Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
The accountant of Royal Products provided the following information for purposes of determining
the optimal product mix:
Product King (K) Product Queen (Q)
Marginal income per unit R30 R36
Machine hours required per unit 6 6
Marginal income per machine hour R5 R6
Ranking in terms of machine hours 2nd 1st
Labour hours required per unit 3 6
Marginal income per labour hour R10 R6
Ranking in terms of labour hours 1st 2nd
Annual demand for product in units 12 000 15 000
Labour hours available per annum 65 000
Machine hours available per annum 82 000
Required:
Determine the optimal product mix by means of the application of linear programming
techniques. (Hint: Start with your objective function and all necessary constraints.)
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