Titus Jr I.
asked • 10/05/23Operations Management question
The Pinewood Furniture Company produces chairs and tables from two resources: labor and wood. The company has 145 hours of labor and 100 board-ft. of wood available each day. Demand for chair is limited to 8 per day. Each chair requires 3 hours of labor and 4 board-ft. of wood, whereas a table requires 22 hours of labor and 9 board-ft. of wood. The profit derived from each chair is $100 and from each table is $500. The company wants to determine the number of chairs and tables to produce each day in order to maximize profit. The correct linear programming model formulation of this problem is:
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2 Answers By Expert Tutors
Mauricio Q. answered • 10/07/23
Tutor
4.8
(4)
MBA Berkeley - EX CEO - Strategy Professor - Business Consultant
FOR SOLVING THIS PROBLEM, THE BEST TOOL IS USING LINEAR PROGRAMMING METHODS, SUCH AS THE SIMPLEX METHOD.
As the answers is limited to 10,000 characters, I will divide my answer in 3 parts:
1. I will explain the how to prepare the problem to solve it
2. I will explain how to solve it using SIMPLEX in excel
3. I will explain how to solve it using SIMPLEX in phyton
1. PREPARE THE PROBLEM TO SOLVE IT:
The Pinewood Furniture Company wants to maximize its profit by determining the optimal number of chairs and tables to produce, given constraints on labor, wood, and demand.
Let:
- C = number of chairs to be produced per day,
- T = number of tables to be produced per day.
OBJECTIVE FUNCTION: The company wants to maximize its profit. Profit from chairs is $100 each and from tables is $500 each, so:
Maximize Z=100C+500T
Subject to Constraints:
1. Labor constraint: Each chair requires 3 hours of labor and each table requires 22 hours. The company has 145 hours available each day. 3C+22T≤145
2. Wood constraint: Each chair requires 4 board-ft. of wood and each table requires 9 board-ft. The company has 100 board-ft. available each day. 4C+9T≤100
3. Demand constraint: The company can only sell up to 8 chairs per day. C≤8
4. Non-negativity constraints: The number of chairs and tables cannot be negative. C≥0,T≥0
In formal mathematical notation, the linear programming model is:
- Maximize Z=100C+500T
- Subject to:
- 3C+22T≤145
- 4C+9T≤100
- C≤8
- C,T≥0
Mauricio Q.
2. SOLVE IT USING EXCEL 1. Go to the "File" tab. 2. Click "Options". 3. In the "Excel Options" dialog box, click "Add-Ins". 4. In the "Manage" box, click "Excel Add-Ins", then click "Go". 5. In the "Add-Ins" box, check "Solver Add-In", then click "OK". After enabling the Solver add-in, you can use the following structure to organize your spreadsheet and solve the problem: 1. Decision Variables: o Cell A1: "Chairs" o Cell A2: "Tables" o Cell B1: (leave this empty for now, it'll be the solution from Solver, this cell represents C) o Cell B2: (leave this empty for now, it'll be the solution from Solver, this cell represents T) 2. Coefficients in Objective Function: o Cell C1: 100 o Cell C2: 500 3. Objective Function Value (Profit): o Cell D1: "Profit" o Cell D2: =B1*C1 + B2*C2 4. Constraints: o Cell E1: "Labor" o Cell E2: "Wood" o Cell E3: "Chair Demand" o Cell F1: 145 o Cell F2: 100 o Cell F3: 8 o Cell G1: =3*B1 + 22*B2 o Cell G2: =4*B1 + 9*B2 o Cell G3: =B1 After populating the cells with the appropriate values and formulas, you can set up the problem in Solver: 10. Go to the "Data" tab and click on "Solver". 11. Set the objective: D2 to "Max". 12. By changing variable cells: B1:B2. 13. Add constraints: G1 <= F1 G2 <= F2 G3 <= F3 14. Choose "Simplex LP" as a solving method. 15. Click "Solve". Solver will find the optimal solution and populate B1 and B2 with the number of chairs and tables to produce, respectively, in order to maximize the profit while adhering to the constraints. Ensure to check if the solution makes logical sense and verify against the original constraints manually as well.
Report
10/07/23
Mauricio Q.
3. SOLVE IT USING PHYTON To solve this linear programming problem using Python, you might use a library like PuLP or SciPy. Below is an example using PuLP, which is widely used for linear programming problems. First, you'll need to install PuLP if you haven't already. You can install it using pip: pip install pulp Then you can use the following code to solve the problem: Plain Bash C++ C# CSS Diff HTML/XML Java Javascript Markdown PHP Python Ruby SQL #Created by Mauricio Quiroga (c) 2023. [email protected] import pulp # Define the Linear Program lp = pulp.LpProblem("Maximize_Profit", pulp.LpMaximize) # Define decision variables C = pulp.LpVariable("C", lowBound=0, cat='Integer') # number of chairs T = pulp.LpVariable("T", lowBound=0, cat='Integer') # number of tables # Objective function: Maximize Z = 100C + 500T lp += 100 * C + 500 * T, "Z" # Constraints lp += 3 * C + 22 * T <= 145, "Labor" lp += 4 * C + 9 * T <= 100, "Wood" lp += C <= 8, "Chair Demand" # Solve the problem lp.solve() # Get the results print("Optimal number of chairs to produce:", pulp.value(C)) print("Optimal number of Here's how Here's Here's how the code works: First, we define our linear program and decision variables, C and T, representing the number of chairs and tables respectively. Then we define our objective function, which is to maximize Z=100C+500T. After that, we define our constraints, ensuring that they are in accordance with the ones you provided. We then solve the linear program and print the results. Run the code, and it will give you the optimal number of chairs and tables to produce in order to maximize profit while satisfying all the constraints.
Report
10/07/23
This linear programming model formula has several parts.
Begin with the sentence "The company wants to determine the number of chairs and tables to produce each day in order to maximize profit."
Writing this formula will tell you what the variables are for the graph that is to come.
The formula comes from the sentence preceding. "The profit derived from each chair is $100 and from each table is $500." This is pretty straightforward.
Let c = number of chairs and t = number of tables
Profit = 100c + 500t
Now set this formula aside to write the constraints formulas. These come from the earlier sentences.
Labor formula: 3c + 22 t <= 145
Wood formula: 4c + 9 t <= 100
There is another constraint in this sentence: "Demand for chair is limited to 8 per day"
So. c <= 8.
Of course we must include the constraints that you cannot produce a negative number of chairs or tables so
c >= 0 and t >= 0 are part of the solution.
The question only asks for the model and does not ask for you to solve it.
So the answer is all the inequalities that I have printed in bold.
Linda B.
tutor
Incidentally, the solution is 8 chairs and 5 tables to maximize the profit of $3300 per day.
Report
10/05/23
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Mark M.
Did you formulate all of the conditions and then plot them?10/05/23