Use the two steps for solving a linear programming problem,...

Use the two steps for solving a linear programming problem, given in the box on page $837,$ to solve the problems in Exercises $17-23 .$ A manufacturer produces two models of mountain bicycles The times (in hours) required for assembling and painting each model are given in the following table: The maximum total weekly hours available in the assembly department and the paint department are 200 hours and 108 hours, respectively. The profits per unit are $\$ 25$ for model $A$ and $\$ 15$ for model $B .$ How many of each type should be produced to maximize profit?

Use the two steps for solving a linear programming problem, given in the box on page $837,$ to solve the problems in Exercises $17-23 .$
A manufacturer produces two models of mountain bicycles The times (in hours) required for assembling and painting each model are given in the following table:
The maximum total weekly hours available in the assembly department and the paint department are 200 hours and 108 hours, respectively. The profits per unit are $\$ 25$ for model $A$ and $\$ 15$ for model $B .$ How many of each type should be produced to maximize profit?

Key Concepts

Corner Point Method

The corner point method is a solution strategy for linear programming problems which is based on the principle that the optimum value of the objective function occurs at one of the vertices (or corner points) of the feasible region. By evaluating the objective function at each vertex, the optimal solution can be identified.

Feasible Region

The feasible region is the set of all possible solutions that satisfy the constraints of a linear programming problem. It is typically a polygon (in two dimensions) or a polyhedron (in higher dimensions) and contains all the candidate solutions where the objective function can be evaluated.

Linear Programming

Linear programming is a mathematical technique used to maximize or minimize a linear objective function, subject to a set of linear inequality or equality constraints. It is widely used for resource allocation problems, where a decision must be made about the allocation of limited resources among competing activities.

Objective Function

The objective function in a linear programming problem is the function that needs to be optimized—either maximized or minimized. It typically represents a quantity such as profit, cost, or efficiency and is expressed as a linear combination of decision variables.

Constraints

Constraints are the conditions or restrictions imposed on a linear programming model based on limitations of resources or other requirements. They are represented by linear equations or inequalities and define the boundaries within which a solution must be found.

Transcript

00:04 We are given a problem and we are asked to solve this problem using when you're programming.

00:13 The problem is manufacturers producing two kinds of bicycles and the time and hours for assembling and painting each model are given in this table.

00:28 So we have that model a takes five hours to assemble.

00:33 Model b takes four hours to assemble.

00:36 Model a takes two hours to paint and model a b takes three hours to paint, and we're told that the maximum total weekly hours available in the assembly department and the paint department are 200 hours and 108 hours respectively.

00:57 Profits per unit are $25 from model a and $15 from model b.

01:03 We are asked to find how many of each type should be produced to maximize profit.

01:10 So the question is how many of each type? let x be the number of model a bikes let y be the number of model bikes and we have that let's z be the total number of bikes then it follows that z is simply equal to x plus y and this is our objective function what are the constraints well we're told the maximum total weekly hours in the assembly department is 200 hours.

02:36 So we have that 200 hours less greater than or equal to.

02:54 How many hours are there in the assembly department? this is going to be the number of hours in the assembly department is going to be the number of hours assembling model a, which is going to be 5 times x plus the number of hours spent and assembling model b, which is 4 times y.

03:18 This must be less than or equal to 200 hours.

03:24 And we're told that the maximum total weekly hours available in the paint department are 108 hours.

03:37 So we have that number of hours painting model a's, which is 2x, plus the number of hours painting model b's, which is 3y, must be less than or equal to 108.

03:52 Hours per week.

04:03 And we have that the profit per unit is 25 for model a and 15 from model b.

04:09 So instead of this being our objective function, instead, let's make z our profit.

04:25 Well, we have that then z is going to be the profit from model a's, which is 25 times a, so, or 25 times x, plus profit from model b's, which is 25 times.

04:45 Is 15 times y.

04:52 So we get z equals 25x plus 15 times y dollars in profit.

04:59 And we have the constraint equations 5x plus 4 y is less than equal to 200.

05:05 2x plus 3y is less than equal to 108.

05:09 And implied in all of this are the constraints that x is greater than are equal to zero and and y is greater than equal to zero.

05:21 This is because you can't work a negative number of hours.

05:27 And so we have now our objective equation and our system of linear inequalities.

05:34 So let's graph this system of linear inequalities.

05:38 Since x is greater than equal to zero and y is greater than equal to zero, we have that the system is in the first quadrant.

05:49 I'll graph this first equation in red, the first inequality in red.

05:53 And to do this, i'll graph 5x plus 4y equals 200.

05:58 If we take y to be 0, we have that x is equal to 200 over 5, which is 40.

06:07 So we have an x intercept of 40, 0.

06:18 And if x is equal to 0, we have that y is equal to 200 over 4 or 50.

06:25 So we have the y intercept at 050, and the line is a solution, so i'll draw a solid line between these two points...

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