PART III: SYSTEM ANALYSIS AND DESIGN EVALUATION
Chapter 5. Optimization in Design and Operations
EBW2409: SYSTEMS APPROACH TO ENVIRONMENTAL MANAGEMENT
5 OPTIMIZATION IN DESIGN AND OPERATIONS
5.1 Introduction
An optimization problem is one requiring the determination of the optimal (i.e., maximum or
minimum) value of a given function, called the objective function, subject to a set of stated
restrictions, or constraints, placed on the variables concerned.
For instance there may be need to optimize a manufacturing related problem subject to
availability of labour, machine time, stocks of raw materials, transport conditions, etc.
Unconstrained optimization: means that no constraints are placed on the function under
consideration. For example, when the total cost of an alternative is a function of increasing
and decreasing cost components, a value may exist for the common variable, or variables,
which will result in a minimum cost for the alternative.
Constrained optimization: In design and operations alike, physical and economic limitations
often exist which act to limit system optimization globally. These limitations arise for a
variety of reasons and generally cannot be removed by the decision maker. Accordingly,
there may be no choice except to find the best or optimum solution subject to the
constraints.
5.2 Constrained Optimization by Linear Programming
Linear programming (LP) is a method of solving certain problems of constrained
optimisation when the objective function and the constraints are linear equations or
inequalities. We should say straight away that linear programming has nothing to do with
computer programming, but its name comes from the more general meaning of planning.
There are three distinct stages to solving a linear programme:
Formulation - getting the problem in the right form.
Solution - finding an optimal solution to the problem.
Sensitivity analysis - seeing what happens when the problem is changed slightly.
5.2.1 Formulation
The first stage of solving a linear programme is to describe the problem in a standard
format. It is easiest to illustrate this formulation with an example, and for this we use a
problem from production planning.
Suppose a small factory makes two types of liquid fertiliser, Growbig and Thrive. It makes
these by similar processes, using the same equipment for blending raw materials, distilling
the mix and finishing (bottling, testing, weighing, etc.). Because the factory has a limited
amount of equipment, there are constraints on the total time available for each process. In
particular, there are only 40 hours of blending available in a week, 40 hours of distilling and
25 hours of finishing. We assume that these are the only constraints and there are none
on, say, sales or availability of raw materials.
The fertilisers are made in batches, and each batch needs the following hours on each
process.
, PART III: SYSTEM ANALYSIS AND DESIGN EVALUATION
Chapter 5. Optimization in Design and Operations
Growbig Thrive
Blending 1 2
Distilling 2 1
Finishing 1 1
If the factory makes a net profit of €300 on each batch of Growbig and €200 on each batch
of Thrive, how many batches of each should it make in a week?
This problem is clearly one of optimising an objective (maximising profit) subject to
constraints (production capacity), as shown in Figure 5.1. The variables that the company
can control are the number of batches of Growbig and Thrive they make, so these are the
decision variables. We can define these as:
G is the number of batches of Growbig made in a week
T is the number of batches of Thrive made in a week.
Figure 5.1: Production problem for growbig and thrive.
Now consider the time available for blending. Each batch of Growbig uses one hour of
blending, so G batches use G hours; each batch of Thrive uses two hours of blending, so
T batches use 2T hours. Adding these together gives the total amount of blending time used
as G + 2T. The maximum amount blending time available is 40 hours, so the time used
must be less than, or at worst equal to, this. So this gives the first constraint:
G + 2T: 40 (blending constraint)
Turning to the distilling constraint, each batch of Growbig uses two hour of distilling, so G
batches use 2G hours; each batch of Thrive uses one hour of distilling, so T batches use T
hours. Adding these together gives the total amount of distilling time used and this must be
, PART III: SYSTEM ANALYSIS AND DESIGN EVALUATION
Chapter 5. Optimization in Design and Operations
less than, or at worst equal to, the amount of distilling time available (40 hours). So this
gives the second constraint:
2G + T: 40 (distilling constraint)
Now the finishing constraint has the total time used for finishing (G for batches of Growbig
plus T for batches of Thrive) less than or equal to the time available (25 hours) to give:
G + T: 25 (finishing constraint)
These are the three constraints for the process - but there is another implicit constraint. The
company cannot make a negative number of batches, so both G and T are positive. This
non-negativity constraint is a standard feature of linear programmes.
G ≥ 0 and T ≥ 0 (non-negativity constraints)
Here the three problem constraints are all 'less than or equal to', but they can be of any
type - less than, less than or equal to, equal to, greater than or equal to, or greater than.
Now we can turn to the objective, which is maximising the profit. The company makes €300
on each batch of Growbig, so with G batches the profit is 300G; they make €200 on each
batch of Thrive, so with T batches the profit is 200T. Adding these gives the total profit that
is to be maximised - this is the objective function.
Maximise 300G + 200T (objective function)
This objective is phrased in terms of maximising an objective. The alternative for LPs is to
minimise an objective (typically phrased in terms of minimising costs).
This completes the linear programming formulation which we can summarise as:
Maximise:
300G + 200T objective function
Subject to:
G + 2T 40
2G + T 40 constraints
G + T 25
With
G ≥ 0 and T ≥ 0 non-negativity constraints
This illustrates the features of all linear programming formulations, which consist of:
Decision variables.
An objective function.
A set of constraints.
A non-negativity constraints.
Chapter 5. Optimization in Design and Operations
EBW2409: SYSTEMS APPROACH TO ENVIRONMENTAL MANAGEMENT
5 OPTIMIZATION IN DESIGN AND OPERATIONS
5.1 Introduction
An optimization problem is one requiring the determination of the optimal (i.e., maximum or
minimum) value of a given function, called the objective function, subject to a set of stated
restrictions, or constraints, placed on the variables concerned.
For instance there may be need to optimize a manufacturing related problem subject to
availability of labour, machine time, stocks of raw materials, transport conditions, etc.
Unconstrained optimization: means that no constraints are placed on the function under
consideration. For example, when the total cost of an alternative is a function of increasing
and decreasing cost components, a value may exist for the common variable, or variables,
which will result in a minimum cost for the alternative.
Constrained optimization: In design and operations alike, physical and economic limitations
often exist which act to limit system optimization globally. These limitations arise for a
variety of reasons and generally cannot be removed by the decision maker. Accordingly,
there may be no choice except to find the best or optimum solution subject to the
constraints.
5.2 Constrained Optimization by Linear Programming
Linear programming (LP) is a method of solving certain problems of constrained
optimisation when the objective function and the constraints are linear equations or
inequalities. We should say straight away that linear programming has nothing to do with
computer programming, but its name comes from the more general meaning of planning.
There are three distinct stages to solving a linear programme:
Formulation - getting the problem in the right form.
Solution - finding an optimal solution to the problem.
Sensitivity analysis - seeing what happens when the problem is changed slightly.
5.2.1 Formulation
The first stage of solving a linear programme is to describe the problem in a standard
format. It is easiest to illustrate this formulation with an example, and for this we use a
problem from production planning.
Suppose a small factory makes two types of liquid fertiliser, Growbig and Thrive. It makes
these by similar processes, using the same equipment for blending raw materials, distilling
the mix and finishing (bottling, testing, weighing, etc.). Because the factory has a limited
amount of equipment, there are constraints on the total time available for each process. In
particular, there are only 40 hours of blending available in a week, 40 hours of distilling and
25 hours of finishing. We assume that these are the only constraints and there are none
on, say, sales or availability of raw materials.
The fertilisers are made in batches, and each batch needs the following hours on each
process.
, PART III: SYSTEM ANALYSIS AND DESIGN EVALUATION
Chapter 5. Optimization in Design and Operations
Growbig Thrive
Blending 1 2
Distilling 2 1
Finishing 1 1
If the factory makes a net profit of €300 on each batch of Growbig and €200 on each batch
of Thrive, how many batches of each should it make in a week?
This problem is clearly one of optimising an objective (maximising profit) subject to
constraints (production capacity), as shown in Figure 5.1. The variables that the company
can control are the number of batches of Growbig and Thrive they make, so these are the
decision variables. We can define these as:
G is the number of batches of Growbig made in a week
T is the number of batches of Thrive made in a week.
Figure 5.1: Production problem for growbig and thrive.
Now consider the time available for blending. Each batch of Growbig uses one hour of
blending, so G batches use G hours; each batch of Thrive uses two hours of blending, so
T batches use 2T hours. Adding these together gives the total amount of blending time used
as G + 2T. The maximum amount blending time available is 40 hours, so the time used
must be less than, or at worst equal to, this. So this gives the first constraint:
G + 2T: 40 (blending constraint)
Turning to the distilling constraint, each batch of Growbig uses two hour of distilling, so G
batches use 2G hours; each batch of Thrive uses one hour of distilling, so T batches use T
hours. Adding these together gives the total amount of distilling time used and this must be
, PART III: SYSTEM ANALYSIS AND DESIGN EVALUATION
Chapter 5. Optimization in Design and Operations
less than, or at worst equal to, the amount of distilling time available (40 hours). So this
gives the second constraint:
2G + T: 40 (distilling constraint)
Now the finishing constraint has the total time used for finishing (G for batches of Growbig
plus T for batches of Thrive) less than or equal to the time available (25 hours) to give:
G + T: 25 (finishing constraint)
These are the three constraints for the process - but there is another implicit constraint. The
company cannot make a negative number of batches, so both G and T are positive. This
non-negativity constraint is a standard feature of linear programmes.
G ≥ 0 and T ≥ 0 (non-negativity constraints)
Here the three problem constraints are all 'less than or equal to', but they can be of any
type - less than, less than or equal to, equal to, greater than or equal to, or greater than.
Now we can turn to the objective, which is maximising the profit. The company makes €300
on each batch of Growbig, so with G batches the profit is 300G; they make €200 on each
batch of Thrive, so with T batches the profit is 200T. Adding these gives the total profit that
is to be maximised - this is the objective function.
Maximise 300G + 200T (objective function)
This objective is phrased in terms of maximising an objective. The alternative for LPs is to
minimise an objective (typically phrased in terms of minimising costs).
This completes the linear programming formulation which we can summarise as:
Maximise:
300G + 200T objective function
Subject to:
G + 2T 40
2G + T 40 constraints
G + T 25
With
G ≥ 0 and T ≥ 0 non-negativity constraints
This illustrates the features of all linear programming formulations, which consist of:
Decision variables.
An objective function.
A set of constraints.
A non-negativity constraints.
