This problem involves optimizing the distribution of specialists among different types of work based on availability and efficiency.

Problem Setup:

1. Input Data:

   • $\vec{a} = (10, 25, 15, 30)$: The number of available specialists for each of the four profiles.
   • $\vec{b} = (35, 10, 15, 10, 20)$: The required number of specialists for five types of work.
   • Matrix $E = \|e_{ij}\|$: Efficiency of specialists from each profile (rows) working on each type of job (columns).

   ( E = \begin{bmatrix} 4 & 7 & 2 & 1 & 4 \ 0 & 4 & 3 & 0 & 0 \ 0 & 8 & 7 & 2 & 4 \ 3 & 5 & 8 & 7 & 2 \end{bmatrix} )

2. Optimization Objective: Distribute the specialists to maximize total efficiency, considering:

   • Each job's requirements ($b_i$).
   • Availability of specialists ($a_i$).
   • Efficiency matrix values ($e_{ij}$).

3. Constraints:

   • A specialist profile $i$ cannot exceed its available number ($a_i$).
   • The total number of specialists allocated to each job must match the requirement ($b_i$).

Approach:

This is a linear programming (LP) problem. The task can be solved as follows:

• Define decision variables $x_{ij}$: Number of specialists of profile $i$ assigned to job $j$.
• Objective: Maximize $Z = \sum_{i=1}^{4} \sum_{j=1}^{5} e_{ij} x_{ij}$.
• Subject to:
  • $\sum_{j=1}^{5} x_{ij} \leq a_i \; \forall i$ (specialist availability).
  • $\sum_{i=1}^{4} x_{ij} = b_j \; \forall j$ (job requirements).
  • $x_{ij} \geq 0$.

Would you like me to:

1. Solve this problem step-by-step?
2. Formulate it into an LP model for a solver?
3. Provide a detailed explanation for each step?