15.2 Production Planning, Forecasting & Inventory Control Models

Production Planning, Forecasting & Inventory Control Models

In high-capital mining operations like Coal India Limited—where millions of tonnes of coal are extracted using fleets of heavy earth-moving machinery (HEMM), dumpers, draglines, and conveyor belts—effective production planning and inventory control ensure continuous equipment uptime while preventing excessive working capital lockup.

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1. Demand Forecasting Models

Forecasting predicts future demand based on historical patterns, trends, and seasonal variations.

Time-Series Forecasting Formulations

1. Simple Moving Average ($n$-period):

   $$F_t = \frac{A_{t-1} + A_{t-2} + \dots + A_{t-n}}{n} = \frac{1}{n}\sum_{i=1}^n A_{t-i}$$

   Characteristics: All past $n$ periods carry equal weight $1/n$. Increasing $n$ smooths out random fluctuations but slows response to real trend shifts.

2. Weighted Moving Average:

   $$F_t = w_1 A_{t-1} + w_2 A_{t-2} + \dots + w_n A_{t-n} = \sum_{i=1}^n w_i A_{t-i}$$

   where $\sum_{i=1}^n w_i = 1$ and higher weights are assigned to more recent periods ($w_1 > w_2 > \dots > w_n$).

3. Simple Exponential Smoothing:

   $$F_t = F_{t-1} + \alpha(A_{t-1} - F_{t-1}) = \alpha A_{t-1} + (1 - \alpha)F_{t-1}$$

   where:

   • $F_t$ = Forecast for current period $t$
   • $F_{t-1}$ = Forecast for previous period $t-1$
   • $A_{t-1}$ = Actual demand in previous period $t-1$
   • $\alpha$ = Smoothing constant ($0 \le \alpha \le 1$)

   Equivalent Moving Average Relationship:

   $$\alpha = \frac{2}{N + 1} \iff N = \frac{2 - \alpha}{\alpha}$$
   • A high $\alpha$ (e.g., $0.7 - 0.9$) makes the forecast highly responsive to recent demand changes.
   • A low $\alpha$ (e.g., $0.1 - 0.3$) provides high stability and dampens noise.

Forecast Error Metrics

Let forecast error for period $t$ be $e_t = A_t - F_t$.

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2. Production Planning Hierarchy and Aggregate Planning

                 AGGREGATE PRODUCTION PLAN (6 - 18 months)
                                     |
                                     v
                 MASTER PRODUCTION SCHEDULE (MPS, Weeks/Months)
                                     |
                                     v
                 MATERIAL REQUIREMENTS PLANNING (MRP-I)
                 (Explodes Bill of Materials & Inventory Status)
                                     |
                                     v
                 MANUFACTURING RESOURCE PLANNING (MRP-II)
                 (Integrates Capacity, Tooling, Finance, Labor)
                                     |
                                     v
                 ENTERPRISE RESOURCE PLANNING (ERP)

Aggregate Planning Strategies

1. Chase Demand Strategy: Match production capacity dynamically to demand in each period by adjusting workforce levels (hiring, firing, overtime, subcontracting). Minimizes inventory holding costs but incurs substantial hiring/firing and workforce instability costs.
2. Level Capacity Strategy: Maintain a constant production output rate and steady workforce. Fluctuations in demand are absorbed by accumulating finished goods inventory during low-demand periods and utilizing backorders/stockouts during peak periods. Ideal for capital-intensive, automated processes.
3. Mixed (Hybrid) Strategy: Combines level baseline production with controlled overtime, temporary workforce shifts, or subcontracting during peak demand.

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3. Deterministic Inventory Control Models

1. Classical Economic Order Quantity (EOQ / Wilson-Harris Model)

Assumptions:

• Annual demand $D$ is constant, continuous, and known with certainty.
• Order quantity $Q$ arrives instantaneously in a single batch (replenishment rate is infinite).
• Lead time is zero or constant.
• Unit purchase cost $C$, ordering cost $C_o$ (per order), and holding cost $C_h$ (per unit per year) are constant.
• Shortages/stockouts are strictly prohibited.

  Inventory ^
    Level   |
          Q |  /|        /|        /|
            | / |       / |       / |
        Q/2 |/--|------/--|------/--|---- Average Inventory = Q/2
            |   |     /   |     /   |
          0 +---+-----+---+-----+---+-----> Time
                |< T >|

Cost Equations:

$$\text{Annual Purchase Cost} = D \cdot C$$ $$\text{Annual Ordering Cost} = \left(\frac{D}{Q}\right) C_o$$ $$\text{Annual Carrying/Holding Cost} = \left(\frac{Q}{2}\right) C_h$$ $$\text{Total Inventory Cost (TIC)} = \left(\frac{D}{Q}\right) C_o + \left(\frac{Q}{2}\right) C_h$$

Equating Annual Ordering Cost to Annual Carrying Cost:

$$\frac{D}{Q} C_o = \frac{Q}{2} C_h \implies Q^{*2} = \frac{2 D C_o}{C_h}$$ $$\mathbf{Q^* = \text{EOQ} = \sqrt{\frac{2 D C_o}{C_h}}}$$

Key Optimal Metrics:

• Minimum Total Inventory Cost: $$\text{TIC}^* = \sqrt{2 D C_o C_h}$$
• Optimal Order Frequency: $$N^* = \frac{D}{Q^*} = \sqrt{\frac{D C_h}{2 C_o}}$$
• Optimal Order Cycle Time: $$T^* = \frac{Q^*}{D} = \sqrt{\frac{2 C_o}{D C_h}}$$

Sensitivity of EOQ to Parameter Estimation Errors

If an order quantity $Q = k Q^*$ is selected instead of the optimal $Q^*$ (where $k > 0$ is an estimation factor):

$$\frac{\text{TIC}(Q)}{\text{TIC}^*} = \frac{1}{2}\left(k + \frac{1}{k}\right)$$

Example: If $Q = 2 Q^*$ ($k = 2$), $\text{TIC} = \frac{1}{2}(2 + 0.5)\text{TIC}^* = 1.25\,\text{TIC}^*$ (only a $25\%$ cost penalty for a $100\%$ order quantity error, illustrating that the total cost curve is relatively flat around the minimum).

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2. Economic Production Quantity / Batch Quantity (EPQ / EBQ)

Used when the manufacturing unit produces components internally at a finite production rate $p$ while demand consumes them at rate $d$ ($p > d$).

  Inventory ^
    Level   |        /\               /\
       Imax |       /  \             /  \
            |      /|   \           /|   \
            |     / |    \         / |    \
            |    /  |     \       /  |     \
          0 +---+---+------+-----+---+------+----> Time
                |tp |  td  |

• Production cycle time: $t_p = Q / p$
• Maximum inventory accumulated: $I_{\max} = t_p(p - d) = Q\left(1 - \frac{d}{p}\right)$
• Average inventory: $I_{\text{avg}} = \frac{I_{\max}}{2} = \frac{Q}{2}\left(1 - \frac{d}{p}\right)$

$$\text{Total Cost} = \left(\frac{D}{Q}\right) C_s + \frac{Q}{2}\left(1 - \frac{d}{p}\right) C_h$$ $$\mathbf{Q^*_{\text{EPQ}} = \sqrt{\frac{2 D C_s}{C_h \left(1 - \frac{d}{p}\right)}} = \sqrt{\frac{2 D C_s}{C_h}}\sqrt{\frac{p}{p - d}}}$$

(where $C_s$ is the setup cost per production run).

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3. Inventory Model with Planned Shortages (Backorders Permitted)

When stockouts are allowed at a backordering cost $C_b$ (per unit short per year):

$$\mathbf{Q^* = \sqrt{\frac{2 D C_o}{C_h} \cdot \left(\frac{C_h + C_b}{C_b}\right)}}$$ $$\mathbf{I_{\max}^* = Q^* \left(\frac{C_b}{C_h + C_b}\right) = \sqrt{\frac{2 D C_o}{C_h} \cdot \left(\frac{C_b}{C_h + C_b}\right)}}$$ $$\text{Maximum Backorder Level } S^* = Q^* - I_{\max}^* = Q^*\left(\frac{C_h}{C_h + C_b}\right)$$ $$\text{Minimum Total Cost } \text{TIC}^* = \sqrt{2 D C_o C_h \left(\frac{C_b}{C_h + C_b}\right)}$$

Since $\frac{C_h + C_b}{C_b} > 1$, permitting shortages increases optimal batch size $Q^*$ while reducing maximum on-hand inventory and total inventory cost.

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4. Reorder Level (ROL) and Safety Stock Under Uncertainty

When lead time demand is uncertain, a safety stock buffer is maintained to prevent stockouts at a specified service level $Z_{\alpha}$.

  Inventory ^
    Level   |
            |   /\        /\
        ROL |--/--\------/--\------------------ Reorder Level (ROL)
            | /    \    /    \
         SS |/------\--/------\---------------- Safety Stock (SS)
            |        \/        \
          0 +-------------------------> Time
                     |< LT >|

Formulations

$$\mathbf{\text{ROL} = \text{Expected Demand During Lead Time} + \text{Safety Stock (SS)}}$$

1. Variable Demand ($d \sim \mathcal{N}(\bar{d}, \sigma_d^2)$) and Constant Lead Time ($LT$):

   $$\sigma_{DL} = \sigma_d \sqrt{LT}$$ $$\text{SS} = Z_{\alpha} \cdot \sigma_d \sqrt{LT}$$ $$\text{ROL} = \bar{d} \cdot LT + Z_{\alpha} \cdot \sigma_d \sqrt{LT}$$

2. Constant Daily Demand ($\bar{d}$) and Variable Lead Time ($LT \sim \mathcal{N}(\bar{LT}, \sigma_{LT}^2)$):

   $$\text{SS} = Z_{\alpha} \cdot \bar{d} \cdot \sigma_{LT}$$ $$\text{ROL} = \bar{d} \cdot \bar{LT} + Z_{\alpha} \cdot \bar{d} \cdot \sigma_{LT}$$

3. Both Demand and Lead Time are Independent Random Variables:

   $$\sigma_{DL} = \sqrt{\bar{LT}\cdot\sigma_d^2 + \bar{d}^2\cdot\sigma_{LT}^2}$$ $$\text{SS} = Z_{\alpha} \cdot \sqrt{\bar{LT}\cdot\sigma_d^2 + \bar{d}^2\cdot\sigma_{LT}^2}$$

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5. Selective Inventory Control Techniques

In mining stores containing tens of thousands of spare parts (from O-rings to diesel engines), selective control allocates managerial attention based on specific operational and financial criteria.