1. **Problem Statement:** Precision Gears Ltd. needs to create an aggregate production plan for 3 months with given demands, capacities, and costs. We use the Transportation Table approach where rows are supply sources (Regular Time, Overtime, Inventory) and columns are monthly demands. 2. **Given Data:** - Demand: Month 1 = 500, Month 2 = 800, Month 3 = 700 units - Initial Inventory (start Month 1) = 100 units - Regular Time Capacity = 600 units/month, Cost = 40/unit - Overtime Capacity = 200 units/month, Cost = 60/unit - Inventory Holding Cost = 5/unit/month - Backordering: Not permitted initially 3. **Transportation Table Setup (No Backordering):** - Rows (Supply Sources): Regular Time, Overtime, Inventory - Columns (Demand): Month 1, Month 2, Month 3 4. **Cost Calculation for Each Cell:** - Regular Time and Overtime costs are constant per unit per month. - Inventory cost applies when inventory from previous month is used to meet current month demand. 5. **Cost Matrix Without Backordering:** - Month 1: - Regular Time: 40 - Overtime: 60 - Inventory: Since initial inventory is available at Month 1 start, no holding cost applies for Month 1 demand, so cost = 0 (using initial inventory) - Month 2: - Regular Time: 40 - Overtime: 60 - Inventory: Holding cost for carrying inventory from Month 1 to Month 2 = 5 - Month 3: - Regular Time: 40 - Overtime: 60 - Inventory: Holding cost for carrying inventory from Month 2 to Month 3 = 5 6. **Matrix Representation (No Backordering):** $$\begin{array}{c|ccc} & \text{Month 1} & \text{Month 2} & \text{Month 3} \\ \hline \text{Regular Time} & 40 & 40 & 40 \\ \text{Overtime} & 60 & 60 & 60 \\ \text{Inventory} & 0 & 5 & 5 \end{array}$$ 7. **If Backordering is Allowed at $20/unit/month$:** - Backordering cost applies when demand is delayed to next month. - Add a new row for Backordering with cost 20 per unit per month. - Backordering cost accumulates if delayed multiple months. 8. **Cost Matrix With Backordering:** - Month 1: - Regular Time: 40 - Overtime: 60 - Inventory: 0 - Backordering: Not applicable for Month 1 demand (cannot delay before Month 1) - Month 2: - Regular Time: 40 - Overtime: 60 - Inventory: 5 - Backordering: 20 (cost to delay Month 2 demand to Month 3) - Month 3: - Regular Time: 40 - Overtime: 60 - Inventory: 5 - Backordering: 20 (cost to delay Month 3 demand to Month 4, if allowed) 9. **Matrix Representation (With Backordering):** $$\begin{array}{c|ccc} & \text{Month 1} & \text{Month 2} & \text{Month 3} \\ \hline \text{Regular Time} & 40 & 40 & 40 \\ \text{Overtime} & 60 & 60 & 60 \\ \text{Inventory} & 0 & 5 & 5 \\ \text{Backordering} & \infty & 20 & 20 \end{array}$$ (Note: Backordering cost for Month 1 demand is infinite or not allowed since no prior month to delay from.)