6.3 Material Requirements Planning (MRP), Bill of Materials & Capacity Planning

Material Requirements Planning (MRP), Bill of Materials & Capacity Planning

Core Principle: Material Requirements Planning (MRP) transforms independent MPS demand into time-phased dependent requirements across multi-echelon Bills of Materials. By coupling inventory status records with lead-time offsetting and work center capacity profiles, MRP ensures components arrive precisely when required while minimizing work-in-process (WIP).

In manufacturing systems, inventory items fall into two distinct demand regimes: independent demand and dependent demand. Independent demand (finished goods, spare parts) is influenced by external market forces and must be forecasted stochastically. Dependent demand (components, subassemblies, raw materials) is derived mathematically from the production schedule of parent assemblies. Applying statistical reorder points to dependent components creates artificial bullwhip amplifications; instead, dependent items must be managed via deterministic, time-phased Material Requirements Planning (MRP).

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1. Core Inputs to the MRP Engine

The MRP processing algorithm integrates three fundamental data streams:

Core MRP Inputs & Outputs Architecture

[Master Production Schedule (MPS)]     [Bill of Materials (BOM)]     [Inventory Status Records]
 (What end-items to make & when)         (Product structure tree)     (On-hand, on-order, lead time)
                │                                   │                              │
                └─────────────────────────┬─────────┴──────────────────────────────┘
                                          ▼
                           [MRP Computational Engine]
                                          │
       ┌──────────────────────────────────┴──────────────────────────────────┐
       ▼                                                                     ▼
[Primary MRP Outputs]                                               [Secondary Exception Outputs]
- Planned Order Releases (Shop & POs)                               - Expedite / De-expedite notices
- Time-phased Planned Order Receipts                                - Cancel order notices
- Rescheduling action advisories                                    - Capacity overload warnings

1. Master Production Schedule (MPS): Defines the quantities and required completion dates for all independent-demand finished items.
2. Bill of Materials (BOM): An engineering database defining the exact hierarchy, parent-child relationships, and unit quantities required to fabricate one parent unit.
3. Inventory Status Records: Contains item-specific operational parameters:
   • On-Hand Inventory: Physical usable stock physically present in the warehouse.
   • Scheduled Receipts ($SR$): Open purchase orders or factory work orders previously committed and due to arrive at the start of a specific time bucket.
   • Safety Stock ($SS$): Buffer inventory reserved to absorb scrap, yield losses, or lead-time variance.
   • Lead Time ($L$): Total elapsed time from order release to physical receipt into inventory (supplier lead time or manufacturing cycle time).
   • Lot-Sizing Rule: Pre-assigned heuristic governing order batch quantities.

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2. Bill of Materials (BOM) & Low-Level Coding (LLC)

A Bill of Materials (BOM) represents product structure hierarchically. A single-level BOM displays only immediate children, whereas an indented multi-level BOM exposes the full explosion path from finished unit to raw materials.

Product Structure Tree with Low-Level Coding

Level 0:                 [Final Product A]
                         /               \
Level 1:           [Sub B] (qty: 2)     [Sub C] (qty: 1)
                   /      \                     \
Level 2:   [Comp D] (3)   [Comp E] (1)         [Comp D] (2)

Low-Level Code (LLC) Assignment:
- Product A: LLC = 0 (Appears only at Level 0)
- Subassembly B: LLC = 1
- Subassembly C: LLC = 1
- Component E: LLC = 2
- Component D: LLC = 2 (Appears at Level 2 under both B and C; LLC = max depth = 2)

The Low-Level Coding (LLC) Algorithm

When a common component appears at multiple levels in different subassemblies (or across multiple products), exploding requirements naively causes fragmentation: orders would be generated at Level 1, and then additional requirements discovered at Level 2 would require duplicate setups and split batches.

To ensure computational efficiency and prevent fragmented lot sizing, MRP systems execute the Low-Level Coding (LLC) algorithm during BOM maintenance:

$\text{LLC}(\text{Item } i) = \max_{p \in \text{Paths}} \left( \text{Depth of Item } i \text{ in Path } p \right)$

• Operational Rule: An end-item is coded as Level 0. Any component used as an immediate child of a Level 0 item is Level 1, unless it also appears deeper in another branch. The item's LLC is assigned the maximum depth level at which it appears anywhere in the company's product structures.
• Explosion Integrity: The MRP processing engine explodes requirements strictly level by level ($LLC = 0, 1, 2, \dots$). No component's net requirements or planned orders are calculated until all parent assemblies possessing a lower LLC number have been fully exploded and their planned order releases aggregated. This ensures all gross requirements for a shared part are consolidated into single time buckets.

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3. The Mathematical Mechanics of MRP Explosion

For each inventory item, the MRP matrix processes a time-phased grid across planning buckets $t = 1, 2, \dots, T$. The row definitions and computational sequence are:

MRP Time-Phased Matrix Structure

Row Parameter                  Mathematical Relationship
─────────────────────────────────────────────────────────────────────────────
Gross Requirements (GR_t)      Parent PORel × Usage Qty + Independent Demand
Scheduled Receipts (SR_t)      Open orders confirmed arriving in period t
Projected Available (PAB_t)    PAB_{t-1} + SR_t + PORc_t - GR_t
Net Requirements (NR_t)        max(0, GR_t + SS - (PAB_{t-1} + SR_t))
Planned Order Receipts (PORc_t) Lot-sized batch to satisfy NR_t (PORc_t ≥ NR_t)
Planned Order Releases (PORel) PORel_{t-L} = PORc_t (Offset by Lead Time L)

Step-by-Step Computational Formulations

1. Gross Requirements ($GR_t$): Total anticipated demand for the component in period $t$:

$GR_t = \sum_{p \in \text{Parents}} \left( PORel_{p, t} \times U_{p} \right) + D_{\text{independent}, t}$

Where $PORel_{p, t}$ is the Planned Order Release of parent $p$ in period $t$, and $U_p$ is the usage multiplier (units of child required per unit of parent).

1. Preliminary Available Balance ($PAB_t^0$): Stock available before receiving any new planned order:

$PAB_t^0 = PAB_{t-1} + SR_t - GR_t$

1. Net Requirements ($NR_t$): Deficit below required safety stock ($SS$):

$NR_t = \begin{cases} GR_t + SS - (PAB_{t-1} + SR_t), & \text{if } PAB_{t-1} + SR_t < GR_t + SS \\ 0, & \text{otherwise} \end{cases}$

1. Planned Order Receipts ($PORc_t$): If $NR_t > 0$, an order receipt is scheduled in period $t$. The quantity is determined by the active lot-sizing rule ($PORc_t \ge NR_t$). If $NR_t = 0$, $PORc_t = 0$.

2. Updated Projected Available Balance ($PAB_t$):

$PAB_t = PAB_{t-1} + SR_t + PORc_t - GR_t$

1. Planned Order Releases ($PORel_{t-L}$): To ensure the order arrives at period $t$, it must be released $L$ periods earlier (lead-time offsetting):

$PORel_{t-L} = PORc_t$

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4. Lot-Sizing Techniques in MRP

Lot-sizing rules dictate how net requirements are grouped into replenishment batches. Choosing an inappropriate lot-sizing rule can either inflate inventory holding costs or create excessive setup overhead.

Part Period Balancing (PPB) Mathematics

PPB evaluates carrying cost in units of Part-Periods (one unit of inventory held for one period). The Economic Part Period ($EPP$) is the ratio of setup cost $S$ to inventory carrying cost per unit per period $h$:

$EPP = \frac{S}{h}$

The algorithm accumulates net requirements for future periods $k = t+1, t+2, \dots$ and computes cumulative part-periods:

$\text{Cumulative Part-Periods} = \sum_{j=t}^{t+m} \left( NR_j \times (j - t) \right)$

The batch is closed at the period $m$ where cumulative part-periods are closest to $EPP$.

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5. Comprehensive Worked Numerical MRP Example

An industrial instrumentation firm manufactures an automated fluid monitor (Item A). Each unit of A requires 2 units of Subassembly B. The operational parameters are:

• Finished Item A: Lot-for-Lot (L4L), Lead Time $L_A = 1$ week, Safety Stock $SS_A = 0$, Initial On-Hand $I_{0, A} = 10$.
• Subassembly B: Fixed Order Quantity $FOQ = 100$ units, Lead Time $L_B = 2$ weeks, Safety Stock $SS_B = 20$, Initial On-Hand $I_{0, B} = 45$, Scheduled Receipt $SR_B = 100$ in Week 1.

The Master Production Schedule requires completing 50 units of Item A in Week 3 and 80 units of Item A in Week 5.

MRP Explosion for Finished Item A (Parent)

• Item A Calculations:
  • In Week 3: $PAB_2 = 10$. Deficit is $50 - 10 = 40$. $NR_3 = 40 \implies PORc_3 = 40$. $PAB_3 = 10 + 40 - 50 = 0$. Offset by $L_A = 1$ week $\implies PORel_2 = 40$.
  • In Week 5: $PAB_4 = 0$. $NR_5 = 80 \implies PORc_5 = 80$. Offset by $L_A = 1$ week $\implies PORel_4 = 80$.

MRP Explosion for Subassembly B (Child, Usage = 2)

Gross requirements for B are driven by Item A's planned releases: $GR_{B, t} = PORel_{A, t} \times 2$:

• Week 2: $40 \times 2 = 80$ units.
• Week 4: $80 \times 2 = 160$ units.

• Subassembly B Step-by-Step Calculations:
  • Week 1: $PAB_1 = I_{0, B} + SR_1 - GR_1 = 45 + 100 - 0 = 145$.
  • Week 2: $GR_2 = 80$. Available before order is $145 - 80 = 65$. Since $65 > SS$ ($20$), $NR_2 = 0$. Updated $PAB_2 = 65$.
  • Week 3: $GR_3 = 0 \implies PAB_3 = 65$.
  • Week 4: $GR_4 = 160$. Required stock is $GR_4 + SS = 160 + 20 = 180$. Prior balance is $PAB_3 = 65$. The net requirement is $NR_4 = 180 - 65 = 115$ units.
  • Because lot size is $FOQ = 100$, the system must order multiples of 100. Sizing for 115 requires $\lceil 115 / 100 \rceil \times 100 = 200$ units. Thus, $PORc_4 = 200$.
  • Updated balance: $PAB_4 = 65 + 200 - 160 = 105$ (note: if safety stock is held separately, $PAB$ is 105 total, or 85 above $SS$).
  • Offset by lead time $L_B = 2$ weeks: order must be released in Week $4 - 2 = 2$. Therefore, $PORel_2 = 200$ units.

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6. Capacity Requirements Planning (CRP) & Closed-Loop MRP

MRP generates planned orders assuming infinite capacity—it assumes machines and labor are always available when needed. In reality, work centers face rigid throughput limits. Capacity Requirements Planning (CRP) acts as the critical reality check, converting planned order releases into machine and labor hours at specific work centers.

MRP to CRP Translation & Load Leveling

[Planned Order Releases (PORel)] × [Standard Routing Sheet (Setup & Run)]
                            │
                            ▼
           [Work Center Capacity Load Profile]

Hours ^
      |              Overload Spikes (Must resolve via OT/rerouting)
      |                  [===]
      |             ┌────┴───┴────┐
      |             │             │
  Cap +-------------+-------------+-------------+------------- Rated Max Capacity
      |    [===]    │             │    [===]    │
      |    │   │    │             │    │   │    │    [===]
      |    │   │    │             │    │   │    │    │   │
      +----+---+----+-------------+----+---+----+----+---+----> Time (Weeks)
           Week 1       Week 2         Week 3       Week 4

CRP Load Calculation

For a specific work center $w$ in time bucket $t$, the required workload in standard hours is:

$\text{Load}_{w, t} = \sum_{j \in J_{w, t}} \left( S_{j, w} + PORel_{j, t} \times U_{j, w} \right)$

Where:

• $J_{w, t}$ is the set of all lots scheduled to be processed at work center $w$ during period $t$.
• $S_{j, w}$ is the setup time in standard hours for lot $j$.
• $PORel_{j, t}$ is the planned lot size.
• $U_{j, w}$ is the run time per unit in standard hours.

Rated Work Center Capacity

Theoretical nominal hours must be discounted by machine availability and worker productivity:

$\text{Rated Capacity} = N \times H_{\text{shift}} \times S \times D \times \text{Utilization} \times \text{Efficiency}$

Where:

• $N$: Number of machines or operators.
• $H_{\text{shift}}$: Working hours per shift.
• $S$: Number of shifts per day.
• $D$: Operating days per period.
• $\text{Utilization}$: $\frac{\text{Actual Operating Hours}}{\text{Scheduled Clock Hours}} \le 1.0$.
• $\text{Efficiency}$: $\frac{\text{Standard Hours Produced}}{\text{Actual Operating Hours}}$.

Resolving Load Discrepancies

When CRP detects an overload condition ($\text{Load}_{w, t} > \text{Rated Capacity}$), closed-loop MRP requires intervention:

1. Short-Term Adjustments: Authorize overtime shifts, re-route operations to alternate work centers, split large production batches, or hire temporary contract labor.
2. Schedule Adjustments: Pull forward planned releases into earlier periods with surplus capacity (load leveling).
3. MPS Revision: If capacity cannot be expanded, the Master Production Schedule must be revised downward in the slushy or liquid zones.