Managing Economies of Scale in the Supply Chain: Cycle Inventory Spring, 2014 Supply Chain Management: Strategy, Planning, and Operation Chapter 10 Byung-Hyun Ha

Contents Introduction Economies of scale to exploit fixed costs Economies of scale to exploit quantity discount Short-term discounting: trade promotions Managing multiechelon cycle inventory

Introduction Cycle inventory Notation D: demand per unit time Q: quantity in a lot or batch size (order quantity) Cycle inventory management (basic) Determining order quantity Q that minimizes total inventory cost with demand D given inventory level time

Introduction Analysis of cycle Average inventory level (cycle inventory) = Q/2 Average flow time = Q/2D  Little’s law: (arrival rate) = (avg. number in system)/(avg. flow time) Example D = 2 units/day, Q = 8 units Average inventory level (7 + 5 + 3 + 1)/4 = 4 = Q/2 Average flow time (0.25 + 0.75 + 1.25 + 1.75 + 2.25 + 2.75 + 3.25 + 3.75)/8 = 2 = Q/2D

Introduction Costs that are influenced by order quantity C: (unit) material cost ($/unit) Average price paid per unit purchased  Quantity discount H: holding cost ($/unit/year) Cost of carrying one unit in inventory for a specific period of time Cost of capital, obsolescence, handling, occupancy, etc. H = hC  Related to average flow time S: ordering cost ($/order) Cost incurred per order Assuming fixed cost regardless of order quantity Cost of buyer time, transportation, receiving, etc.  10.2 Estimating cycle inventory-related costs in practice SKIP!

Introduction ? ? Assumptions Cycle optimality regarding total cost Constant (stable) demand, fixed lead time, infinite time horizon Cycle optimality regarding total cost Order arrival at zero inventory level is optimal. Identical order quantities are optimal. ? ?

Introduction ? Determining optimal order quantity Q* Economy of scale vs. diseconomy of scale, or Tradeoff between total fixed cost and total variable cost Q1 D ? Q2 D

Economies of Scale to Exploit Fixed Costs Lot sizing for a single product Economic order quantity (EOQ) Economic production quantity (EPQ) Production lot sizing Lot sizing for multiple products Aggregating multiple products in a single order Lot sizing with multiple products or customers

Economic Order Quantity (EOQ) Assumption Same price regardless of order quantity Input D: demand per unit time, C: unit material cost S: ordering cost, H = hC: holding cost Decision Q: order quantity D/Q: average number of orders per unit time Q/D: order interval Q/2: average inventory level Total inventory cost per unit time (TC) TO: total order cost TH: total holding cost TM: total material cost

Economic Order Quantity (EOQ) Total cost by order quantity Q Optimal order quantity Q* that minimizes total cost Opt. order frequency Avg. flow time TC Q Q*

Economic Order Quantity (EOQ) Robustness around optimal order quantity (KEY POINT) Using order quantity Q' = Q* instead of Q*  0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 1/2( + 1/) 1.250 1.133 1.064 1.025 1.006 1.000 1.017 1.057 1.113 1.178 TC' = 1.25TC* TC*  0.5 1 2

Economic Order Quantity (EOQ) Robustness regarding input parameters Mistake in indentifying ordering cost S' = S instead of real S Misleading to  0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 1/2( + 1/) 1.061 1.033 1.016 1.006 1.001 1.000 1.004 1.014 1.028 1.043 TC' = 1.061TC* TC*  What does mean by robust?  0.5 1 2

Economic Order Quantity (EOQ) Sensitivity regarding demand (KEY POINT) Demand change from D to D1 = kD Opt. order frequency Avg. flow time

Economic Order Quantity (EOQ) Reducing flow time by reducing ordering cost (KEY POINT) Efforts on reducing S to S1 = S Hoping Q1* = kQ* How much should S be reduced? (What is ?)   = k2 (ordering cost must be reduced by a factor of k2)

Economic Production Quantity (EPQ) Production of lot instead of ordering P: production per unit time Total cost by production lot size Q Optimal production quantity Q*  When P goes to infinite, Q* goes to EOQ. Q x D (P – D) Q/P Q/D – Q/P = Q(1/D – 1/P) 1/(D/Q) = Q/D

Aggregating Products in a Single Order Multiple products m products D: demand of each product S: ordering cost regardless of aggregation level All the other parameters across products are the same. All-separate ordering All-aggregate ordering  Impractical supposition for analysis purpose

Lot Sizing with Multiple Products Multiple products with different parameters m products Di, Ci, hi: demand, price, holding cost fraction of product i S: ordering cost each time an order is placed Independent of the variety of products si: additional ordering cost incurred if product i is included in order Ordering each products independently? Ordering all products jointly Decision n: number of orders placed per unit time Qi = Di /n: order quantity of item i Total cost and optimal number of orders

Lot Sizing with Multiple Products Example 10-3 and 10-4 Input Common transportation cost, S = $4,000 Holding cost fraction, h = 0.2 Ordering each products independently ITC* = $155,140 Ordering jointly n* = 9.75 JTC* = $155,140 i LE22B LE19B LE19A Di Ci si 12,000 $500 $1,000 1,200 120 i LE22B LE19B LE19A Qi* ni* TCi* 1,095 11.0 $109,544 346 3.5 $34,642 110 1.1 $10,954 i LE22B LE19B LE19A Qi* 1,230 123 12.3

Lot Sizing with Multiple Products How does joint ordering work? Reducing fixed cost by enjoying robustness around optimal order quantity Is joint ordering is always good? No! Then, possible other approaches? Partially joint NP-hard problem (i.e., difficult) A heuristic algorithm Subsection: “Lots are ordered and delivered jointly for a selected subset of the products” SKIP!

Exploiting Quantity Discount Total cost with quantity discount Types of quantity discount Lot size-based All unit quantity discount Marginal unit quantity discount Volume-based Decision making we consider Optimal response of a retailer Coordination of supply chain TO: total ordering cost TH: total holding cost TM: total material cost

All Unit Quantity Discount Pricing schedule Quantity break points: q0, q1, ..., qr , qr+1 where q0 = 0 and qr+1 =  Unit cost Ci when qi  Q  qi+1, for i=0,...,r where C0  C1    Cr  It is possible that qiCi  (qi + 1)Ci Solution procedure 1. Evaluate the optimal lot size for each Ci. 2. Determine lot size that minimizes the overall cost by the total cost of the following cases for each i. Case 1: qi  Qi*  qi+1 , Case 2: Qi*  qi , Case 3: qi+1  Qi* average cost per unit C0 C1 C2 ... Cr ... q0 q1 q2 q3 ... qr

All Unit Quantity Discount Example 10-7 r = 2, D = 120,000/year S = $100/lot, h = 0.2 Q* = 10,000 i 1 2 qi Ci $3.00 5,000 $2.96 10,000 $2.92

All Unit Quantity Discount Example 10-7 (cont’d) Sensitivity analysis Optimal order quantity Q* with regard to ordering cost (no discount) C = $3 (discount) (original) S = $100/lot 6,324 10,000 (reduced) S' = $4/lot 1,256

Marginal Unit Quantity Discount Pricing schedule Quantity break points: q0, q1, ..., qr , qr+1 where q0 = 0 and qr+1 =  Marginal unit cost Ci when qi  Q  qi+1, for i=0,...,r where C0  C1    Cr Price of qi units Vi = C0(q1 – q0) + C1(q2 – q1) + ... + Ci–1(qi – qi–1) Ordering Q units Suppose qi  Q  qi+1 . marginal cost per unit C0 C1 C2 ... Cr ... q0 q1 q2 q3 ... qr

Marginal Unit Quantity Discount Example 10-8 r = 2, D = 120,000/year S = $100/lot, h = 0.2 Q* = 16,961 i 1 2 qi Ci Vi $3.00 $0 5,000 $2.96 $15,000 10,000 $2.92 $29,800

Marginal Unit Quantity Discount Example 10-8 (cont’d) Sensitivity analysis Optimal order quantity Q* with regard to ordering cost  Higher inventory level (longer average flow time)? (no discount) C = $3 (discount) (original) S = $100/lot 6,324 16,961 (reduced) S' = $4/lot 1,256 15,775

Why Quantity Discount? 1. Improve coordination to increase total supply chain profit Each stage’s independent decision making for its own profit Hard to maximize supply chain profit (i.e., hard to coordinate) How can a manufacturer control a myopic retailer? Quantity discounts for commodity products Quantity discounts for products for which firm has market power 2. Extraction of surplus through price discrimination Revenue management (Ch. 15)  Other factors such as marketing that motivates sellers Munson and Rosenblatt (1998) Manufacturer (supplier) Retailer customers supply chain

Coordination for Total Supply Chain Profit Quantity discounts for commodity products Assumption Fixed price and stable demand  fixed total revenue  Max. profit  min. total cost Example case Two stages with a manufacture (supplier) and a retailer Manufacturer (supplier) Retailer customers SS = 250 hS = 0.2 CS = 2 SR = 100 hR = 0.2 CR = 3 D = 120,000

Coordination for Total Supply Chain Profit Quantity discounts for commodity products (cont’d) No discount Retailer’s (local) optimal order quantity ( supply chain’s decision) Q1 = (212,000100/0.23)1/2 = 6,325 Total cost (without material cost) TC1 = TC1S + TC1R = $6,008 + $3,795 = $9,803 Minimum total cost, TC*, regarding supply chain (coordination) Q* = 9,165 TC* = TC*S + TC*R = $5,106 + $4,059 = $9,165 Dilemma? Manufacturer saving by $902, but retailer cost increase by $264 How to coordinate (decision maker is the retailer)?

Coordination for Total Supply Chain Profit Quantity discounts for commodity products (cont’d) Lot size-based quantity discount offering by manufacturer q1 = 9,165, C0 = $3, C1 = $2.9978 Retailer’s (local) optimal order quantity (considering material cost) Q2 = 9,165 Total cost (without material cost) TC2 = TC2S + TC2R = $5,106 + $4,057 = $9,163 Savings (compared to no discount) Manufacturer: $902 Retailer: $264 (material cost) – $262 (inventory cost) = $2 KEY POINT For commodity products for which price is set by the market, manufacturers with large fixed cost per lot can use lot size-based quantity discounts to maximize total supply chain profit. Lot size-based discount, however, increase cycle inventory in the supply chain.

Coordination for Total Supply Chain Profit Quantity discounts for commodity products (cont’d) Other approach: setup cost reduction by manufacturer Retailer’s (local) optimal order quantity Q3 = Q1 = 6,325 Total cost (without material cost): no need to discount! TC3 = TC3S + TC3R = $3,162 + $3,795 = $6,957  Same with optimal supply chain cost when material cost is considered  Expanding scope of strategic fit Operations and marketing departments should be cooperate! Manufacturer (supplier) Retailer customers S'S = 100 hS = 0.2 CS = 2 SR = 100 hR = 0.2 CR = 3 D = 120,000

Coordination for Total Supply Chain Profit Quantity discounts for products with market power Assumption Manufacturer’s cost, CS = $2 Customer demand depending on price, p, set by retailer D = 360,000 – 60,000p  Profit depends on price. D = 360,000 – 60,000p Manufacturer (supplier) Retailer customers CS = 2 CR = ? p = ?

Coordination for Total Supply Chain Profit Quantity discounts for products with market power (cont’d) No coordination (deciding independently) Manufacturer’s decision on CR Expected retailer’s profit, ProfR ProfR = (p – CR)(360 – 60p) Retailer’s optimal price setting (behavior) when CR is given p1 = 3 + 0.5CR Demand by p1 (supplier’s order quantity) D = 360 – 60p1 = 180 – 30CR Expected manufacturer’s profit, ProfS ProfS = (CR – CS)(180 – 30CR)  CR1 that maximizes ProfR (manufacturer’s decision) CR1 = $4 Retailer’s decision on p1 with given CR1 p1 = $5 (D1 = 360,000 – 60,000p1 = 60,000) Supply chain profit, Prof01 Prof01 = ProfR1 + ProfS1 = $120,000 + $60,000 = $180,000

Coordination for Total Supply Chain Profit Quantity discounts for products with market power (cont’d) Coordinating supply chain Optimal supply chain profit, Prof0* Prof0 = (p – CS)(360 – 60p) p* = $4 D* = 120,000 Prof0* = $240,000  Double marginalization problem (local optimization) But how to coordinate? i.e., ProfS* = ?, ProfR* = ?

Coordination for Total Supply Chain Profit Quantity discounts for products with market power (cont’d) Two pricing schemes that can be used by manufacturer Two-part tariff Up-front fee $180,000 (fixed) + material cost $2/unit (variable) Retailer’s decision ProfR = (p – CR)(360 – 60p) p2 = 3 + 0.5CR = $4 Prof02 = ProfR2 + ProfS2 = $180,000 + $60,000 = $240,000  Retailer’s side: larger volume  more discount Volume-based quantity discount q1 = 120,000, C0 = $4, C1 = $3.5 p3 = $4 Prof03 = ProfR3 + ProfS3 = $180,000 + $60,000 = $240,000

Coordination for Total Supply Chain Profit Quantity discounts for products with market power (cont’d) KEY POINT For products for which the firm has market power, two-part tariffs or volume-based quantity discounts can be used to achieve coordination in the supply chain and maximizing supply chain profits. For those products, lot size-based discounts cannot coordinate the supply chain even in the presence of inventory cost. In such a setting, either a two-part tariff or a volume-based quantity discount, with the supplier passing on some of its fixed cost to the retailer, is needed for the supply chain to be coordinated and maximize profits. Lot size-based vs. volume-based discount Lot size-based: raising inventory level  suitable for supplier’s high setup cost  Hockey stick phenomenon & rolling horizon-based discount

Short-term Discounting: Trade Promotion Trade promotion by manufacturers Induce retailers to use price discount, displays, or advertising to spur sales. Shift inventory from manufactures to retailers and customers. Defend a brand against competition. Retailer’s reaction? Pass through some or all of the promotion to customers to spur sales. Pass through very little of the promotion to customers but purchase in greater quantity during the promotion period to exploit the temporary reduction in price. Forward buy  demand variability increase  inventory & flow time increase  supply chain profit decrease

Short-term Discounting: Trade Promotion Analysis Determining order quantity with discount Qd Unit cost discounted by d (C' = C – d) Assumptions Discount is offered only once. Customer demand remains unchanged. Retailer takes no action to influence customer demand. Qd Q* Qd/D 1 – Qd/D

Short-term Discounting: Trade Promotion Analysis (cont’d) Optimal order quantity without discount Q* = (2DS/hC)1/2 Optimal total cost without discount TC* = CD + (2DShC)1/2 Total cost with Qd Example 10-9 C = $3  Q* = 6,324 d = $0.15  Qd* = 38,236 (forward buy: 31,912  500%) KEY POINT Trade promotions lead to a significant increase in lot size and cycle inventory, which results in reduced supply chain profits unless the trade promotion reduces demand fluctuation.

Short-term Discounting: Trade Promotion Retailer’s action of passing discount to customers Example 10-10 Assumptions Customer demand: D = 300,000 – 60,000p Normal price: CR = $3 Ignoring all inventory-related cost Analysis Retailer’s profit, ProfR ProfR = (p – CR)(300 – 60p) Retailer’s optimal price setting with regard to CR p = 2.5 + 0.5CR No discount (CR1 = $3) p1 = $4, D1 = 60,000 Discount (CR2 = $2.85) p2 = $3.925, D2 = 64,500 (p1 – p2 = 0.075 < 0.15 = CR1 – CR2)

Short-term Discounting: Trade Promotion Retailer’ response to short-term discount Insignificant efforts on trade promotion, but High forward buying Not only by retailers but also by end customers Loss to total revenue because most inventory could be provided with discounted price KEY POINT Trade promotions often lead to increase of cycle inventory in supply chain without a significant increase in customer demand.

Short-term Discounting: Trade Promotion Some implications Motivation for every day low price (EDLP) Suitable to high elasticity goods with high holding cost e.g., paper goods strong brands than weaker brand (Blattberg & Neslin, 1990) Competitive reasons  Sometimes bad consequence for all competitors Discount by not sell-in but sell-though Scanner-based promotion

Managing Multiechelon Cycle Inventory Configuration Multiple stages and many players at each stage General policy -- synchronization Integer multiple order frequency or order interval Cross-docking (Skip!)