Question

A Typing and Printing store wants to determine the optimal replacement policy for its photocopier during the next six years. The store does not keep photocopiers for longer than three years. The current price of brand XYZ is $3,500. The price is expected to rise by 4% per year. However, the store can trade in the machine with the same brand. The trade-in value of a one-year-old device is 68%, a two-year-old machine is 18%, and a three-year-old machine is 7%.

a) Develop a network flow for this problem. Make sure to include replacement costs in the network.
b) Write out the linear programming formulation for this problem and solve it to find the best replacement options.

Answer 1

The network flow model for this problem involves nodes, edges, costs, and trade-in values. The linear programming formulation involves decision variables, an objective function, and constraints. Solving the linear programming formulation will provide the best replacement options for the store.

Step-by-step explanation:

Network Flow Model

The network flow model for this problem can be represented as follows:

• Nodes: Start Node, Year 1 Node, Year 2 Node, Year 3 Node, End Node
• Edges: Start Node to Year 1 Node (with capacity 1), Year 1 Node to Year 2 Node (with capacity 1), Year 2 Node to Year 3 Node (with capacity 1), Year 3 Node to End Node (with capacity 1)
• Costs: Start Node to Year 1 Node (replacement cost of brand XYZ for 1 year), Year 1 Node to Year 2 Node (replacement cost of brand XYZ for 2 years), Year 2 Node to Year 3 Node (replacement cost of brand XYZ for 3 years)
• Trade-in Values: Year 1 Node to End Node (trade-in value of a one-year-old device), Year 2 Node to End Node (trade-in value of a two-year-old device), Year 3 Node to End Node (trade-in value of a three-year-old device)

Linear Programming Formulation

The linear programming formulation for this problem can be written as follows:

• Decision Variables: x1, x2, x3 (representing the number of years the photocopier is kept)
• Objective Function: Minimize (Replacement Cost of brand XYZ * x1 + Replacement Cost of brand XYZ * 2 * x2 + Replacement Cost of brand XYZ * 3 * x3 - 0.68 * Trade-in Value of one-year-old device * x1 - 0.18 * Trade-in Value of two-year-old device * x2 - 0.07 * Trade-in Value of three-year-old device * x3)
• Constraints: x1 + x2 + x3 <= 6 (photocopier cannot be kept for longer than 3 years), x1, x2, x3 >= 0 (non-negativity)

Solving this linear programming formulation will give you the best replacement options for the store.