A company needs to rent space for its office employees both in the suburbs and downtown. Space is available in suburbia at a rate of $120 per square metre (per annum), while downtown space rents for $250 per square metre (per annum). In suburbia, only 30% of the space is “executive” quality, while the rest is ordinary quality. At the downtown location, 70% of the space is executive quality, while the rest is ordinary quality. The company needs a total of at least 1000 square metres of space, of which at least 510 square metres must be executive quality. No more than three quarters of the entire space is to be at either location. They wish to know how much space they should rent in each place so as to minimize the total expenditure on rent. a) Show the decision variables b) Show the objective function c) Show the constraints in standard form (i.e. suitable for using Excel’s Solver) d) Solve the linear programming problem by graphing
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.
A company needs to rent space for its office employees both in the suburbs and downtown. Space is available in suburbia at a rate of $120 per square metre (per annum), while downtown space rents for $250 per square metre (per annum). In suburbia, only 30% of the space is “executive” quality, while the rest is ordinary quality. At the downtown location, 70% of the space is executive quality, while the rest is ordinary quality. The company needs a total of at least 1000 square metres of space, of which at least 510 square metres must be executive quality. No more than three quarters of the entire space is to be at either location. They wish to know how much space they should rent in each place so as to minimize the total expenditure on rent.
a) Show the decision variables
b) Show the objective function
c) Show the constraints in standard form (i.e. suitable for using Excel’s Solver)
d) Solve the linear programming problem by graphing
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