MGT 420 ASSIGNMENT 1
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MGT 420 ASSIGNMENT 1 1. Logan, Ltd. makes two products, tables and chairs, which must be processed through assembly and finishing departments. Each table requires 5 hours to assemble and 3 hours to finish, but each chair requires 2 hours to assemble and 5 hours to finish. Assembly department has 8 0 hours available, and finishing department can handle up to 9 0 hours of work. The company wants to see at least 5 chairs produced during the production. The production of chairs should be more than or equal to the production of table s . Each table yields a profit of $8, and each chair can be sold for a profit of $4. a. Show the feasible region graphically (Draw a graph by using MS WORD) . b. What are the extreme points of the feasible region? c. Find the optimal solution using the graphical method. d. Are there any slack values? Are there any surplus values? e. Compute the shadow price s of each constraint. Answer: Problem Breakdown: Logan, Ltd. produces two products — tables and chairs. We need to determine the number of tables (T) and chairs (C) to produce to maximize profit while considering various constraints. Here's the information provided: • Time requirements for processing: o Tables: 5 hours to assemble, 3 hours to finish o Chairs: 2 hours to assemble, 5 hours to finish • Available time: o Assembly department: 80 hours o Finishing department: 90 hours • Profit: o Profit per table = $8 o Profit per chair = $4 • Constraints: o At least 5 chairs should be produced: C≥5C \ geq 5 o The number of chairs produced should be greater than or equal to the number of tables: C≥TC \ geq T o Time constraints: ▪ 5T + 2C ≤ 80 (Assembly department constraint) ▪ 3T + 5C ≤ 90 (Finishing department constraint) a. Feasible Region Graphically: To create the graph, plot the inequalities on a graph. Use the following steps: 1. Graph the assembly constraint : 5T+2C≤805T + 2C \ leq 80 o Rewrite for CC: C=80−5T2C = \ frac{80 - 5T}{2} 2. Graph the finishing constraint : 3T+5C≤903T + 5C \ leq 90 o Rewrite for CC: C=90−3T5C = \ frac{90 - 3T}{5} 3. Add the other constraints : o C≥5C \ geq 5 o C≥TC \ geq T 4. Plot the feasible region formed by these inequalities. You can use MS Word or Excel to plot these graphs. The feasible region will be the area bounded by the lines of these inequalities. b. Extreme Points of the Feasible Region: To determine the extreme points of the feasible region, solve the system of equations formed by the intersection of the boundary lines of the constraints. The extreme points represent potential solutions. These points are where the constraints intersect. The system of equations that we need to solve to find the intersections is: 1. 5T+2C=805T + 2C = 80 (Assembly constraint)Page 2
MGT 420 ASSIGNMENT 1 1. Logan, Ltd. makes two products, tables and chairs, which must be processed through assembly and finishing departments. Each table requires 5 hours to assemble and 3 hours to finish, but each chair requires 2 hours to assemble and 5 hours to finish. Assembly department has 8 0 hours available, and finishing department can handle up to 9 0 hours of work. The company wants to see at least 5 chairs produced during the production. The production of chairs should be more than or equal to the production of table s . Each table yields a profit of $8, and each chair can be sold for a profit of $4. a. Show the feasible region graphically (Draw a graph by using MS WORD) . b. What are the extreme points of the feasible region? c. Find the optimal solution using the graphical method. d. Are there any slack values? Are there any surplus values? e. Compute the shadow price s of each constraint. Answer: Problem Breakdown: Logan, Ltd. produces two products — tables and chairs. We need to determine the number of tables (T) and chairs (C) to produce to maximize profit while considering various constraints. Here's the information provided: • Time requirements for processing: o Tables: 5 hours to assemble, 3 hours to finish o Chairs: 2 hours to assemble, 5 hours to finish • Available time: o Assembly department: 80 hours o Finishing department: 90 hours • Profit: o Profit per table = $8 o Profit per chair = $4 • Constraints: o At least 5 chairs should be produced: C≥5C \ geq 5 o The number of chairs produced should be greater than or equal to the number of tables: C≥TC \ geq T o Time constraints: ▪ 5T + 2C ≤ 80 (Assembly department constraint) ▪ 3T + 5C ≤ 90 (Finishing department constraint) a. Feasible Region Graphically: To create the graph, plot the inequalities on a graph. Use the following steps: 1. Graph the assembly constraint : 5T+2C≤805T + 2C \ leq 80 o Rewrite for CC: C=80−5T2C = \ frac{80 - 5T}{2} 2. Graph the finishing constraint : 3T+5C≤903T + 5C \ leq 90 o Rewrite for CC: C=90−3T5C = \ frac{90 - 3T}{5} 3. Add the other constraints : o C≥5C \ geq 5 o C≥TC \ geq T 4. Plot the feasible region formed by these inequalities. You can use MS Word or Excel to plot these graphs. The feasible region will be the area bounded by the lines of these inequalities. b. Extreme Points of the Feasible Region: To determine the extreme points of the feasible region, solve the system of equations formed by the intersection of the boundary lines of the constraints. The extreme points represent potential solutions. These points are where the constraints intersect. The system of equations that we need to solve to find the intersections is: 1. 5T+2C=805T + 2C = 80 (Assembly constraint)