Inventory Management

1
Inventory Management

• A Basic Introduction for
• Operations Supply Chain Management
• Dr. Mark P. Van Oyen
• file inven-lec.ppt

2
Types of Inventory

• 1. Raw materials and purchased parts.
• 3. Finished goods.
• Supplier
• Distributor
• Retailer
• 4. Replacement parts, tools, and supplies.
• We are NOT talking about Work in process (WIP).
• Shop-floor control of partially completed goods.
• Often dealt with using PUSH (MRP II, ERP) or PULL
  (Kanban, CONWIP, Bucket Brigades, etc.)
• Goldratt warns about using Economic Batch
  Quantity for shop floor lot-sizing which is our
  EOQ as youll see.

3
Customers, demand centers sinks
Field Warehouses stocking points
Sources plants vendors ports
Regional Warehouses stocking points
Supply
Inventory warehousing costs
Production/ purchase costs
Transportation costs
Transportation costs
Inventory warehousing costs
4
Perspectives on Inventory

• Inventory becomes an increasingly large part of
  total assets and managerial focus as we move down
  the supply chain.

• Suppliers
• Manufacturers
• Wholesalers
• Distributors
• 4. Retailers

5
Goals Reduce Cost, Improve Service

• By effectively managing inventory
• Xerox eliminated 700 million inventory from its
  supply chain
• Wal-Mart became the largest retail company
  utilizing efficient inventory management (many
  new business processes and distribution,
  logistics, IS, warehousing innovations all for
  the net impact of excellent inventory management)
• GM has reduced parts inventory and transportation
  costs by 26 annually

6
Functions of Inventory

• To meet anticipated demand.
• Display, provide customer hands-on experience
  (dealer inventory)
• To protect against stockouts. (variation,
  unanticipated demand)
• To smooth production requirements.
• Level aggregate production plan is an example.
• To take advantage of order cycles.
• Efficiency in fixed costs of ordering or
  producing (e.g. share delivery truck or economy
  of scale in batch production)
• To de-couple operations, providing smooth flow
  between operations
• WIP inventory between successive manufacturing
  steps or supply chain echelons (buffer stock)
  permits operations to continue during periods of
  breakdowns/strikes/storms.
• To hedge against price increases, or to take
  advantage of quantity discounts. caution
  hazardous to corporate health
• Despite Zero Inventory zealots, an appropriate
  amount of inventory is a necessary evil in almost
  all systems!

7
Tradeoffs in the Size of Inventories

• Inventories that are too high are expensive to
  carry, and they tie up capital.
• Warehousing, transportation
• Greater risk of defects, even when carefully
  inventoried
• Market shifts leave seller with many unwanted
  parts
• Inventories that are too low can result in
• reduced operational efficiency (machine
  starvation)
• poor service (delayed service, product
  substitution)
• lost sales (customer refuses raincheck, finds a
  more reliable supplier)

8
Objectives/Controls of Inventory Control

• Achieve satisfactory levels of customer service
  while keeping inventory costs under control.
• Fill rate (probability of meeting a demand from
  inventory)
• Inventory turnover ratio Annual Sales / Average
  Inventory Level D / WIP
• Others not focused upon in this class Average
  Backlog, Mean time to fill order (Cycle Time)
• Profit maximization by achieving a balance in
  stocking, avoiding both over-stocking and
  under-stocking.

• It boils down to 2 control decisions
• Decide how much to order/produce (Q)
• Decide when to order/produce (ROP)

9
Costs

• h Carrying (or holding) cost (applied per unit
  of inventory)
• Associated with keeping inventory for a period of
  time.
• Capital tied up
• Warehousing and transportation
• Perishable products -OR- expected risk of
  damage to product
• Note that easiest calculation is (Holding cost
  rate/unit/yr) (Average annual inventory level)
• S Ordering OR production costs (Fixed order set
  up variable unit costs)
• For ordering and receiving inventory if ordering
  from a supplier, (Delivery charges, postage
  handling, cost of purchasing dept. labor)
• Independent of order quantity (Q) from supplier.
• OR view this as production costs if we are
  the manufacturer

10
Costs

• Purchase cost per unit of inventory (P)
• We model only a variable cost from supplier
• If we make the parts ourselves, S captures
  production setup cost, while P captures the
  variable cost of production
• Shortage costs. (used in Newsvendor Model with
  demand uncertainty when demand exceeds supply of
  inventory and a stockout occurs.
• Opportunity costs (lost sales and LOST
  CUSTOMERS!)
• Loss of good will, or, may need to substitute
  higher-cost item!
• Do not underestimate shortage costs, or you will
  settle for less market share than you really want

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The Inventory Management Problem

• Determine Inventory Policy
• How much to order or make? (Q)
• When to order or make? (Reorder point)
• How much to store in safety stock?
• To
• Minimize the cost of the inventory system

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Part I How Much to Order Economic Order
Quantity (EOQ)

• Objective Identify the optimal order quantity
  that minimizes the sum of certain annual costs
  that vary with order size.
• Assumption Fixed order quantity systems (dont
  change order size dynamically over time)
• CERTAIN demand (no random market behavior)
• We will consider 2 models

• EOQ all items delivered as a batch

• EOQ-QD EOQ with quantity discounts (uses EOQ
  assumptions)

13
Model EOQ (all items delivered as a batch)

• How Many Parts to Make at Once by Ford Harris
• Original 1913 version of this model.
• Very simple - makes a lot of restrictive
  assumptions. Its not realistic at first glance,
  but it is! Its usefulness keeps it alive!
• Nicely illustrates basic tradeoffs that exist in
  any inventory management problem.
• Basic Model Scenario
• Own a warehouse from which parts (brake pads) are
  demanded by customers.
• Periodically run out of parts and have to
  replenish inventory by ordering from suppliers.

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KEY The Inventory Cycle
15
EOQ Model - Assumptions

• Demand, D, is known and constant.
• Purchase Price, P, is known and constant.
• Fixed setup cost per order, A, independent of Q
• Holding cost h (or C) per unit per year.
• Lots of size Q are delivered in full (Production
  Perspective produce items and hold them in FGI
  until production is completed, at which time we
  ship them out).
• There will be no stockouts, no backorders, no
  uncertainties!
• Lead Time is known and constant.

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Basic EOQ Model Assumptions

• D 3120 (units/yr) Annual demand rate e.g.
  Sell brake pads with D 60 pads/wk 52
  wks/yr.
• P 2 /pad Unit production/purchase cost - not
  counting setup or inventory costs (/unit)
• A 28.85 /order Order set-up cost, constant per
  order, for any order size
• h (or C) Average annual carrying cost per unit
  (/unit/yr.) 25 of purchase price 0.50 /yr
• Q order quantity decision variable e.g.
  How many brake pads to order

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Inventory Cycle for Basic EOQ
D 6052 pads/yr
Q 600
Inventory
Level
Time
Q/D Length of Order Cycle
10 wk.
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Total Annual Cost Calculation

• Order Frequency F
• D/Q reciprocal of period e.g. (60
  pads/wk) 52 /600 pads 52/10. (orders/yr)
• Annual ordering cost
• ( orders/year) (order setup cost) (unit
  purch. cost) (annual demand) (D/Q)A PD
• Average inventory per order cycle
• (Q0)/2 Q/2. why? use geometry to get it (vs.
  calc.)
• Annual carrying cost
• (Q/2) h.
• Total annual Stocking Cost (TSC)
• annual carrying cost annual ordering
  cost
• TSC (D/Q)A PD (Q/2) h

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Inventory Cycle for Basic EOQ
Q 600
Inventory Level
D 6052 pads/yr
Q/2 Ave Inventory 300.
Time
Q/D Length of Order Cycle 10 wk.
20
Cost Minimization Goal
The Total Stocking Cost (TSC) Curve is U-Shaped
Annual Cost
h q/2
Ordering Costs
PD
PD, cost to purchase stock
0
Order Quantity Used (q)
Q (optimal)
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Economic Order Quantity (EOQ)

• There is a tradeoff between carrying costs and
  ordering costs!
• EOQ is the value, Q, that minimizes TSC.
• In this sense, the Q determined by the EOQ
  formula is OPTIMAL given a simplistic model.
• EOQ is found by either
• Using calculus and solving (d/dQ)TSC 0, the
  points of graph with zero slope are either local
  maxima minima or,
• Observing that the EOQ occurs where carrying and
  ordering costs are equal, i.e., by solving (Q/2)
  h (D/Q)A. this is not a general method!

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EOQ Model - Development

• It turns out that
• Costs are minimized where ordering and carrying
  costs are equal.
• Thus,

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EOQ Model - Example

• Demand 60 pads/wk
• Ordering Cost, A 28.85/order
• Unit Carrying Cost, h 25 of purchase price/yr.
• Unit purchase price, P 2.00/pad

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EOQ-QD EOQ with All Units at Quantity Discount

• Purchase Price is known but varies with the
  amount purchased. The price is the same for all
  units. Also, holding cost may be fraction of
  price!
• Demand and Lead time both known and constant.
• Lots are delivered in FULL as in Model I.
• The problem is solved as a series of problems -
  one for each price break.
• (1) solve for EOQ for each possible price and
  modify the Q value to be feasible at that price,
  then
• (2) compute resulting TSC and finally
• (3) search for the lowest cost.

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EOQ-QD Quantity Discounts (All Units)

• Quantity discounts offered by the supplier to the
  buyer occur when the unit purchase price of the
  product (actual cost ac) decreases as the
  quantity purchased (Q) increases.
• Assume orders are received all at once (Model I).
  
• Total Cost (Q/2)h (D/Q)A (D)(ac).
• Careful h may be some percent of (ac) .
• Find Q EOQ that minimizes total cost.

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Example

• D 10,000, A 5.50, h 0.2(ac).
• Price Schedule
• Note all-units quantity discount.
• E.g. at Q 524, ac2.00 applies to all Q units
  (not just 400-524!)

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Computing EOQ for each range

• ac 2.20 yields optimal EOQ at a level that
  deserves a better price/volume, so this is the
  one situation in which we can disqualify the
  possibility of this range containing our answer!
• ac 2.00 yields EOQ 524.4 which is feasible.
• ac 1.80 yields EOQ 552.8 which is NOT
  feasible. We then take the closest quantity that
  will give us that price range
• Q 700 for ac 1.80.
• Next step cost these out - answer has to be one
  of them!

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Computing EOQ for Example

• TC(Q) (Q/2)h (D/Q)A (D)ac.
• TC(Q 524.4) (524.4/2)(0.40)
  (10,000/524.4)(5.5) 10,000(2.00) 20,209.76.
• TC(Q 700) (700/2)(0.36) (10,000/700)(5.5)
  10,000(1.8) 18,204.57 (min).
• Conclusion EOQ 700. (even though the EOQ
  solution was not feasible at that price!)

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Part II When to Order? Inventory Management
Under Uncertainty

• Demand or Lead Time or both uncertain
• Even good managers are likely to run out once
  in a while (a firm must start by choosing a
  service level/fill rate)
• When can you run out?
• Only during the Lead Time if you monitor the
  system.
• Solution build a standard ROP system based on
  the probability distribution on demand during the
  lead time (DDLT), which is a r.v. (collecting
  statistics on lead times is a good starting
  point!)

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The Typical ROP System
Average Demand
ROP set as demand that accumulates during
lead time
ROP ReOrder Point
Lead Time
31
The Self-Correcting Effect- A Benign Demand
Rate after ROP
Hypothetical Demand
Average Demand
ROP
Lead Time
Lead Time
32
What if Demand is brisk after hitting the ROP?
Hypothetical Demand
Average Demand
ROP EDDLT SS
ROP gt
EDDLT
Safety Stock
Lead Time
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When to Order A (Q,r) Policy

• The basic EOQ models address how much to order Q
• Now, we address when to order.
• Re-Order point (ROP) occurs when the inventory
  level drops to a predetermined amount, which
  includes expected demand during lead time (EDDLT)
  and a safety stock (SS)
• ROP EDDLT SS.
• ROP is referred to as r in the (Q,r) policy

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When to Order r ROP

• SS is additional inventory carried to reduce the
  risk of a stockout during the lead time interval
  (think of it as slush fund that we dip into when
  demand after ROP (DDLT) is more brisk than
  average)
• ROP depends on
• Demand rate (forecast based).
• Length of the lead time.
• Demand and lead time variability.
• Degree of stockout risk acceptable to management
  (fill rate, order cycle Service Level)

DDLT, EDDLT Std. Dev.
35
The Order Cycle Service Level, (SL)

• SL of demand during the lead time ( of
  DDLT) the firm wishes to satisfy. SL
  probability that random customer during lead time
  is served
• SL must be set according to Shortage Cost
• This is NOT annual service level (or fill rate),
  since that averages over all time periods and
  will be a larger number than SL.
• SL should not be 100 for most firms. (90?
  95? 98?) SL increases as Safety Stock
  increases, but with diminishing returns (due to
  shape of demand distribution)

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Safety Stock
Quantity
Maximum probable demand during lead time (in
excess of EDDLT) defines SS
Expected demand during lead time (EDDLT)
ROP
Safety stock (SS)
Time
LT
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Variability in DDLT and SS

• Variability in demand during lead time (DDLT)
  means that stockouts can occur.
• Variations in demand rates can result in a
  temporary surge in demand, which can drain
  inventory more quickly than expected.
• Variations in delivery times can lengthen the
  time a given supply must cover.
• We will emphasize Normal (continuous)
  distributions to model variable DDLT, but
  discrete distributions are common as well.
• SS buffers against stockout during lead time.

38
Service Level and Stockout Risk

• Target service level (SL) determines how much SS
  should be held.
• Remember, holding stock costs money.
• SL probability that demand will not exceed
  supply during lead time (i.e. there is no
  stockout then).
• Service level stockout risk 100.

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Reminder The Normal Distribution
Standard Deviation 5
Standard Deviation 10
Average 30
40
Computing SS from SL for Normal DDLT

• Example DDLT is normally distributed a mean of
  693. and a standard deviation of 139.
• EDDLT 693.
• s.d. (std dev) of DDLT ? 139.
• As computational aid, we need to relate this to Z
  standard Normal with mean0, s.d. 1
• Z (DDLT - EDDLT) / ?

41
Reorder Point (ROP)
42
Area under standard Normal pdf from - ? to z
Z standard Normal with mean0, s.d. 1Z (X
- ? ) / ?See GF Appendix ASee Stevenson,
second from last page
P(Z ltz)
43
Computing SS from SL for Normal DDLT to provide
SL 95.

• ROP EDDLT SS EDDLT z (?).
• z is the number of standard deviations SS is
  set above EDDLT, which is the mean of DDLT.
• z is read from Appendix B Table B2. Of Stevenson
  -OR- Appendix A (p. 768) of Gaither Frazier
• Locate .95 (area to the left of ROP) inside the
  table (or as close as you can get), and read off
  the z value from the margins z 1.64.
• Example ROP 693 1.64(139) 921
• SS ROP - EDDLT 921 - 693. 1.64(139) 228
• If we double the s.d. to about 278, SS would
  double!
• Lead time variability reduction can same a lot of
  inventory and (perhaps more than lead time
  itself!)

44
Analysis of Std Dev (DDLT)

• The reorder point (s) must account for deviations
  from average (call this safety stock - SS).
  There are two standard ways
• View 1 Summarize the std. deviation of Demand
  During Lead Time (DDLT) as ?.
• View 2 (more detailed) STD std. dev. of
  demand for 1 day or 1 unit of time, so std.
  deviation of Demand During Lead Time (DDLT)
  STD ? ?LT
• View 2 has SS z ? (STD ? ?LT) where z is
  chosen from statistical tables to ensure that the
  probability of stockouts during leadtime is
  100-SL.
• ROP EDDLT SS LT?AVG z ?STD??LT

45
Summary View

• Holding Cost h Q/2 SS
• Order trigger by crossing ROP
• Order quantity up to (SS Q)

QSS Target
Not full due to brisk Demand after trigger
ROP EDDLT SS
ROP gt
EDDLT
Safety Stock
Lead Time
46
The (s,S) Policy Continuous Review

• (s, S) Policy Whenever the inventory position
  drops below a certain ReOrder Point, s (aka ROP),
  we order to raise the inventory position to level
  S s Q.
• The reorder point is a function of
• The Lead Time
• Average demand
• Demand variability
• Service level

47
A View of (s, S) Policys ROP, S ROP Q
S
Inventory Position
Lead Time
Lead Time
Inventory Level
s
0
Time
48
Inventory Supply Chain

• So far, weve approached the analysis from the
  perspective of Purchasing, ordering from a
  supplier
• This is local optimization
• Concept The battle between supply chains has
  more impact on your success than your battles
  with your direct competitors!?!?

49
Risk Pooling

• Consider these two systems

Market One
Warehouse One
LT W1
Ship Times
Supplier
Market Two
Warehouse Two
LT W2
Ship Times
Market One
Centralized Warehouse
Ship Times
Supplier
Market Two
50
Risk Pooling

• For the same service level, which system will
  require more inventory? Why?
• For the same total inventory level, which system
  will have better service? Why?
• What are the factors that affect these answers?

51
Risk PoolingImportant Observations

• Centralizing inventory control reduces both
  safety stock and average inventory level for the
  same service level.
• This works best for
• High coefficient of variation, which reduces
  required safety stock.
• Negatively correlated demand. Why?
• Do you see other kinds of risk pooling? (hp,
  Bennetton, )

52
Inventory Supply Chain

• How does our classic inventory management relate
  to the supply chain?
• It depends is the supply chain centralized
  (coordinated centrally as in vertical
  integration, or decentralized with local control)
• The particular inventory management
  implementation at each operation and each level
  of the supply chain combines to result in a
  hard-to-predict system dynamic behavior for the
  S. Chain!

53
Centralized Systems
This is just a story-telling device for class.
The simplest supply chain is one with centralized
control (e.g., a vertically integrated supply
chain).

• Centralized Decision

Supplier
Warehouse
Retailers
54
Centralized Distribution Systems

• Question How much inventory should management
  keep at each location?
• A good strategy (based on current research,
  called echelon inventory systems) is the keep
  track of the total inventory at your level or
  below as follows
• retailer raises inventory to level Sretail upon
  hitting its Reorder Point (ROPretail)
• supplier raises to Ssupplier the sum of (1) its
  own inventory, (2) inven. in transit to
  retailers, and (3) inventory in their set of
  retailers. If there is not enough inventory in
  the warehouse to meet all demands from retailers,
  it is allocated so that the service level at each
  of the retailers will be equal.

55
Details Echelon Inventory

• For the Wholesaler, the ordering process can be
  governed using inventory position counted as
  echelon inventory (e.g. wholesaler inventory
  shipped to retailers retailer inventory)
  reorder from Mfr.
• For this to work, Demand must be seen in real
  time as demand faced by all retailers (need Real
  Time Environment).
• Simple approach requires information sharing

56
Inventory Management Best Practices

• Periodic inventory reviews dynamic order
  quantity sizing helps identify slow-moving
  products, also clearer inventory reduction
  measurements
• Many companies use continuous review systems,
  which tend to be more efficient in theory (but
  also require greater effort or investment in
  inventory tracking).
• As of 2000, the Army, Air Force, and Navy used
  the following methods
• Cts. Review Fixed Q (as in our theory here)
• Periodic Review Order up to Q SS (we called
  it (s,S) policy)
• Instantaneous Reorder use 1, reorder 1

57
Inventory Management Best Practices

• Periodic inventory reviews dynamic order
  quantity sizing helps identify slow-moving
  products, also clearer inventory reduction
  measurements
• Tight tracking of usage rates, management of lead
  times and safety stock
• ABC approach top rev. 20, next 15, low
  review. Each category receives appropriate
  inventory review frequency based on unit volume
  and/or sales volume.
• Reduced safety stock levels LT reduction,
• Shift more inventory, or inventory ownership, to
  suppliers (Vendor Managed Inventory VMI,
  Continuous Replenishment Programs - CRP)
• Quantitative approaches

58
Changes In Inventory Turnover

• Inventory turnover ratio (annual sales in
  units)/(avg. inventory level)
• Inventory turns increased by 30 from 1995 to
  1998
• Inventory turns increased by 27 from 1998 to
  2000
• Overall the increase is from 8.0 turns per year
  to over 13 per year over a five year period
  ending in year 2000. (varies by sector)

59
Part III Single-Period Model Newsvendor

• Used to order perishables or other items with
  limited useful lives.
• Fruits and vegetables, Seafood, Cut flowers.
• Blood (certain blood products in a blood bank)
• Newspapers, magazines,
• High Fashion clothing (long lead time -gt 1 shot)
• Unsold or unused goods are not typically carried
  over from one period to the next rather they are
  salvaged or disposed of.
• Model can be used to allocate time-perishable
  service capacity.
• Two costs shortage (short) and excess (long).

60
Single-Period Model

• Shortage or stockout cost may be a charge for
  loss of customer goodwill, or the opportunity
  cost of lost sales (or customer!)
• Cs Revenue per unit - Cost per unit.
• Excess (Long) cost applies to the items left over
  at end of the period, which need salvaging
• Co Overage Cost Original cost per unit -
  Salvage value per unit.

61
The Single-Period Model Newsvendor

• How do I know what service level is the best one,
  based upon my costs?
• Answer Assuming my goal is to maximize profit
  (at least for the purposes of this analysis!) I
  should satisfy SL fraction of demand during the
  next period (DDLT)
• If Cs is shortage cost/unit, and Co is excess
  cost/unit, then

(derived via the magic of Leibnitzs Rule)
62
Single-Period Model for Normally Distributed
Demand

• Computing the optimal stocking level differs
  slightly depending on whether demand is
  continuous (e.g. normal) or discrete. We begin
  with continuous case.
• Suppose demand for apple cider at a downtown
  street stand varies continuously according to a
  normal distribution with a mean of 200 liters per
  week and a standard deviation of 100 liters per
  week
• Revenue per unit 1 per liter
• Cost per unit 0.40 per liter
• Salvage value 0.20 per liter.

63
Single-Period Model for Normally Distributed
Demand

• Cs 60 cents per liter
• Co 20 cents per liter.
• SL Cs/(Cs Co) 60/(60 20) 0.75
• To maximize profit, we should stock enough
  product to satisfy 75 of the demand (on
  average!), while we intentionally plan NOT to
  serve 25 of the demand.
• The folks in marketing could get worried! If
  this is a business where stockouts lose long-term
  customers, then we must increase Cs to reflect
  the actual cost of lost customer due to stockout.

64
Single-Period Model for Continuous Demand

• demand is Normal(200 liters per week, variance
  10,000 liters2/wk) so ? 100 liters per week
• Continuous example continued
• 75 of the area under the normal curve must be to
  the left of the stocking level.
• Appendix shows a z of 0.67 corresponds to a
  left area of 0.749
• Optimal weekly purchase/ stocking level mean
  z (?) 200 (0.67)(100) 267. liters.

65
Single-Period Discrete Demand Lively Lobsters

• Lively Lobsters (L.L.) receives a supply of
  fresh, live lobsters from Maine every day. Lively
  earns a profit of 7.50 for every lobster sold,
  but a day-old lobster is worth only 8.50. Each
  lobster costs L.L. 14.50.
• (a) what is the unit cost of a L.L. stockout?
• Cs 7.50 lost profit
• (b) unit cost of having a left-over lobster?
• Co 14.50 - 8.50 cost salvage value 6.
• (c) What should the L.L. service level be?
• SL Cs/(Cs Co) 7.5 / (7.5 6) .56
  (larger Cs leads to SL gt .50)
• Demand follows a discrete (relative frequency)
  distribution as given on next page.

66
Lively Lobsters SL Cs/(Cs Co) .56

• Demand follows a discrete (relative frequency)
  distribution
• Result order 25 Lobsters, because that is the
  smallest amount that will serve at least 56 of
  the demand on a given night.