BUS2 190 Quantitative Business Analysis, Spring 2016 Homework 7 Enter your answers to Canvas. Avoid rounding errors between steps. Round your nal answers to 4 decimal places as needed. 7.1. True of False. 7.1.a (0.3 pt) A linear programming problem may have more than one optimal solution. 7.1.b (0.3 pt) In a linear program, an infeasible solution violates all constraints. 7.1.c (0.3 pt) A feasible solution point does not have to lie on the boundary of the feasible region. 7.2. (0.3 pt) Use graphical method to solve the following linear program. max 8x + 7y subject to: 3x + y 13 (1) 5x + 3y 27 (2) x+y 3 (3) x 0 (4) y 0 (5) What is the optimal value of the objective function? [Hint: Find the optimal values of x and y, then plug in those values to the objective function. For example, when (x, y) = (3, 4), then the objective function value = 8(3) + 7(4) = 52. Note that the optimal solution of (x, y) is NOT (3, 4) because you can further move the objective function upwards and nd other feasible solutions that are better than this point.] 7.3. Tony Stark's company makes two types of bullet proof \"iron suits\": Super Red and Invincible Blue. These iron suits are made of two types of special metals: metal A and metal B. Each Super Red suit requires 3 pounds of metal A and 1 pound of metal B. Each Invincible Blue suit requires 1 pound of metal A and 2 pounds of metal B. 1 BUS2 190 Quantitative Business Analysis, Spring 2016 The prot for each Super Red suit is $100 dollars and the prot for each Invincible Blue suit is $120. Due to government regulations, Tony can only obtain 450 pounds of metal A and 300 pounds of metal B per month. Tony also needs to produce at least 50 Invincible Blue suits per month to fulll a government contract. The demand for the iron suits is high and all the suits made will be sold. As Tony's Chief Operations Ocer, Tony wants you to nd a production plan (i.e., the number of suits for each type to produce) to maximize his monthly prot. You decide to formulate the production planning problem as a linear program, and dene the decision variable as follows. Let x be the number of Super Red suits to make per month. Let y be the number of invincible Blue suits to make per month. 7.3.a (0.3 pt) What is the objective function? A) max x + y B) max 100x + 120y C) max 120x + 100y D) max 300x + 450y Next, choose answers for 7.3.b-7.3.e from the following equations A) x + y 450 B) x + y 300 C) 3x + y 450 D) x + 2y 300 E) y 50 F) x 0 G) x = 0 7.3.b (0.3 pt) Which constraint represents metal A's availability? 7.3.c (0.3 pt) Which constraint represents metal B's availability? 7.3.d (0.3 pt) Which constraint represents the minimum requirement of Invincible Blue suits? 7.3.e (0.3 pt) Which constraint represents the non-negativity constraint for the quantity of Super Red suits? 2 BUS2 190 Quantitative Business Analysis, Spring 2016 After you answer the above questions, you would have the complete formulation (objective function and constraints) of the linear program. The following steps will help you nd the optimal solution for the production plan. Draw the constraint lines. Shade the feasible region of this linear program Draw a line that has a same slop as the objective function [hint: draw the prot line that equal to 12000] Utilize the objective function line to nd the optimal solution of x and y. Don't simply eyeball the answer. You should nd the exact solution by solving the two equations (constraints) for the intersection point. 7.3.f (0.3 pt) Report the maximum prot at the optimal solution. 7.4. Island Water Sports is a business that provides rental equipment and instruction for a variety of water sports in a resort town. On one particular morning, a decision must be made of how many Wildlife Raft Trips and how many Group Sailing Lessons should be scheduled. Each Wildlife Raft Trip requires one captain and one crew person, and can accommodate six passengers. The revenue per raft trip is $120. Ten rafts are available, and at least 30 people are on the list for reservations this morning. Each Group Sailing Lesson requires one captain and two crew people for instruction. Two boats are needed for each group. Four students form each group. There are 12 sailboats available, and at least 20 people are on the list for sailing instruction this morning. The revenue per group sailing lesson is $160. The company has 12 captains and 18 crew available this morning. The company would like to maximize the number of customers served while generating at least $1800 in revenue and honoring all reservations. Dene the decision variables as follows and formulate the problem as a linear program. Let x be the number of Wildlife raft Trips to be scheduled. Let y be the number of Group Sailing Lessons to be scheduled. 7.4.a What is the objective function? A) max x + y B) max 120x + 160y 1800 C) max 120x + 160y D) max x + y 1800 E) min 1800x + 1800y 3 BUS2 190 Quantitative Business Analysis, Spring 2016 F) max 6x + 4y The following constraints complete the problem: 120x + 160y 1800 (1) x 10 (2) 2y 12 (3) 6x 30 (4) 4y 20 (5) x + y 12 (6) x + 2y 18 (7) 7.4.b (0.3 pt) What does constraint (1) represent? A) The maximum number of captions who can be deployed. B) The maximum number of crews who can be deployed. C) The minimum revenue requirement. D) The maximum quantity of the rafts available. E) The maximum quantity of the sailboats available. F) The demand of raft trips that must be satised. G) The demand of sailing lessons that must be satised. 7.4.c (0.3 pt) What does constraint (2) represent? A) The maximum number of captions who can be deployed. B) The maximum number of crews who can be deployed. C) The minimum revenue requirement. D) The maximum quantity of the rafts available. E) The maximum quantity of the sailboats available. F) The demand of raft trips that must be satised. G) The demand of sailing lessons that must be satised. 7.4.d (0.3 pt) What does constraint (4) represent? A) The maximum number of captions who can be deployed. B) The maximum number of crews who can be deployed. C) The minimum revenue requirement. D) The maximum quantity of the rafts available. 4 BUS2 190 Quantitative Business Analysis, Spring 2016 E) The maximum quantity of the sailboats available. F) The demand of raft trips that must be satised. G) The demand of sailing lessons that must be satised. 7.4.e (0.3 pt) What does constraint (6) represent? A) The maximum number of captions who can be deployed. B) The maximum number of crews who can be deployed. C) The minimum revenue requirement. D) The maximum quantity of the rafts available. E) The maximum quantity of the sailboats available. F) The demand of raft trips that must be satised. G) The demand of sailing lessons that must be satised. 5