Chapter 3. Managing economies of scale in a supply chain: cycle inventory Learning objectives: Balance the appropriate costs to choose the optimal amount of cycle inventory in a supply chain Understand the impact of quantity discount on lot size and cycle inventory Devise appropriate discounting schemes for a supply chain Understand the impact of trade promotions on lot size and cycle inventory Identify managerial levers that reduce lot size and cycle inventory in a supply chain without increasing cost

Role of cycle inventory

Why do companies hold inventory? Why might they avoid doing so? To take advantage of economic purchase order size : economy of scale (cycle inventory) To meet anticipated customer demand To account for differences in production timing (smoothing) To protect against uncertainty (demand surge, price increase, lead time slippage) To maintain independence of operations (buffering) WHY NOT? Requires additional space Opportunity cost of capital Spoilage / obsolescence

The role of cycle inventory in a supply chain A lot or batch size is the quantity that a stage of a SC either produces or purchases at a time. The lot size is usually larger than the quantities demanded by the customer. Cycle inventory is the average inventory in a SC due to this difference. Key point : Cycle inventory exists in a SC bcs different stages exploit the economies of scale to lower total cost. The costs considered include: material cost, fixed ordering cost, and holding cost.

The role of cycle inventory in a supply chain Example: Consider a computer store selling an average of D = 4 printers a day but ordering Q = 80 printers from the manufacturer each time. Cycle inventory = lot size/2 = Q/2 = 40 Average flow time = cycle inventory/demand rate = 40/4 = 10 days (inventory holding time) Inventory turnover (taux de rotation), inventory coverage (taux de couverture) On-hand Inventory Time Q Demand Rate, D Average Cycle Inventory, Q/2 Q/2

Two Decisions in Inventory Management When is it time to reorder? If it is time to reorder, how much? Example to motivate EOQ. Call a student, give a scenario (say, demand for boxes of champagne): D = 1,000/year; K = $400/order; h = $20/unit/year Then, for each value of Q (you give the values to the student), ask him/her to compute number of orders per year, avg inventory, and the associated costs. They will immediately realize that there is a tradeoff. Q D/Q AvgInv Ordering $ Holding $ Total $ 1,000 1 500 400 10,000 10,400 2 250 800 5,000 5,800 5 100 2,000 2,000 4,000 100 10 50 4,000 1,000 5,000

Economies of scale to exploit fixed costs: Economic Order Quantity Model

Economies of scale to exploit fixed costs: Economic Order Quantity Model On-hand Inventory Time Q Demand Rate, D Time Between Orders (Cycle Time) T = Q/D Average Cycle Inventory, Q/2 Reorder Point, R Place Order Receive order Lead Time, L Q/2 Point out that average inventory in a cycle is equal to Q/2, and can be calculated using calculus.

Basic EOQ Assumptions Constant Demand Rate Constant Lead Time Orders received in full after lead-time. Constant Unit Price (no discounts)

Economic Order Quantity Cost Model: Constant Demand, No Shortages TC = total annual inventory cost D = annual demand (units / year) Q = order quantity (units) K = cost of placing an order or setup cost ($) c = cost per unit I = annual interest rate Total Annual Inventory Cost = Annual Ordering Cost Annual Holding Cost + TC = (D / Q) K + (Q / 2) Ic

Cost Relationships for Basic EOQ (Constant Demand, No Shortages) Total Cost Ordering Cost TC – Annual Cost Carrying Cost Q* EOQ balances carrying costs and ordering costs in this model. Order Quantity (how much)

Trade-off in EOQ Model: Inventory Level vs. Number of Orders Many orders, low inventory level On-hand Inventory Q Time Q Few orders, high inventory level On-hand Inventory Time

EOQ Results (How Much to Order) (Constant Demand, No Shortages) 2 D K Ic Economic Order Quantity Number of Orders per year Length of order cycle Total cost = D / Q* T = Q* / D = TC = (D / Q*) K + (Q* / 2) Ic

Determining When to Reorder Quantity to order (how much…) determined by EOQ Reorder point (when…)determined by finding the inventory level that is adequate to protect the company from running out during delivery lead time With constant demand and constant lead time, (EOQ assumptions), the reorder point is exactly the amount that will be sold during the lead time. This slide pretty much speaks by itself. Example:

EOQ Example HOW MUCH TO ORDER? WHEN TO ORDER? D = 1,000 units per year S = $20 per order IC = $8.33 per unit per month BE CAREFUL! HOW MUCH TO ORDER? WHEN TO ORDER? Number of orders per year = Length of order cycle = T = Total cost = Make sure you point out about UNITS. Costs can be in any unit (per month or per year) BUT you need to use consistent units for demand and holding costs.

Exercise Question: What if the company can only order in multiples of 12? (That is, order either 0 or 12 or 24 or 36, etc…)?

Robustness of EOQ model Annual Cost Very Flat Curve - Good!! Total Cost DTC Q*-DQ Q*+DQ Q* Order Quantity You can use one example (next slide) Would have to mis-specify Q* by quite a bit before total annual inventory costs would change significantly.

Example: EOQ Robustness Suppose that in the last problem, you have mis-specified the order costs by 100% and the holding costs by 50%. That is, S used in the computation = $40/order (actual cost = $20 / order) IC used in computation = $150 / unit / year (actual = $ 100) Then, using these wrong costs, you would have gotten Your actual TC (computed substituting Q’ into TC, using correct costs of S = $20, and h = $100: Only 1% above minimum TC!

Key points KP1 : Total ordering and holding costs are relatively stable around the economic order quantity. A firm is often better served by ordering a convenient lot size close to the EOQ rather than the precise EOQ (robustness). KP2 : If the demand increases by a factor of k, the optimal lot size increases by a factor of k0,5 . The number of orders placed per year increases by a factor of k0,5. Flow time due to cycle inventory decreases by a factor of k0,5. KP3 : To reduce the optimal lot size by a factor of k, the fixed cost K must be reduced by a factor of k2. KP4 : Aggregating replenishment across products, retailers, or suppliers in a single order allow for a reduction of lot size of individual products bcs the fixed costs are now spread across differents aggregated entities.

Lot sizing with multiple products or customers

In general, the ordering, transportation, and receiving cost of an order grows with the variety of products or pickup points. A portion of the fixed cost of an order can be related to transportation (this depends only on the load but not on the product variety) A portion of the fixed cost is related to loarding and receiving (this cost increases with variety on the truck) Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

similar to EOQ model except the followings. Assumptions : similar to EOQ model except the followings. Di : annual demand for product i S: order cost incurred each time an order is placed, independent of the variety of products included si: additional order cost incurred if product i is included in the order. Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Each product manager orders his model independently (highest cost) Three approaches : Each product manager orders his model independently (highest cost) The product managers jointly order every product in each lot (easy to administer and implement, but not selective enough and expensive joint ordering if product specific order cost high) Product managers order jointly but not every order contains every product, i.e. each lot contains a selected subset of products. Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

The annual demands are DL = 12000, DM = 1200, DH = 120. Example: Best Buy sells 3 models of computers, the Litepro, the Medpro, the Heavypro. The annual demands are DL = 12000, DM = 1200, DH = 120. Each model costs Best Buy 500$. A fixed transportation cost of 4000$ is incurred each time an order is delivered. For each model ordered and delivered on the same truck, an additional fixed cost of 1000$ is incurred for receiving and storage. Best Buy has an annual holding cost of 20%. Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Approach 1 : Independent ordering QL = 1095, QM = 346, QH = 110. Oder frequencies : 11/year, 3,5/year, 1.1/year. Total inventory cost = 155140 $ Other measures of interest : cycle inventory, annual holding cost/prod, annual ordering cost, flow time. Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Approach 2 : Lots ordered and delivered for all Combined fixed order cost/order : K = S +  si The optimal order frequency is (to explain, express total cost in T): Example : n* = 9.75, annual inventory cost = 136528$, i.e. a reduction of 13% over approach 1. Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Approach 3 : Lots ordered and delivered jointly for a selected subset of products Step 1. Identify most frequently ordered product assuming each being ordered independently. The most frequently order products i* is included each time an order is placed Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Step 2. Identify the frequency with which other products are included. Calculate the order frequency as a multiple of As the most frequently ordered product is in each order, the inclusion of a product i incurs an additional product specific fixed order cost of si. Product i is included once every mi orders Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Why? (order cycle T for n, order cycle miT for i) Step 3. Recalculate the order frequency of the most frequently order product n. Why? (order cycle T for n, order cycle miT for i) Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Step 4. For each product, evaluate the order frequency ni = n/mi and the total cost of such an ordering policy. Example : n = 11.47, mL = 1, mM = 2, mH = 5, annual total inventory cost = 130767$, a reduction of 4% over approach 2. Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

A key to reducing cycle inventory is the reduction of lot size. Key point: A key to reducing cycle inventory is the reduction of lot size. A key to reducing lot size without increasing costs is to reduce the fixed cost associated with each lot. This may be achieved by reducing the fixed cost itself or by aggregating lots across products, customers, suppliers. When aggregating, tailored aggregation is best, especially if product specific costs are large. Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Economies of scale to exploit quantity discounts

Introduction Pricing schedule often displays economies of scale, with prices decreasing as lot size increases. A discount is lot size based if the pricing schedule offers discounts based on the quantity ordered in a single lot. A discount is volume based if the discount is based on the total quantity purchased over a given period. Two commonly used lot size based discount schemes : all unit quantity discounts, marginal unit quantity discount or multiblock tarriffs Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Two basic questions Given a pricing schedule with quantity discount, what is the optimal purchasing decision for a buyer? How does this affect the SC in terms of lot size, cycle inventories, flow times? Under what conditions should a supplier offer quantity discounts? What are appropriate pricing schedules that a supplier should offer? Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

EOQ with all quantity discount Pricing schedule : The unit purchase cost is Ci if the order quantity is at least qi with q0 = 0 < q1 < q2 < … < qr = ∞ and c0 > c1 > c2 > … The retailer’s objective is to maximise its profit, i.e. minimise the sum of material, order, and holding costs. Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Example Drug Online (DO) is an online retailer of prescription drugs. Demand for vitamins is 10000 bottles per month. DO incurs a fixed order cost of 100$ each time an occurs is placed with the manufacturer. DO has an annual holding cost of 20%. The pricing schedule of the manufacturer is the all unit discount schedule: Order quantity Unit Price ($) 0-5000 3 5000-10000 2,96 10000 or more 2,92 Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Solution Step 1. Determine the EOQ Qi for each price Ci Step 2. Determine the total annual cost TCi for each price range Case 1: Qi >= qi+1, ignored as it is considered for Qi+1 Case 2: Qi < qi, Case 3: qi <= Qi < qi+1, Step 3. Determine the optimal order quantity. Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Example (draw TC(Q)) Step 1: Q0 = 6324, Q1 = 6367, Q2 = 6410 Step 2: Q0 >= 5000, TC0 ignore 5000 < Q1 < 10000, TC1 = 358969 $ Q2 < 10000, TC2 = 354520$ Step 3: Optimal order size = q2 = 10000, TC = TC2. Remarks : Presence of quantity discount leads to Larger order size of 10000 units than the normal EOQ = 6324 If S = 4$, order size under all unit discount schedule is still 10000 units and is 8 times the normal EOQ = 1265. Order quantity Unit Price ($) 0-5000 3 5000-10000 2,96 10000 or more 2,92 Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

V0 = 0, Vi+1 = Vi + ci (q i+1 – qi), for i = 0, 1, … EOQ with marginal quantity discount Pricing schedule : The pricing schedule contains specified break points q0 = 0 < q1 < q2 < … < qr = ∞. The marginal cost of a unit decreases at the break points to ci if the order quantity is at least qi with c0 > c1 > c2 > … The purchasing cost Vi of qi units is determined as follows: V0 = 0, Vi+1 = Vi + ci (q i+1 – qi), for i = 0, 1, … Purchasing cost of an order of Q such that qi <= Q < qi+1 is: C(q) = Vi + ci(Q-qi) Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Solution Step 1. Determine the EOQ Qi for each price range Ci (why?) Step 2. Determine the total annual cost TCi for each price range Case 1: Qi < qi, Qi* = qi Case 2: Qi > qi+1, Qi* = qi+1 Case 3: qi <= Qi < qi+1, Qi* = Qi Step 3. Determine the optimal order quantity. Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Example (draw TC(Q)) V0 = 0, V1 = 15000, V2 = 29800 Step 1: Q0 = 6324, Q1 = 11028, Q2 = 16961 Step 2: Q0 >= 5000, TC0 = 363900$ Q1 > 10000, TC1 = 361780 $ 10000 < Q2, TC2 = 360365$ Step 3: Optimal order size = Q2 = 16961, TC = TC3. Remarks : Much larger order size of 16961 units than the normal EOQ = 6324 If S = 4$, order size 15755 is 12,5 times the normal EOQ = 1265. Order quantity Unit Price ($) 0-5000 3 5000-10000 2,96 10000 or more 2,92 Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Value of quantity discount in a supply chain? Key point There can be significant increase of order size and cycle inventory in the absence of fixed order costs as long as quantity discounts are offered. Quantity discounts lead to a significant buildup of cycle inventory in a supply chain. In many SC, quantity discounts contribute more to cycle inventory than fixed ordering cost. Value of quantity discount in a supply chain? Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Why quantity discount?

Coordination to increase total SC profits Quantity discount for commodity products For commodity products, a competitive market exists, the market sets the price, the firm’s objective is the lower costs. Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

A two-stage supply chain example Retailer Drug Online (DO) is an online retailer of prescription drugs. Demand for vitamins is 10000 bottles per month. DO incurs a fixed order cost of 100$ each time an occurs is placed with the manufacturer. DO has an annual holding cost of 20%. Normal wholesale price of the manufacturer cr = 3$ per unit. Manufacturer : processing, packing & shipping DO orders A line packing bottles at a steady rate matching the demand. Fixed setup cost Km = 250$ / order Production cost cm = 2$/bottle Holding cost = 20% Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts. DO makes its lot sizing decision based on costs it faces.

Manufacturer Inventory DO Inventory production for next DO delivery Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Coordination to increase total SC profits Quantity discount for commodity products DO Local optimum face wholesale unit price cr = 3$ DO : D = 120000/year, Kr = 100$, I = 20%, cr = 3$ EOQ = 6324, TC-DO = 3795 $ (DO cost excluding purchasing cost crD) Manufacturer : Setup cost Km = 250$ / order, unit production cost cm = 2$ Annual setup cost = KmD/EOQ = 4744$ Annual holding cost = (Icm)EOQ/2 = 1265$ TC-M = 6009$ (Manuf cost excluding sales revenue crD) Total SC cost = TC-DO + TC-M = 6009 + 3795 = 9804$ Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Coordination to increase total SC profits Quantity discount for commodity products Supply Chain Global Optimum DO cost : TC-DO = KrD/Q + (Icr)Q/2 Manufacturer cost: TC-M = KmD/Q + (Icm)Q/2 SC cost: TC-SC = TC-DO + TC-M = (Km+Kr)D/Q + I(cr+cm)Q/2 SC lot size Q = [2D(Km+Kr)/ (Icr+Icm)]0,5=9165 Opt SC cost = 9165 $ Gain of SC optimum = 9804 – 9165 = 638$ Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Coordination to increase total SC profits Quantity discount for commodity products Pricing scheme for achieving opt SC profit: C = 3$/bottle if Q < 9165, C = 2.9978$ otherwise. DO : has an incentive to order EOQ = 9165, material cost reduction just enough to offset the increase of ordering & holding cost Total SC cost = opt SC cost = 9165 $ In practice, the manufacturer may have to share the increase of SC profit of 638$. Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Coordination to increase total SC profits Quantity discount for commodity products Key point For commodity products for which price is set by the market, manufacturer with large fixed costs per lot can use lot-size quantity discounts to maximise total SC profits. Lot size-based discounts, however, increase cycle inventory in the SC. The benefit of quantity discount decreases as the setup cost of the manufacturer decreases. (Importance of coordination between marketing & production) Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Coordination to increase total SC profits Quantity discount for products for which the firm has market power Consider the scenario in which the manufacturer has invented a new vitamin pill, vitaherb, for which few competitors exist. The price p at which DO sells vitaherb influence demand. Assume that: D = 360000 – 60000p. Production cost Cs = 2$/bottle The manufacturer decides the price Cr to charge DO Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Coordination to increase total SC profits Quantity discount for products for which the firm has market power When decisions are coordinated: SC profil : Prof_SC = (p – Cs)(360000 – 60000p) Results: p = 3 + Cs/2 = 4 D = 120000, Prof_SC = 240000$ Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Coordination to increase total SC profits Quantity discount for products for which the firm has market power When decisions are made independently: DO sets p : MAXp Prof_r = (p – Cr)(360000 – 60000p) p = 3+0.5Cr Manufacturer sets Cr: MAXCr Prof_m Prof_m = (Cr – Cs)(360000 – 60000p) = (Cr – Cs)(180000 – 30000Cr) Cr = 3 + 0.5Cs = 4  p = 5, D = 60000 Results: Prof_m = 120000$, Prof_r = 60000$, SC profil = 180000 Loss of 60000$ due to independent price setting, phenomenon known as double marginalization Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Coordination to increase total SC profits Quantity discount for products for which the firm has market power Key point The supply chain profit is lower if each stage of the supply chain makes its pricing decisions independently, with the objective of maximizing its own profit. A coordinated solution results in higher profit. Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Coordination to increase total SC profits Quantity discount for products for which the firm has market power Pricing schemes to achieve the coordinated solution Two-part tariff : The manufacturer charges its entire profit as an up-front franchise fee and then sells to the retailer at cost. It is then optimal for the retailer to price as though the two stages are coordinated. Example : Opt Prof_SC = 240000 $, Prof_DO = 60000$ (when no coordination) Pricing scheme: charge the DO of the franchise fee of 180000$ and material cost of Cr = 2$/bottle. DO maximises its profit if it sets p = 4$ (Why?). Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Coordination to increase total SC profits Quantity discount for products for which the firm has market power Pricing schemes to achieve the coordinated solution Volume-based quantity discount: The two-tariff is a volume-based quantity discount as the average material cost of DO declines as the purchase increases. Design discount scheme to encourage DO the purchase the opt quantity 120000. Pricing scheme : Cr = 4$ if the purchase < 120000, and Cr = 3.5$ otherwise. DO optimal solution: p = 4, Prof_DO = 60000$, D = 120000, Prof_SC = 240000$ (Why?). Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Coordination to increase total SC profits Quantity discount for products for which the firm has market power Key point For products for which the firm has market power, two-part tariffs or volume-based discounts can be used to achieve SC coordination and maximize SC profits. Lot size-based discounts are not optimal even in the presence of inventory costs. In such as setting, either two-part tariff or a volume-based discount, with the supplier passing on some of its fixed cost to the retailer, is needed for the SC to be coordinated. Lot size based discount tends to raise the cycle inventory. In contrast, volume based discounts are compatible with small lots. Use lot size based discount only when the supplier has high fixed cost. Volume-based discounts suffer from orders peak toward the end of financial horizon. Volume discount based on a rolling horizon could help. Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Short-term discounting: trade promotions

Introduction Manufacturers use trade promotions to offer a discounted price and a time period over which the discount is effective. Ex: 10% off for any purchase from 12/15 to 01/25. The goal is to influence retailers to act in a way that helps the manufacturer achieve its objectives. Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Introduction Key goals (from the manufacturer perspective) Induce retailers to use price discount, displays or advertising to spur sales Shift inventory from the manufacturer to the retailer and the customer Defend a brand against competition Need to understand the impact of trade promotion on the behaviour of a retailer and SC performances. Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Introduction Retailer’s options facing a trade promotion Pass through some or all of the promotion to customers to spur sales (increase the sales of the whole SC) Pass through very little of the promotion to customer but purchase in greater quantity during promotion period to exploit the temporary reduction in price (forward buy and no increase of sales) Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Forward buy d$: discount per product offered Q* : EOQ at normal price Qd : lot size ordered at discounted price Assumptions: Discount is offered once Retailer takes no action to influence demand Qd is an integer multiple of Q*. Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Qd Q* Time

Forward buy Forward buy = Qd – Q* Why? Profit maximization (gain in fix cost, gain in purchase, loss of inventory cost). Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Forward buy Let T = Q/D be the period covered by short-term promotion Cost during T without promotion Cost during T with promotion Cost Gain during T Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Forward buy The optimal Q is obtained from Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Forward buy Example: DO is a retailer selling vitaherb. Demand is 120000 bottles/year. The manufacturer currently charges 3$/bottle and DO has an annual holding cost of 20%. Fixed order cost K = 1000 $. What is the current lot size Q* of DO, cycle time, average flow time? The manufacturer has offered a discount of 0.15$ for all bottles purchased by the retailer over the coming month. How many bottles should DO order given the promotion? Answer: Q* = 6324, Qd = 38236, Forward buy = 31912 Remark: 5% discount causes the lot size to increase by 500+%. Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Forward buy Key point : Trade promotions lead to a significant increase in lot size and cycle inventory because of forward buying by retailer. This generally results in reduced SC profits unless the trade promotion reduces demand fluctuations. Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Impact on the demand Example: Assume DO selling at price p faces a demand of D = 300000 – 60000p. The normal price charged by the manufacturer is Cr = 3$/bottle. Ignoring the inventory related costs, evaluate the optimal response of DO to a discount of 0.15$ per bottle. Answer: Without discount and Cr = 3$, p = 4$, D = 60000 With discount of 0.15$ and Cr = 2.85$, p = 3.925$, D = 64500. 7.5% increase in demand, DO pass only half of the trade promotion discount to Customers. Why: Maxp Profit-DO = (p – Cr)*(300000 – 60000p)  p = 2.5+0.5Cr Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Impact on the demand Key point Faced with a short term discount, it is optimal for retailers to pass through only a fraction of the discount to the customer, keeping the rest for themselves. Simultaneously, it is optimal for retailers to increase the purchase lot size and forward buy for future periods. Thus, trade promotions often lead to an increase of cycle inventory in a SC without a significant increase in customer demand. Trade promotion should be designed so that retailers limit their forward buying and pass along more of the discount to end customers. Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Managing multiechelon cycle inventory Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

A mutliechelon distribution supply chain stage 4 stage 1 Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts. stage 3 stage 2 A multiechelon supply chain has multiple stages and possibly many players et each stage. Goal: decrease the total costs by coordinating orders across the SC

One manufacturer supplying one retailer (Instantaneuous production, lotsize Q) No synchronization : production right after delivery, average INV = 3Q/2 mfg inventory shipping production retailer inventory Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts. Synchronization : production after before delivery, average INV = Q/2

Simple multiechelon with one player at each stage Integer replenishment policy: lot size at each stage = integer multiple of the lot size of its immediate customer Coordination of ordering across stages allows for a portion of the delivery to a stage to be cross-docked on to the next stage Extent of cross-docking depends on the ratio of fixed ordering cost S and holding cost H at each stage. The closer the ratio, the higher the optimal percentage of cross-docked product. Shown to be quite close to optimal. Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

distributor replenshment order arrives Distributer replenishes every two weeks retailer shipment is cross-docked Retailer replenishes every week retailer shipment is from inventory Retailer replenishes every two weeks retailer shipment is cross-docked Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts. retailer shipment is cross-docked Retailer replenishes every four weeks

One distributor supplies multiple retailers Integer replenishment policy: Distinguish retailers with high demand from those with low demand Group retailers such that all retailers in one group order together fr = n*fd or fd = n*fr, for each retailer r where n is an integer and fr and fd are retailer and distributor order frequencies Each player orders periodically with reorder interval equal to an integer multiple of some base period Shown to be near optimal by Roundy. Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Integer replenishment policies Divide all parties within a stage into groups such that all parties of a group order from the same supplier and have the same reorder interval Set reorder intervals across stages such that the receipt of a replenishment order at any stage is synchronized with the shipment of a replenishment order to at least one of its customers. The synchronized portion can be cross-docked. For customers with a longer reorder interval than the supplier, make the customer reorder interval an integer multiple of the suppliers' interval and synchronize their replenishment to facilitate cross-docking For customers with a shorter reorder interval, make the supplier's reorder interval an integer multiple of the customer's interval and synchronize the replenishment The relative frequency of reordering depends on the setup cost, holding cost and demand at different parties. Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Key points Integer replenishment policies can be synchronized in multiechelon supply chains to keep cycle inventory and order costs low. Under such policies, the reorder interval at any stage is an integer multiple of a base reorder interval. Synchronized integer replenishment policies facilitate a high level of cross-docking. Whereas the integer policies synchronize replenishment and decrease cycle inventories, they increase safety inventories because of the lack of flexibility with the timing of a reorder These policies make the most sense for supply chains in which cycle inventories are large and demand is relatively predictable. Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Integer replenishment policies Divide all parties within a stage into groups such that all parties of a group order from the same supplier and have the same reorder interval Set reorder intervals across stages such that the receipt of a replenishment order at any stage is synchronized with the shipment of a replenishment order to at least one of its customers. The synchronized portion can be cross-docked. For customers with a longer reorder interval than the supplier, make the customer reorder interval an integer multiple of the suppliers' interval and synchronize their replenishment to facilitate cross-docking For customers with a shorter reorder interval, make the supplier's reorder interval an integer multiple of the customer's interval and synchronize the replenishment The relative frequency of reordering depends on the setup cost, holding cost and demand at different parties. Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts.

Echelon inventory Ordering policies based on echelon inventory (s, S), (r, Q) Problems: where to locate the inventory, how to allocate the inventory warehouse echelon inventory supplier warehouse Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts. warehouse echelon lead time

Echelon inventory Local installation based-stock policy: Order-up-to local base stock Sj if the local inventory position < Sj Local inventory position of j = inventory on hand at stage j + on order of stage j + in transit to stage j – backorder (on order) of stage j-1 On order Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts. On hand L3 L4 Demand D L2 L1 1 4 3 2 in transit

Echelon inventory Echelon 3 Echelon based-stock policy: Order-up-to echelon base stock ESj if the echelon inventory position < ESj Echelon inventory position of j = on order of stage j + inventories at stages (j, j-1, …, 1) + in transit to stages (j, j-1, …, 1) – backorder of stage 1 Point out that we have seen the basic method. Now, we are going to study variations of the EOQ, where we relax the assumptions made so far. We will start with quantity discounts. L4 Echelon 3 Demand D L3 L2 L1 3 2 1 4