Model Answer
0 min readIntroduction
Aggregate planning is a crucial intermediate-term planning process in operations management that aims to align production output with anticipated demand over a specified timeframe, typically 3 to 18 months. Its primary objective is to balance production, workforce, and inventory levels to minimize overall costs while meeting customer demand and organizational policies. This involves strategic decisions regarding production rates, inventory management, and the utilization of different capacity options like regular time, overtime, and subcontracting. Effective aggregate planning bridges the gap between long-term strategic goals and short-term operational schedules, ensuring resource optimization and cost-efficiency in dynamic market conditions.
Understanding the Aggregate Planning Problem
The core of an aggregate planning problem lies in managing the trade-offs between various cost factors (production, inventory holding) and capacity options (regular time, overtime, subcontracting) to satisfy fluctuating demand. Shortages are typically penalized heavily or, as in this problem, not permitted, making careful planning essential.
Problem Parameters:
- Planning Horizon: 4 months
- Initial Inventory: 50 units
- Carrying Cost: ₹1.60 per unit per month
- Shortages: Not permitted
Production Capacities and Costs:
| Production Capacity/Month | Units | Production Cost/Unit |
|---|---|---|
| Regular time | 900 units | ₹6/- |
| Overtime | 300 units | ₹8/- |
| Subcontracting | 200 units | ₹9/- |
Forecasted Demand:
- Month 1: 1300 units
- Month 2: 1400 units
- Month 3: 1225 units
- Month 4: 1475 units
Aggregate Plan Calculation
To provide an aggregate plan, we will adopt a cost-minimization strategy, prioritizing the cheapest production options first (regular time, then overtime, then subcontracting) to meet demand, while also considering inventory carrying costs. Since shortages are not permitted, production must always meet or exceed demand plus any required ending inventory.
Let's calculate the plan month by month:
Month 1:
- Beginning Inventory: 50 units
- Forecasted Demand: 1300 units
- Net Demand to be Met: 1300 - 50 = 1250 units
- Regular Time Production: Max 900 units @ ₹6/unit.
Cost = 900 * ₹6 = ₹5400 - Remaining Demand: 1250 - 900 = 350 units
- Overtime Production: Max 300 units @ ₹8/unit.
Cost = 300 * ₹8 = ₹2400 - Remaining Demand: 350 - 300 = 50 units
- Subcontracting Production: 50 units @ ₹9/unit.
Cost = 50 * ₹9 = ₹450 - Total Production: 900 (Regular) + 300 (Overtime) + 50 (Subcontracting) = 1250 units
- Ending Inventory: 0 units
- Total Cost (Month 1): ₹5400 + ₹2400 + ₹450 = ₹8250
Month 2:
- Beginning Inventory: 0 units
- Forecasted Demand: 1400 units
- Net Demand to be Met: 1400 units
- Regular Time Production: Max 900 units @ ₹6/unit.
Cost = 900 * ₹6 = ₹5400 - Remaining Demand: 1400 - 900 = 500 units
- Overtime Production: Max 300 units @ ₹8/unit.
Cost = 300 * ₹8 = ₹2400 - Remaining Demand: 500 - 300 = 200 units
- Subcontracting Production: Max 200 units @ ₹9/unit.
Cost = 200 * ₹9 = ₹1800 - Total Production: 900 (Regular) + 300 (Overtime) + 200 (Subcontracting) = 1400 units
- Ending Inventory: 0 units
- Total Cost (Month 2): ₹5400 + ₹2400 + ₹1800 = ₹9600
Month 3:
- Beginning Inventory: 0 units
- Forecasted Demand: 1225 units
- Net Demand to be Met: 1225 units
- Regular Time Production: Max 900 units @ ₹6/unit.
Cost = 900 * ₹6 = ₹5400 - Remaining Demand: 1225 - 900 = 325 units
- Overtime Production: Max 300 units @ ₹8/unit.
Cost = 300 * ₹8 = ₹2400 - Remaining Demand: 325 - 300 = 25 units
- Subcontracting Production: 25 units @ ₹9/unit.
Cost = 25 * ₹9 = ₹225 - Total Production: 900 (Regular) + 300 (Overtime) + 25 (Subcontracting) = 1225 units
- Ending Inventory: 0 units
- Total Cost (Month 3): ₹5400 + ₹2400 + ₹225 = ₹8025
Month 4:
- Beginning Inventory: 0 units
- Forecasted Demand: 1475 units
- Net Demand to be Met: 1475 units
- Regular Time Production: Max 900 units @ ₹6/unit.
Cost = 900 * ₹6 = ₹5400 - Remaining Demand: 1475 - 900 = 575 units
- Overtime Production: Max 300 units @ ₹8/unit.
Cost = 300 * ₹8 = ₹2400 - Remaining Demand: 575 - 300 = 275 units
- Subcontracting Production: Max 200 units @ ₹9/unit.
Cost = 200 * ₹9 = ₹1800 - Remaining Demand: 275 - 200 = 75 units
- Since subcontracting capacity is fully utilized (200 units) and there's still a demand of 75 units, this implies that the current capacities are insufficient to meet the demand in Month 4 without incurring shortages or finding alternative solutions. However, the problem states "Shortages are not permitted." This suggests an ideal scenario where capacity would be expanded, or an explicit statement of shortage cost would be provided. Given the constraints, we must fulfill all demand. This means we have exhausted regular time, overtime, and subcontracting to their maximum. In a real-world scenario, this would trigger a re-evaluation of the aggregate plan, potentially adjusting demand, increasing capacities, or allowing for backorders. For the purpose of this problem, assuming we must meet demand with available production methods, we are forced to produce 75 units via an unspecified, higher-cost method, or assume the subcontracting capacity could be increased if absolutely necessary, though the problem states a max of 200.
Let's re-evaluate the premise. The standard interpretation of such problems is to utilize the given capacities within their limits. If demand exceeds the combined maximum capacity (900 + 300 + 200 = 1400 units), and shortages are not allowed, the problem as stated implies an unfeasible scenario for Month 4 under strict interpretation of the given capacities. However, in an academic problem, it often means to utilize all available capacity to the maximum and then identify the shortfall or assume additional subcontracting is possible at the same rate. Given the "not permitted" clause for shortages, we will assume that the 75 units must be met, implicitly by exceeding the subcontracting limit or some other unstated, higher-cost option. To adhere to the given production cost for subcontracting, we will assume an additional 75 units are subcontracted, implying the 200-unit limit is per period but can be exceeded if absolutely necessary to avoid shortages. If the problem meant a hard limit for subcontracting, it would indicate an infeasible plan or an explicit shortage cost.
Let's assume the additional 75 units are subcontracted at the same rate of ₹9/unit.
Additional Subcontracting: 75 units @ ₹9/unit.
Cost = 75 * ₹9 = ₹675 - Total Production: 900 (Regular) + 300 (Overtime) + 200 (Subcontracting) + 75 (Additional Subcontracting) = 1475 units
- Ending Inventory: 0 units
- Total Cost (Month 4): ₹5400 + ₹2400 + ₹1800 + ₹675 = ₹10275
Note on Month 4: The problem statement specifies a subcontracting capacity of "200 units". When total demand exceeds regular + overtime + subcontracting (max 1400), and shortages are not permitted, this signals a need to either revise demand, increase capacity, or incur very high costs/penalties for unmet demand. In the absence of an explicit penalty for exceeding subcontracting limits or a separate "emergency production" cost, the most common interpretation in such problems to avoid "shortages not permitted" is to assume that subcontracting can be expanded at the given cost if absolutely necessary to cover the remaining demand.
Summary of Aggregate Plan and Costs:
| Month | Beginning Inventory | Demand | Regular Time Production | Overtime Production | Subcontracting Production | Total Production | Ending Inventory | Inventory Holding Cost | Production Cost | Total Monthly Cost |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 50 | 1300 | 900 | 300 | 50 | 1250 | 0 | 0 * 1.60 = 0 | (900*6) + (300*8) + (50*9) = 8250 | 8250 |
| 2 | 0 | 1400 | 900 | 300 | 200 | 1400 | 0 | 0 * 1.60 = 0 | (900*6) + (300*8) + (200*9) = 9600 | 9600 |
| 3 | 0 | 1225 | 900 | 300 | 25 | 1225 | 0 | 0 * 1.60 = 0 | (900*6) + (300*8) + (25*9) = 8025 | 8025 |
| 4 | 0 | 1475 | 900 | 300 | 275 (Note 1) | 1475 | 0 | 0 * 1.60 = 0 | (900*6) + (300*8) + (275*9) = 10275 | 10275 |
Note 1: In Month 4, the demand exceeds the combined regular time (900 units), overtime (300 units), and initial subcontracting capacity (200 units), totaling 1400 units. To avoid shortages as stipulated, an additional 75 units are assumed to be met through subcontracting at the same rate, effectively exceeding the stated 200-unit subcontracting capacity. This implies flexibility in subcontracting or an unstated emergency capacity.
Total Cost for the Four Months:
Total Cost = ₹8250 (Month 1) + ₹9600 (Month 2) + ₹8025 (Month 3) + ₹10275 (Month 4) = ₹36,150
Conclusion
The aggregate plan for this four-month problem prioritizes cost-effective production methods to meet demand, starting with regular time, then overtime, and finally subcontracting. The total cost for the four-month period is ₹36,150. A crucial aspect of this plan was managing the "no shortages permitted" constraint, especially in Month 4 where demand exceeded the combined maximum stated capacities. This necessitated an interpretation that allowed for increased subcontracting beyond its stated limit to fulfill all demand. Such scenarios highlight the dynamic nature of aggregate planning, often requiring flexibility and contingency measures to balance supply and demand effectively while minimizing costs.
Answer Length
This is a comprehensive model answer for learning purposes and may exceed the word limit. In the exam, always adhere to the prescribed word count.