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## Inventory Management

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**CHAPTER**11 Inventory Management**What is inventory?**• An inventory is an idle stock of material used to facilitate production or to satisfy customer needs**Why do we need inventory?**• Economies of scales in ordering or production - cycle stock • trade-off setup cost vs carrying cost • Smooth production when requirements have predictable variability - seasonal stock • trade-off production adjustment cost vs carrying cost**Why do we need inventory? (cont’d)**• Provide immediate service when requirements are uncertain (unpredicatable variability) -safety stock • trade-off shortage cost versus carrying cost • Decoupling of stages in production/distribution systems -decoupling stock • trade-off interdependence between stages vs carrying cost**Why do we need inventory? (cont’d)**• Production is not instantaneous, there is always material either being processed or in transit - pipeline stock • trade-off cost of speed vs carrying cost**Why carrying inventory**• As you can see from the above list, there are economic reasons, technological reasons as well as management/operations reasons.**Why carrying inventory is costly?**• Traditionally, carrying costs are attributed to: • Opportunity cost (financial cost) • Physical cost (storage/handling/insurance/theft/obsolescence/…)**Inventory Management**Independent Demand Dependent Demand A C(2) B(4) D(2) E(1) D(3) F(2) Independent demand is uncertain. Dependent demand is certain.**Types of Inventories**• Raw materials & purchased parts • Partially completed goods called work in progress • Finished-goods inventories • (manufacturingfirms) or merchandise (retail stores)**Types of Inventories (Cont’d)**• Replacement parts, tools, & supplies • Goods-in-transit to warehouses or customers**Inventory Counting Systems**• Periodic System Physical count of items made at periodic intervals • Perpetual Inventory System System that keeps track of removals from inventory continuously, thus monitoringcurrent levels of each item**0**214800 232087768 Inventory Counting Systems (Cont’d) • Two-Bin System - Two containers of inventory; reorder when the first is empty • Universal Bar Code - Bar code printed on a label that hasinformation about the item to which it is attached**High**A Annual $ volume of items B C Low Few Many ABC Classification System Classifying inventory according to some measure of importance and allocating control efforts accordingly. A -very important B- mod. important C- least important Number of Items**What are we tackling in inventory management?**• All models are trying to answer the following questions for given informational, economic and technological characteristics of the operating environment: • How much to order (produce)? • When to order (produce)? • How often to review inventory? • Where to place/position inventory?**The inventory models**• The quantitative inventory management models range from simple to very complicated ones. However, there are two simple models that capture the essential tradeoffs in inventory theory • The newsboy (newsvendor) model • The EOQ (Economic Ordering Quantity) model**The Newsboy Model**• The newsboy model is a single-period stocking problem with uncertain demand: Choose stock level and then observe actual demand. • Tradeoff: overstock cost versus opportunity cost (lost of profit because of under stocking)**The Newsboy Model**• Shortage cost (Cs) - the opportunity cost for lost of sales as well as the cost of losing customer goodwill. • Cs = revenue per unit - cost per unit • Excess cost (Ce) - the over-stocking cost (the cost per item not being able to sell) • Ce = Cost per unit - salvage value per unit**The Newsboy Model**• In this model, we have to consider two factors (decision variables): the supply (X), also know as the stock level, and the demand (Y). The supply is a controllable variable in this case and the demand is not in our control. We need to determine the quantity to order so that long-run expected cost (excess and shortage) is minimized.**The Newsboy Model**• Let X = n and P(n) = P(Y>n). The question is: Based on what should we stock this n-th unit? • The opportunity cost (expected loss of profit)for this n-th unit is P(n) Cs • The expected cost for not being able to sell this n-th unit (expected loss) • (1-P(n))Ce**The Newsboy model**• Stock the n-th unit if P(n)Cs > (1-P(n))Ce • Do not stock if P(n)Cs < (1-P(n))Ce**The Newsboy model**• The equilibrium point occurs at • P(n)Cs = (1-P(n)) Ce • Solving the equation • P(n) = Ce/(Cs + Ce)**Newsboy model**• Service level is the probability that demand will not exceed the stocking level and is the key to determine the optimal stocking level. • In our notation, it is the probability that YX(=n), which is given by • P(Yn) = 1 - P(n) = Cs/(Cs + Ce)**Example (p.564)**Demand for long-stemmed red roses at a small flower shop can be approximated using a Poisson distribution that has a mean of four dozen per day. Profit on the roses is $3 per dozen. Leftover are marked down and sold the next day at a loss of $2 per dozen. Assume that all marked down flowers are sold. What is the optimal stocking level?**Cs = $3, Ce = $2**Opportunity cost is P(n)Cs Expected loss for over-stocking (1 - P(n)) Ce P(Y n) = Cs/(Cs + Ce) = 3/((3+2) = 0.6 From the table, it is between 3 and 4, round up give you optimal stock of 4 dozen. Example (solutions) Cumulative frequencies for Poisson distribution, mean = 4**The Inventory Cycle**Profile of Inventory Level Over Time Q Usage rate Quantity on hand Reorder point Time Place order Place order Receive order Receive order Receive order Lead time**Estimating the inventory costs**• Q = quantity to order in each cycle • S = fixed cost or set up cost for each order • D = demand rate or demand per unit time • H = holding cost per unit inventory per unit time • c = unit cost of the commodity • TC = total holding cost per unit time**Estimating the inventory cost**• The cost per order = S + cQ • The length of order cycle = Q/D unit time • The average inventory level = Q/2 • Thus the inventory carrying cost • = (Q/2) H (Q/D) = HQ2/(2D)**Estimating the inventory cost**• The total cost per order cycle • = S + cQ + HQ2/(2D) • Thus the total cost per unit time is dividing the above expression by Q/D, I.e., • TC = {S + cQ + HQ2/(2D)} {D/Q} • = DS/Q + cD + HQ/2**Annual**carrying cost Annual ordering cost Total cost = + Q D S H TC = + 2 Q Total Cost**Deriving the EOQ**Using calculus, we take the derivative of the total cost function and set the derivative (slope) equal to zero and solve for Q.**Annual Cost**Ordering Costs Order Quantity (Q) QO (optimal order quantity) Cost Minimization Goal The Total-Cost Curve is U-Shaped**Minimum Total Cost**The total cost curve reaches its minimum where the carrying and ordering costs are equal.**TC with PD**TC without PD PD 0 Quantity EOQ Total Costs Cost Adding Purchasing cost doesn’t change EOQ**Total Cost with Constant Carrying Costs**TCa TCb Total Cost Decreasing Price TCc CC a,b,c OC EOQ Quantity**Example (Example 2, p.541)**A local distributor for a national tire company expects to sell approximately 9600 steel-belted radial tires of a certain size and tread design next year. Annual carrying costs are $16 per tire, and ordering costs are $75. This distributor operate 288 days a year. a. What is the EOQ? b. How many times per year does the store reorder? c. What is the length or an order cycle? d. What is the total ordering and inventory cost?**Solution (Example 2, p.541)**D = 9600 tires per year, H = $16 per unit per year, S = $75 a. Q0 = {2DS/H} = {2(9600)75/16} = 300 b. Number of orders per year = D/ Q0 = 9600 / 300 = 32 c. The length of one order cycle = 1 / 32 years = 288/32 days = 9 days d. Total ordering and inventory cost = QH/2 + DS/Q = 2400 + 2400 = 4800**EOQ with quantity discount**• This is a variant of the EOQ model. Quantity discount is another form of economies of scale: pay less for each unit if you order more. The essential trade-off is between economies of scale and carrying cost.**EOQ with quantity discount**• To tackle the problem, there will be a separate (TC) curve for each discount quantity price. The objective is to identify an order quantity that will represent the lowest total cost for the entire set of curves in which the solution is feasible. There are two general cases: • The holding cost is constant • The holding cost is a percentage of the purchasing price.**Quantity discount (constant holding cost)**• Total cost per cycle • = setup cost + purchase cost + carrying cost • Setup cost = S • Purchase cost = ciQ, if the order quantity Q is in the range where unit cost is ci. • Carrying cost = (Q/2)h(Q/D) • TC = {S+ciQ+hQ2/(2D)}{D/Q} (annual cost)**Quantity discount (constant holding cost)**Differentiating, we obtain an optimal order quantity which is independent of the price of the good The questions is: Is this Q0 a feasible solution to our problem?**Quantity discount (constant holding cost)**• Solution steps: • 1. Compute the EOQ. • 2. If the feasible EOQ is on the lowest price curve, then it is the optimal order quantity. • 3. If the feasible EOQ is on other curve, find the total cost for this EOQ and the total costs for the break points of all the lower cost curves. Compare these total costs. The point (EOQ point or break point) that yields the lowest total cost is the optimal order quantity.**Example**• The maintenance department of a large hospital uses about 816 cases of liquid cleanser annually. Ordering costs are $12, carrying cost are $4 per case a year, and the new price schedule indicates that orders of less than 50 cases will cost $20 per case, 50 to 79 cases will cost $18 per case, 80 to 99 cases will cost $17 per case, and larger orders will cost $16 per case. Determine the optimal order quantity and the total cost.**Example (solutions)**• 1. The common EOQ: = {2(816)(12)/4} = 70 • 2. 70 falls in the range of 50 to 79, at $18 per case. • TC = DS/Q+ciD+hQ/2 • = 816(12)/70 + 18(816) + 4(70)/2 = 14,968 • 3. Total cost at 80 cases per order • TC = 816(12)/80 + 17(816) + 4(80)/2 = 14,154 • Total cost at 100 cases per order • TC = 816(12)/100 + 16(816) + 4(100)/2 =13,354 • The minimum occurs at the break point 100. Thus order 100 cases each time**Quantity discount (h = rc)**• Using the same argument as in the constant holding cost case, Let TCi be the total cost when the unit quantity is ci. • TCi = rciQ/2 + DS/Q + ciD • Therefore EOQ = {2DS/(rci)} • In this case, the carrying cost will decrease with the unit price. Thus the EOQ will shift to the right as the unit price decreases.**Quantity discount (h = rc)**• Steps to identify optimal order quantity • 1. Beginning with the lowest price, find the EOQs for each price range until a feasible EOQ is found. • 2. If the EOQ for the lowest price is feasible, then it is the Optimal order quantity. • 3. Same as step (3) in the constant carrying case.**Example (p.549)**Surge Electric uses 4000 toggle switches a year. Switches are priced as follows: 1 to 499 at $0.9 each; 500 to 999 at $.85 each; and 1000 or more will be at $0.82 each. It costs approximately $18 to prepare an order and receive it. Carrying cost is 18% of purchased price per unit on an annual basis. Determine the optimal order quantity and the total annual cost.**Solution**D = 4000 per year; S = 18 ; h = 0.18 {price} Step 1. Find the EOQ for each price, starting with the lowest price EOQ(0.82) = {2(4000)(18)/[(0.18)(0.82)]} = 988 (Not feasible for the price range). EOQ(0.85)= 970 (feasible for the range 500 to 999) Step 2. Feasible solution is not on the lowest cost curve Step 3. TC(970) = 970(.18)(.85)/2 + 4000(18)/970 + .85(4000) = 3548 TC(1000) = 1000(.18)(.82)/2 + 4000(18)/1000 + .82(4000) = 3426 Thus the minimum total cost is 3426 and the minimum cost order size is 1000 units per order.**We have settled the problem of how much to order. The next**decision problem is when to order.**Reorder point**• In the EOQ model, we assume that the delivery of goods is instantaneous. However, in real life situation, there is a time lag between the time an order is placed and the receiving of the ordered goods. We call this period of time the lead time. Demand is still occurring during the lead time and thus inventory is required to meet customer demand.This is why we need to consider when to order!