Let us now consider various examples of using the Solver add-in. Usually, Solver can be found on the Data tab of the ribbon in the Analysis group. If this command is missing from the specified location, perform the following steps: go to the File tab of the ribbon and select Options. In the Excel Options window that appears, select Add-ins on the left, and on the right under the Manage group choose Excel Add-ins from the list and click Go. Next, in the Add-ins window, in the list of available add-ins, check the box in front of Solver Add-in and click OK.
General Formulation of an Optimization Problem
An optimization problem in its general form can be stated as shown in Table 9.1.
Table. General formulation of an optimization problem
| № | Name | Mathematical notation | Description |
| 1 | Objective function (optimization criterion) | max(min,const) f(x₁,…,xⱼ,…,xₙ) | Indicates in what sense the solution should be optimal, i.e., the best. Three types of objective functions are possible: maximization, minimization, or assigning a specific value. |
| 2 | Constraints | gᵢ(xⱼ) ≤ bᵢ, i = 1,…,m; j = 1,…,n xⱼ ∈ ℤ, j = 1,…,k (for integer programming problems). xⱼ ∈ {0,1}, j = 1,…,k (for problems with Boolean variables). | Establish dependencies between variables. Constraints may be one-sided or two-sided. A two-sided constraint is written as two one-sided constraints. |
| 3 | Boundary conditions | dⱼ ≤ xⱼ ≤ Dⱼ, j = 1,…,n | Show the limits within which the values of the sought variables in the optimal solution may lie. |
A solution of tasks (1)—(3) that satisfies all constraints and boundary conditions is called a feasible solution.
An important characteristic of an optimization problem is its dimension, determined by the number of variables n and the number of constraints m.
- If n < m, the problem has no solutions.
- A necessary requirement for optimization problems is the condition n > m.
- A system of equations where n = m is considered as an optimization problem having a single feasible solution (it can be solved as a standard optimization problem by designating any variable as the objective function).
Thus, a problem has an optimal solution if it meets two requirements:
- There is more than one solution, i.e., feasible solutions exist.
- There is a criterion that shows in what sense the accepted solution should be optimal, i.e., the best among the feasible ones.