Lot-Sizing Problems

You are an expert in multi-period lot-sizing models and production/inventory planning optimization. Your goal is to help determine optimal order or production quantities across multiple time periods to minimize total costs including setup, holding, and sometimes shortage costs.

Initial Assessment

Before solving lot-sizing problems, understand:

1. Planning Context

   • Planning horizon? (weeks, months, quarters)
   • Rolling or fixed horizon?
   • Demand pattern? (deterministic, forecast, actual orders)
   • Lead time considerations?

2. Cost Structure

   • Setup/ordering cost per order ($)?
   • Holding cost per unit per period ($/unit/period)?
   • Production/purchase cost per unit?
   • Backorder or shortage costs?
   • Cost structure time-varying?

3. Capacity and Constraints

   • Production capacity limits per period?
   • Storage capacity limits?
   • Minimum lot sizes or batch constraints?
   • Multiple items sharing capacity?

4. Product Characteristics

   • Single item or multiple items?
   • Bill of materials structure?
   • Product substitutability?
   • Shelf life or obsolescence?

5. Operational Requirements

   • Can backorders occur?
   • Must demand be satisfied immediately?
   • Multi-level production (BOM)?
   • Supplier constraints (MOQ, lead times)?

---

Lot-Sizing Problem Fundamentals

Problem Statement

Given:

• Planning horizon: T periods (t = 1, 2, ..., T)
• Demand in each period: D_t (known deterministically)
• Setup cost: S (fixed cost incurred when ordering/producing)
• Holding cost: h per unit per period
• Unit cost: c (often ignored if constant)

Decision:

• When to order/produce?
• How much to order/produce in each period?

Objective:

• Minimize total cost = setup costs + holding costs

Key Lot-Sizing Policies

1. Lot-for-Lot (L4L)

• Order exactly what is needed each period: Q_t = D_t
• Minimizes holding cost (zero inventory)
• Maximizes setup costs
• Use when setup costs are very low

2. Fixed Order Quantity (FOQ)

• Order the same quantity Q every time
• Simple to implement
• May not match demand patterns well

3. Economic Order Quantity (EOQ)

• Use EOQ formula with average demand
• Assumes constant demand rate

4. Period Order Quantity (POQ)

• Order every N periods (N determined by EOQ)
• Combines multiple periods' demand

5. Least Unit Cost (LUC)

• Choose lot size that minimizes cost per unit
• Forward-looking heuristic

6. Least Total Cost (LTC) / Part-Period Balancing

• Balance setup and holding costs
• Choose lot when cumulative holding cost ≈ setup cost

7. Silver-Meal Heuristic

• Minimize average cost per period
• Popular practical heuristic

8. Wagner-Whitin Algorithm

• Dynamic programming approach
• Finds optimal solution for uncapacitated problem
• Polynomial time complexity O(T²)

---

Python Implementation: Lot-Sizing Models

Basic Lot-Sizing Heuristics

import numpy as np
import pandas as pd
from typing import List, Dict, Tuple
import matplotlib.pyplot as plt

class LotSizingProblem:
    """
    Multi-period lot-sizing problem solver

    Implements various heuristics and optimal algorithm
    """

    def __init__(self, demands: List[float], setup_cost: float,
                 holding_cost: float, unit_cost: float = 0):
        """
        Parameters:
        -----------
        demands : list
            Demand for each period [D1, D2, ..., DT]
        setup_cost : float
            Fixed cost incurred when placing order/production run
        holding_cost : float
            Cost per unit held in inventory per period
        unit_cost : float
            Variable cost per unit (often omitted if constant)
        """
        self.demands = np.array(demands)
        self.T = len(demands)
        self.S = setup_cost
        self.h = holding_cost
        self.c = unit_cost

    def lot_for_lot(self) -> Dict:
        """
        Lot-for-Lot (L4L) policy: Order exactly demand each period

        Minimizes inventory but incurs setup cost every period
        """

        orders = self.demands.copy()
        inventory = np.zeros(self.T + 1)  # Inventory at end of each period
        setup_costs = np.zeros(self.T)
        holding_costs = np.zeros(self.T)

        for t in range(self.T):
            # Order arrives at start of period
            inventory[t] += orders[t]

            # Satisfy demand
            inventory[t] -= self.demands[t]

            # Costs
            if orders[t] > 0:
                setup_costs[t] = self.S
            holding_costs[t] = inventory[t] * self.h

            # Carry inventory forward
            inventory[t + 1] = inventory[t]

        total_setup = setup_costs.sum()
        total_holding = holding_costs.sum()
        total_cost = total_setup + total_holding

        return {
            'method': 'Lot-for-Lot',
            'orders': orders,
            'inventory': inventory[:-1],
            'setup_cost': total_setup,
            'holding_cost': total_holding,
            'total_cost': total_cost,
            'num_orders': (orders > 0).sum()
        }

    def fixed_order_quantity(self, Q: float) -> Dict:
        """
        Fixed Order Quantity: Order Q units whenever inventory insufficient

        Parameters:
        -----------
        Q : float
            Fixed order quantity
        """

        orders = np.zeros(self.T)
        inventory = np.zeros(self.T + 1)
        setup_costs = np.zeros(self.T)
        holding_costs = np.zeros(self.T)

        for t in range(self.T):
            # Check if order needed
            if inventory[t] < self.demands[t]:
                orders[t] = Q
                inventory[t] += Q

            # Satisfy demand
            inventory[t] -= self.demands[t]

            # Handle backorders if negative
            if inventory[t] < 0:
                # Need more orders
                num_orders = int(np.ceil(-inventory[t] / Q))
                orders[t] += num_orders * Q
                inventory[t] += num_orders * Q

            # Costs
            if orders[t] > 0:
                setup_costs[t] = self.S * (orders[t] / Q)  # Proportional setups
            holding_costs[t] = max(0, inventory[t]) * self.h

            # Carry forward
            inventory[t + 1] = inventory[t]

        total_setup = setup_costs.sum()
        total_holding = holding_costs.sum()
        total_cost = total_setup + total_holding

        return {
            'method': f'FOQ (Q={Q})',
            'orders': orders,
            'inventory': inventory[:-1],
            'setup_cost': total_setup,
            'holding_cost': total_holding,
            'total_cost': total_cost,
            'num_orders': (orders > 0).sum()
        }

    def silver_meal(self) -> Dict:
        """
        Silver-Meal Heuristic

        Iteratively add periods to lot, stop when average cost per period increases
        """

        orders = np.zeros(self.T)
        inventory = np.zeros(self.T + 1)
        t = 0

        while t < self.T:
            # Determine lot size starting at period t
            best_periods = 1
            min_avg_cost = float('inf')

            for k in range(1, self.T - t + 1):
                # Consider covering periods t through t+k-1
                # Cost = Setup cost + holding cost for carrying inventory

                total_demand = sum(self.demands[t:t+k])
                holding_cost = sum((j - t) * self.demands[t+j] * self.h
                                  for j in range(k))

                total_cost = self.S + holding_cost
                avg_cost = total_cost / k

                if avg_cost < min_avg_cost:
                    min_avg_cost = avg_cost
                    best_periods = k
                else:
                    # Average cost increased, stop
                    break

            # Place order for best_periods worth of demand
            order_qty = sum(self.demands[t:t+best_periods])
            orders[t] = order_qty

            # Update inventory levels
            inv = order_qty
            for j in range(best_periods):
                if t + j < self.T:
                    inventory[t + j] = inv
                    inv -= self.demands[t + j]

            t += best_periods

        # Calculate costs
        setup_costs = np.where(orders > 0, self.S, 0)
        holding_costs = inventory[:-1] * self.h

        return {
            'method': 'Silver-Meal',
            'orders': orders,
            'inventory': inventory[:-1],
            'setup_cost': setup_costs.sum(),
            'holding_cost': holding_costs.sum(),
            'total_cost': setup_costs.sum() + holding_costs.sum(),
            'num_orders': (orders > 0).sum()
        }

    def least_unit_cost(self) -> Dict:
        """
        Least Unit Cost (LUC) Heuristic

        For each order, add periods until cost per unit increases
        """

        orders = np.zeros(self.T)
        inventory = np.zeros(self.T + 1)
        t = 0

        while t < self.T:
            min_unit_cost = float('inf')
            best_periods = 1

            for k in range(1, self.T - t + 1):
                # Total demand covered
                total_demand = sum(self.demands[t:t+k])

                # Holding cost
                holding_cost = sum((j - t) * self.demands[t+j] * self.h
                                  for j in range(k))

                # Total cost
                total_cost = self.S + holding_cost

                # Unit cost
                unit_cost = total_cost / total_demand

                if unit_cost < min_unit_cost:
                    min_unit_cost = unit_cost
                    best_periods = k
                else:
                    break

            # Place order
            order_qty = sum(self.demands[t:t+best_periods])
            orders[t] = order_qty

            # Update inventory
            inv = order_qty
            for j in range(best_periods):
                if t + j < self.T:
                    inventory[t + j] = inv
                    inv -= self.demands[t + j]

            t += best_periods

        # Calculate costs
        setup_costs = np.where(orders > 0, self.S, 0)
        holding_costs = inventory[:-1] * self.h

        return {
            'method': 'Least Unit Cost',
            'orders': orders,
            'inventory': inventory[:-1],
            'setup_cost': setup_costs.sum(),
            'holding_cost': holding_costs.sum(),
            'total_cost': setup_costs.sum() + holding_costs.sum(),
            'num_orders': (orders > 0).sum()
        }

    def wagner_whitin(self) -> Dict:
        """
        Wagner-Whitin Algorithm

        Dynamic programming approach - finds optimal solution
        for uncapacitated lot-sizing problem

        Time complexity: O(T²)
        """

        T = self.T
        # F[t] = minimum cost for periods 1..t
        F = np.full(T + 1, np.inf)
        F[0] = 0

        # Predecessor: which period did we last order from?
        pred = np.zeros(T + 1, dtype=int)

        # DP forward pass
        for t in range(1, T + 1):
            for j in range(t):
                # Consider ordering in period j to cover demand through period t
                # Cost = F[j] + setup cost + holding cost

                # Holding cost for covering periods j+1 through t
                holding_cost = 0
                for k in range(j + 1, t + 1):
                    # Hold demand[k-1] for (k - j - 1) periods
                    holding_cost += (k - j - 1) * self.demands[k - 1] * self.h

                cost = F[j] + self.S + holding_cost

                if cost < F[t]:
                    F[t] = cost
                    pred[t] = j

        # Backtrack to find order periods
        orders = np.zeros(T)
        t = T
        while t > 0:
            j = pred[t]
            # Order in period j to cover through period t
            order_qty = sum(self.demands[j:t])
            orders[j] = order_qty
            t = j

        # Reconstruct inventory
        inventory = np.zeros(T + 1)
        for t in range(T):
            inventory[t] += orders[t]
            inventory[t] -= self.demands[t]
            inventory[t + 1] = inventory[t]

        # Calculate costs
        setup_costs = np.where(orders > 0, self.S, 0)
        holding_costs = inventory[:-1] * self.h

        return {
            'method': 'Wagner-Whitin (Optimal)',
            'orders': orders,
            'inventory': inventory[:-1],
            'setup_cost': setup_costs.sum(),
            'holding_cost': holding_costs.sum(),
            'total_cost': setup_costs.sum() + holding_costs.sum(),
            'num_orders': (orders > 0).sum(),
            'optimal': True
        }

    def compare_methods(self) -> pd.DataFrame:
        """Compare all lot-sizing methods"""

        methods = [
            self.lot_for_lot(),
            self.silver_meal(),
            self.least_unit_cost(),
            self.wagner_whitin()
        ]

        # Add FOQ with EOQ-based quantity
        avg_demand = self.demands.mean()
        if avg_demand > 0:
            eoq = np.sqrt(2 * avg_demand * self.T * self.S / self.h)
            methods.append(self.fixed_order_quantity(eoq))

        results = []
        for method in methods:
            results.append({
                'Method': method['method'],
                'Total Cost': method['total_cost'],
                'Setup Cost': method['setup_cost'],
                'Holding Cost': method['holding_cost'],
                'Num Orders': method['num_orders'],
                'Avg Inventory': method['inventory'].mean()
            })

        df = pd.DataFrame(results)
        df = df.sort_values('Total Cost')

        return df

    def plot_solution(self, solution: Dict):
        """Visualize lot-sizing solution"""

        fig, (ax1, ax2, ax3) = plt.subplots(3, 1, figsize=(14, 10))

        periods = np.arange(1, self.T + 1)

        # Plot 1: Demand and Orders
        ax1.bar(periods, self.demands, alpha=0.6, label='Demand', color='blue')
        order_periods = periods[solution['orders'] > 0]
        order_qtys = solution['orders'][solution['orders'] > 0]
        ax1.bar(order_periods, order_qtys, alpha=0.8, label='Orders', color='green')

        ax1.set_xlabel('Period', fontsize=11)
        ax1.set_ylabel('Quantity', fontsize=11)
        ax1.set_title(f"{solution['method']}: Demand and Order Pattern",
                     fontsize=12, fontweight='bold')
        ax1.legend()
        ax1.grid(True, alpha=0.3)

        # Plot 2: Inventory Levels
        ax2.plot(periods, solution['inventory'], marker='o', linewidth=2,
                color='orange', label='Ending Inventory')
        ax2.fill_between(periods, 0, solution['inventory'], alpha=0.3, color='orange')
        ax2.axhline(y=solution['inventory'].mean(), linestyle='--',
                   color='red', label=f"Avg Inv = {solution['inventory'].mean():.1f}")

        ax2.set_xlabel('Period', fontsize=11)
        ax2.set_ylabel('Inventory Level', fontsize=11)
        ax2.set_title('Inventory Over Time', fontsize=12, fontweight='bold')
        ax2.legend()
        ax2.grid(True, alpha=0.3)

        # Plot 3: Costs
        setup_costs = np.where(solution['orders'] > 0, self.S, 0)
        holding_costs = solution['inventory'] * self.h

        width = 0.35
        ax3.bar(periods - width/2, setup_costs, width, label='Setup Cost',
               color='red', alpha=0.7)
        ax3.bar(periods + width/2, holding_costs, width, label='Holding Cost',
               color='blue', alpha=0.7)

        ax3.set_xlabel('Period', fontsize=11)
        ax3.set_ylabel('Cost ($)', fontsize=11)
        ax3.set_title('Period Costs', fontsize=12, fontweight='bold')
        ax3.legend()
        ax3.grid(True, alpha=0.3)

        plt.tight_layout()
        return plt


# Example Usage
def example_lot_sizing():
    """Example: Multi-period lot-sizing problem"""

    print("\n" + "=" * 70)
    print("MULTI-PERIOD LOT-SIZING PROBLEM")
    print("=" * 70)

    # 12-period planning horizon with varying demand
    demands = [50, 60, 40, 80, 100, 90, 70, 60, 50, 80, 90, 100]

    problem = LotSizingProblem(
        demands=demands,
        setup_cost=200,      # $200 per order
        holding_cost=2,      # $2 per unit per period
        unit_cost=10         # $10 per unit (often ignored)
    )

    print("\nProblem Data:")
    print(f"  Planning Horizon: {problem.T} periods")
    print(f"  Setup Cost: ${problem.S}")
    print(f"  Holding Cost: ${problem.h}/unit/period")
    print(f"  Total Demand: {problem.demands.sum():.0f} units")
    print(f"  Average Demand: {problem.demands.mean():.1f} units/period")

    print("\n  Period-by-period demand:")
    for t, d in enumerate(demands, 1):
        print(f"    Period {t:2d}: {d:3.0f} units")

    # Compare all methods
    print("\n" + "=" * 70)
    print("COMPARISON OF LOT-SIZING METHODS")
    print("=" * 70)

    comparison = problem.compare_methods()
    print("\n" + comparison.to_string(index=False))

    # Analyze optimal solution
    optimal = problem.wagner_whitin()

    print("\n" + "=" * 70)
    print(f"OPTIMAL SOLUTION: {optimal['method']}")
    print("=" * 70)

    print(f"\n{'Total Cost:':<25} ${optimal['total_cost']:,.2f}")
    print(f"{'Setup Cost:':<25} ${optimal['setup_cost']:,.2f}")
    print(f"{'Holding Cost:':<25} ${optimal['holding_cost']:,.2f}")
    print(f"{'Number of Orders:':<25} {optimal['num_orders']}")
    print(f"{'Average Inventory:':<25} {optimal['inventory'].mean():.1f} units")

    print("\n  Order Schedule:")
    for t in range(problem.T):
        if optimal['orders'][t] > 0:
            print(f"    Period {t+1}: Order {optimal['orders'][t]:.0f} units")

    # Visualize
    problem.plot_solution(optimal)
    plt.savefig('/tmp/lot_sizing_optimal.png', dpi=300, bbox_inches='tight')
    print(f"\nOptimal solution plot saved to /tmp/lot_sizing_optimal.png")

    return problem, optimal


if __name__ == "__main__":
    example_lot_sizing()

---

Capacitated Lot-Sizing

Single-Item Capacitated Lot-Sizing Problem (CLSP)

from pulp import *

class CapacitatedLotSizing:
    """
    Capacitated Lot-Sizing Problem (CLSP)

    Production capacity constraints in each period
    """

    def __init__(self, demands: List[float], setup_cost: float,
                 holding_cost: float, production_cost: float,
                 capacity: List[float]):
        """
        Parameters:
        -----------
        demands : list
            Demand for each period
        setup_cost : float
            Fixed setup cost per period
        holding_cost : float
            Holding cost per unit per period
        production_cost : float
            Variable production cost per unit
        capacity : list
            Production capacity each period
        """
        self.demands = np.array(demands)
        self.T = len(demands)
        self.S = setup_cost
        self.h = holding_cost
        self.c = production_cost
        self.capacity = np.array(capacity)

    def solve_mip(self) -> Dict:
        """
        Solve using Mixed-Integer Programming

        Decision variables:
        - x_t: production quantity in period t
        - y_t: binary, 1 if production occurs in period t
        - I_t: inventory at end of period t
        """

        # Create problem
        prob = LpProblem("Capacitated_Lot_Sizing", LpMinimize)

        # Decision variables
        x = [LpVariable(f"x_{t}", lowBound=0) for t in range(self.T)]
        y = [LpVariable(f"y_{t}", cat='Binary') for t in range(self.T)]
        I = [LpVariable(f"I_{t}", lowBound=0) for t in range(self.T + 1)]

        # Objective: minimize total cost
        prob += (lpSum([self.S * y[t] + self.c * x[t] + self.h * I[t]
                       for t in range(self.T)]))

        # Constraints

        # Initial inventory
        prob += I[0] == 0

        # Inventory balance
        for t in range(self.T):
            prob += I[t + 1] == I[t] + x[t] - self.demands[t]

        # Production capacity
        for t in range(self.T):
            prob += x[t] <= self.capacity[t] * y[t]

        # Solve
        prob.solve(PULP_CBC_CMD(msg=0))

        # Extract solution
        production = np.array([x[t].varValue for t in range(self.T)])
        setup_decisions = np.array([y[t].varValue for t in range(self.T)])
        inventory = np.array([I[t].varValue for t in range(self.T + 1)])

        # Calculate costs
        setup_cost = self.S * setup_decisions.sum()
        prod_cost = self.c * production.sum()
        holding_cost = self.h * inventory[:-1].sum()
        total_cost = value(prob.objective)

        return {
            'status': LpStatus[prob.status],
            'production': production,
            'inventory': inventory[:-1],
            'setups': setup_decisions,
            'total_cost': total_cost,
            'setup_cost': setup_cost,
            'production_cost': prod_cost,
            'holding_cost': holding_cost,
            'num_setups': int(setup_decisions.sum())
        }


# Example: Capacitated Lot-Sizing
def example_capacitated():
    """Example: Lot-sizing with capacity constraints"""

    print("\n" + "=" * 70)
    print("CAPACITATED LOT-SIZING PROBLEM")
    print("=" * 70)

    demands = [40, 60, 80, 50, 70, 90]
    capacity = [100, 100, 100, 100, 100, 100]  # Max production per period

    problem = CapacitatedLotSizing(
        demands=demands,
        setup_cost=300,
        holding_cost=2,
        production_cost=10,
        capacity=capacity
    )

    print("\nProblem Data:")
    print(f"  Periods: {problem.T}")
    print(f"  Setup Cost: ${problem.S}")
    print(f"  Holding Cost: ${problem.h}/unit/period")
    print(f"  Production Cost: ${problem.c}/unit")
    print(f"  Capacity per Period: {capacity[0]} units")

    print("\n  Period Demands:")
    for t, d in enumerate(demands, 1):
        print(f"    Period {t}: {d} units")

    # Solve
    solution = problem.solve_mip()

    print(f"\n{'=' * 70}")
    print(f"OPTIMAL SOLUTION")
    print("=" * 70)

    print(f"\n{'Status:':<25} {solution['status']}")
    print(f"{'Total Cost:':<25} ${solution['total_cost']:,.2f}")
    print(f"{'Setup Cost:':<25} ${solution['setup_cost']:,.2f}")
    print(f"{'Production Cost:':<25} ${solution['production_cost']:,.2f}")
    print(f"{'Holding Cost:':<25} ${solution['holding_cost']:,.2f}")
    print(f"{'Number of Setups:':<25} {solution['num_setups']}")

    print("\n  Production Schedule:")
    for t in range(problem.T):
        if solution['production'][t] > 0.01:
            print(f"    Period {t+1}: Produce {solution['production'][t]:.0f} units, "
                  f"Ending Inv: {solution['inventory'][t]:.0f}")

    return problem, solution


if __name__ == "__main__":
    example_capacitated()

---

Tools & Libraries

Python Libraries

Optimization:

• pulp

  : Linear/mixed-integer programming

• pyomo

  : Optimization modeling

• scipy.optimize

  : General optimization

• ortools

  : Google OR-Tools

Numerical:

• numpy

  ,

  pandas

  : Data manipulation

Commercial Software

Production Planning:

• SAP APO: Advanced Planning & Optimization with lot-sizing
• Oracle Demantra: Demand and supply planning
• Blue Yonder: Supply chain planning
• Kinaxis RapidResponse: Integrated planning

MRP Systems:

• Most ERP systems have lot-sizing rules (SAP, Oracle, Microsoft Dynamics)

---

Common Challenges & Solutions

Challenge: Capacity Constraints

Problem:

• Production capacity insufficient in some periods
• Cannot produce when needed

Solutions:

• Use capacitated lot-sizing MIP model
• Consider overtime production (higher cost)
• Build inventory in advance during low-demand periods
• Outsource production for peak periods

Challenge: Setup Time vs. Setup Cost

Problem:

• Setups consume both time (capacity) and cost
• Simple models consider only cost

Solutions:

• Include setup time in capacity constraints
• Use CLSP with setup times
• Sequence-dependent setup times → more complex models

Challenge: Multi-Item Lot-Sizing

Problem:

• Multiple products share same production capacity
• Joint setup costs or time

Solutions:

• Multi-item CLSP formulation
• Proportional Lot-Sizing (PROPLS) heuristic
• Priority-based allocation
• See multi-item examples in code

Challenge: Uncertainty in Demand

Problem:

• Lot-sizing assumes deterministic demand
• Real demand is uncertain

Solutions:

• Use rolling horizon planning (replan each period)
• Add safety stock to demands
• Robust optimization with demand scenarios
• See stochastic-inventory-models and dynamic-lot-sizing

---

Related Skills

• economic-order-quantity: Single-period lot-sizing
• dynamic-lot-sizing: Time-varying parameters and stochastic demand
• stochastic-inventory-models: Probabilistic inventory models
• master-production-scheduling: Aggregate production planning
• capacity-planning: Long-term capacity decisions
• production-scheduling: Short-term scheduling