7.1 Deterministic Inventory Control: Classical EOQ, EPQ & Quantity Discounts

Deterministic Inventory Control: Classical EOQ, EPQ & Quantity Discounts

Inventory management is a fundamental discipline in industrial and systems engineering, balancing working capital investment against production continuity and customer satisfaction. Inventory represents a major asset on an organization's balance sheet, often comprising 20% to 50% of total capital. Operating without sufficient inventory risks catastrophic line shutdowns and lost customer goodwill, whereas holding excessive stock ties up valuable working capital and incurs warehousing, taxation, insurance, and obsolescence costs.

Deterministic inventory control models evaluate scenarios where demand rates, production rates, ordering lead times, and unit purchase costs are known with certainty. These foundational formulations provide the theoretical and mathematical baseline for modern aggregate planning, Material Requirements Planning (MRP), and advanced Enterprise Resource Planning (ERP) systems.

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1. Inventory Taxonomy and Cost Structures

Functional Classifications of Industrial Inventory

Industrial inventory is categorized according to its position across the transformation value stream:

1. Raw Materials: Unprocessed purchased materials, raw commodities, and standard hardware awaiting introduction into manufacturing operations.
2. Work-in-Process (WIP): Semi-finished parts, subassemblies, and batches physically situated between intermediate machining, fabrication, or assembly operations.
3. Finished Goods: Completely fabricated and inspected products packaged at the factory or warehouse awaiting customer distribution.
4. Maintenance, Repair, and Operating (MRO) Supplies: Consumables, cutting tooling, machine lubricants, spare parts, and personal protective equipment required to support operations without becoming part of the end product.
5. Pipeline / Transit Inventory: Goods in transit between supply chain nodes (e.g., loaded on trucks, railcars, or container ships).
6. Decoupling Inventory: Buffers positioned between adjacent manufacturing stages having unequal cycle times, preventing upstream disruptions from idling downstream workstations.

Economic Cost Components

Every lot-sizing decision balances three antagonistic financial cost vectors:

• Unit Purchasing Cost ($C$): The acquisition price paid to a supplier or the variable direct manufacturing cost per unit ($/unit).
• Ordering / Setup Cost ($S$): The fixed cost incurred each time a replenishment order is placed or a production line is changed over ($/order or $/setup). For external purchases, $S$ captures requisition processing, purchase order generation, freight administrative fees, receiving dock handling, and incoming QA inspections. In manufacturing, $S$ captures teardown, die replacement, machine recalibration, purging, scrap from test runs, and lost line capacity.
• Holding / Carrying Cost ($H$): The annual cost to carry one unit of inventory for one full year ($/unit-year). It is universally formulated as a percentage $i$ of the unit acquisition cost: $H = i \times C$ Where the carrying charge rate $i$ (typically 15% to 35% annually) aggregates:
  • Cost of Capital / WACC (10%–20%): Opportunity cost of working capital tied up in stock.
  • Storage & Warehousing (2%–5%): Building depreciation, lease expenses, utility power, climate control, and material handling equipment leases.
  • Risk & Obsolescence (3%–10%): Product expiration, engineering design revisions rendering stock scrap, and inventory write-downs.
  • Taxes & Insurance (1%–3%): Property taxes on stored assets and comprehensive casualty/theft insurance premiums.
• Shortage / Stockout Cost ($C_s$ or $p_s$): The penalty incurred when customer demand exceeds available on-hand stock ($/unit short per unit time), incorporating administrative expediting, overtime premiums, special air freight charges, lost gross margin, and contractual late delivery penalties.

2. The Classical Economic Order Quantity (EOQ) Model

Originally derived by Ford Whitman Harris in 1913, the classical Economic Order Quantity (EOQ) model determines the replenishment lot size $Q$ that minimizes the total annual cost of ordering and holding inventory.

Foundational Assumptions

1. Demand rate $D$ is constant, continuous, and deterministic (units/year).
2. Replenishment is instantaneous; the entire order quantity $Q$ arrives simultaneously in a single batch (infinite replenishment rate, $p \to \infty$).
3. Replenishment lead time $L$ is fixed, constant, and known with certainty.
4. Unit purchasing cost $C$ is strictly constant, independent of order quantity (no volume discounts).
5. No stockouts or shortages are permitted ($C_s \to \infty$).
6. The planning horizon is infinite, and inventory involves a single independent product.

Mathematical Derivation of EOQ

Under instantaneous batch replenishment and steady demand rate $D$, the on-hand inventory traces a repeating sawtooth profile fluctuating between a maximum of $Q$ and a minimum of $0$. The average inventory level held across the operational cycle is:

$\bar{I} = \frac{Q + 0}{2} = \frac{Q}{2}$

The number of replenishment cycles initiated per year is:

$N = \frac{D}{Q}$

The Total Annual Cost function $TC(Q)$ comprises annual material acquisition cost, annual ordering cost, and annual inventory holding cost:

$TC(Q) = D \cdot C + \frac{D}{Q}S + \frac{Q}{2}H$

Because annual acquisition cost $D \cdot C$ is constant with respect to lot size $Q$, the lot-sizing optimization focuses strictly on the Total Relevant Cost ($TRC$):

$TRC(Q) = \frac{D}{Q}S + \frac{Q}{2}H$

To find the minimizing lot size, compute the first derivative of $TRC(Q)$ with respect to $Q$ and set it to zero:

$\frac{d(TRC)}{dQ} = -\frac{D}{Q^2}S + \frac{H}{2} = 0$

Equating annual ordering cost to annual holding cost:

$\frac{D}{Q}S = \frac{Q}{2}H$

Solving for $Q$ yields the celebrated classical EOQ formula:

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

To verify that $Q^*$ is a global minimum, examine the second derivative:

$\frac{d^2(TRC)}{dQ^2} = \frac{2DS}{Q^3} > 0 \quad \forall Q > 0$

Because the second derivative is strictly positive for all physical quantities $Q > 0$, the total relevant cost curve is strictly convex, confirming that $Q^*$ uniquely minimizes total annual inventory cost.

Cost (\$/Year)
     ^
     |        / Total Relevant Cost TRC(Q)
     |       /       .__-—--_.
     |      /      /          \ 
     |     /     /             \     Holding Cost (Q/2 * H)
     |    /    /                \   /
     |   /   /                   \ /
     |  |   /      EOQ            X
     |  |  /        |            / \
     |  | /         |           /   \
     |  |/          v          /     \  Ordering Cost (D/Q * S)
     +-------------------------------------------------------> Order Quantity (Q)
                    Q*

Operational Metrics at EOQ

When operating at the optimal order quantity $Q^* = \sqrt{2DS/H}$:

1. Equivalence of Costs: Annual ordering cost equals annual holding cost exactly: $\text{Annual Ordering Cost} = \frac{D}{Q^*}S = \sqrt{\frac{DSH}{2}}$ $\text{Annual Holding Cost} = \frac{Q^*}{2}H = \sqrt{\frac{DSH}{2}}$
2. Minimum Total Relevant Cost: $TRC(Q^*) = \frac{D}{Q^*}S + \frac{Q^*}{2}H = 2 \sqrt{\frac{DSH}{2}} = \sqrt{2DSH}$
3. Optimal Order Frequency: The number of orders placed per year is: $N^* = \frac{D}{Q^*} = \sqrt{\frac{DH}{2S}}$
4. Optimal Cycle Time ($T^*$): The duration between successive replenishment orders: $T^* = \frac{Q^*}{D} = \sqrt{\frac{2S}{DH}} \text{ (years)}$ $T^* = \frac{Q^*}{D} \times N_{days} \text{ (operating working days)}$
5. Reorder Point ($ROP$): Under deterministic lead time $L$ (expressed in years): $ROP = d \times L = \left( \frac{D}{\text{Operating Days/Year}} \right) \times L_{\text{days}}$ Note: If lead time $L$ exceeds the cycle time $T^*$, use effective lead time $L_e = L - m T^*$, where $m = \lfloor L / T^* \rfloor$.

3. Parametric Sensitivity and the Flat Cost Curve

A critical insight for practicing industrial engineers is the mathematical robustness of the EOQ formula. In real-world factory operations, parameter inputs ($D, S, H$) are estimates subject to forecasting error or accounting variability. Furthermore, logistics realities (such as master carton counts, skid/pallet quantities, or truckload capacities) often force engineers to round the computed lot size $Q^*$ to a convenient packaging multiple.

Sensitivity Formulation

Let $Q$ represent an arbitrary, non-optimal order quantity, and let $Q^*$ represent the true theoretical EOQ. The ratio of total relevant cost at $Q$ relative to the optimal cost $TRC(Q^*)$ is derived algebraically:

$\frac{TRC(Q)}{TRC(Q^*)} = \frac{\frac{D}{Q}S + \frac{Q}{2}H}{\sqrt{2DSH}}$

Substituting $S = \frac{(Q^*)^2 H}{2D}$ into the numerator:

$\frac{TRC(Q)}{TRC(Q^*)} = \frac{\frac{D}{Q} \left[ \frac{(Q^*)^2 H}{2D} \right] + \frac{Q}{2}H}{\sqrt{2D \left[ \frac{(Q^*)^2 H}{2D} \right] H}} = \frac{\frac{(Q^*)^2 H}{2Q} + \frac{QH}{2}}{Q^* H} = \frac{1}{2} \left( \frac{Q^*}{Q} + \frac{Q}{Q^*} \right)$

This simple, elegant equation reveals the fundamental cost ratio relationship:

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

Quantitative Impact of Lot-Sizing Deviations

4. The Economic Production Quantity (EPQ / POQ) Model

In internal manufacturing environments, items are produced incrementally over time rather than arriving instantaneously. The Economic Production Quantity (EPQ) model—also designated the Production Order Quantity (POQ) model—extends classical EOQ to account for a finite production rate $p$.

Model Assumptions & Mechanics

• Production rate is $p$ units per day (or annual production capacity $P$).
• Consumption demand rate is $d$ units per day (or annual demand $D$).
• The manufacturing rate strictly exceeds the demand rate: $p > d$ ($P > D$). If $p \le d$, the production facility can never satisfy demand, producing continuously with infinite backlogs.

The Operational Production Cycle

Each replenishment cycle $T$ is divided into two distinct operating intervals:

1. Run Time ($t_1$): The production equipment operates at rate $p$ while demand simultaneously depletes stock at rate $d$. Inventory accumulates at the net buildup rate of $(p - d)$: $t_1 = \frac{Q}{p}$
2. Pure Depletion Time ($t_2$): Production stops ($p = 0$). Demand continues at rate $d$ until inventory drops to zero: $t_2 = \frac{I_{max}}{d}$

The total cycle duration is:

$T = t_1 + t_2 = \frac{Q}{p} + \frac{I_{max}}{d} = \frac{Q}{d}$

Inventory Level
     ^
Imax |           /\                     /\ 
     |          /  \                   /  \ 
     |         /    \                 /    \ 
     |        /      \               /      \ 
     |       / Net    \  Pure       /        \ 
     |      / Buildup  \ Depletion /          \ 
     |     /  (p - d)   \  (-d)   /            \ 
   0 +----+--------------+-------+--------------+---> Time
          |<--- t1 ----->|<- t2 >|
          |<------- T ---------->|

Maximum and Average Inventory Levels

The maximum inventory level $I_{max}$ is reached at the exact moment production ceases at time $t_1$:

$I_{max} = t_1 (p - d) = \left( \frac{Q}{p} \right) (p - d) = Q \left( 1 - \frac{d}{p} \right) = Q \left( 1 - \frac{D}{P} \right)$

Because inventory rises linearly from $0$ to $I_{max}$ and falls linearly back to $0$, the average inventory held across the entire cycle is:

$\bar{I} = \frac{I_{max}}{2} = \frac{Q}{2} \left( 1 - \frac{d}{p} \right)$

Derivation of Optimal Production Lot Size

The Total Relevant Cost equation balances annual machine setup costs with annual holding costs:

$TRC(Q) = \frac{D}{Q}S + \frac{Q}{2} H \left( 1 - \frac{d}{p} \right)$

Differentiating with respect to $Q$ and equating to zero:

$\frac{d(TRC)}{dQ} = -\frac{D}{Q^2}S + \frac{H}{2} \left( 1 - \frac{d}{p} \right) = 0$

$\frac{D}{Q}S = \frac{Q}{2} H \left( 1 - \frac{d}{p} \right)$

Solving for $Q$ yields the optimal Economic Production Quantity ($EPQ$):

$EPQ = Q_p^* = \sqrt{\frac{2DS}{H \left( 1 - \frac{d}{p} \right)}} = \sqrt{\frac{2DS}{H \left( 1 - \frac{D}{P} \right)}}$

Comparing EPQ and EOQ

Defining the production buildup factor $\psi = \sqrt{1 - \frac{d}{p}} < 1$:

$EPQ = \frac{EOQ}{\sqrt{1 - \frac{d}{p}}}$

Because $(1 - d/p) < 1$, the divisor is less than 1, proving that $EPQ > EOQ$. Under finite production replenishment, batches are larger than purchase orders, yet maximum inventory $I_{max}$ and average inventory $\bar{I}$ are strictly lower than their EOQ counterparts due to concurrent consumption during production.

5. Quantity Discount Valuation Models

Suppliers frequently offer price discounts to incentivize larger order sizes. Lower purchase prices reduce material acquisition expenditures and annual ordering frequency, but elevate average inventory and annual holding costs.

All-Units vs. Incremental Discounts

Industrial pricing contracts follow two primary structures:

1. All-Units Quantity Discounts: The discounted unit price applies to every single unit purchased in the order, provided the lot size meets or exceeds the price breakpoint threshold $q_k$.
2. Incremental Quantity Discounts: The discounted price applies only to units ordered in excess of the breakpoint threshold, preserving higher prices for baseline units (similar to marginal income tax brackets).

All-Units Discount Cost Curve:                 Incremental Discount Cost Curve:
Total Purchase Cost ($)                        Total Purchase Cost ($)
      ^
      |         / Tier 1: C1                         ^
      |        /                                     |                   / Slope C2
      |       /                                      |                  / 
      |      /    / Tier 2: C2                       |       Breakpoint / (C2 < C1)
      |     /    /                                   |      .          /
      |    /    /                                    |     /          /
      |   /    /                                     |    / Slope C1 /
      |  /    /                                      |   /          /
      +------+---------------------> Q               +--+----------+----------> Q
             q1 (Price Drops)                           q1         q2

The 4-Step All-Units Discount Algorithm

To identify the global cost-minimizing order quantity under an all-units discount schedule with price tiers $C_1 > C_2 > \dots > C_m$ and breakpoints $q_1 = 0 < q_2 < \dots < q_m$:

1. Calculate Feasible EOQ for Each Tier: Starting at the lowest price tier $m$ (highest volume discount), compute: $EOQ_k = \sqrt{\frac{2DS}{i C_k}}$
2. Check Feasibility: Determine whether $EOQ_k$ falls within its valid interval $[q_k, q_{k+1})$:
   • If $EOQ_k \ge q_{k+1}$, discard it (a lower price tier is available and better).
   • If $q_k \le EOQ_k < q_{k+1}$, $EOQ_k$ is feasible. Proceed to step 3.
   • If $EOQ_k < q_k$, $EOQ_k$ is infeasible at price $C_k$. The minimum quantity required to qualify for price $C_k$ is the breakpoint quantity $q_k$.
3. Compute Total Annual Cost ($TC$): The total cost must include acquisition purchase price, because $D \cdot C_k$ varies between tiers: $TC(Q) = D \cdot C_k + \frac{D}{Q}S + \frac{Q}{2}(i C_k)$ Calculate $TC(Q)$ for:
   • The lowest-price feasible $EOQ_k$.
   • All price breakpoints $q_j$ that offer lower unit prices than the feasible $EOQ_k$.
4. Select Global Optimum: The candidate order quantity yielding the lowest total annual cost $TC(Q)$ is the optimal order quantity $Q^*$.

6. Planned Backorders / Shortage Models

In certain industrial environments—such as custom capital machinery, aerospace components, or high-value engineered assemblies—stockouts do not result in lost sales. Instead, customers accept delayed delivery, creating planned backorders.

Model Parameters

• Shortage cost rate $C_b$ ($/unit backordered per year).
• $Q$ = Total replenishment batch quantity.
• $B$ = Maximum backorder quantity accumulated at the end of a cycle.
• $I_{max} = Q - B$ = Maximum on-hand physical inventory.

Inventory Level
     ^
Imax |\                     /\ 
     | \                   /  \ 
     |  \                 /    \ 
   0 +---\---------------/------\-------------------> Time
     |    \             /        \ 
  -B |     \___________/          \ 
     v      Max Backorder

Cost Formulation and Optimal Values

Over the cycle $T = Q/D$, inventory is positive for time $t_1 = (Q - B)/D$ and in backorder for time $t_2 = B/D$. The Total Relevant Cost equation is:

$TRC(Q, B) = \frac{D}{Q}S + \frac{(Q - B)^2}{2Q}H + \frac{B^2}{2Q}C_b$

Taking partial derivatives with respect to $Q$ and $B$, setting them to zero, and solving simultaneously yields:

$Q^* = \sqrt{\frac{2DS}{H}} \times \sqrt{\frac{H + C_b}{C_b}} = EOQ \times \sqrt{1 + \frac{H}{C_b}}$

$B^* = Q^* \left( \frac{H}{H + C_b} \right) = \sqrt{\frac{2DS}{C_b}} \times \sqrt{\frac{H}{H + C_b}}$

$I_{max}^* = Q^* - B^* = Q^* \left( \frac{C_b}{H + C_b} \right)$

Engineering Insights

• Because $\sqrt{(H + C_b)/C_b} > 1$, the optimal lot size under planned backorders is strictly larger than classical EOQ ($Q^* > EOQ$).
• As shortage penalty $C_b \to \infty$, $\sqrt{(H + C_b)/C_b} \to 1$ and $B^* \to 0$, collapsing directly back to the classical EOQ model without shortages.
• Permitting backorders reduces average physical holding inventory, thereby lowering total operating costs whenever carrying high physical inventory is more costly than customer delay penalties.

7. Worked Engineering Examples

Example 1: EPQ Optimization for a Precision Machining Center

Problem Statement: A precision automotive components plant operates 250 days per year. A dedicated CNC machining cell produces cast iron engine mounts with the following parameters:

• Annual demand ($D$): 10,000 units/year
• Daily production rate ($p$): 200 units/day
• Cell setup cost ($S$): $200 per production changeover
• Unit manufacturing cost ($C$): $25.00 per unit
• Annual inventory carrying charge ($i$): 20% per year ($H = 0.20 \times \text{\textdollar}25.00 = \text{\textdollar}5.00\text{/unit-year}$)

Calculate:

1. Daily demand rate $d$ and the buildup ratio $(1 - d/p)$.
2. The optimal Economic Production Quantity ($EPQ$).
3. The maximum on-hand inventory level ($I_{max}$).
4. Total annual setup and holding costs.
5. Length of the production run ($t_1$) and total cycle time ($T$).

Step-by-Step Solution:

1. Daily demand rate and buildup factor: $d = \frac{D}{\text{Operating Days}} = \frac{10,000}{250} = 40\text{ units/day}$ $\left( 1 - \frac{d}{p} \right) = 1 - \frac{40}{200} = 1 - 0.20 = 0.80$

2. Economic Production Quantity ($EPQ$): $EPQ = \sqrt{\frac{2DS}{H(1 - d/p)}} = \sqrt{\frac{2(10,000)(200)}{5.00(0.80)}} = \sqrt{\frac{4,000,000}{4.00}} = \sqrt{1,000,000} = 1,000\text{ units}$

3. Maximum on-hand inventory level ($I_{max}$): $I_{max} = EPQ \left( 1 - \frac{d}{p} \right) = 1,000 \times 0.80 = 800\text{ units}$

4. Annual setup and holding costs: $\text{Annual Setup Cost} = \frac{D}{EPQ} \times S = \frac{10,000}{1,000} \times \text{\textdollar}200 = 10 \times \text{\textdollar}200 = \text{\textdollar}2{,}000\text{/year}$ $\text{Annual Holding Cost} = \frac{I_{max}}{2} \times H = \frac{800}{2} \times \text{\textdollar}5.00 = 400 \times \text{\textdollar}5.00 = \text{\textdollar}2{,}000\text{/year}$ $\text{Total Relevant Cost} = \text{\textdollar}2{,}000 + \text{\textdollar}2{,}000 = \text{\textdollar}4{,}000\text{/year}$

5. Production duration and cycle time: $t_1 = \frac{EPQ}{p} = \frac{1,000}{200} = 5\text{ operating days}$ $t_2 = \frac{I_{max}}{d} = \frac{800}{40} = 20\text{ operating days}$ $T = t_1 + t_2 = 5 + 20 = 25\text{ operating days}$ $\text{Or directly: } T = \frac{EPQ}{d} = \frac{1,000}{40} = 25\text{ operating days}$

The CNC cell runs production for 5 days, accumulating 800 units of maximum stock, and then shuts down for 20 days while assembly consumes the buffer at 40 units/day.

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Example 2: All-Units Quantity Discount Evaluation

Problem Statement: An industrial plant requires $D = 5,000$ specialty valves annually. Ordering cost is $S =$ $180 per order, and inventory holding rate is $i = 20\%$. The supplier provides the following pricing tier:

• Tier 1: Orders of 1 to 999 units $\implies C_1 = \text{\textdollar}50.00$ per unit
• Tier 2: Orders of 1,000 or more units $\implies C_2 = \text{\textdollar}45.00$ per unit

Determine whether the company should take advantage of the discount.

Solution:

Step 1: Compute EOQ for Tier 2 ($C_2 = \text{\textdollar}45.00$): $H_2 = i \times C_2 = 0.20 \times \text{\textdollar}45.00 = \text{\textdollar}9.00\text{/unit-year}$ $EOQ_2 = \sqrt{\frac{2(5,000)(180)}{9.00}} = \sqrt{\frac{1,800,000}{9.00}} = \sqrt{200,000} \approx 447.21\text{ units}$

Because $447 < 1,000$, $EOQ_2$ is infeasible; the firm cannot buy 447 units at $45.00. The lowest order quantity to qualify for Tier 2 is the breakpoint $q_2 = 1,000$ units.

Step 2: Compute EOQ for Tier 1 ($C_1 = \text{\textdollar}50.00$): $H_1 = i \times C_1 = 0.20 \times \text{\textdollar}50.00 = \text{\textdollar}10.00\text{/unit-year}$ $EOQ_1 = \sqrt{\frac{2(5,000)(180)}{10.00}} = \sqrt{\frac{1,800,000}{10.00}} = \sqrt{180,000} \approx 424.26\text{ units}$

Because $1 \le 424 < 1,000$, $EOQ_1 = 424$ is feasible in Tier 1.

Step 3: Compare Total Annual Costs including acquisition costs:

Total Cost at $EOQ_1 = 424$ units ($C_1 = \text{\textdollar}50.00$): $TC(424) = D \cdot C_1 + \frac{D}{Q}S + \frac{Q}{2}H_1$ $TC(424) = 5,000(50.00) + \frac{5,000}{424}(180) + \frac{424}{2}(10.00)$ $TC(424) = 250,000 + 2,122.64 + 2,120.00 = \text{\textdollar}254{,}242.64$

Total Cost at Breakpoint $Q = 1,000$ units ($C_2 = \text{\textdollar}45.00$): $TC(1,000) = D \cdot C_2 + \frac{D}{Q}S + \frac{Q}{2}H_2$ $TC(1,000) = 5,000(45.00) + \frac{5,000}{1,000}(180) + \frac{1,000}{2}(9.00)$ $TC(1,000) = 225,000 + 900.00 + 4,500.00 = \text{\textdollar}230{,}400.00$

Conclusion: Ordering $Q^* = 1,000$ units at the breakpoint saves the company $254,242.64 - $230,400.00 = $23,842.64 per year (a 9.4% total cost reduction), driven overwhelmingly by the $25,000 direct purchase discount which easily overcomes the $2,380 increase in annual inventory carrying cost.