Quantitative Analysis BA 452 Homework 1 Questions Homework 1 covers the theory and applications in Lessons I-1 to I-3. This document has four parts: Objectives of doing your homework. Assignment of homework questions, with suggestions about which other questions may help you understand the homework questions. Homework 1 Supplemented Questions listing 58 questions: 4 of them are your homework, others may help you understand the homework questions, and the rest may are there help just you in understand case you find fine them points.
Some Some supplemental supplemental questions questions refer torefer a section to a section of a chapter of a chapter in the textbook in the (for textbook example,(for Section example, 2.1 means 2 Section Chapter 1 of2your Section textbook). 1 of your textbook). Homework 1 Supplemented Answers listing answers to all 58 Homework questions excluding 1 Supplemented your 4 homework Answersquestions. listing answers to all 58 questions including your 4 homework questions.
1 Quantitative Analysis BA 452 Homework 1 Questions Objectives By working through the homework questions and the supplemental questions, you will: 1. Obtain an overview of the kinds of problems linear programming has been used to solve. Learn how to develop linear programming models for simple problems. Be able to identify the special features of a model that make it a linear programming model.
Learn how to solve two variable linear programming models by the graphical solution procedure. Understand the importance of extreme points in obtaining the optimal solution. Know the use and interpretation of slack and surplus variables. Be able to interpret the computer solution of a linear programming problem.
Understand how alternative optimal solutions, infeasibility and unboundedness can occur in linear programming problems. Understand the following terms: problem formulation feasible region constraint function slack variable objective function standard form solution redundant constraint optimal solution extreme point nonnegativity constraints surplus variable mathematical model alternative optimal solutions linear program infeasibility linear functions unbounded feasible solution 2 Quantitative Analysis BA 452 Homework 1 Questions Assignment Questions 11, 13, 27, and 53 is your homework assignment. Questions 11, 13, and 27 should be answered without referring to notes or using computers (Hint: Define decision variables I = Internet fund investment in thousands, B = Blue Chip fund investment in thousands. Then, the objective is to maximize the projected annual return 0.) Question 53 can be answered with notes and computers.
To supplement those homework questions, you should consider (but not turn in) the following questions. Questions answered without notes or computers: 1-13, 17-19, 21-26, 30-31, 34-36, 38, 42-46. Questions answered with notes and computers: 14-15, 20, 27-29, 33, 37, 39-41, 49-54. Tip: Those homework questions and supplementary questions are grouped into sets of similar type.
Once you have mastered the questions in a set, you can skip the rest of the questions in that set. Tip: Some of your Exam 1 questions will be variations of some of those homework questions. 3 Quantitative Analysis BA 452 Homework 1 Questions Homework 1 Supplemented Questions 1. Which of the following mathematical relationships could be found in the linear programming model, and which could not? For the relationships that are unacceptable for linear programs, state why.
Find the solution that satisfy the following constraints: a. Show a separate graph of the constraint lines and the solutions that satisfy each of the following constraints: a. Show a separate graph of the constraint lines and the solutions that satisfy each of the following constraints: a. Show a separate graph of the constraint lines and the solutions that satisfy each of the following constraints: a.
Three objective functions for linear programming problems are 7𝐴𝐴 + 10𝐵𝐵, 6𝐴𝐴 + 4𝐵𝐵, and −4𝐴𝐴 + 7𝐵𝐵. Show the graph of each for objective function values equal to 420. Identify the feasible region for the following set of constraints: 0.5𝐵𝐵 ≤ 50 𝐴𝐴, 𝐵𝐵 ≥ 0 4 Quantitative Analysis BA 452 Homework 1 Questions 8. Identify the feasible region for the following set of constraints: 2𝐴𝐴 − 1𝐵𝐵 ≤ 0 −1𝐴𝐴 + 1.
Identify the feasible region for the following set of constraints: 3𝐴𝐴 − 2𝐵𝐵 ≥ 0 2𝐴𝐴 − 1𝐵𝐵 ≤ 200 1𝐴𝐴 ≤ 150 𝐴𝐴, 𝐵𝐵 ≥ 0 10. For the linear program Max 2𝐴𝐴 + 3𝐵𝐵 s. 1𝐴𝐴 + 2𝐵𝐵 ≤ 6 5𝐴𝐴 + 3𝐵𝐵 ≤ 15 𝐴𝐴, 𝐵𝐵 ≥ 0 find the optimal solution using the graphical solution procedure. What is the value of the objective function at the optimal solution? 11.
Solve the following linear program using the graphical solution procedure: Max 5𝐴𝐴 + 5𝐵𝐵 s. Consider the following linear programming problem: Max 3𝐴𝐴 + 3𝐵𝐵 s. Find the optimal solution using the graphical solution procedure. If the objective function is changes to 2A + 6B, what will the optimal solution be? c.
How many extreme points are there? What are the values of A and B at each extreme point? 5 Quantitative Analysis BA 452 Homework 1 Questions 13. Consider the following linear program: Max 1𝐴𝐴 + 2𝐵𝐵 s. Show the feasible region. What are the extreme points of the feasible region? c.
Find the optimal solution using the graphical procedure., is a small firm that produces a variety of chemical products. In a particular production process, three raw materials are blended (mixed together) to produce two products: a fuel additive and a solvent base. Each ton of fuel additive is a mixture of 2/5 ton of material 1 and 3/5 of material 3. A ton of solvent base is a mixture is a mixture of ½ ton of material 1, 1/5 ton of material 2 and 3/10 ton of material 3.
After deducting relevant costs, the profit contribution is $40 for every ton of fuel additive and $30 for every ton of solvent base. RMC’s production is constrained by the limited availability of the three raw materials. For the current production periods, RMC has available the following quantities of each new material: Raw Material Amount Available for Production Material 1 20 tons Material 2 5 tons Material 3 21 tons Assuming the RMC is interested in maximizing the total profit contribution, answer the following: a. What is the linear programming model for the problem? b.
Find the optimal solution using the graphical solution procedure. How many tons of each product should be produced, and what is the projected total profit contribution? c. Is there any unused material? If so, how much? d. Are any of the constraints redundant? Is do, which ones? 15.
Refer to the Par, Inc., problem described in Section 2. Suppose that Par’s management encounters the following situations: a. The accounting department revises its estimate of the profit contribution for the deluxe bag to $18 per bag. A new low-cost material is available for the standard bag, and the profit contribution per standard bag can be increased to $20 per bag.
New sewing equipment is available that whole increase the sewing operations capacity to 750 hours. (Assume the 10A +9B is the appropriate objective function.) If each of these situations is encountered separately, what is the optimal solution and the total profit contribution? 6 Quantitative Analysis BA 452 Homework 1 Questions 16. Refer to the feasible region for Par, Inc., problem in Figure 2. Develop an objective function that will make extreme point 5 the optimal extreme point.
What is the optimal solution for the objective function you selected in part (a)? c. What are the values of the slack variables associated with this solution? 17. Write the following linear program in standard form: Max 5𝐴𝐴 + 2𝐵𝐵 s. For the linear program Max 4𝐴𝐴 + 1𝐵𝐵 s.
Write the problem in standard form. Solve the problem using the graphical solution procedure. What are the values of the three slack variables at the optimal solution? 19. Given the linear program Max 3A + 4B s.
Write the problem in standard form. Solve the problem using the graphical solution procedure. What are the values of the three slack variables at the optimal solution? 7 Quantitative Analysis BA 452 Homework 1 Questions 20. For the linear program Max 3A + 2B s.
Write the problem in standard form. Solve the problem using the graphical solution procedure. What are the values of the three slack variables at the optimal solution? 21. Consider the following linear program: Max 2A + 3B s.
5A + 5B ≤ 400 Constraint 1 -1A + 1B ≤ 10 Constraint 2 1A + 3B ≥ 90 Constraint 3 𝐴𝐴, 𝐵𝐵 ≥ 0 a. Graph the constraints, and place a number (1, 2, or 3) next to each constraint line to identify which constraint it represents. Shade in the feasible region on the graph. Identify the optimal extreme point.
What is the optimal solution? d. Which constraints are binding? Explain. How much clack or surplus is associated with the nonbinding constraint? 8 Quantitative Analysis BA 452 Homework 1 Questions 22. Reiser Sports Products wants to determine the number of All-Pro (A) and College (C) footballs to produce in order to maximize profit over the next four-week planning horizon.
Constraints affecting the production quantities are the production capacities in three departments: cutting and dyeing; sewing; and inspection and packaging. For the four-week planning period, 340 hours of cutting and dyeing time, 420 hours of sewing time, and 200 hours of inspection and packaging time are available. All-Pro footballs provide a profit of $5 per unit and College footballs provide a profit of $4 per unit. The liner programming model with production times expressed in minutes is as follows: Max 5A + 4C s.
12A + 6C ≤ 20,400 Cutting and dyeing 9A + 15C ≤ 25, 200 Sewing 6A + 6C ≤ 12,000 Inspection A, C ≥ 0 a) Graph all constraints, then shade the feasible region for this problem. b) Determine the coordinates of each extreme point and the corresponding profit. Which extreme point generates the highest profit. c) Draw the profit line corresponding to a profit of $4000.
Move the profit line as far from the origin as you can in order to determine which extreme point will provide the optimal solution. Compare your answer with the approach you used in part (b). d) Which constraints are binding? Explain. e) Suppose that the values of the objective function coefficients are $4 for each All-Pro model produced and $5 for each college model.
Use the graphical solutions procedure to determine the new optimal solution and the corresponding value of profit. Embassy motorcycles (EM)manufactures two lightweight motorcycles designed for easy handling and safety. The EZ-Rider model has a new engine and a low profile that make it easy to balance. The Lady- Sport model is slightly larger, uses a more traditional engine, and is specifically designed to appeal to women riders.
Embassy produces the engines for both models at its Des Moines, Iowa plant. Each EZ- Rider requires 6 hours of manufacturing time and each Lady-Sport engine requires 3 hours of manufacturing time. The Des Moines plant has 2100 hours of engine manufacturing time available for the next production period. Embassy’s motorcycle frame supplier can supply as many EZ-Rider frames as needed.
However, the Lady-Sport frame is more complex and the supplier can only provide up to 280 Lady-Sport frames for the next production period. Final assembly and testing requires 2 hours for each EZ-Rider model and 2.5 hours for each Lady-Sport model. A maximum of 1000 hours of assembly and testing time are available for the next production period. The company’s accounting department projects a profit contribution of $2400 for each EZ-Rider produced and $1800 for each Lady-Sport produced.
Formulate a linear programming model that can be used to determine the number of units of each model that should be produced in order to maximize the total contribution to profit. Solve the problem graphically. What is the optimal solution? c. Which constraints are binding? 9 Quantitative Analysis BA 452 Homework 1 Questions 24.
Kelson Sporting Equipment, Inc., makes two different types of baseball gloves: A regular model and a catcher’s model.