TPPE37 Manufacturing Control Agenda Lecture 2 Inventory Management 1. Inventory System Defined 2. Inventory Costs 3. Inventory classification 4. Economic order quantity model 5. Newsboy problem 6. Reorder point system 1 2 The Planning Hierarchy A typical production process Master planning Demand management Final assembly schedule Sales and operations planning Production planning Master prod. schedule Resource planning Master (production) scheduling Rough-cut capacity plan Front end Feedback Material planning Detailed capacity planning Engine Primary material flow Vendor Shop floor Back end Raw material supplier Manufacturer End-product manufacturer Distributor Retailer 3 4
Inventory Why we hold inventory? Replenishment (Production or purchasing) The acumulated result when Demand replenishmnet Demand (external or internal demand) Economies of scale Uncertainty in supply and demand Speculation Transportation Smoothing production/purchasing Logistics Cost of controlling inventory Inventory 5 6 Different types of inventory Inventory Theoretical inventory to to maintain a process flow Cycle inventory for for economies of of scale Seasonal inventory for for uneven demand Speculation inventory for for price variation Where do we hold inventory? suppliers and manufacturers warehouses and distribution centers retailers Types of inventory WIP and subassemblies raw materials finished goods Safety inventory to to avoid stockout 7 8
Why inventory management is important? Independent vs. Dependent Demand Total investment in inventory is 20-25% of the total annual GNP. In the last quarter of 1999, total investment on inventories is $1.37 trillion in US 34% in Manufacturing 26% in Retail 22% in Wholesale 8% in Farm 10% in Other Independent Demand (Demand for the final end-product or demand not related to other items) Finished product E(1) Dependent Demand (Derived demand items for component parts, subassemblies, raw materials, etc) 9 Component parts 10 Classification of inventory Inventory Costs Deterministic Dependent demand Stochastic Inventory Deterministic Independent demand Stochastic Holding costs Physical holding cost (out-of-pocket) Financial holding cost (opportunity cost) Setup (or production change) costs Ordering costs Shortage costs Multi echelon MRP, etc Multi echelon System nervousness rescheduling, etc EOQ, lot sizing, etc Newsboy, control (S, s), (S, Q), etc 11 12
Economic order quantity Decision problem Constant rate of demand. (D) Immediate replenishment (no lead time, L=0) No shortage allowed Constant ordering cost (K) Constant inventory holding cost (H) Q inventory T Slope=-D time Objective function (Ordering costs + Inventory holding costs) Decision variable Order quantity Q Parameter Orderingcost(K) and holding cost (H) Constraint Positive order quantity 13 14 Cost Minimization EOQ model The Total-Cost Curve is U-Shaped Annual Cost Q D TC = H + K 2 Q EOQ = Q* = 2KD H Ordering Costs Q* (optimal order quantity) Order Quantity (Q) 15 16
EOQ Example EOQ Example A Swedish tea shop imports green tea from China Demand = 2000 units/year Cost to place an order = 1000 SEK Value of tea = 250 SEK/unit Holding cost per unit per year = 25 SEK/year 2KD 2(2000 )(1000) Q* = = = 400 units H 25 Total ordering cost = 1000 2000/400 =5000 Total holding cost = 25 400/2 =5000 17 18 Stochastic inventory The newsboy problem Single period problems Newsboy model Multi-period problems Continuous review (s, Q) Periodic review (R, S) Major assumptions Single selling season One order placed prior to or at the beginning of that period Uncertainty of demand, but the distribution of demand is known Costs are known Applications Newsvendor: newspaper, magazines Style goods: fashion clothes, some seasonable/holiday goods, sports fan items Perishable good: Christmas trees, food and medical items 19 20
Description Notation A newspaper vendor wishes to determine the number of copies of newspaper he should purchase each day. A study of historical data showed that the demand during any day is a random variable that approximately normally distributed, with mean 11.73 and standard deviation 4.74. Each copy is purchased for 2.5 SEK and sold for 7.5 SEK, and he is paid 1 SEK for each unsold copy by his supplier. D: demand, a continuous nonnegative random variable Q: order quantity, decision variable f(x): probability density function (pdf) of random variable D F(x): cumulative distribution function (cdf), df(x)/dx=f(x) h: overage cost for the remaining product, SKE/unit b: underage cost for unsatisfied demand, SEK/unit. 21 22 Overage and underage costs Optimal policy Cost of overage h: purchase cost - salvage value The optimal solution, Q*, occurs when F ( Q*) = b /( b + h) Cost of underage b: lost profit: selling price purchase cost or, lost profit + any additional penalty f(x) area = F ( Q * ) µ Q* 23 24
Area under the normal distribution Numerical example A newspaper vendor (Cont.) Demand is normally distributed with mean µ = 11.73 and standard deviation σ = 4.74. inventory cost is h = 2.5-1 = 1.5 SEK. The stockout cost is b = 7.5 2.5 = 5 SEK The critical ratio is b/(b+h) =.50 /.65 =.77. Using the table of normal distribution, we obtain a standardised value of Z = 0.74. The optimal number of purchasing copy is Q* = Zσ + u= 0.74 4.74 + 11.73 = 15.24 15 25 26 Numerical example -results More examples Results Hence, the number of copies should be able to satisfy all of the daily demand with probability 0.77. This is also called type 1 service level. The optimal Q* is the 77th percentile of the demand distribution. f(x) Serv1 The Christmas tree vendor: how many trees should be purchased? The cafeteria manager: how many hot meal should be prepared? The blood bank: how many donations of blood should be sought? The farmer: what quantity of a particular crop should be planted?... area = F ( Q * ) µ Q* 27 28
Inventory Control Systems Continuous review A replenishment control system is to address the following issues How often the inventory status should be determined? When a replenishment order should be placed? How large the replenishment order should be? Assumptions Inventory status is always known It provides stable service level and uses less safety stock It is expensive for fast-moving items 29 30 Periodic review Order-point, order-quantity (s, Q) system Assumptions Inventory status is not updated until the next review time There is considerable uncertainty between review interview Review interval is often determined by the physical constraints It is easy to coordinate orders of different items It is easy to estimate workloads s+q s 0 This is a continuous review model L Inventory level 31 32
(s, Q) system - ordering policy Periodic-review, order-up-to-level (R, S) system Policy: A fixed quantity Q is ordered when inventory position drops to or below the reorder point s It is also called a two-bin system S This is a periodic review review model Advantage It is very simple The production of supplier is predictable Disadvantage It does not cope effectively with sudden-large orders 0 L R L R L Net inventory/inventory position 33 34 (R, S) system - ordering policy Buffer mechanism - safety stock Policy: At every review interval R, enough is ordered to raise inventory position to level S. The reorder point has a close relation with safety stock s= xˆ + SS= xˆ + Z σ L L L It is also called replenishment cycle system Advantage It takes advantage when items are sharing resource, or have a coordination opportunity It is easy to adjust the level S Disadvantage It often has a higher inventory holding cost compared with the continuous system xˆ -expected demand over a lead time L SS - Safety stock Z - Safety factor σ L - Standard deviation of the forecasting error over lead time Methods to determine Z: Simple-minded method Minimising costs Service level 35 36
Safety stock - service level approach Example the tea shop again Cycle service level rule: Select safety factor Z so that the area under normal distribution= Serv1 Demand D = 2000 units/year = 38.5 units/week Lead time L = 1 week Standard deviatino of demand s L = 12 Serv1 = 0.95 From the table, Z=1.64 SS = Zs L = 1.64µ12=19.7 20 s= 38.5 µ 1 + 20 = 40.5 41 37 38 The tea shop how to manage it? Multi-item management Using an (s, Q) model Auditing your inventory level When it is reduced to 41 units, place a new order = 400 units ABC classification (Pareto analysis) 39 40