Minimization and Maximization Problems Discussion Essay

Minimization and Maximization Problems Discussion Essay

Chapter for the week should be read prior to answering Initial Response and should reflect and reference the text and/or another research.  Must include 2-3 paragraphs per question.

1.Discuss the similarities and differences between minimization and maximization problems using the graphical solution approaches of linear programming.

2.Develop your own original product mix LP problem with two constraints and two real variables.  Follow the four steps to formulating LP problems on page 241. Explain the meaning of the numbers on the right-hand side of each of your constraints. Also, explain the significance of the technological coefficients.

replies:

replies

Leeshawn– The similarities between minimization and maximization begin with the solutions graphs. Both types of problems use constraint lines to develop a solution. They both can also use corner methods to solve the graphical solution approaches to linear programming. Minimization and maximization are also similar in the fact that they both can use is profit line solutions.

The differences between maximization and minimization are as follows:

Maximization’s solutions will be in the upper right of the graph at the farthest point from zero. While, minimizations solutions will be in the bottom left of the graph, at the closest point to the zero.

Adria– There are many differences between the minimization and maximization problems. Maximization problems, once the lines have been drawn, the best choice solution will be bounded to the upper right when using the isoprofit line method. The maximum profit will have the longest distance to zero. For the minimization problems the minimum cost will be the point closest to the zero. The minimization beset solution will be located to the lower left.

For the similarities in both the minimization and the maximization problems, they will be using the the graphical approach of the LP. These graphs will be used to develop constraint line in the graph to find the region for the solution for each of the problems. The corner method can also be used for both types of problems. Although they each use a form of the isoprofit line solution, the maximization problems use the isoprofit linear solution while the minimization problems will use the isocost line solution.

dana– Let’s assume that there are two machines M1 M2 produce canvas bag and leather bag.

Capacities of machines per hour in below table.Canvas shoesLeather shoesM164M239

M1 and M2 can produce 300 and 500 units in week.

Selling price of Canvas bag is $60 and selling price of Leather bag is $90.

Consider 8 hours a day and 5 days a week.

Calculate the optimal mix of Canvas bags and Leather bags to maximize the contribution.

Solution:

Variables :

Let x1 be the number of units of canvas shoes produced by M1 .y1 be the number of units of leather shoes produced by M1 in a week.

X2 be the number of canvas shoes produced by M2 in a week.

Y2 be the number of Leather shoes produced by M2 in a week.

Contribution:

$60( X1+ X2) + $90*(Y1+ Y2) is maximum

Constraints are:

X1+ Y1<=300

X2+Y2<=500

Molly–

Four steps should be followed when creating a linear programming (LP) problem. The first step is understanding the problem being face by management (Render et at., 2019, p. 7.2). For my example, I will use a fictitious example in my husband’s woodshop. Due to other business facets, my husband has a limited amount of time he can spend in his workshop weekly. This week, he wants to determine how many cutting boards and coasters to make to maximize his overall profit.

To solve this LP problem, I must determine the objective as well as its constraints (Render et at., 2019, p. 7.2). The objective is maximizing my husband’s weekly sales of homemade coasters and cutting boards. However, due to my husband’s limited available time, he can only spend 15 hours a week gluing and sanding and an additional 5 hours staining and finishing the coasters and cutting boards.

The decision variables must be determined next (Render et at., 2019, p. 7.2). The actual decisions I will make revolves around the optimal number of coasters and cutting boards to make in a week. These decision variables, coasters and cutting boards, along with the constraints will now be formulated into mathematical expressions allowing me to solve the objective (Render et at., 2019, p. 7.2).

Decision Variables: C1 (coasters) & C2 (cutting boards)

Maximizing: Profit = 15C1 + 20C2

Constraints: C1 ? 0

C2 ? 0

1C1 + 3C2 ? 15 (glue & sand)

1C1 + 1.5C2 ? 5 (stain & finish)

The numbers on the right-hand side of my constraints (bolded values) are the number of hours per week my husband can spend gluing and sanding or staining and finishing. For example, each week my husband has allotted 15 hours to gluing and sanding as it’s the most time-consuming process of woodworking. He also has an additional 5 hours allotted to stain and finish the items he created that week. Each week he can only spend so much time sanding or staining final products.

The amount of time he has allotted for each constraint has the potential to be drastically changed by technological coefficients. Changes in technological coefficients can cause the shape of the feasible region to shift, which can potentially cause a change in the optimal value (Render et at., 2019, p. 7.7). For example, if my husband purchased a new electric sander that brought the sanding time for cutting boards down to 1.5 hours, the feasible region would shift but the optimal value would remain the same.

Reference

Render, B., Stair, R. M., Hanna, M. E., & Hale, T. S. (2019). Quantitative analysis for management (13th ed.). Upper Saddle River, NJ: Pearson

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