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Contois Carpets is a small manufacturer of carpeting for home and office installations. Production capacity, demand, production cost per square yard (in dollars), and inventory holding cost per square yard (in dollars) for the next four quarters are shown in the network diagram below. Min s.t. Beginning Inventory Flow X16 x26 Quarter 1 Demand Flow x37 Quarter 1 Production Flow Quarter 2 Demand Flow x 48 50 Quarter 2 Production Flow Quarter 3 Demand Flow X59 X67 600 Quarter 4 Demand Flow X78 X89 300 Quarter 3 Production Flow Quarter 4 Production Flow 910 500 = $ 400 = $ Production Capacities = $ = $ = $ Ending Inventory Flow Xij ≥ 0 for all i, j. Solve the linear program to find the optimal solution. = $ = $ = $ Production Nodes = $ 2 Quarter 1 Production 3 Quarter 2 Production Beginning Inventory 4 Quarter 3 Production Quarter 4 Production Develop a linear programming model to minimize cost and meet demand exactly. (Let x; be the number of square yards of carpet which flows from node i to node j.) Production Cost Per Square Yard 0 2 5 Inventory Cost per Square Yard 3 3 Production (arcs) Demand Nodes Report the cost (in dollars) associated with the optimal solution. $ 6 Quarter 1 Demand 0.25 7 Quarter 2 Demand. 0.25 8 Quarter 3 Demand 0.25 9 Quarter 4 Demand 0.25 10 Ending Inventory 400 500 400 400 100 Demand

Contois Carpets is a small manufacturer of carpeting for home and office installations. Production capacity, demand, production cost per square yard (in dollars), and inventory holding cost per square yard (in dollars) for the next four quarters are shown in the network diagram below. Min s.t. Beginning Inventory Flow X16 x26 Quarter 1 Demand Flow x37 Quarter 1 Production Flow Quarter 2 Demand Flow x 48 50 Quarter 2 Production Flow Quarter 3 Demand Flow X59 X67 600 Quarter 4 Demand Flow X78 X89 300 Quarter 3 Production Flow Quarter 4 Production Flow 910 500 = $ 400 = $ Production Capacities = $ = $ = $ Ending Inventory Flow Xij ≥ 0 for all i, j. Solve the linear program to find the optimal solution. = $ = $ = $ Production Nodes = $ 2 Quarter 1 Production 3 Quarter 2 Production Beginning Inventory 4 Quarter 3 Production Quarter 4 Production Develop a linear programming model to minimize cost and meet demand exactly. (Let x; be the number of square yards of carpet which flows from node i to node j.) Production Cost Per Square Yard 0 2 5 Inventory Cost per Square Yard 3 3 Production (arcs) Demand Nodes Report the cost (in dollars) associated with the optimal solution. $ 6 Quarter 1 Demand 0.25 7 Quarter 2 Demand. 0.25 8 Quarter 3 Demand 0.25 9 Quarter 4 Demand 0.25 10 Ending Inventory 400 500 400 400 100 Demand

Practical Management Science
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter2: Introduction To Spreadsheet Modeling
Section: Chapter Questions
Problem 20P: Julie James is opening a lemonade stand. She believes the fixed cost per week of running the stand...
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**Production and Demand Optimization for Contois Carpets** Contois Carpets, a small manufacturer for home and office carpeting, is looking to optimize production and meet demands cost-effectively over four quarters. Below is a detailed analysis derived from a network diagram, capturing production capacities, costs, demands, and inventory management. ### Diagram Explanation The network diagram consists of multiple nodes representing production capacities and demand points across four quarters. The nodes are connected by arcs which indicate the flow of square yards of carpet. Here's a breakdown of the key components: #### Production Nodes 1. **Node 1**: Beginning Inventory - Capacity: 50 square yards - Cost: $0 per square yard 2. **Nodes 2 to 5**: Quarterly Production - Quarter 1 Production (Node 2): 600 sq. yards with a cost of $2/sq. yard - Quarter 2 Production (Node 3): 300 sq. yards with a cost of $5/sq. yard - Quarter 3 Production (Node 4): 500 sq. yards with a cost of $3/sq. yard - Quarter 4 Production (Node 5): 400 sq. yards with a cost of $3/sq. yard #### Demand Nodes 6. **Quarterly Demand Nodes** - Quarter 1 Demand (Node 6): 400 sq. yards - Quarter 2 Demand (Node 7): 400 sq. yards - Quarter 3 Demand (Node 8): 400 sq. yards - Quarter 4 Demand (Node 9): 400 sq. yards #### Inventory Management - **Inventory Holding Cost**: $0.25 per square yard is applied to any inventory saved for future quarters. #### Ending Inventory - **Node 10**: Designed to accommodate a demand of 100 square yards by the end of the fourth quarter. ### Linear Programming Model The goal is to develop a linear programming model that minimizes costs and exactly meets demand for all quarters. - **Objective Function**: Minimize costs by optimizing production and inventory flow to meet demand requirements. - **Constraints**: - Ensure non-negativity: \( x_{ij} \geq 0 \) for all \( i, j \) - Control flows from production nodes to demand nodes following capacity and cost guidelines. ### Solution Parameters
Transcribed Image Text:**Production and Demand Optimization for Contois Carpets** Contois Carpets, a small manufacturer for home and office carpeting, is looking to optimize production and meet demands cost-effectively over four quarters. Below is a detailed analysis derived from a network diagram, capturing production capacities, costs, demands, and inventory management. ### Diagram Explanation The network diagram consists of multiple nodes representing production capacities and demand points across four quarters. The nodes are connected by arcs which indicate the flow of square yards of carpet. Here's a breakdown of the key components: #### Production Nodes 1. **Node 1**: Beginning Inventory - Capacity: 50 square yards - Cost: $0 per square yard 2. **Nodes 2 to 5**: Quarterly Production - Quarter 1 Production (Node 2): 600 sq. yards with a cost of $2/sq. yard - Quarter 2 Production (Node 3): 300 sq. yards with a cost of $5/sq. yard - Quarter 3 Production (Node 4): 500 sq. yards with a cost of $3/sq. yard - Quarter 4 Production (Node 5): 400 sq. yards with a cost of $3/sq. yard #### Demand Nodes 6. **Quarterly Demand Nodes** - Quarter 1 Demand (Node 6): 400 sq. yards - Quarter 2 Demand (Node 7): 400 sq. yards - Quarter 3 Demand (Node 8): 400 sq. yards - Quarter 4 Demand (Node 9): 400 sq. yards #### Inventory Management - **Inventory Holding Cost**: $0.25 per square yard is applied to any inventory saved for future quarters. #### Ending Inventory - **Node 10**: Designed to accommodate a demand of 100 square yards by the end of the fourth quarter. ### Linear Programming Model The goal is to develop a linear programming model that minimizes costs and exactly meets demand for all quarters. - **Objective Function**: Minimize costs by optimizing production and inventory flow to meet demand requirements. - **Constraints**: - Ensure non-negativity: \( x_{ij} \geq 0 \) for all \( i, j \) - Control flows from production nodes to demand nodes following capacity and cost guidelines. ### Solution Parameters
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