# Lecture 7: Inventory Control 2 - Advanced Topics and Risk Pooling **Date:** July 31 **Duration:** 1.5 hours **Instructor:** Prof. Vivek Farias before [[lecture 6]] --- ## manual notes [[2025-07-31|25-07-31-09]] not loosing customer but item, but moving onto next item - underage cost is not $60, as customer profile is - 1 product match with each customer (different size) same thing in the positive direction () compelling assortment so you'll buy another, if you don't see compelling one, 👗60$ multiplied by probability of you won't buy anything from me (due to repeated visit and assortment) virtuous cycle: rapid rotation, ⭐️formula modularize thoughts portfolio existis in particular point of time, long term value thinking (opportunity to calibrate numbers) 🔄rapid rotation -(reason to revisit)-> revisits -(lower u, lower safety stock)-> understock -()-> being exposed to supply risk is esp. bad fast turning business models 질문할때 짧고 핵심을. I HAVE QUESTION AND OBSERVATION. long term customer behavior (substitue over time), tractically important super sexy, they just flow out of the tounge understanding the levers of supply chain costs when you're at R, stockout when placing order should be larger than ddlt probability of having inventory P(no stockout) = .95 expected inventory you have should exceed probability of running out of inventory before being replenished leadtime -> shouldn't wait till inventory become zero, place order, what should that level be? if deterministic (demand beat leadtime) surge in demand, run out early with great variability comes great inventory - R = E[DDLT] + k_95 sigma(DDLT) - our decision (promise level) affects the marginal event probability but doesn't affect the pdf of underlying demand variable itself. P(DDLT <=R) = alpha R = E[DDLT] +k sigma sqrt(DDLT) mean = 400, sd = 80, LT=2 (service level of 95%), inventory cost of 45% DDLT is a random variable of demand happening during two weeks with mean 2 * 400 and sd sq(2) * 80 as leadtime goes up, my sigma goes up; volatility increases safety stock (friction of variability) is k sigma sqrt(ddlt) inventory tools 1. nv + eoq heuristics 2. LT (PS, SS), sigma (impacts SS), service level(SS); uncertainty and lt 3. amzn, wm 🚨 show students how increasing LT, sigma, service level affect q* (what vivek showed in picture, turn this into claude artifact (simulation)) probability of stock is not 0 but 5%. in reality hard to track demand (mean, sigma), leadtime distribution network (ppl's expectation; expect everything fast), - competing on the dimension of CHOICE (giving you what you want) + TIME (fast) - tailwinds make this ripe for innovation (mobile, sharing economy/last mile logistics liquidity) 1. four states, two actions, one metric ($); 1. (modeling) what are the four states, two actions, one metric 2. what's the first order tradeoff? 1. speed for Macdonald: 2. mile range for tesla: 1. 3. what are the cost variables associated with state and actions? 3. (statistics and probability) what are the probability of each state and action 4. (optimization) what is the decision 2. problem i'd like to solve with you is, how the competition between time and choice (demand modeling should have both fast and accurate - how to put this in time metric) challenge - mom and pop - incentive misalignment (paying for inventory) retailers themselves will compass tied to this (bear inventory), 👀eyeballs (advertising) effect you can uncensor carefully via distribution?? service level is super high - so censoring may not be a great problem but for those that need scarcity and high cost (ferrari, etc) --- ## Learning Objectives - Master advanced inventory models under uncertainty - Understand risk pooling concepts and benefits - Analyze multi-location inventory systems - Apply service level optimization techniques ## Key Concepts ### Stochastic Inventory Models - **Demand Uncertainty:** Random variables and distributions - **Lead Time Variability:** Supply uncertainty effects - **Service Level Constraints:** Probability-based targets - **Safety Stock Optimization:** Balancing cost and service ### (R,Q) Inventory Policy - **Reorder Point (R):** Inventory level triggering new order - **Order Quantity (Q):** Fixed amount ordered each time - **Policy Logic:** Order Q units when inventory hits R - **Continuous Review:** Monitor inventory continuously ### (R,Q) Model Parameters **Reorder Point Calculation:** - R = μL + zα × σL - Where: - μL = Expected demand during lead time - σL = Standard deviation of demand during lead time - zα = Safety factor for service level α **Order Quantity:** - Use EOQ formula: Q* = √(2DK/h) - Or modified EOQ for uncertain demand **Safety Stock:** - SS = zα × σL - Higher service levels require more safety stock - Cost increases exponentially with service level ### Service Level Types Revisited #### Type I Service Level (Cycle Service Level) - **Definition:** Probability of not stocking out during lead time - **Formula:** P(DL ≤ R) = α - **Application:** Appropriate when stockout cost is very high - **Calculation:** R = F^(-1)(α) where F is CDF of lead time demand #### Type II Service Level (Fill Rate) - **Definition:** Fraction of demand satisfied from stock - **Formula:** β = 1 - E[Shortage]/Q - **Application:** More relevant for ongoing operations - **Calculation:** More complex, involves expected shortage function ## Risk Pooling Fundamentals ### Definition and Concept - **Risk Pooling:** Combining uncertain demands to reduce total variability - **Mathematical Basis:** Variance of sum < Sum of variances (when demands not perfectly correlated) - **Central Limit Theorem:** Aggregated demand becomes more predictable ### Types of Risk Pooling #### Location Pooling - **Centralization:** Consolidate inventory at fewer locations - **Benefit:** √n reduction in safety stock (for n identical locations) - **Trade-off:** Response time vs. inventory cost #### Product Pooling - **Generic Products:** Design for late customization - **Common Components:** Share parts across products - **Postponement:** Delay differentiation until demand known #### Time Pooling - **Shorter Review Periods:** More frequent ordering - **Cross-docking:** Reduce storage time - **Just-in-Time:** Minimize time in system #### Supplier Pooling - **Multiple Sources:** Reduce supply risk - **Flexible Contracts:** Option to source from different suppliers - **Supply Diversification:** Geographic and capacity spreading ### Quantitative Risk Pooling Analysis #### Single Location Model - **Demand:** D ~ N(μ, σ²) - **Safety Stock:** SS = zα × σ - **Total Inventory:** μ + zα × σ #### Pooled Model (n identical locations) - **Aggregate Demand:** nD ~ N(nμ, nσ²) - **Aggregate Std Dev:** √n × σ - **Safety Stock:** SS_pooled = zα × √n × σ - **Reduction Factor:** √n (significant for large n) #### Correlation Effects - **Perfect Correlation (ρ = 1):** No pooling benefit - **Zero Correlation (ρ = 0):** Maximum pooling benefit - **Negative Correlation (ρ < 0):** Even greater benefits - **Real World:** Usually positive but less than perfect correlation ### Advanced Inventory Topics #### Multi-Echelon Systems - **Supply Chain Structure:** Multiple levels of inventory - **Coordination:** Upstream-downstream interactions - **Bullwhip Effect:** Demand amplification up supply chain - **Information Sharing:** Reduce distortion through transparency #### Substitution and Commonality - **Product Substitution:** Use similar products when stockout - **Component Commonality:** Share parts across products - **Design Strategy:** Build flexibility into product architecture - **Inventory Benefits:** Reduce total safety stock requirements #### Capacity Constraints - **Storage Limitations:** Physical or financial constraints - **Optimal Allocation:** Distribute limited capacity across items - **Lagrange Multipliers:** Mathematical optimization approach - **ABC Analysis:** Focus resources on high-value items ## Performance Measurement ### Inventory Metrics - **Inventory Turnover:** COGS / Average Inventory - **Days Sales Outstanding:** Average Inventory / Daily Sales - **Fill Rate:** Orders Filled Complete / Total Orders - **Service Level:** Various measures of stockout avoidance ### Cost Analysis - **Total Inventory Cost:** Ordering + Holding + Shortage costs - **Trade-off Optimization:** Balance competing objectives - **Sensitivity Analysis:** How parameters affect optimal policies - **What-if Scenarios:** Compare alternative strategies ## Strategic Inventory Decisions ### Centralization vs. Decentralization **Centralization Benefits:** - Risk pooling advantages - Economies of scale in purchasing - Better demand visibility - Reduced total safety stock **Decentralization Benefits:** - Faster response time - Lower transportation costs - Local market responsiveness - Reduced complexity **Hybrid Approaches:** - Regional distribution centers - Cross-docking facilities - Virtual centralization through information sharing ### Postponement Strategies - **Manufacturing Postponement:** Delay final assembly - **Logistics Postponement:** Delay shipment until orders received - **Product Postponement:** Design for late customization - **Time Postponement:** Delay commitment decisions ## Discussion Questions for Class 1. **Risk Pooling Applications:** In what situations is risk pooling most beneficial? When might it not work well? 2. **Centralization Trade-offs:** How should a company decide between centralized vs. decentralized inventory? 3. **Service Level Setting:** How should service level targets be set? Should they be the same for all products? 4. **Technology Impact:** How do modern information systems change inventory management strategies? ## Real-World Applications ### E-commerce - **Amazon:** Risk pooling through fulfillment centers - **Drop-shipping:** Virtual inventory pooling - **Prime delivery:** Service level differentiation ### Automotive Industry - **Component commonality:** Share parts across models - **Dealer networks:** Multi-echelon inventory - **Just-in-time:** Minimize inventory investment ### Healthcare - **Blood banks:** Perishable inventory pooling - **Hospital networks:** Equipment sharing - **Pharmaceutical distribution:** Service level requirements ## Key Takeaways - **Risk Pooling Power:** Combining uncertain demands can dramatically reduce inventory requirements - **Strategic Design:** Inventory systems should be designed to enable pooling benefits - **Service Trade-offs:** Higher service levels come at increasing inventory costs - **System Thinking:** Inventory decisions affect entire supply chain performance ## Preparation for Next Class - Read [[Assignment_Yedioth_Case]] - **Focus:** Newsvendor model application and risk pooling analysis - **Graded Assignment:** Team case write-up due August 1 - Review quantitative techniques for pooling calculations ## Teaching Notes - Build on inventory fundamentals from previous lecture - Use mathematical examples to demonstrate pooling benefits - Connect theoretical concepts to Zara case insights - Prepare students for quantitative analysis in Yedioth case - Emphasize strategic implications of inventory design decisions