Supply Chain Basics and Inventory Analytics

Components of a Supply Chain

A supply chain is the network of entities and activities that deliver a product to the end customer. Intuitively: every morning’s milk carton has a long journey behind it — from farm to plant to shop. Understanding the components of that journey reveals the three universal building blocks of any supply chain.

The Mother Dairy Example (Worked Example)

Consider how fresh milk reaches Delhi households at 6 a.m.:

• Procurement (Inbound): Mother Dairy sources raw milk from hundreds of cooperatives across Punjab, Rajasthan, Uttar Pradesh, Haryana, and Gujarat. The milk is transported to the Patparganj plant in East Delhi.
• Processing (In-house): The plant homogenises, pasteurises, and stores milk in large tanks. Capacity: 650,000 litres per day. Products include skimmed, toned, double-toned, full cream milk (in half- and one-litre packs), plus 30+ ice‑cream flavours and other dairy items.
• Distribution (Outbound): Nearly 100 tankers criss‑cross Delhi, supplying:
  • 600 booths
  • 200 manually operated containers (loose milk in congested areas)
  • 400 delivery agents (home delivery)
  • 850 retail shops (polythene packs)

These three stages — inbound, in-house, and outbound — form the supply chain.

Three Generic Components of Any Supply Chain

Exam tip: Any supply chain, regardless of industry (manufacturing, healthcare, services), can be decomposed into these three components. Memorise the labels and their typical issues — they are a frequent exam frame.

Key takeaways

• A supply chain consists of three components: inbound (sourcing), in-house (processing), outbound (distribution).
• The Mother Dairy example illustrates each component with concrete numbers and entities.
• Inbound issues: supplier development, strategic sourcing, supply management.
• In-house issues: master scheduling, MRP, capacity planning, material handling.
• Outbound issues: warehousing, distribution, channel management, third‑party logistics.

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Supply Chain Structure

Supply chain structure refers to the entities involved in delivering goods and services to the ultimate customer, their relative positioning, roles, and responsibilities. It also determines the nature and volume of information flow and material flow across the chain.

Layers of a Supply Chain

A typical structure has multiple layers (or tiers) from the original supplier to the final customer. In the diagram below, there are six layers (supplier + 4 intermediaries + customer):

• Upstream (material flows toward the customer): from supplier to customer.
• Downstream (information flows toward the supplier): customer → retailer → distributor → … → supplier.

Note: In made‑to‑order or turnkey environments, retailers and distributors may be absent — the product goes directly from factory to customer. However, the three‑component decomposition still holds.

Information and Material Flows

In a multi‑layer chain, each layer operates with a review cycle and processing delay, causing cumulative lead time.

Information flow (upstream) — example timings:

Material flow (downstream) — example timings:

The total lead time to fulfil a customer order is the sum of all delays across layers — information flows upstream, then material flows downstream. Each layer adds its own delay, so the structure directly impacts responsiveness.

Exam tip: Be ready to compute total lead time by adding all review, processing, and transit times. A common trap is forgetting that information must flow upstream before material can flow downstream — both contribute to the delay.

Additional Issues in Supply Chain Management

Beyond the three components, managers must address:

• Designing an appropriate supply chain structure (number and role of layers).
• Inventory planning and control.
• Performance metrics for the chain.

Key takeaways

• Structure defines the number of layers (tiers) and the flow of information (upstream) and material (downstream).
• Each layer introduces its own review frequency and processing delay, accumulating total lead time.
• Information must travel from customer to supplier before material can return — this double flow creates the total order‑to‑delivery delay.
• Supply chain management also encompasses structure design, inventory control, and performance measurement.

Bullwhip Effect in Supply Chains

The bullwhip effect describes how a small change in downstream demand (e.g., from customers) becomes increasingly amplified as orders travel upstream through the supply chain — like the crack of a whip. This distortion causes excessive inventory, poor demand management, and operational inefficiency.

Observations from the Beer Game

The Beer Game is a simulation (24–40 weeks) that reproduces real-world supply chain fluctuations. A typical result:

• Customer order: 4 crates/week (weeks –2 to 4) → increases to 8/week from week 5.
• Retailer orders from wholesaler: peaks at 20.
• Wholesaler orders from factory warehouse: peaks at 45.
• Factory orders from raw-material supplier: peaks at 68.

Plot of order quantities across layers:

Backlogs also vary both in magnitude and timing (phase lag): e.g., factory backlog peaks at week 16, next layer at week 17.

Key Observations

1. Downstream demand is amplified upstream (4 → 68).
2. All levels experience the effect, though in different magnitudes.
3. The farther from the end customer, the greater the amplitude.
4. Phase lag exists — peaks occur at different times across layers.
5. Demand becomes mixed with noise as it travels upstream.

Causes of the Bullwhip Effect

Three structural drivers:

• Number of layers – more layers → more amplification.
• Delay – finite transit and communication delays between layers.
• Rate of change – larger demand fluctuations trigger stronger whip.

Behavioural and policy factors:

• Each layer uses its own forecast (not shared).
• Order timing differs (e.g., weekly vs. bi‑weekly reviews).
• Price fluctuations and promotions create artificial demand surges.
• Rationing of supply during shortages – buyers inflate orders to secure a larger share.

Exam tip: The bullwhip effect is tested by explaining why a small consumer change (4→8) leads to huge upstream swings (68). Memorise the Beer Game numbers and the five observations.

Methods of Reducing the Bullwhip Effect

Exam tip: “Everyday low pricing” is a classic real‑world countermeasure – link it directly to removing promotion‑induced bullwhip.

Key takeaways

• Bullwhip effect: small downstream changes → large upstream swings (4→68).
• Caused by layers, delays, rate of change, and behavioural factors (own forecasts, different order timing, promotions, rationing).
• Remedies: reduce layers (3PL, e‑commerce), cut delays (EDI, POS), share data, and stabilise pricing.

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Inventory in Supply Chains

Inventory exists throughout a supply chain (manufacturing, retail, hospitals, hotels) due to both external and internal factors. Managing these six types is critical for operational efficiency.

Reasons for Inventory

External Factors

Internal Factors

Key takeaways

• Inventory exists for six distinct reasons: seasonal, hedging, cyclic, safety, pipeline, decoupling.
• External drivers: demand and price fluctuations.
• Internal drivers: operational policies (ordering patterns, uncertainty buffers, lead‑time stocks) and system design (decoupling).
• Understanding each type helps tailor inventory policies to the specific source of variability.

Why organizations hold inventory

Inventory is deliberately placed in supply chains to manage complexity, uncertainty, and timing differences. Six distinct categories emerge from the discussion.

1. Decoupling inventory – Splits a long production process into independent stages. A 10‑station production line is complex: a problem at station 3 disrupts downstream stations. By grouping stations into stages (stations 1–3, 4–7, 8–10) and holding inventory between stages, each stage can operate semi‑independently. This reduces the impact of local disruptions and simplifies control.

Decoupling inventory is a design choice – the system is deliberately split.

2. Cyclic inventory – Arises from ordering in batches rather than continuously. Every time an order of size $Q$ arrives, it is consumed at a steady rate. A new order is placed when stock reaches a reorder point, creating a repeating sawtooth pattern. Cyclic inventory is the stock that cycles through between order arrivals.

3. Pipeline inventory – Exists because of lead time. When an order is placed, it takes time (lead time) to arrive. During that interval, consumption continues. Pipeline inventory is the quantity already in transit or in process; its average equals the demand during the lead time.

4. Safety stock – Buffers against uncertainty in demand, supply quantity, lead time, or quality. Safety stock is held above the normal cycle stock. The sawtooth pattern sits on top of a baseline of safety stock that is never intentionally consumed under normal conditions.

5. Hedging inventory – Used to protect against price fluctuations (e.g., in international commodity markets). The goal is to ensure material availability at reasonable prices over a long horizon.

6. Seasonal inventory – Absorbs predictable demand peaks and troughs. Production can be smoothed by building inventory during low‑demand periods to meet high‑demand periods.

From an operational control perspective, the last three categories – cyclic inventory, safety stock, and pipeline inventory – are the most relevant for day‑to‑day planning.

Key takeaways

• Decoupling inventory reduces production complexity; it is a system design choice.
• Cyclic inventory results from batch ordering; its average is $\frac{Q}{2}$.
• Pipeline inventory is due to lead time; its average equals lead‑time demand.
• Safety stock handles uncertainty in demand, supply, or lead time.
• Hedging and seasonal inventory address price volatility and demand seasonality, respectively.
• For routine inventory planning, focus on cyclic, pipeline, and safety stock.

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The two fundamental decisions

Every inventory planning problem must answer:

1. How much to order? – the order quantity $Q$.
2. When to order? – the reorder point (time or stock level).

Without structured methods, organizations risk both excess inventory (blocking capital, obsolescence) and shortages (stoppages, lost customer goodwill, rush purchases at higher cost).

The three key costs

Exam tip: Shortage cost is the hardest to quantify. In many models it is treated as a penalty per unit short or per stockout event.

The planning problem: choose $Q$ and the reorder point so that the total of these three costs is minimised.

Key takeaways

• The two decisions are order quantity (how much) and reorder point (when).
• Three costs: ordering, holding, shortage.
• Ordering cost is incurred per order; holding cost is proportional to average inventory; shortage cost is uncertain but real.
• Unplanned inventory management leads to excess stock, shortages, and emergency costs.

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Intuition

The EOQ model answers “how much to order” when demand is known and constant. It ignores uncertainty for now. The insight: ordering in large batches reduces ordering cost but increases holding cost; ordering in small batches reduces holding cost but increases ordering cost. An optimal quantity balances the two.

Assumptions

• Demand rate $D$ is constant and known (e.g., annual demand = 10,000 units).
• Lead time is known and constant (for now).
• No quantity discounts.
• Instantaneous replenishment.
• Only ordering and holding costs matter (shortage cost is avoided by ordering exactly in time).

Derivation of total cost

Let:

• $D$ = annual demand (units)
• $Q$ = order quantity (units per order)
• $C_o$ = cost per order ($)
• $C_h$ = holding cost per unit per year ($)

Number of orders per year = $D / Q$ Average inventory = $Q/2$ (assuming consumption from $Q$ to 0)

Total annual cost: $TC(Q) = \frac{D}{Q} C_o + \frac{Q}{2} C_h$

The cost curve is U‑shaped:

The minimum occurs where the two cost components are equal (derivative = 0):

$\frac{dTC}{dQ} = -\frac{D C_o}{Q^2} + \frac{C_h}{2} = 0$

Solving: $Q^* = \sqrt{\frac{2 D C_o}{C_h}}$

This is the Economic Order Quantity (EOQ).

Worked example

Given:

• $D = 10{,}000$ units/year
• $C_o = 200$ dollars per order
• Unit cost = $400$ dollars; holding cost = $16\%$ of unit cost = $0.16 \times 400 = 64$ dollars per unit per year

$Q^* = \sqrt{\frac{2 \times 10{,}000 \times 200}{64}} = \sqrt{\frac{4{,}000{,}000}{64}} = \sqrt{62{,}500} = 250 \text{ units}$

Derived decisions:

• How much? Order 250 units each time.
• When? Number of orders per year $= 10{,}000 / 250 = 40$. Working days per year = 320, so order every $320 / 40 = 8$ working days.
• Total cost: Ordering cost: $40 \times 200 = 8{,}000$ dollars Holding cost: $(250/2) \times 64 = 125 \times 64 = 8{,}000$ dollars Total = $16,000

Exam tip: At the EOQ, ordering cost exactly equals holding cost. This is a quick check for correctness.

Robustness of the model

The total cost curve has a flat bottom – small deviations from $Q^*$ do not increase total cost significantly. This means the model is robust: even if parameters are slightly misestimated, the cost penalty is small.

Summary table of the example

Key takeaways

• EOQ balances ordering and holding costs: $Q^* = \sqrt{2DC_o/C_h}$.
• At $Q^*$, ordering cost = holding cost.
• The total cost curve is flat near the optimum – deviations matter little.
• The model assumes known, constant demand; it provides a foundation for more realistic models (with uncertainty, discounts, etc.).

Intuition: The Ordering vs. Holding Trade-off

A firm sources a product from a supplier to meet customer demand. It must decide how many units to order each time (order quantity (Q)).

• Large (Q) → fewer orders per year (low ordering cost) but high average inventory (high holding cost).
• Small (Q) → low average inventory (low holding cost) but many orders (high ordering cost).

The economic order quantity (EOQ) is the (Q) that minimises total annual inventory cost — the sum of ordering and holding costs.

Cost Components

Let:

• (D) = annual demand (units/year)
• (S) = ordering cost per order ($$$/order)
• (H) = unit holding cost ($$$/unit/year)
• (C) = unit purchase cost ($$$/unit) — fixed, independent of (Q)

Total annual relevant cost (TAC) = ((D/Q)S + (Q/2)H).

Deriving the EOQ Formula

Minimise TAC with respect to (Q). Set the derivative to zero:

[ \frac{d\text{TAC}}{dQ} = -\frac{DS}{Q^{2}} + \frac{H}{2} = 0 ]

Solving:

[ \frac{H}{2} = \frac{DS}{Q^{2}} \quad\Rightarrow\quad Q^{2} = \frac{2DS}{H} ]

[ \boxed{Q^{*} = \sqrt{\frac{2DS}{H}}} ]

Exam tip: The same formula is obtained by setting annual ordering cost equal to annual holding cost: (\frac{D}{Q}S = \frac{Q}{2}H) → solve for (Q).

Sensitivity: What changes EOQ?

Inventory Buildup (Sawtooth) Pattern

• Order of size (Q) arrives → inventory jumps to (Q).
• Demand depletes inventory linearly at rate (D/365) per day.
• When inventory reaches zero, a new order arrives immediately.
• Cycle repeats.

Key Takeaways

• EOQ balances ordering cost and holding cost.
• Formula: (Q^{*} = \sqrt{2DS/H}).
• Larger demand or ordering cost → larger EOQ; larger holding cost → smaller EOQ.
• Purchase cost is not part of the EOQ decision (it is constant).
• The trade-off is visualised by the sawtooth inventory diagram and the U‑shaped total cost curve.

Intuition and Setup

When a firm manufactures goods in-house rather than sourcing from a supplier, it must decide how many units to produce in each production run (the lot size). Consider:

• Annual demand $D$ (constant over the year).
• Daily demand $\bar{d} = D / 365$ (assuming 365 operating days).
• Daily production capacity $\bar{c}$ (units per day).

The firm can run production lots of size $Q$. For example, with $D = 20{,}000$ and $\bar{c}=100$ per day:

• Produce all $20{,}000$ in one run → run equipment for 200 days, then stop. Huge inventory builds up.
• Produce in lots of $2{,}000$ → 10 runs, each taking 20 days. Inventory is lower but more changeovers are needed.

Tradeoff: Large $Q$ → high holding cost (inventory). Small $Q$ → many setups → high setup cost (changeover cost). This mirrors the EOQ tradeoff, but now production happens at a finite rate $\bar{c}$ while demand $\bar{d}$ is continuous.

Inventory Buildup Pattern

During a production run of $t$ days ($t = Q / \bar{c}$), inventory accumulates at rate $\bar{c} - \bar{d}$ because production outpaces consumption. After the run ends, inventory is consumed at rate $\bar{d}$ until zero, then the next run begins.

Thus the inventory level follows a sawtooth pattern with peak at the end of the production run.

Average Inventory

• Maximum inventory = $(\bar{c} - \bar{d}) \times t = (\bar{c} - \bar{d}) \times \frac{Q}{\bar{c}} = Q \left(1 - \frac{\bar{d}}{\bar{c}}\right)$.
• Minimum inventory = 0.
• Since the pattern is triangular, average inventory = $\frac{\text{max}}{2} = \frac{Q}{2} \left(1 - \frac{\bar{d}}{\bar{c}}\right)$.

Cost Components (Annual)

Total annual cost:

$$TC(Q) = \frac{D}{Q}S + \frac{Q}{2}\left(1 - \frac{\bar{d}}{\bar{c}}\right)H + D C_m$$

Optimisation: The EPQ Formula

Take derivative w.r.t. $Q$ and set to zero:

$$\frac{dTC}{dQ} = -\frac{DS}{Q^2} + \frac{H}{2}\left(1 - \frac{\bar{d}}{\bar{c}}\right) = 0$$

Solve for $Q$:

$$Q^* = \sqrt{\frac{2 D S}{H \left(1 - \frac{\bar{d}}{\bar{c}}\right)}}$$

This is the Economic Production Quantity (EPQ) – the lot size that minimises total annual cost.

Intuition of Parameter Effects

Exam tip: When $\bar{c} \to \infty$ (instantaneous production), $\left(1 - \frac{\bar{d}}{\bar{c}}\right) \to 1$, and EPQ reduces to the EOQ formula: $Q^* = \sqrt{\frac{2DS}{H}}$. The EPQ is always larger than the EOQ for the same $D$, $S$, $H$ because the finite production rate reduces the average inventory.

Key takeaways

• EPQ model decides lot size for in-house production under continuous demand and finite production rate.
• Average inventory = $\frac{Q}{2} \left(1 - \frac{\bar{d}}{\bar{c}}\right)$, which is smaller than the EOQ average ($Q/2$) because consumption occurs during production.
• Optimal lot size: $Q^* = \sqrt{\frac{2DS}{H (1 - \bar{d}/\bar{c})}}$.
• Increased capacity ($\bar{c}$) reduces $Q^*$; increased demand ($D$) increases $Q^*$.
• The tradeoff is between setup cost (fixed per run) and holding cost (proportional to average inventory).
• Manufacturing cost is a constant add‑on and does not affect the optimal $Q$.

EPQ Estimation – Worked Example

Economic Production Quantity (EPQ) determines the lot size that minimises total setup and holding costs when production and consumption occur simultaneously. Unlike the EOQ model (instantaneous replenishment), EPQ accounts for a finite production rate – inventory builds up gradually while demand is drawn down concurrently.

The formula is:

$Q_{EPQ} = \sqrt{\frac{2DS}{H \left(1 - \frac{d}{p}\right)}}$

where

• $D$ = annual demand
• $S$ = setup cost per production run
• $H$ = annual holding cost per unit
• $d$ = daily demand rate
• $p$ = daily production (capacity) rate

Below is a worked‑through example that applies the formula and then computes the resulting annual setup cost, annual holding cost, and production run length.

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Data

A scooter manufacturer needs seat cushions. Relevant data:

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1. Compute EPQ

Plug into the EPQ formula:

$$\begin{aligned} Q_{EPQ} &= \sqrt{\frac{2 \times 60\,000 \times 6\,000}{12 \times \left(1 - \frac{200}{1\,000}\right)}} \\[6pt] &= \sqrt{\frac{720\,000\,000}{12 \times 0.8}} \\[6pt] &= \sqrt{\frac{720\,000\,000}{9.6}} \\[6pt] &= \sqrt{75\,000\,000} \\[6pt] &\approx 8\,660 \text{ units} \end{aligned}$$

Each production run therefore consists of 8,660 cushions.

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2. Annual Setup Cost

Number of setups per year = $\dfrac{D}{Q_{EPQ}} = \dfrac{60\,000}{8\,660} \approx 6.928$ runs.

Annual setup cost = (Number of setups) × $S$

$= 6.928 \times 6\,000 \approx \text{Rs. }41\,570$

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3. Annual Holding Cost

The inventory profile in one cycle is:

Average inventory during a cycle is:

$I_{avg} = \frac{ \left(1 - \frac{d}{p}\right) \times Q_{EPQ} }{2}$

Proof: maximum inventory = $(p - d) \times t$, where $t = Q_{EPQ}/p$ is the production time. Hence maximum inventory = $(p - d) \times \frac{Q_{EPQ}}{p} = \left(1 - \frac{d}{p}\right) Q_{EPQ}$. Since minimum inventory is zero, average inventory equals half of the maximum.

Plug values:

$$\begin{aligned} I_{avg} &= \frac{ \left(1 - \frac{200}{1\,000}\right) \times 8\,660 }{2} \\ &= \frac{0.8 \times 8\,660}{2} \\ &= 3\,464 \text{ units} \end{aligned}$$

Annual holding cost = $I_{avg} \times H = 3\,464 \times 12 \approx \text{Rs. }41\,570$.

Note that the annual holding cost equals the annual setup cost – a property of the optimal lot size.

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4. Production Time per Cycle

Production time $t$ is the number of days the production line runs per cycle:

$t = \frac{Q_{EPQ}}{p} = \frac{8\,660}{1\,000} = 8.66 \text{ days}$

Interpretation: Each cycle starts a production run that lasts 8.66 days, during which 8,660 cushions are produced. After production stops, the accumulated inventory is consumed by demand until stock hits zero, at which point the next 8.66‑day run begins.

Exam tip: Always check that the EPQ denominator $1 - d/p$ remains positive. If $d \ge p$, production cannot keep up with demand – the system is not sustainable without overtime or subcontracting.

Key takeaways

• EPQ extends EOQ by replacing instantaneous replenishment with a finite production rate $p$.
• The optimal lot size is $Q_{EPQ} = \sqrt{ \frac{2DS}{H(1 - d/p)} }$.
• Average inventory in EPQ equals $\frac12 \left(1 - d/p\right) Q_{EPQ}$.
• At the optimum, annual setup cost equals annual holding cost.
• Production time per cycle = $Q_{EPQ} / p$; the full cycle length = $Q_{EPQ} / d$.

Buy Strategy (Sound Max Case)

The Sound Max case evaluates whether to continue buying a critical plastic enclosure from a supplier or to produce it in‑house. This section covers the buy‑strategy analysis: given demand, ordering cost, purchase cost, and holding cost, determine the optimal order quantity and the resulting annual total cost.

Problem Setup

• Sound Max buys the component from a supplier.
• Annual demand is known, and the firm can order in any lot size. Larger lots reduce ordering frequency but increase inventory holding; smaller lots reduce holding but increase ordering cost.
• The goal is to find the Economic Order Quantity (EOQ) that minimises total annual cost (ordering + holding + purchase cost).

Data Summary

Applying EOQ: The Trade-off

The Economic Order Quantity (EOQ) is the lot size that balances ordering and holding costs:

$Q^* = \sqrt{\frac{2DS}{H}}$

Worked Example: Optimal Order Quantity

Plug in the values:

$Q^* = \sqrt{\frac{2 \times 24000 \times 1000}{117}} = \sqrt{\frac{48,000,000}{117}} \approx \sqrt{410,256.41} \approx 641 \text{ units}$

Thus Sound Max should order in batches of 641 units each time.

Annual Cost Breakdown

1. Annual Ordering Cost

Number of orders per year = $\frac{D}{Q^*} = \frac{24000}{641} \approx 37.44$ Cost = $\frac{D}{Q^*} \times S = 37.44 \times 1000 = \text{Rs. }37,440$ (rounded to ₹37,470)

The example uses ₹37,470. The exact calculation is $\frac{24000}{641} \times 1000 \approx 37,444$; the difference is rounding.

2. Annual Holding Cost

Average inventory = $\frac{Q^*}{2} = \frac{641}{2} = 320.5$ units Cost = $\frac{Q^*}{2} \times H = 320.5 \times 117 = \text{Rs. }37,498.5$.

3. Annual Purchase Cost

Total units bought = $D = 24,000$ Cost = $D \times C_s = 24,000 \times 650 = \text{Rs. }15,600,000$

4. Total Annual Cost (Buy Strategy)

$\text{Total Cost} = \text{Ordering Cost} + \text{Holding Cost} + \text{Purchase Cost}$ $= 37,470 + 37,470 + 15,600,000 = \text{Rs. }15,674,940$

(The example uses ₹15,647,940; a small arithmetic difference does not change the logic.)

Exam tip: The purchase cost dominates the total, but it is independent of lot size. The EOQ minimises only the ordering + holding cost. Do not confuse total cost with inventory‑related cost when comparing strategies.

Key Takeaways

• The buy strategy uses the EOQ model: $Q^* = \sqrt{2DS/H}$.
• Optimal order quantity = 641 units (rounded), balancing ordering cost (₹1,000/order) and holding cost (₹117/unit/year).
• Annual ordering cost ≈ ₹37,470; annual holding cost ≈ ₹37,470; annual purchase cost = ₹15,600,000.
• Total annual cost under the buy strategy ≈ ₹15.67 million.
• This baseline will be compared with the proposed in‑house production strategy to decide which is cheaper.

Make Strategy (In-House Production)

The make strategy evaluates whether SoundMax should produce the plastic enclosure in-house rather than buy from an external supplier. The key operational decision is the production lot size (how many units per production run), because it creates a trade-off between setup cost (cost per run) and holding cost (cost of carrying inventory). The Economic Production Quantity (EPQ) model finds the lot size that minimises total annual cost.

Data for the make-scenario

The EPQ Model

When producing in-house, inventory accumulates at rate $\bar{c} - d$ during a production run and is consumed at rate $d$ during the rest of the cycle. The optimal lot size that balances setup and holding costs is:

$$Q_{EPQ} = \sqrt{\frac{2 D S}{H \left(1 - \frac{d}{\bar{c}}\right)}}$$

Plugging in the numbers:

$$Q_{EPQ} = \sqrt{\frac{2 \times 24\,000 \times 6\,000}{111.6 \times \left(1 - \frac{80}{300}\right)}} = \sqrt{\frac{288\,000\,000}{111.6 \times 0.7333}} \approx 1\,876 \text{ units}$$

Thus the optimal production batch size is 1,876 units per run.

Inventory Profile

The cycle works as follows:

where $t = Q_{EPQ} / \bar{c}$ is the production run length. Average inventory in the system:

$$\text{Avg inventory} = \frac{\text{max inventory}}{2} = \frac{Q_{EPQ}}{2} \left(1 - \frac{d}{\bar{c}}\right)$$

For SoundMax:

$$\text{Avg inventory} = \frac{1\,876}{2} \times \left(1 - \frac{80}{300}\right) \approx 938 \times 0.7333 \approx 688 \text{ units}$$

Total Annual Cost Under Make Strategy

The total annual cost has three components:

1. Annual setup cost Number of runs per year $= D / Q_{EPQ}$ $\text{Setup cost} = \frac{D}{Q_{EPQ}} \times S = \frac{24\,000}{1\,876} \times 6\,000 \approx \text{Rs. }76\,760$

2. Annual holding cost $\text{Holding cost} = \text{Avg inventory} \times H = 688 \times 111.6 \approx \text{Rs. }76\,760$

3. Annual production cost $\text{Production cost} = D \times C_m = 24\,000 \times 620 = \text{Rs. }1\,48\,80\,000$

Total make cost = ₹1,48,80,000 + ₹76,760 + ₹76,760 = ₹1,50,33,520 per year.

Comparison with Buy Strategy

The total annual cost under the buy strategy is ₹1,56,74,940.

Because ₹1,50,33,520 < ₹1,56,74,940, making in-house is the optimal decision.

Exam tip: In EPQ, the average inventory expression $\frac{Q}{2}(1 - d/\bar{c})$ is the only difference from the EOQ formula. When $d/\bar{c}$ is small (capacity much larger than demand), the term approaches 1 and EPQ behaves like EOQ. Always check whether production capacity is binding.

Key takeaways

• EPQ determines lot size for in-house production, trading off setup cost vs. holding cost.
• Formula: $Q_{EPQ} = \sqrt{2DS / [H(1 - d/\bar{c})]}$.
• Average inventory depends on the ratio $d/\bar{c}$; the closer demand is to capacity, the lower the average inventory.
• Total cost = setup cost + holding cost + production cost (material cost).
• Compare make vs. buy total cost to choose the optimal strategy.