Inventory- a stock or store of goods Dependent demand items- components or sub-assemblies (In a Roland piano, the bench, for example). Forecast is based on # of related finished goods Independent demand items- finished goods that have their own demand curve (subject to randomness we discussed during forecasting section

Types of inventories- piano example Raw materials & parts (e.g. piano keys) Work In Process (keyboard assembly) Finished Goods (keyboard, stand and bench) Replacement items (keyboard cover handle) In-transit inventory

Why keep inventory if it costs so much? There are times in which the cost of keeping inventory is less than the benefits derived: smooth production requirements as seen in Agg. Planning examples decouple operations A distribution company wants to keep distributing even if the ship carrying the next shipment is late!

Why keep inventory if it costs so much? To meet our stockout goals. Software is quick-decision purchase- many companies have 0% stockout strategies as a result (I.e. opportunity cost = 100%; inventory cost may equal 50%) To capitalize on opportunities. If we have excess warehouse and staff capacity, we may save by buying a lot at a great price.

Ordering: quantity & timing Realities in the real world Your order quantity may have to be done for political reasons (new product the president is behind- Edirol example) We may not be able to affect the timing of orders. Distribution companies usually have to place 3 or even 6 month orders for highly technological products to smooth production planning. So fixed interval models are developed.

Counting Inventory Periodic systems count physically at regular intervals and re-order when necessary. Your accounting audit will require this. Perpetual systems (that count inventory as it changes in real time and re-ordering when we hit a reorder point) are almost universally used as the cost of computing has decreased. Most companies combine use of both.

Adding math models to your tool kit What is the lead time of your order (time between submission & receipt) What is your holding cost (includes interest, insurance obsolescence, theft, wear, warehousing, etc.) What is your ordering cost (including the cost of the transaction and receipt What is your Shortage cost (opportunity)

What inventory do we evaluate? Pareto principle tells us that 20% of our items will account for 80% of our orders/ supply requests So, use the ABC system to classify value Item demand Unit Cost Annual $ value Class 1 10 50 500 2 100 1 100 3 75 300 22500 4 5 2500 12500 5 50 35 1750 6 130 50 6500

More on ABC System Can be used to determine number of re-counts in physical counts (e.g. A’s get 3; B’s get 2; C’s get 1) Can also be used to determine who does counts (A’s counted by controller, staff & warehouse; B’s by staff & warehouse; C’s by warehouse only)

Profile of Inventory Level Over Time The Inventory Cycle Profile of Inventory Level Over Time Q Usage rate Quantity on hand Reorder point Time Receive order Place order Receive order Place order Receive order Lead time

So we’ve evaluated the right inventory. Now let’s order. EO Q Model minimizes the sum of holding and ordering costs by finding the optimal order quantity. Assumptions: 1) one product at a time; 2) we’re confident in our annual demand forecast; 3) demand is even; 4) lead time is constant (management issue); 5) orders received in one delivery; 6) no qty discounts

Getting to EOQ: we’re balancing... ANNUAL CARRYING COST = (Q/2)*H (Q= order quantity units; H- carrying cost/unit) ANNUAL ORDERING COST = (D/Q)*S (D= annual unit demand; Q= order size; and S= ordering cost calculus then gives us EO Q, the optimal order quantity

Total cost = annual carrying cost + annual order cost The Total-Cost Curve is U-Shaped Ordering Costs QO Order Quantity (Q) Annual Cost (optimal order quantity) Carrying Costs

Given that demand = 405/month Carrying cost = $30/yr/unit Order Cost = $4/order 1) EOQ= SQR(2*(405*12)*4)/30)= 36 2)What is average # of bags on hand? Q/2= 18 3) # of orders per year= (405*12)/36 =135 4) Carrying cost = (36/2)*30=540; ordering cost= (4860/36)*4=540 total cost = 540 +540 =1080 **We need figures represented as annual costs.

Determining the economic run quantity of production When company is producer and user, determines optimum production run size (since production usually happens faster than usage) When we’re producing our own goods, assumes setup costs are the same as order costs in formula so total cost = carrying cost +setup cost TC = (Max. Inventory/2)*H + (D/Q)*S Economic Run quantity = SQRRT(2DS/H)* SQRRT(p/(p-u)) where p=prod. Rate u=usage rate cycle time =Q/u run time = Q/p

Quantity discounts if carrying costs are constant Goal: minimize total cost, where TC = (Q/2)*H + (D/Q)*S + PD where P= unit price Step 1: compute the common EOQ (if carrying cost is a constant $ figure, it won’t vary) Step 2: compute total cost at EOQ and price breaks and compare

Quantity discounts if carrying costs are constant Assume: D5000,/yr h= $2/unit/yr s=$48 Units Price 1-399 $10 400-599 $9 600+ $8 STEP 1: compute the common EOQ= SQR ((2DS)/H) = SQR((2*5000*48)/2)= 489.90 STEP 2: compute the TC @ EOQ (490) = (Q/2)*H + (D/Q)*S + PD = (490/2)*2 + 5000/490)*48 + (9*5000) = $45980 (with rounding) STEP 3: compare with TC at discount levels TC = (Q/2)*H + (D/Q)*S + PD TC @600 = (600/2)*2 + (5000/600)*48 + (8*5000)= 41000 600 is the optimum order quantity account for discounts

We know how much to order… now, when do we reorder? ROP: predetermined inventory level of an item at which a reorder is placed. Demand (d) and Lead TIME (LT) ROP= d*LT Example: Monthly demand is 400. Lead Time is two weeks (.50 months). ROP= 400 *.50 =200 Reorder when inventory level reaches 200. This model assumes static d and LT

What if demand or lead time is variable? Then we add a safety stock to help us satisfy orders if demand is higher than expected. Company policy: What is our service level? It is the number: 1- stock-out risk. “Our service level goal is 95%. In other words, there’s a 95% probability we won’t stock out.

Handling variability, 2 We assume the variability is characterized by the normal distribution. Turn to page 889. The shaded area under the curve represents the probability of us having inventory, given the variability in the average demand or average lead time. So let’s say we have a service level goal of 95%. What is the Z score that characterizes 95% of the area under the normal curve? About 1.645

When lead time is variable: First example: LEAD TIME variable. When lead time is variable, ROP= d* avgLT + z*d(LT) where d= demand rate; LT= lead time; LT=std. Dev. Of lead time Get the z score (based on your service level goal) from the table as we saw on the last slide based on company’s stockout policy..

ROP= d* avgLT + z*d(LT) Given: demand during lead time =400/day Lead Time = 5 days,  =2 acceptable stockout risk= 5% STEP 1: get your Z score 1-.05 = .95 z (.95) =1.65 STEP 2: plug in 400*5 + 1.65*400*2= 3320 Reorder when inventory = 3320

If demand rate is variable: ROP= avgd* LT + z* sqr.root of LT * (d) assume: avg d =1000; d= 14; LT=4; company stockout policy = 10% risk. Z score for .90 = 1.28 1000*4 + 1.28* 2 * 14= 4000+ 35.84= 4036 in real world, d is derived by managers keeping careful records to determine it.

For next time PROBLEMS (not questions) Ch 12 #s 1,6,13,19, Page 587- know models 1,2,3, and 4a,b,c

Problem 1 Item Usage Unit Cost Value Class 4021 90 1400 126000 A

Problem 6 D=800/MO @ $10/UNIT S=$28 H= 35% OF UNIT COST/YR D- 9600/YR H= $3.50/UNIT/YR CURRENT TC = (q/2)*H + (D/Q)*S CURRENT TC= (800/2)*3.50 + (9600/800)*28 =1736 EOQ= square root of ((2DS)/H) EOQ = SQR ((2*9600*28)/3.50) =SQR 153600 =391.91= 392 TC at 392= (392/2 )*3.5 + (9600/392)*28=1371.71 Cost savings =1736-1372=364

Problem 13: carrying costs are constant D=18000 H= $0.60/yr S=$96 STEP 1: Common EOQ= SQR ((2DS)/H) = SQR((2*18000*96)/.6)= 2400 STEP 2: TC @2400 = (Q/2)*H + (D/Q)*S + PD =(2400/2)*0.60 + (18000/2400)*96 + (1.20*18000)=$23040 TC @5000 = (5000/2)*0.60 + (18000/5000)*96 + (18000*1.15)=$22545.60 TC@10000= (10000/2)*.60 + (18000/10000)*96 + 18000*1.10=22972.80 5000 is the optimum order quantity account for discounts

Problem 19, page 597 see page 573, equation 12-12. The estimate of standard deviation of lead time demand is available, so you can use this simpler equation Expected demand during LT = 300 Std dev of LT demand = 30 a) Step 1 z=2.33 a) step 2 300+(2.33*30)=69.9=370 b) from a)--> 70 units c)less safety stock is required because we’d be carrying an amount of inventory causing us to stock out more often.

Problem 23 625= 85*6 + z*85*1.10 solving for z, z=1.22 Hint: plot the information you do have under the equation, then solve for what you don’t have. When the book says “the delivery time is normal” that means we’ve got a variable lead time problem. When lead time is variable, ROP= d* avgLT + z*d(LT) where d= demand rate; LT= lead time; LT=std. Dev. Of lead time 625= 85*6 + z*85*1.10 solving for z, z=1.22 from table on p. 883, that shows an 89% probability of supply, implying an 11% probability the supply will be exhausted.