What Operations Managers Do? 10 OM Strategy Decisions: • • • • • • • • • •
Design of Goods & Services Managing Quality Process Strategy Location Strategies Layout Strategies Human Resources Supply Chain Management Inventory Management Scheduling Maintenance
10 Decision Areas: • service & product design • quality management • process & capacity design • location • layout design • human resources & job design • supply chain management • inventory, MRP, and J-I-T • intermediate, short-term, and project scheduling • maintenance
Scope of Operations Management SYSTEM DESIGN – DESIGN – involves
decisions relating to the system capacity, geographic locations of facilities, arrangement of departments, layout of equipment, product or service planning, and acquisition of equipment. SYSTEM OPERATION – OPERATION – involves scheduling, inventory planning & control, management of personnel, project management, and quality assurance
Introduction to Inventory Management
Good inventory management is important for the successful operation of most businesses and their supply chains Operations, marketing and finance have interests in good inventory management, the importance of which is due to the ff: Inventory is necessary for operations Inventory contributes to customer satisfaction Inventory is a vital part of business; Inventory investment is a company’s company’s largest asset (30% of current assets and 90% of working capital) Managerial Performance → ROI = Profit After Tax Total Assets
INVENTORY
Definition: STOCK or STORE of GOODS
Types of Inventory 1. RM (raw materials and purchased parts) 2. WIP (work-in-process, partially completed
goods) 3. FG (finished-goods inventory) 4. MRO (maintenance/repair/operating supplies) 5.
Pipeline Inventory (goods-in-transit to warehouses or customers)
FUNCTIONS of INVENTORY
To meet anticipated demand (walk-ins & provide selection ) To smooth production requirements ( seasonally seasonally high demand ) To decouple production process (buffers vs. m/c breakdowns or supplies fluctuations) To protect against stockouts (delivery delay and variability increase risk of shortages) To take advantage of order cycles (economic lot sizes for production or periodic periodic ordering for purchases) To hedge against inflation and price increases (quantity discounts) To permit operations ( production production operations are not instantaneous, need intermediate stocking of RM, WIP, FG)
OBJECTIVES of INVENTORY CONTROL
Inadequate control of inventories result in: Under-stocking (missed deliveries, lost sales, dissatisfied customers, and production bottlenecks Overstocking (funds tied-up, high inventory holding costs) 2 concerns of inventory in ventory management 1) level of customer service 2) costs of ordering and carrying inventories Overall objective of inventory inventor y management to achieve a BALANCE in stocking, i.e., achieve achi eve satisfactory levels of customer service while keeping inventory costs within reasonable bounds to make 2 fundamental decisions: TIMING and SIZE of
INVENTORY MANAGEMENT REQUIREMENTS for effective I.M.
A system to keep track of the inventory on inventory on hand and on order A reliable forecast of demand that demand that includes an indication of possible forecast error Knowledge of lead of lead times and lead time variability Reasonable estimates of inventory holding costs, ordering costs, and shortage costs A classification system for inventory items
PERFORMANCE MEASURES
Customer Satisfaction (number and quantity of backorders and/or customer complaints) Inventory Turnover (ratio of goods sold to average
Inventory Counting Systems
PERIODIC SYSTEM : a physical count of items in inventory is made at periodic intervals (e.g. weekly, monthly) in order to decide how much to order of each item
Advantages:
Accurate on-hand quantity known (basis for ordering) Order for many items results in economies in processing pr ocessing and shipping orders at the same time
Disadvantages:
Lack of control between reviews Need to protect against shortages between review periods by carrying extra stock
Inventory Counting Systems
PERPETUAL SYSTEM : (continual system) system that keeps track of removals from inventory continuously, thus monitoring current levels of each item
Advantages:
Control provided by continuous monitoring of withdrawals Fixed-order quantity is ordered when amount on hand reaches a pre-determined minimum level
Disadvantages:
Added cost of record keeping Physical count must still be performed (to verify records because of errors, pilferages)
Inventory Counting Systems Periodic System
1.
Supermarkets, discount stores and department stores have always been major users of periodic counting systems Today, most have switched to bar coding
computerized checkout system laser scanning device universal product code (UPC) or bar code
Perpetual System
2.
Ranges from very simple to very sophisticated A two-bin system is an elementary example:
Uses two (2) containers for inventory; items are withdrawn from the first bin until its contents are exhausted It is then time to reorder; sometimes an order card is placed at the bottom of the first bin The second bin contains enough stock to satisfy demand until the order is
Inventory Counting Systems Perpetual System
1.
two-bin system (cont’d)
Advantage : no need to record each withdrawal from inventory Disadvantage : reorder card may not be turned in for a variety of reasons (e.g. misplaced, person responsible forgets to turn it in)
can be either batch or on-line Batch systems Inventory records are collected periodically and entered into the system On-Line systems Transactions are recorded immediat immediately ely Advantage : they are always up-to-date unlike in batch systems, a sudden surge in demand could result r esult in reducing the amount of inventory below the reorder point between periodic read-ins (frequent batch collections can minimize that problem).
Inventory Classification System Inventory items are not of equal importance
in terms of dollars involved, profit potential, sales or usage volume, or stockout penalties Example : Producer of Electrical Equipment Inventory carried includes generators, coils of wire, and miscellaneous nuts and bolts among other items in stock Unrealistic to devote equal attention to each of these items Instead, a more reasonable approach would be to allocate control efforts according to the relative importance of various items in inventory A-B-C approach
classifies inventory items according to some measure of importance, usually dollar usage (i.e. dollar value per unit un it multiplied by annual usage rate), and allocating control efforts accordingly
Inventory Classification System ABC Analysis divides on-hand inventory into three (3) classifications on the basis of annual dollar value
It is an inventory application of the Pareto principle which states that there are a “critical few and trivial man.”
Idea is to establish inventory policies that focus resources on the few critical inventory parts and not the many trivial ones.
It is not realistic to monitor inexpensive items with the same intensity as very expensive items
ABC Classification
Annual dollar volume = annual demand x cost per unit
Typically three (3) classes of items are a re used: A (very important), B (moderately important), and C (least important)
The actual number of categories may vary from organization to organization, organization, depending on the extent to which a firm wants to differentiate control c ontrol efforts.
Inventory Classification System
ABC Classification
Class A items generally account for about only 15% to 20% of the number of items in inventory but represent 70% to 80% of the total dollar usage Class C items may represent only 5% to 10% of the annual dollar volume but about 55% to 60% of the total inventory items Class B items are those inventory items i tems with medium annual dollar volume, about 15% to 20% of the total value representing about 30% of the the number of inventory items Percentages vary from firm to firm, but in most instances a relatively small number of items will account for a large
Inventory Classification System
ABC Classification
Class A items should receive a close attention through frequent reviews of amounts on hand and control over withdrawals Class C items should receive only loose control (two-bin system, bulk orders) Class B items should have controls that lie between A & C Note that Class C items are not necessarily unimportant ; incurring a stockout of C items such as the nuts and bolts used to assemble manufactured goods can result in a costly shutdown of an assembly line (due to low annual dollar volume, larger orders or ordering in advance will help without incurring much additional cost)
Inventory Classification System Example No.1 Item 1 2 3 4 5 6 7 8 9 10 11 12
Annual Demand 1,000 5,000 1,900 1,000 2,500 2,500 400 500 200 1,000 3,000 9,000
x
Unit Cost $4,300 720 500 710 250 192 200 100 210 35 10 3
Annual Dollar = Value $4,300,000 3,600,000 950,000 710,000 625,000 480,000 80,000 50,000 42,000 35,000 30,000 27,000
Class A
Class B
Class C
Inventory Classification System Example No.2 Item % of No. Stock Of Items Number Stocked # 10286 20% # 11526 # 12760 # 10867 # 10500 # 12572 # 14075 # 01036 # 01307 # 10572
Annual Volume x Unit (Units) Cost $90.00 1,000 154.00 500
Annual Percent of Dollar Annual $ = Volume Value 38.8% $90,000 33.2% 77,000
30%
1,550 350 1,000
17.00 42.86 12.50
26,350 15,001 12,500
11.3% 6.4% 5.4%
50%
600 2,000 100 1,200 250
14.17 0.60 8.50 0.42 0.60
8,502 1,200 850 504 150
3.7% .5% .4% .2% .1%
Class 72%
23%
A
B
5% C
Record Accuracy
Good inventory policies are meaningless if management does not know what inventory is on hand. Accuracy of records is a critical ingredient in i n production and inventory systems Record accuracy allows organizations to focus on those items that are needed, rather than settling for being sure that “some of everything” is in inventory Only when an organization can determine accurately what it has on hand can it make precise decisions about a) ordering
b) scheduling
c) shipping
To ensure accuracy,
Incoming & outgoing record keeping must be good Stockroom security must also be good
Record Accuracy
A well-organized stockroom will have
Limited access Good housekeeping Storage areas that hold fixed amounts of inventory Bins, shelf space, and parts that are labeled accurately
Cycle Counting Records must still be verified through a continuing audit (cycle count) Is a continuing reconciliation of inventory with inventory records Is a physical count of items in inventory, the purpose pur pose of which is to reduce discrepancies between records and actual quantity
Record Accuracy
Cycle Counting
Procedures: 1) Items are counted 2) Records are verified 3) Inaccuracies are periodically documented 4) Cause of inaccuracies is traced 5) Appropriate remedial action is taken to ensure integrity of the inventory system Cycle counting uses ABC classification as a guide: • • •
A items will be counted frequently perhaps once a month B items will be counted less frequently, perhaps once a quarter C items will be counted perhaps once every ev ery 6 months
Example : Cycle Counting Counting Cole’s Trucks, Inc., a builder of high-quality refuse trucks, has about 5,000 items in its inventory. After hiring Matt Jones, a bright young industrial engineering student, for the summer, the firm determined that it has 500 A items, 1,750 B items, and 2,750 C items. Company policy is to count A items every month (every 20 working days), all B items every quarter (every 60 working days), and all C items every 6 months (every 120 working days). How many items should be counted each day? SOLUTION:
Item Class Quantity 500 A 1,750 B 2,750 C
Number of Item Cycle Counting Policy Counted per Day Each month (20 working days) 500 / 20 = 25/day Each quarter (60 working days) 1,750 / 60 = 29/day Every 6 months (120 working days) 2,750 / 120 = 23/day Seventy-seven items are counted each day
Demand Forecasts and Lead-Time Information
Inventories are used to satisfy demand requirements, so it is essential to have reliable estimates of the amount and timing of demand . It is also essential to know how long it will take orders to be delivered or the leadtime (the time between submitting an order and receiving it) Managers need to know the extent of variability of demand and leadtime
The greater the potential variability, the greater the need for additional stock to reduce the risk of a shortage between deliveries. Thus, there is a crucial link between forecasting and inventory management
Point-of-sale (POS) systems electronically electronically record actual sales. Knowledge of actual sales can greatly enhance forecasting and inventory inven tory management. management. Real time information on actual demand enables management to make necessary changes to restocking decisions.
Cost Information
Three (3) basic information are associated with inventories, namely: 1. Holding Cost 2. Ordering Cost
Set-Up Cost
Shortage Cost
Holding Cost
Holding or carrying costs are the costs associated with holding or “carrying ” inventory over time, usually a year
Relates to physically having items in storage, and therefore include interest, insurance, taxes, depreciation, obsolescence, deterioration, spoilage, pilferage, breakage, and warehousing costs (heat, light, rent, security); also include opportunity costs associated with funds tied up
Is stated in 2 ways:
Percentage of unit price Dollar amount per unit
) )
usually ranging from 20% to 40% of the item’s VALUE
Ordering Cost
Is the cost of ordering and receiving inventory
Varies with the actual placement of an order
Is the cost of the ordering process that includes determining how much is needed, preparing invoices, shipping costs, inspecting goods upon arrival for quality and quantity, and moving the goods to temporary storage
Is generally expressed as a fixed dollar amount per order , regardless of order size (or quantity ordered)
Set-Up Cost
When orders are being b eing manufactured by the firm itself instead of ordering it from a supplier, the t he cost of machine set-up is the cost to prepare a machine or process (time in preparing equipment for the job by adjusting the machine, changing cutting tools, etc.) for manufacturing an orders Is analogous to ordering cost, i.e., it is expressed as a fixed charge per production run , regardless of the size of the run
Shortage Cost
Results when demand exceeds the supply of inventory on hand.
Includes the opportunity cost of o f not making a sale, loss of customer goodwill, late charges, and expediting costs
Is considered as cost of lost production or downtime, if the shortage occurs in an item carried for internal use (e.g. to supply su pply an assembly line)
Is sometimes difficult to measure and may be subjectively estimated
INVENTORY MODELS for INDEPENDENT DEMAND
Two most important questions in Inventory Management: a) how much to order b) when to order
Three independent demand models are: 1. Basic economic order quantity (EOQ) model 2. Production order quantity model (EPQ) 3. Quantity discount model (QD)
Basic EOQ Model
Used to identify the order size that will minimize the sum of the annual costs of holding inventory and ordering inventory Assumptions: 1. Only one product is involved (spr ead 2. Annual demand requirements are known, constant (spread evenly throughout the year), and independent 3. Leadtime is known and does not vary 4. Receipt of inventory is instantaneous and complete, i.e., inventory from an order arrives in one batch at a t one time in a single delivery 5. Quantity discounts are not possible 6. Stockouts (shortages) can be completely avoided if orders
Basic EOQ Model
Q = 350 units
Inventory ordering and Usage occurs in CYCLES
Order size, Q = 350 units Usage rate = 50 units per day Leadtime = 2 days Reorder Point = 100 units (2 days’ supply)
Usage Rate = 50 units / day
Qty on hand
ROP = 100 0 Receive order
5 Place order
7 Receive order
12 Place order
14 Receive order
Day
Basic EOQ Model
The optimal order quantity reflects a trade-off between carrying costs and ordering costs. As order size varies, one type of cost will increase while the other decreases. If order size is relatively small, the average inventory will be low, resulting in low carrying costs. However, a small order size will necessitate frequent orders, which will drive up annual ordering costs. Conversely, ordering large quantities at infrequent intervals can hold down annual ordering costs, but that would result in higher average inventory levels and therefore increased carrying costs.
Basic EOQ Model Q
Many orders produce a low average inventory Average Q = Inventory
2
0
1 year
Time
Q
Q Average = Inventory 2
Basic EOQ Model
Ideal Solution = order size s ize that causes neither a few very large orders nor many small orders but one that lies somewhere in between
The exact amount to order will depend on the relative magnitudes of the carrying and ordering costs
The objective is to minimize total costs
Significant costs are ordering (or set-up) cost and holding (or carrying) cost.
All other costs, such as the cost of the inventory itself , are constant
Minimize ordering cost & holding cost = Minimizing TC
Basic EOQ Model Annual carrying cost =
Q 2
H
Where, Q = order quantity in units H = holding (carrying) cost
t s o C l a u n n A
Q H 2
Order Qty
Annual ordering cost =
D Q
S
Where,
t s o C l a u n n A
D S Q
D = Annual Demand, in units S = ordering cost Order Qty
Basic EOQ Model The total annual cost associated with carrying and ordering or dering inventory when Q units are ordered each time is Annual Annual TC = carrying + ordering = cost cost t s o C l aTCmin u n n A
Q H + DS 2 Q
TC QH 2 D S Q Order Qty
The optimal order quantity Q* is when annual ordering cost equals annual holding cost, thus Q H = D S 2 Q
Basic EOQ Model Solving for Q*, Q H = D S 2 Q 2DS = Q2H Q2 = 2DS (EOQ) Q* =
H 2DS
Expected numberHof orders = N =
Demand Order Quantity
=
D Q*
Length of order cycle = Q* (the time between orders) or D No. of Working Days per Year Expected time between orders = T = N
Basic EOQ Model –Example # 1 Sharp, Inc. a company that markets painless hypodermic needles to hospitals, would like to reduce its inventory cost by determining the optimal number of hypodermic needles to obtain per order. The annual annu al demand is 1,000 units; the ordering cost is $10 per order; and the holding cost per unit per year year is $0.50. Using these figures, c) Calc Calcula ulate te he he optim optimal al num numbe berr of uni units ts per per ord order er d) Det Determ ermine ine the expe expecte cted d number number of order orderss place placed d during during the the year e) Det Determ ermine ine the expe expecte cted d time time betw between een orde orders rs (ass (assum umee 250 250 working days per year
Basic EOQ Model –Example # 1 SOLUTION: a) Q* = b)
N =
c)
T = = T =
d) TC =
√
2DS
=
D H = Q*
√
1,000 200
2 (1,000) (10) 0.50
= 200 units
= 5 orders per year
No. of Working Days per Year N 250 working days per year 5 orders per year 50 days between orders
Q H + DS 2 Q
= =
200 (0.50) 2 50 + 50
+
1,000 (10)
200order & carrying Note that
Basic EOQ Model –Example # 2 A local distributor for a national n ational tire company expects to sell approximately 9,600 steel-belted radial tires of a certain size and tread design next year. Annual carrying cost is $16 per tire, and ordering cost is $75. The distributor operates 288 days a year. a) b) c) d)
What is the EOQ? Wha How ma many ny times times per per year year does does the the store store reorde reorder? r? Whatt is th Wha thee leng length th of of an an orde orderr cycl cycle? e? Whatt is the Wha the total total annua annuall cost cost if the the EOQ EOQ is orde ordered red??
Basic EOQ Model –Example # 2 SOLUTION: a) Q* = b)
N =
c)
T = = T =
d) TC = =
√
2DS
=
D H = Q*
√
2 (9,600) 75 16
= 300 tires
9,600 tires/year = 32 orders per year 300 tires/order
No. of Working Days per Year N 288 working days per year 32 orders per year 9 working days between orders
Q H + DS 2 Q 2,400 + 2,400
=
300 (16 ) 2
+
9,600 (75 )
300 costs Note that order & carrying
Basic EOQ Model –Example # 3 Piddling Manufacturing assembles security monitors. It purchases 3,600 black-and-white cathode ray tubes a year at $65 each. Ordering costs are $31, and annual carrying costs are 20 percent of the purchase price. Compute (a) the optimal order quantity and (b) the total annual cost of ordering and carrying the inventory. Solution:
a) Q* =
b) TC =
D = 3,600 cathode ray tubes per year S = $31 H = (given as % of purchase price) = 0.20($65) = $13 2DS
=
√
H Q H + DS 2 Q
2 (3,600) 31 13
=
131 (13 ) 2
= 131 cathode ray tubes
+
3,600 (31 ) 131
Economic Production Quantity (EPQ) Model
The batch mode of production is widely used in production. (Even in assembly operations, portions of the work are done in batches)
Reason for Batch Production: the capacity to produce a part exceeds the part’s usage or demand rate. (As long as production continues, inventory will grow, so it makes sense to periodically produce such items in batches, or lots)
The assumptions of the EPQ model are similar to those of EOQ model, except that instead of orders received in a single delivery, units are received incrementally during production.
Economic Production Quantity (EPQ) Model
An economic order quantity technique applied to production orders Assumptions: 1. 2. 3. 4. 5. 6. 7.
Only one item is involved Annual demand is known The usage rate is constant Usage occurs continually, but production occurs periodically The production rate is constant Lead time does not vary There are no quantity discounts
Economic Production Quantity (EPQ) Model
During the production phase of the cycle, inventory builds up at a rate equal to the difference between production and usage rates.
Example: if daily production rate is 20 units and the daily usage rate is 5 units, inventory will build up at the rate of 20 – 5 = 15 units per day.
As long as production occurs, the inventory level will continue to build; when production ceases, the inventory level will begin to decrease The inventory level will be at a maximum at the point where production ceases When the amount of inventory on hand is exhausted, production is resumed, and the cycle repeats itself
Economic Production Quantity (EPQ) Model
Run size Q* l e v e l max y r o t n e v n I
Production and Usage
Usage only Cumulative Production
I
Amount on hand Part of inventory cycle during w/c production is
Demand part of cycle with no
Time
Economic Production Quantity (EPQ) Model
Since the company makes the product (or components) itself, there are no ordering costs as such. Instead, there are set-up costs with every production run (batch). Set-up costs are the costs required to prepare the equipment for the job, such as cleaning, adjusting, and changing tools & fixtures. Set-up costs are analogous to ordering costs because they are independent of the lot (run) size.
They are treated in the formula in exactly the same way The larger the run size, the fewer the number of runs needed, and hence the lower the annual set-up cost
Economic Production Quantity (EPQ) Model Number of production runs or batches = D Q Annual set-up cost = number of runs per year x set-up cost per run = D S Q I max H + DS Total cost = TC = Carrying Cost + Setup cost = 2 Q* where, Imax = maximum inventory = Q* ( p – u ) or Q* ( 1 – u / p ) p Economic run quantity, Q* =
2DS
p p - u
H ( 1 – u / p )
H Where, p = daily production or delivery rate u = usage rate per day
Cycle time (time between beginnings of runs) = Run time time (production phase of the cycle) cycle) =
or
2DS
Q*
Q* u
EPQ Model - Example # 1 A toy manufacturer uses 48,000 rubber wheels per year for its popular dump truck series. The firm makes its own wheels, w heels, which it can produce at a rate of 800 per day. The toy tucks are assembled uniformly over the entire year. year. Holding (carrying) cost is $1 per wheel a year. Setup cost for a production run of wheels is $45. The firm operates 240 days per year. Determine a) Optimal run size b) Minimum total annual cost for carrying and setup c) Cycle time for the optimal size d) Run time SOLUTION:
D S H p
= 48,000 wheels per year = $45 = $1 per wheel per year = 800 wheels per day 48,000 wheels per 240 days, or 200 wheels per day
EPQ Model - Example # 1 a) Q* =
√ √ ( ) 2DS H
b) TCmin =
I max 2
√
√
800 = 2,400 2(48,000)4 = 800 - 200 wheels 5 1 1,800 x $1 + 48,000 x $45 H + DS = 2 2,400 Q* = $900 + $900 = $1,800 p p - u
(800 0 – 200 200 ) = 1,800 wheels Compute first for I max = Q* ( p – u ) = 2,400 (80 p 800 Thus, a run of wheels will be 2,400 wheels Q* c) Cycle time = = = 12 days made every 12 200 wheels per day u days
d) Run time =
Q* p
=
2,400 wheels 800 wheels per day
= 3 days
Each run will require 3days
EPQ Model - Example # 2 Nathan manufacturing, Inc., makes and sells specialty hubcaps for the retail r etail automobile market. Nathan’s forecast for its wire-wheel hubcap is 1,000 units next year, with an average daily demand of 4 units. However, the production process is most efficient at 8 units per day. So the company produces 8 per day but uses only 4 per day. Given the following values, solve for the optimum number of units per order. ( Note o f this hubcap Note: This plant schedules production of only as needed, during the 250 days per year the shop operates.)
EPQ Model - Example # 2 SOLUT UTIION:
Annual demand = D = 1,0 1,000 units Setup cost = S = $10 Holding cost = H = $0.50 per unit per year Daily production rate = p = 8 units daily Daily demand rate = u = 4 units daily
Economic run quantity Q* = = =
2DS
p p - u
2DS
or
H ( 1 – u / p )
H 2 (1,000) 10 0.50 ( 1 – 4 / 8) 20,000
=
80,000
0.50 ( 1/2 ) = 282.3 hubcaps or 283 hubcaps
Quantity Discount Model Quantity Discounts
– are price reductions for large orders offered to customers to induce them to buy in large quantities.
If quantity discounts are offered, the buyer must weigh the potential benefits of reduced purchase price and fewer l arge quantities against orders that will result in buying in large the increase in carrying costs caused by higher average inventories. The buyer’s goal with quantity discounts is to select the order quantity that will minimize total cost, where total cost is the sum the sum of carrying cost, ordering cost, and a nd purchasing
Quantity Discount Model TC = Carrying cost + Ordering Cost + Purchasing Cost =
Q H + DS + PD 2 Q t s o C l a u n n A
where P = Unit price TC with PD
TC without PD Q H 2
PD D S Q
EOQ*
Order Qty
Quantity Discount Model Order Quantity 1 to 44 45 to 69 70 or more
Price per Box $2.00 1.70 1.40
Note that no one curve applies to the entire range of quantities; each curve applies to only a portion of the feasible total cost is range. Hence, the applicable or feasible initially on the curve with the highest unit price and drops down, curve by curve, at the price breaks, which are the minimum quantities needed to obtain the discounts.
TC @ $2.00 each
Cost Though each curve has a minimum, those points are not necessarily feasible.
TC @ $1.70 each TC @ 1.40 each PD @
The actual total$2.00 each cost curve is denoted by the solid lines; only those price-quantity
PD @ $1.70 each
PD @ 1.40 each
Order Qty
Quantity Discount Model
The objective of the QD model is to identify an order quantity that will represent the lowest total cost for the entire set of curves
Two (2) general cases of the QD model:
t s o C
Carrying costs are constant (fixed amount per unit)
Carrying costs are stated as a percentage of purchase pu rchase price TCa TCb TCc
CC a,b,c OC Quantity
t s o C
TCa TCb TCc CCa CC b CCc OC
Quantity
Quantity Discount Model Procedures when CC = constant 1.
Compute the common minimum EOQ* point
2.
Only one of the unit prices will have the th e minimum point in its feasible range since the ranges do not no t overlap. Identify that range.
If the feasible minimum point is on the lowest price range, that is the optimal order quantity.
If the feasible minimum point is in any other range, compute the total cost for the minimum point and for the price p rice breaks of all lower unit costs. Compare the total costs; the quantity (minimum point or price break) that yields the lowest total cost is the optimal order quantity.
QD Model - Example No. 1 The maintenance department of a large hospital uses about 816 cases of liquid cleanser annually. Ordering costs are $12, carrying costs 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. SOLUTION: D = 816 cases per year S = $1 2 H = $4 per case per year
Range 1 to 49 50 to 79 80 to 99
Price $ 20 18 17
QD Model - Example No. 1 1.
Compute the common EOQ* =
2.
The 70 cases can be bought at $18 per case because 70 falls in the range of 50 to 79 cases. The total cost to purchase 816 cases a year, at the rate of 0 cases per order will be TC70
= Carrying cost + Order cost
+ Purchase cost
=
+
= 6.
2DS/ H = 2(816)12/ 4 = 70 cases
(Q/2) H +
(D/Q) S
PD
(70 / 2) 4 + (816 / 70 ) 12 + 18 ( 816 ) = $14,968
Because lower cost ranges ($17 and $16) exist, each must be checked against the minimum cost generated by 70 cases at $18 each: TC80
= (80/2)4
+ (816/80)12
+ 17(816) = $14,154
TC100
= (100/2)4 + (816/100)12 + 16(816) = $13,354
Therefore, because 100 cases per order yields the th e lowest total cost,
Quantity Discount Model Procedures when CC = percentage of price For each discount, calculate a value for the optimal order size Q*, using the formula Q* = 2DS/ (% x P) 2. For any discount, if the order quantity is too low to qualify for the discount, adjust the order quantity upward to the lowest quantity (or price break) that will qualify for the discount. 3. Compute the total cost (holding + ordering + purchase costs) for every every Q* determined in steps 1 and 2. Be sure to use the adjusted value for Q* and the respective discounted price and holding cost. 4. Select the Q* that has the lowest total cost, as computed in step 3. It will be the quantity that will minimize the total inventory cost 1.
QD Model - Example No. 2 Wohl’s Discount Store stocks toy race cars. Recently, the store has been given a quantity discount schedule for these cars. This quantity schedule is shown in the table below. Ordering cost is $49.00 per order, annual demand is 5,000 race cars, and inventory inventor y carrying charge is 20% of the price. What order quantity will minimize the total inventory cost? Discount Quantity 0 to 900 1,000 to 1,999 2,000 and over
Discount (%) no discount 4 5
Discount Price (P) $5.00 $4.80 $4.75
QD Model - Example No. 2 The first step is to compute Q* for every discount: Q*5.00 = 2(5,000)(49)/ (.2)(5.00) = 700 cars order (FEASIBLE) Q*4.80 = 2(5,000)(49)/ (.2)(4.80) = 714 cars order (Not in range) Q*4.75 = 2(5,000)(49)/ (.2)(4.75) = 718 cars order (Not in range) 5) The second step is to adjust upward those values of Q* that are below the allowable discount range: Q*5.00 = 700 - as is (because it is in the range) Q*4.80 = 1,000 - adjusted Q*4.75 = 2,000 - adjusted each Q* [Note: TC = (D/Q)S + (Q/2)H + PD]: 9) Compute total costs for each Price Quantity OC CC PD Total $5. 00 700 $350 $ 3 50 $ 2 5 , 0 0 0 $25, 700 $4. 80 1,000 $ 2 45 $ 4 80 $ 2 4 , 0 0 0 $24, 725 $4. 75 2,000 $ 1 22 . 5 0 $9 5 0 $23, 750 $ 2 4 , 8 22 . 5 0 1)
QD Model - Example No. 3 Surge Electric uses 4,000 toggle switches sw itches a year. Switches are priced as follows: 1 to 499, 90 cents each; 500 to 999, 85 cents each; and 1,000 or more, 80 cents each. It costs approximately appro ximately $30 to prepare an order and receive it, and carrying costs are 40 % of purchase price per unit on an annual basis. Determine the optimal order quantity and the total annual cost. SOLUTION: D = 4,000 switches per year S = $30 H = 0. 40P
Minimum Point
0.80
=
Range 1 to 499 500 to 999 1,000 or more
Price $0. 90 $0. 85 $0. 80
H 0.40(0.90) = 0.36 0.40(0.85) = 0.34 0.40(0.80) = 0.32
2(4,000)30/0.32 = 866 switches ( NOT
FEASIBLE)
Minimum Point 0.85 = 2(4,000)30/0.34 = 840 switches (FEASIBLE) TC840 = (840 / 2)0.34 + (4,000/ 840)30 + 0.85(4,000) = $3,686
When to REORDER with EOQ Ordering
EOQ models answer the question of how much to order, The question of when to order is the function of models that identify the reorder point (ROP) in terms of a quantity. ROP occurs when the quantity on hand drops to a predetermined amount which:
generally includes expected demand during leadtime plus an extra cushion of stock serves to reduce the probability of experiencing a stockout during leadtime requires perpetual inventory in order to know if it has been reached
The goal in ordering is to place an order when the amount of inventory on hand is sufficient to satisfy demand during the time it takes to receive that order (i.e., leadtime).
REORDER POINT
Four determinants of the reorder point quantity 1.
The rate of demand (usually based on forecast)
2.
The leadtime
3.
The extent of demand and/or leadtime variability
4.
The degree of stockout risk acceptable to mgmt.
If demand and leadtime are both constant, then ROP = d x LT , where d = demand rate (units per day or week)
LT = leadtime in days or weeks
REORDER POINT - Example Tingly takes Two-a-Day vitamins, which are delivered to his home by a routeman seven days after an order is called in. At what poit should Tingly reorder? Usage = 2 vitamins a day Leadtime = 7 days ROP = usage x leadtime = 2 vitamins vitamins per day x 7 days days = 14 vitamins Thus, Tingly should reorder when 14 vitamin tablets
PROBABILISTIC MODELS with CONSTANT LEADTIME
PROBABILISTIC MODEL is a statistic model applicable PROBABILISTIC when product demand or any other variable is not known, but can be specified by means of a probability distribution An important concern in inventory management is maintaining an adequate SERVICE LEVEL in the face of uncertain demand SERVICE LEVEL = Complement of the probability of a stockout, i.e., i.e., (SL = 1 – probability of a stockout). For instance, if the probability of a stockout is 0.05, then the service level is 0.95 Uncertain demand raises the probability of a stockout; one method of reducing stockouts is to hold extra units in inventory, referred to as SAFETY STOCK.
Safety Stock
Reduces the risk of running out of inventory (a stockout ) during leadtime Acts as a buffer when added to the reorder r eorder point, thus ROP = ( d x L ) + ss The amount of safety stock maintained depends on the cost of incurring a stockout and the cost of holding the extra inventory. Annual stock out cost is computed as follows: = the sum of the units short x the probability x the stockout cost/unit x the number of orders per per year
ROP with SS - Example David Rivera Optical has determined that its reorder point for eyeglass frames is 50 ( d x LT) units. Its carrying cost per frame per year is $5, and stockout (or lost sale) cost is $40 per frame. The store has experienced the following probability distribution for inventory demand during the reorder period. The optimum number of orders per year is six. Number of Units Probability 30 .2 40 .2 ROP → 50 .3 60 .2 70 .1 1.0 How much safety stock should David Rivera keep on hand?
ROP with SS - Example SOLUTION: The objective ob jective is to find the amount of safety stock that minimizes the sum of the additional inventory holding costs and stockout cost. The annual holding cost is simply the holding cost per unit multiplied by the units added to the ROP. For example, a safety stock of 20 frames, which implies that the new ROP, with safety stock, is 70 (= 50 + 20), raises the addnual carrying cost by $5(20) = $100. However, computing annual stockout cost is more interesting. For any level of safety safety stock, stockout cost cost is the expected expected cost of stocking out. It is computed by multiplying the number of frames short by the probability of demand at that level, by the stockout cost, by the number of times per year the stockout can occur (which in this case is the number of orders per year). Then add stockout costs for each possible stockout level for a given ROP. For zero safety stock, for example, a shortage of 10 frames will occur if demand is 60, and a shortage of 20 frames f rames will occur if the demand is 70.
ROP with SS - Example SOLUTION: (continued) Thus the stockout costs for zero safety stock are (10 frames short)(.2)($40 per stockout)(6 possible stockouts per year) + (20 frames short)(.1)($40)(6) = $960 The following table summarizes the total costs for each alternative: Safety Additional Total Stock Holding Cost Stockout Cost Cost 20 (20)($5) = $100 = 0 $100 10 (10)($5) = $ 50 (10)(.1)($40)(6) = $ 240 $290 0 0 (10)(.2)($40)(6) + (20)(.1)($40)(6) = $ 9 60 $960 The safety stock with the lowest lowest total cost is 20 frames. Therefore, this
Service Level
The cost of inventory policy increases dramatically with an increase in service levels.
Inventory costs, indeed, increases exponentially as service level increases
Because it costs money to hold safety stock, the cost of carrying safety stock must be weighed against the reduction in stockout risk it provides.
The customer service service level increases as the risk of stockout decreases
Order cycle service level can be defined as the probability that demand will not exceed supply during the leadtime
Service Level
Service level when defined as meeting 95% of the demand implies a probability of 95 percent that demand will not exceed supply during leadtime Conversely, 95% service level implies having stockout risk of only 5% of the time SL = 100 percent - Stockout risk The amount of safety stock depends on the following factors: 1. 2. 3.
The average demand rate and average leadtime Demand and leadtime variability The desired service level
Service Level
For a given order cycle service level , the greater the variability in either demand rate or leadtime, the greater the amount of safety stock that will be needed to achieve that service level
Similarly, for a given amount of variation in demand rate or leadtime, achieving an increase in the service level will require increasing the amount of safety stock.
Selection of a service level may reflect stockout costs (e.g. lost sales, customer dissatisfaction) or it might simply be a policy variable (e.g. the manger wants to achieve a specified service level for a certain item)
ROP based on Normal Distribution of LT demand Expected demand ROP = during leadtime + z Service Level
d LT LT
where,
z = number of standard deviations
(Probability of no strockout)
d LT LT
= the standard deviation of leadtime demand
Expected Demand
ROP
Quantity
Safety stock 0
z
z-axis
Use Appendix B, Table B to obtain z-values,
The value of z depends on the stockout risk that the manager is willing to accept. The smaller the risk the manager is willing to accept, the greater the
ROP based on Normal Distribution of LT demand - Example No. 1 Example : Suppose that the manager of a construction supply house determined from historical records that demand for sand during leadtime averages 50 tons. In addition, suppose the manager determined that the demand during leadtime could be described by a normal distribution that has a mean of 50 tons and a standard deviation deviation of 5 tons. tons. Answer the the following questions, assuming that the manager is willing to accept a stockout risk of no more than 3 percent: a) What value of z is appropriate? b) How much safety stock should be held? c) What reorder point should be used?
ROP based on Normal Distribution of LT demand - Example No. 1 SOLU SO LUTIO TION: N: Ex Expe pect cted ed le leadt adtim imee de dema mand nd = 50 to tons ns
σd LT LT = 5 tons Stockout risk = 3 percent
From Appendix B, Table B, using a service level of 1 - .03 .03 = .9700, a value of z = +1.88 is obtained.
Safety stock =
ROP = Expected leadtime demand + safety stock
zσd LT LT = 1.88(5) = 9.40 tons
= 50 + 9.40 = 59.40 tons
ROP based on Normal Distribution of LT demand
If only demand is variable, then σd LT LT = √ LT σd and ROP = d x LT + z √ LT σd
d = Average daily or weekly demand where, If only leadtime is variable, then σd LT LT = d σ LT and
ROP = d x LT + z d σd where,
9)
d = daily or weekly demand LT = Average leadtime in days or weeks in days or weeks σ LT = Standard deviation of lead time in
If both demand and leadtime are variable, 2 2 2 then σd LT LT = √ LT σd + d σ LT and ROP = d x LT + z√ LT σd 2 + d 2 σ2 LT
NOTE: Each of these models assumes that demand and leadtime
ROP based on Normal Distribution of LT demand - Example No. 2 A restaurant uses an average of 50 jars of a special sauce each week. Weekly usage of sauce sauce has a standard deviation of 3 jars. The manager is willing to accept no more than a 10 percent risk of stockout during leadtime, which is two weeks. Assume the distribution of usage is normal. a) Whic Which h of the abov abovee formu formula lass is app appropr ropriat iatee for for this this situ situati ation? on? Why? Why? b)) De b Detter ermi mine ne th thee val value ue of z. c) Determine the ROP. d = 50 jars per week SOLUTION: LT = 2 weeks SL = 1 - .10 = .90 σd = 3 jars per week • Because demand is variable (i.e., has a standard deviation), d eviation), formula no. 1 is appropriate. • From Appendix B, Table B, using a service level of .9000, z = +1.28 ROP = d x LT + z √ LT σd = 50 x 2 + 1.28 √ 2 (3) = 100 + 5.43 = 105.43 •
Shortages and Service Levels
The ROP computation does not reeal the expected amount of shortage for a given leadtime service level. The expected number of units short can be determined d etermined using the same information used in ROP computations, with one additional piece of information, E(z), (from the Table) E(n) = E(z) d LT LT , where E(n) = Expected number of units short per order cycle E(z) = Standardized number of units short (from Table) Table) σd LT LT = standard deviation of leadtime demand
Shortages – Example No.1 Suppose the standard deviation of leadtime demand is known to be 20 units. Leadtime demand is approximately normal. b) For a leadt leadtime ime serv service ice leve levell of 90 perce percent, nt, dete determine rmine the the expect expectee number of units short for any order cycle. c) Wh What at leadt leadtime ime se servi rvice ce leve levell would would be an an expected expected shor shortag tagee of 2 units imply? SOLUTION : σd LT LT = 20 units • •
Leadtime (cycle) service level = .90. From table, E(z) = 0.048 E(n) = 0.048(20 units) = 0.96 unit, or about 1 unit For the case where E(n) = 2, solve for E(z) and determine from the table the corresponding service level. Thus, E(z)=E(n)/ σd LT LT
Shortages – Example No.2
Expected number of units short per year is D E(N) = E(n) Q
Example:
Given the following information, determine the expected number of units short per year. D = 1,000
Q = 25 0
E(n) = 2.5
Then, E(N) = 2.5 (1,000 / 250 ) = 10.0 units per year
Annual Service Level
Annual Service Level is the percentage of demand filled directly from inventory (also known as fill rate) Formula:
SLannual
E(N) = 1 – D
E(N) = E(n)D/Q = E(z)
SLannual
=
1 – E(z) Q
d LT LT
d LT LT
D/Q
Annual Service Level – Example Given a leadtime service level of 90, D = 1,000, Q = 250, and σd LT LT = 16, determine the annual service level and the amount of cycle safety stock that would provide an annual service level of .98. From the table, E(z) = 0.048 for a 90 percent leadtime service level. SLannual = 1 – 0.048(16)/250 = .997 • SLannual = 1 – E(z) σd LT • LT / Q 0.98 = 1 – E(z)(16)/250 Solving, E(z) = 0.312; From the table, with E(z)=0.312, it can be seen that this value if E(z) E(z) is a little more more than the value of 0.307. So it appears that an acceptable value of z might be 0.19. 0. 19. The necessary safety stock to achieve the specified annual service level is equal to z σd LT LT. Hence, the safety stock is 0.19(16) = 3.04, or approximately 3 units
How Much to Order: Fixed-Order-Interval Model
FOI Model is used when orders must be placed at fixed time intervals (weekly, twice a month, etc.)
Is widely used by retail businesses
Question to be answered at each order point is: How much should be ordered for the next (fixed)interval?
If demand is variable, the order size will tend to vary from fr om cycle to cycle.
Different from EOQ/ROP approach in which the order size generally remains fixed from cycle to cycle, while the length of the cycle varies (shorter if demand is above average, and longer if demand is below average)
How Much to Order: Fixed-Order-Interval Model
Reasons to use FOI Model: 1) A supplier’s policy might encourage orders at
fixed intervals 2) Grouping orders for items from the same supplier
can produce savings in shipping costs 3) An alternative for retail operations which do not
lend itself to continuous monitoring of inventory levels (only periodic checks will do with the use of fixed-interval ordering)
Differences between Fixed-Quantity Systems & Fixed-Order-Interval Models FIXED-QUANTITY MODEL Orders are triggerred by quantity (ROP) Stockout protection only during leadtime Higher than normal demand causes a shorter time between orders Close monitoring of inventory is required to know when ROP is reached
FIXED-ORDER-INTERVAL Orders are triggered by time Stockout protection for leadtime plus the next order cycle Result of higher than normal demand is a larger order size Only a periodic review (physical inspection) of inventory on hand is needed prior to ordering
Fixed-Order-Intervall Model Fixed-Order-Interva
Determining the amount to order
Expected demand Amount Safety = during protection + to order stock interval = d (OI (OI + LT) + z
d
OI + LT -
Amount on hand at reorder time A
where whe re OI = Ord Order er inte interva rvall (le (length ngth of tim timee bet betwee ween n orde orders) rs) A = Amount on hand at reorder time As in previous models, demand during the protection interval is normally distributed.
Fixed-Order-Interval Model Example Given the following information, determine the amount to order. d = 3 30 0 units per day σd = 3 uni units ts per day LT = 2 days
Desired service level = 99 99 percent Amount Am ount on han hand d at reo reorde rderr tim timee = 71 uni units ts OI = 7 days
Z = 2.33 for 99 percent service level Amount = d (OI (OI + LT) + z σd √ OI + LT - A to order = 30 (7 + 2) + 2.33(3) √ 7 + 2 - 71 = 220 units
Fixed-Order-Intervall Model Fixed-Order-Interva
An issue related to FIO system is the risk of stockout
Stockout can occur at two points in the order cycle: 1) Shortly after the order is placed, while waiting to
receive the current order 2) Near the end of the cycle, while waiting to receive
the next order
Fixed-Order-Intervall Model Fixed-Order-Interva
Finding the Risk of Stockout
To find the initial risk of a stockout, use the ROP formula ROP = d x LT + z √ LT σd , setting ROP equal to the quantity on hand when the order o rder is placed, and solve for Z, then obtain the service level for that value of Z from Appendix B, Table B and subtract it from 1.0000 to get the risk of a stockout To find the risk of a stockout at the end of the order cycle, use the fixed-interval formula, Amount to Order = d (OI (OI + LT) + z σd √ OI + LT - A , and solve for z. Then obtain the service level for that value of z from Appendix B, Table B and subtract it from 1.0000 to get
Fixed-Order-Interval Model Example – Risk of Stockout Given the following information: LT = 4 days A = 43 units OI = 12 days Q = 171 units σd = 2 units/day d = 10 units/day Determine the risk of a stockout at f) Th Thee end end of the ini nittia iall lea leadt dtim imee g) Th Thee en end d of th thee se seco cond nd le lead adti time me SOLUTION; i) For the ris risk k of sto stocko ckout ut for for the the firs firstt leadt leadtime ime,, ROP = 10 x 4 + z(2) z(2)(2) (2) = 43 Solving, z = +.75. From Appendix B, Table B, the service level is .7734 The risk is 1 - .7734 = .2266, which is fairly high. l) For the ris risk k of a stoc stockout kout at the end of the the seco second nd lea leadti dtime me,, subst substitu itutin ting g the the given values, 171 = 10 x (4 + 12) + z(4)(2) – 43 . Solving, z = +6.75 . This value is way out in the right tail of the normal distribution, making service level virtually 100 percent, and thus, the risk of a stockout at this point is
Fixed-Order-Interval Model
Benefits Results in the tight control needed for A items in an ABC classification due to the periodic reviews it requires Grouping orders can yield savings in ordering, packing, and shipping costs (when 2 or more items are ordered from the same supplier) Practical approach if inventory withdrawals cannot be closely monitored Disadvantages Necessitates a larger amount of safety stock for a given risk of stockout because of the need to protect against shortages during the entire order interval plus lead time (instead of leadtime only)
Single-Period Model
Sometimes referred to as the newsboy problem
Used for ordering of perishables (fresh ( fresh fruits, vegetables, seafood, cut flowers) and other items with limited lives (newspapers, magazines, spare parts for specialized equipment)
Analysis of single-period situations focuses on two costs: shortage and excess
Shortage cost – generally the t he unrealized profit per unit
Excess cost – difference between purchase cost and a nd salvage value of items left over at the end of the period
Single-Period Model Continuous Stocking Levels
Formula for costs:
Cshortage = Cs = Revenue per unit unit – Cost per unit Cexcess
= Ce = Original cost per unit – Salvage value per unit
Continuous Stocking Levels Optimal stocking level = balance between shortage & excess Service Level = probability that demand will not exceed the stocking level (SL computation is the key to determining the Ce optimal stocking level, So) Cs Service level Service Level = Cs Cs + Ce Quantity → where Cs = shortage cost per unit
S
Single-Period Model - Example Continuous Stocking Levels Sweet cider is delivered weekly to Cindy’s Cider Bar. Demand varies uniformly between 300 liters and 500 liters per week. Cindy pays 20 cents per liter for the cider and charges 80 cents per liter for it. Unsold cider has no slavage value and cannot be carried over into the next week due to spoilage. Find the optimal stocking level and its stockout risk for that quantity. SOLUTION: Ce = cost per unit – Salvage value per unit = $0.20 - $0 = $0.20 per liter Cs = Revenue per unit – Cost per unit = $0.80 - $0.20 = $0.60 per liter SL = Cs / (Cs + Ce) = $0.60/($0.60+$0.20) = 0.75 So = 300 + 0.75(500 – 300) = 450 liters 75%
