Five basic Linear Programming problems
ORSA MAC I NAME: ____________________________ DATE: 9 Oct 2009 1. A farmer grows corn and wheat on a farm of 40 acres. The farmer has 120 hours of labor available for planting. An acre of corn requires 2 hours of labor to plant and an acre of wheat requires 4 hours of labor to plant. An acre of corn produces 25 bushels, while an acre of wheat produces only 20 bushels. The farmer makes $5 per bushel of corn and $7.50 per bushel of wheat. How many acres of each should the farmer plant if he wants to maximize his profit? Formulate the algebraic model. Do not waste your time solving the problem. 2. Given the LP model shown below. Min 5X + 7Y + 10W ST Constraint 1) 2X + 2Y + 2W ≤ 50 Constraint 2) 2X + 7Y + 5W ≤ 100 Constraint 3) 2X + 1Y ≥ 25 X, Y, W ≥ 0 a. Convert the model to standard form. b. In creating the initial Simplex tableau from a model in standard form, why aren’t the excess variables, Ei, included in the initial set of basic variables? 3. Given the problem statement, LP model, and Simplex tableau shown below. Problem Statement: The ORSA MAC Widget Company produces two types of widgets used in the Air Force’s new anti-gravity machine. Each widget 1 produced consumes three ounces of silver and two ounces of a base metal. Each widget 2 produced consumes two ounces of silver and four ounces of the base metal. The production line is allocated 25 ounces of silver and 30 ounces of the base metal per day. The Air Force requires at least four widget 1 per day and five widget 2 per day. ORSA MAC Widget Company charges the Air Force $40 for each widget 1 and $20 for each widget 2. The company wants to develop a daily production plan that maximizes its revenue. The Model in Standard Form: Let Wi = the number of widgets of type i, where i = 1 & 2 produced on a daily basis. Max Z = 40W1 + 20W2 + 0S1 + 0S2 + 0E1 + 0E2 – 1000A1 – 1000A2 ST Silver) 3W1 + 2W2 + S1 = 25 Base Metal) 2W1 + 4W2 + S2 = 30 Widget 1 Demand) W1 – E1 + A1 = 4 Widget 2 Demand) W2 – E2 + A2 = 5 Non-Negativity) W1, W2, S1, S2, E1, E2, A1, A2 ≥ 0 The Simplex Tableau: a. Is the simplex tableau the final optimal tableau? Why or why not? b. If the tableau is not optimal, identify the entering and leaving variables. If the tableau is optimal, write “Optimal Tableau.” c. What is the current solution? Identify the value of all of the variables (basic and non-basic) and the objective function value. d. Given the table shown below, E1 (i.e. the number of excess widget 1 units produced) could be an entering variable since it has a positive reduced cost. If the value of E1 is set at 1 (i.e. E1 = 1), what happens to the values of the current set of basic variables and the objective function value? 4. Assume that in the above Widget problem the amount of silver has been increased by 3 ounces and the amount of base medal by 2. The revised algebraic model is shown below. Max Z = 40W1 + 20W2 ST Silver) 3W1 + 2W2 ≤ 28 Base Metal) 2W1 + 4W2 ≤ 32 Widget 1 Demand) W1 ≥ 4 Widget 2 Demand) W2 ≥ 5 Non-Negativity) W1, W2 ≥ 0 Given the Excel Solver solution for that model, answer the following questions. a. How many of each type of widget should ORSA MAC Widget produce and what is the total revenue? b. Identify the binding constraints. 5. In the out-of-class problem Iam Smart, the assembly plant manager for ORSA MAC Auto, was working on the production schedule for the following month. He had to decide how many Family Thrillseeker and Classy Cruisers to assemble in the plant that month to maximize profit for the company. He was faced with both labor and resource material constraints. Iam developed a linear programming model and determined that he should assemble 3800 Family Thrillseekers and 2400 Classy Cruisers for a profit of $26,640,000. Iam knows that he can increase the following month’s plant capacity by using overtime. If he does, the production schedule calls for the assembly of 3250 Family Thrillseekers and 3500 Classy Cruisers for a potential profit of $30,600,000. This calculation did not consider the cost of overtime and Iam knows that overtime labor does not come without an extra cost. What is the maximum amount that Iam should be willing to pay for this overtime labor? 1 ORSA MAC I FOUO- Sensitive Examination Materials Math Programming Exam 2 Examination Materials – Sensitive in Nature Math Programming Exam 1 – Out-of-Class Problem NAMES: ___________________________ ___________________________ ___________________________ ___________________________ DUE DATE: _____________________ Raw Score: ______________________ ADMINISTRATIVE NOTES: 1. This problem is worth 25 points and will count as part of your Exam 1 grade. 2. You will work this problem as a member of a team (see attached team assignments). Turn in only one problem report. This report does not have to be typed. Hand written answers are fine. Please insure that it is neatly written and organized. I must be able to read it to grade it. 3. Turn in a copy of your Excel file(s) along with the report. Identify your file by student names (e.g. Smith-Harding). 4. Submit your report and Excel file(s) to me at
[email protected] NLT 0800 on Thursday 17 June 2021. Problem Scenario: ORSA MAC Auto, a large automobile manufacturing company, organizes the vehicles it manufactures into three families: a family of trucks, a family of small cars, and a family of midsize and luxury cars. One plant located in Nowhere, VA assembles two models from the family of midsized and luxury cars. The first model, the Family Thrillseeker, is a four-door sedan marked as a great buy for middle-class families with tight budgets. Each Family Thrillseeker sold generates a profit of $3,600 for the company. The second model, the Classy Cruiser, is a two-door luxury car marketed as a privilege of affluence for upper-middle-class families and each Classy Cruiser generates a healthy profit of $5,400 for the company. Iam Smart, the assembly plant manager, is working on the production schedule for next month. Specifically, he must decide how many Family Thrillseekers and Classy Cruisers to assemble in the plant to maximize profit for the company. Currently, the plant has a labor capacity of 48,000 labor-hours and it takes six labor-hours to assemble one Family Thrillseeker and ten and a half hours to assemble one Classy Cruiser. Because the plant is simply an assembly plant, the required parts are manufactured elsewhere and shipped to the assembly plant. For the next month there are no apparent parts issues with the exception of doors. Because of a recent strike the doors supplier will not be at full production. Iam knows that he will only be able to obtain a maximum of 20,000 doors (10,000 left-hand doors and 10,000 right-hand doors) during the coming month. A recent company forecast of monthly demands for different automobile models suggest that the demand for the Classy Cruiser is limited to 3,500 cars. There is no limit on the demand for the Family Thrillseeker within the capacity limits of the assembly plant. Requirements: A. (5 points) Formulate and solve a linear programming problem to determine the number of Family Thrillseekers and Classy Cruisers to assemble during the month. Provide a copy of your Excel model as support for your solution. B. (10 points) The marketing department knows that it can pursue a targeted $500,000 advertising campaign that will increase demand for the Classy Cruiser next month by 20%. Additionally, Iam knows that using overtime he can increase next month’s plant labor capacity by 25%. The estimated costs for the maximum usage of overtime labor-hours is $1,600,000 beyond the regular time rates. Should Iam employ either of these options? Justify your recommendation. Once again, provide a copy of your Excel model. C. (10 points) ORSA MAC Auto discovered that dealerships are actually heavily discounting the price of the Family Thrillseeker to move them off the lot. Because of a profit-sharing agreement with its dealers, the company is therefore only making a profit of $2,800 on the Family Thrillseeker. Through a random testing of the Family Thrillseeker at the end of the assembly process Iam discovered that in 60% of the cases two of the four doors did not seal properly. Because the percentage of defective cars is so high, Iam decided to perform quality control tests on every Family Thrillseeker at the end of the line. Because of the added tests, the time it takes to assemble a Family Thrillseeker will increase from six hours to seven and a half hours. What impact will these two issues have on the original production plan (see requirement A)? Would implementing the advertising campaign or using overtime lessen the impact of these problems? Justify your recommendation and provide a copy of your Excel model. NAME TEAM ASSIGNMENT BUFORD DUSTIN 2 CANNON, DANIEL 3 CHOI PEACE P 5 COOPER JED R