2 1. A bakery makes coffee cakes and Danish pastries in large pans. The main ingredients are flour and sugar. There are 60 pounds of flour and 32 pounds of sugar available. The demand for coffee cakes is less than or equal to 10 pans. Five pounds of flour and 2 pounds of sugar are required to make a pan of coffee cakes, and 5 pounds of flour and 4 pounds of sugar are required to make a pan of Danish pastries. A pan of coffee cakes has a profit of $3.50 and a pan of Danish pastries has a profit of $4.50. Determine the number of pans of cakes and Danish pastries to produce each day so that profit will be maximized. (a) Formulate a linear programming model for this problem by defining the decision variables, objective function and all the constraints. (b) Find the optimal solution of this model by hand using the corner points graphical method. 2. Solve the following linear programming model graphically. In addition, write the problem in standard form and do a constraint analysis for the optimal solution. Maximize 5x + 10y Subject to 2x + 3y 120 5x + 2y 40 x, y > 0 3. Love My Pet Foods produces dog food, made from beef products and grain. Each pound of beef costs $1.35, and each pound of grain costs $0.82. A pound of the dog food must contain at least 10 units of Vitamin 1 and 9 units of Vitamin 2. A pound of beef contains 8 units of Vitamin 1 and 11 units of Vitamin 2. A pound of grain contains 12 units of Vitamin 1 and 7 units of Vitamin 2. How many pounds of beef and grain should be included in each pound of dog food to minimize total cost? Formulate a linear programming model for the above situation by determining (a) The decision variables (b) Determine the objective function. What does it represent? (c) Determine all the constraints. Briefly describe what each constraint represents. Note: Do NOT solve the problem after formulating. 4. Determine whether the following linear programming problem is infeasible, unbounded, or has multiple optimal solutions. Draw a graph and explain your conclusion. Maximize 1020x + 1250y Subject to 2x + 1y > 40 -3x + 4y < 60 x, y > 0 5. An investment company has 25 million dollars to invest this year. Currently, the company is considering different ways to invest the money. Their objective is to have maximum return on the 1 investment. What are the different investment alternatives that may be used to invest the 25 million dollars and what are the important factors in each alternative that should be considered in making the decision? 5b. Each student must comment on another student's investment alternative to discuss its important aspects. Reply regarding below answer. The investment company should invest 15 million dollars in its Investment Management System for its portfolio management, and 10 million dollars in brand awareness. Investing in a web-based portfolio would allow the company to implement a tracking software that will offer real-time price updates. These updates would sync directly to investment brokers. Investing in an updated investment system will allow the company to manage more accounts and react to unforeseen market risk. Updating the system will lower risk due to its real-time price updates. The 10 million dollars will be used to advertise the companies updated system and its capability to maximize its accounts by diversifying its selection, which in return lowers risk. 6. For this discussion topic, each student is required to have at least 2 postings: One explaining at least one of the following lettered keywords and a second responding to another student's posting. Please select a lettered keyword that has not been explained by the time you post your response. You can select any keyword to explain once all keywords are explained. Please do not copy the definition of a keyword from the textbook or the lecture notes. Use your own words to explain the meaning of each keyword. You can also give examples to clarify your definition. f. Model Selection and Development g. Model Solution and Implementation 7. A cereal company makes a cereal from two ingredients: wheat and oats. Each ingredient provides two essential nutrients: vitamin A and vitamin B. The company wants to know how many ounces of wheat and oats to include in each box of cereal so that the minimum requirements of 50 milligrams of vitamin A and 15 milligrams of vitamin B are satisfied. An ounce of wheat provides 10 milligrams of vitamin A and 2 milligrams of vitamin B. An ounce of oats provides 5 milligrams of vitamin A and 1 milligram of vitamin B. An ounce of wheat costs $0.03 and an ounce of oats costs $0.06. The company wants to minimize the total cost of production. Formulate the above situation as a linear programming problem. Do NOT solve after formulating. 8. Infeasible Problem: explain in your words and do not copy the definition of keywords from any textbook or google.com - Some of the questions above require drawing graphs. Here is a list of options for you to include a graph in your answer. Draw the graph using MS Paint or CorelDraw or some other software. Copy the graph in a MS Word file and post the file on the course website in appropriate category. OR Draw the graph by hand on a paper, scan it or take a photo and post it with your answer. OR Describe the graph in words in your answer. - Please post your answers in assignments section of the course website. 2 - A team should post their answers in the assignments section of the course website by any one of the team members. The submission must contain names of all team members. A member whose name is missing from the assignment will not receive any credit. A team submission will be graded based on the answers submitted by the team. If answer to one or more questions is missing because a member of the team did not do his/her part, then the team will not receive any credit for those questions. - Please submit the homework answers in Microsoft Word or PDF format. Scans or photos of the answers are acceptable but they must be clear and legible. Other formats are not acceptable. 3