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linear programming

linear programming

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linear programming

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  1. linear programming

  2. Introduction • A linear programming problem may be defined as the problem of maximizing or minimizinga linear function subject to linear constraints. • The constraints may be equalitiesor inequalities. • The function to be maximized or minimized is called the objective function. • A vector, x for the standard maximum problem or y for the standard minimum problem, is said to be feasible if it satisfies the corresponding constraints. The set of feasible vectors is called the constraint set.

  3. A linear programming problem is said to be feasible if the constraint set is not empty; • Otherwise it is said to be infeasible. • A feasible maximum (resp. minimum) problem is said to be unbounded if the objective function can assume arbitrarily large positive (resp. negative) values at feasible vectors; otherwise, it is said to be bounded. • Thus there are three possibilities for a linear programming problem. It may be bounded feasible, it may be unbounded feasible, and it may be infeasible. • The value of a bounded feasible maximum (resp., minimum) problem is the maximum (resp. minimum) value of the objective function as the variables range over the constraint set. A feasible vector at which the objective function achieves the value is called optimal.

  4. Illustrative example • LP could be used for the planning of working activities of the three academic lecturers who are working on two types of tasks at present time:- Either they work on their teaching processes- or they work in scientific research • We can suppose that the value of the contract for one taught subject is CZK 150,000. However, these three academic staff are involved in the value of 2 / 3, namely:

  5. The value of a published paper is 60,000 CZK (income from the Ministry of Education of the scientific results application). • In this table 1 the total time capacity of a lecturer is determined by using the formula: • The total capacity of the lecturer = 3 × 20 × 8 = 480 hours (3 months of the teaching semester, 20 working days per month and 8 hours per working day). • The challenge is to design the load time for each of three academic staff so that in the next semester they can contribute to the maximum output value of their teaching and research activities.

  6. For maximization of their performance contributions of both teaching and research activities it is necessary to respect the three limitations:

  7. Limitation of the total time capacity for each academic staff member (max 480 hours). • Limitation of the minimum required output from the grant (at least three scientific papers submitted or grant applications during 7 months). • Limitation of the minimum required output from their teaching (at least one course taught during 3 months).

  8. Formulation of tasks: • Limitation by the total time capacity: Course supervisor: 60x1 + 25x2  480 (1)Grant researcher : 55x1 + 48x2  480 (2) Doctoral student-teacher: 40x1 + 75x2  480 (3) • Limitation of the minimum required output of the (number of publications): x1 3 (4) Limitation of the minimum required output of the educational activity x2 1 (5) • It is possible to formulate a objective function that characterizes the distribution of time-efficient fund of academics for the effective evaluation of their intellectual capital: •    CV = 100 000 x1 + 60 000 x2= MAX (6)

  9. Where is:

  10. Determination of the optimal output of the research activities and teaching • Now it is necessary to identify the optimal value of the outputs x1 and x2. • These data can then be put into the inequalities (1), (2), (3). • It could be used for the determination of time load of academics for parallel implementation of two types of work activities (teaching and research activities). • Optimum values ​​of x1 and x2 are given by position of X point. This position is determined as the intersection of lines given by equations (1) and (2):

  11. The value of the maximum possible output value of the university from the work of three academics is then:

  12. Illustration of the situation X1 (5) The area of admissible solutions F and P 12 10 8,7 x 8 Poptim (4) (1) 3 1 Foptim X2 19,2 0 6,4 10 20 (3) (2)

  13. Individual workers time proportion corresponding to the optimal output: • Course supervisor: 420 Nh research 50 Nh teaching (10 Nhod time reserve) ---------------------------------------------------------------------------------------------- • Grant researcher : 385 Nh research 96 Nh teaching (1 Nhod hours overtime)

  14. Individual workers time proportion corresponding to the optimal output: • PhD student-teacher: 280Nh research150 Nhteaching (50Nhod timereserve) ----------------------------------------------------------------------------------------------- • Discussion:The illustrative example deals with optimizing the distribution of working time between tasks in the educational and scientific activities. In the following time period the procedure would be analogous, only there will be changes in the restrictive conditions (e.g. due to changes in capacity, due to vacation time, etc.).