. The objective function or constraints, if already defined in the model, won't be affected as they will be naturally extended as if the new variables were not referred during their definition (e.g., for a linear objective function the coefficients for the new variables would be set to zero). If the new variables should enter any already defined parts of the problem, you should modify them with the appropriate functions from the suite.

See Section 4.1 in the E04 Chapter Introduction for more details about the NAG optimization modelling suite.

4 References

None.

5 Arguments

1: $handle$ – void * Input: On entry: the handle to the problem. It needs to be initialized (e.g., by e04rac) and must not be changed between calls to the NAG optimization modelling suite.
2: $nadd$ – Integer Input: On entry: $n_{add}$ , the number of decision variables to add to the problem.

Constraint: $nadd > 0$ .
3: $nvar$ – Integer * Output: On exit: $n$ , the new total number of the variables in the problem.
4: $fail$ – NagError * Input/Output: The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_HANDLE: The supplied handle does not define a valid handle to the data structure for the NAG optimization modelling suite. It has not been properly initialized or it has been corrupted.
NE_INT: On entry, $nadd = ⟨ value ⟩$ .
Constraint: $nadd > 0$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_PHASE: The problem cannot be modified right now, the solver is running.

7 Accuracy

Not applicable.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

e04tac is not threaded in any implementation.

9 Further Comments

None.

10 Example

This example demonstrates, using a simple production-planning problem, how to use the functions of the NAG optimization modelling suite to edit and solve different variants of the problem.

We consider a situation where a factory can manufacture two different chemicals

A_{1}

and

A_{2}

. The goal for the factory is to determine the quantities

x_{1}

and

x_{2}

of each chemical to maximize profit under the following circumstances:

•a unit of $A_{1}$ weighs $40$ kg and a unit of $A_{2}$ weighs $80$ kg;
•the total daily production cannot exceed $16000$ kg to match the transport capabilities;
•the factory generates $ $2$ profit for each unit of $A_{1}$ and $ $4.5$ profit for each unit of $A_{2}$ ;
•both products need to use the same machine as part of their respective processes; a unit of $A_{1}$ requires $1.2$ minutes of machine time while a unit of $A_{2}$ requires $3$ minutes; the machine can only function for $1500$ minutes daily;
•a unit of $A_{1}$ uses $6$ square metres of packing material while a unit of $A_{2}$ uses $10$ square metres; $6000$ square metres of packing materials are available each day;
•production of $A_{2}$ is limited to $100$ units per day.

Note that since the chemicals are considered fluid, the quantities

x_{1}

and

x_{2}

are not limited to integer values.

We can now formulate the problem as a linear program:

\begin{array}{l} \underset{x \in ℝ^{n}}{maximize} & 2 x_{1} + 4.5 x_{2} \\ subject to & 1.2 x_{1} + 3 x_{2} \leq 1500 & (machine time constraint) \\ 6 x_{1} + 10 x_{2} \leq 6000 & (packaging material constraint) \\ 40 x_{1} + 80 x_{2} \leq 16000 & (transport constraint) \\ 0 \leq x_{1} & (capacity constraint) \\ 0 \leq x_{2} \leq 100 & (capacity constraint) \end{array}

If the factory expands its capabilities and is now capable of producing a new chemical

A_{3}

with:

•a unit of $A_{3}$ takes $5$ minutes on the common machine;
•a unit of $A_{3}$ takes $12$ square metres of packaging material;
•a unit of $A_{3}$ weighs $120$ kg;
•a unit of $A_{3}$ generates $ $7$ profit;
•production of $A_{3}$ is limited to $50$ units per day.

The problem becomes:

\begin{array}{l} \underset{x \in ℝ^{n}}{maximize} & 2 x_{1} + 4.5 x_{2} + 7 x_{3} \\ subject to & 1.2 x_{1} + 3 x_{2} + 5 x_{3} \leq 1500 & (machine time constraint) \\ 6 x_{1} + 10 x_{2} + 12 x_{3} \leq 6000 & (packaging material constraint) \\ 40 x_{1} + 80 x_{2} + 120 x_{3} \leq 16000 & (transport constraint) \\ 0 \leq x_{1} & (capacity constraint) \\ 0 \leq x_{2} \leq 100 & (capacity constraint) \\ 0 \leq x_{3} \leq 50 & (capacity constraint) \end{array}

At a later date, regulation changes require that products

A_{2}

and

A_{3}

follow a rigorous quality assurance test before being sent to market. Now the factory is only able to process a total of 100 units per day which amounts to adding the following constraint to our linear program: