: where

Example:

15 Problem decomposition

• Each feasible solution (P) is a feasible solution of exact one sub-problem.

• A feasible solution of a sub-problem is a feasible solution (P).

d. h.

The decomposition is used for splitting an integer variable to 0 and 1.

The original problem (P) is decomposed to sub-problems under the following conditions:

n-th generation

(n+1)-the generation

n-th generation

(n+1)-the generation

16 For the example:

The integer variables will be splitted one after another. This is expressed by a family tree.

17

Solution of (P): -variables are integer. Solution of (RP): -Variablen are not necessarily integer.

Relaxation of the problem Definition of a relaxation:

We have

i. e. • When (RP) has no solution, (P) has no solution.

• The solution of both problems have the relation:

Original problem (P) relaxed problem (RP)

Solution of (P) Solution of (RP)

i.e. is always the lower bound of .

The relaxation of the integer variables:

• When the integer variables are integer at a solution of (RP), this solution is a solution of (P).

i. e.

18

19 Fathoming

Three criteria to fathoming (deactiviation):

Should the current node of the sub-problem be deactivited (no longer be separated)?

• When it is clear that there is no better solution than the best solution till now (value of the objective function ).

• When an optimal solution is found in the feasible region.

• When the relaxed problem has no feasible solution, has also no feasible solution and then is fathomed (deactivited).

• When , the solution can not be improved, then it is fathomed.

• When at the solution of the -variables are integer, it must be a solution of and then is fathomed.

A sub-problem or a node

20 Algorithm „Branch-and-Bound“ Step 1: Relaxation of the problem and solution with LP (the lower bound)

Step 2: Deaktiviation of a node, when

Step 3: Branching a sub-problem at an active node

Step 4: Backtracking, when the node is deactivated.

• When at the solution the -variables are integer, STOP.

• Otherwise generate two new sub-problems by branching a y-variable

• All y-variables are integer.

• The sub-problem is infeasible.

• At the solution:

• Depth-first search

• Breadth-first search

21

Depth-first search Breadth-first search

22 Example: Rucksack (Transport Problem)

Aim: Maximization of the value in the rucksack Question: Welche articles should be selected into the rucksack?

• Relaxation of the problems to LP Solution:

• Branch und Solution:

23

• Branch Solution:

und Solution

• Branch Solution:

und Solution

infeasible

24 Tree description of the Branch-and-Bound

25 Structure optimization of a separation process

26

Composition (mol fraction):

Economic data and heat exchange coefficient:

A 0.15 B 0.3

Feed flow:

Cost of the extra materials:

Cooling water:

Heating steam:

C 0.35 D 0.2

27 Analysis of the total costs of a column k

• Cooling and heating cost (T€/a)

Total costs of a column:

Total costs of the super structure:

The optimization variables:

• Investition cost (T€/a)

• Operation cost (T€/a)

where are the prices of the cooling and heating (T€/kJ)

28 The equality constraints (balance equations): • Total feed flow (kmol/h)

• Flow between two columns (BCD)

• Flow between two columns (ABC)

• Flow between two columns (AB)

• Flow between two columns (BC)

29 • Flow between two columns (CD)

• Heating and cooling energy:

The inequality constraints: • Constraints of the flows:

Parameters:

30 The optimal solution (total costs: 3308 T€/a)

The second optimal solution: A new restriction is added:

It leads to the total costs: 3927 T€/a

31 The third optimal optimal solution:

Another new restriction is introduced:

Then the solution has the total costs: 4102 T€/a.