EIN 6392 Manufacturing Management

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EIN 6392Manufacturing Management

• Catalog Description Variety and importance of
  management decisions. Total quality management,
  just-in time manufacturing, concurrent
  engineering, material requirements
  planning,production scheduling, and inventory
  control.

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Manufacturing History

• What can we learn from history?
• First Industrial Revolution (mid-1700s)
• Steam Engine
• Mass production
• Vertical Integration
• Interchangeable parts (and workers)
• Economies of Scale

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Manufacturing History

• Second Industrial Revolution (Late 1800s)
• Transport and Communications Infrastructure
• Allowed for creation of mass markets
• Mass Retailers (Sears)
• Horizontal and Vertical Integration
• Carnegie Rail, Steel, Mining
• High volume production

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Manufacturing History

• Henry Ford Emphasis on speed of production
• Turn of the century (early 1900s)
• Assembly-line production
• Fast labor times
• Repetitive, standardized processes
• Speed of output impacts cost per unit

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Scientific Management

• Frederick W. Taylor (late 1800s/early 1900s)
• Measured workers speed
• Emphasized the best way to perform tasks
• Mathematical models
• Worker incentives
• Accounting principles
• Management planning systems

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Manufacturing in the 20th Century

• Pierre Du Pont (early 1900s)
• Installed Taylors management systems at Du Pont
• E.I. Du Pont de Nemours Co. was a collection of
  explosives companies
• Du Pont first used the metric ROI (Return on
  Investment) to measure performance
• Du Pont succeeded W. Durant, who consolidated
  Buick with Cadillac, Oldsmobile, and Oakland to
  form GM in 1908

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General Motors

• The Du Pont Company invested heavily in GM, and
  forced Durant out in 1920
• GM was performing poorly and had little
  management structure
• Du Pont asked Alfred P. Sloan to help restructure
  GM
• Sloan devised a central corporate structure to
  oversee independent operating divisions of GM

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Sloans Innovations

• Sloan saw the value in focusing divisions on
  target markets
• Chevrolet targeted low-end while Buick and Olds
  went after middle-market.
• Sloan used ROI and developed scientific
  forecasting, inventory management, and market
  share estimation systems.
• Sloan planned obsolescence and emphasized variety
  while Ford used little customization

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Modern Manufacturing Corporation

• Sloans collection of scientific management,
  organizational structuring, and market emphasis
  created a model for the modern U.S. manufacturing
  corporation
• U.S. manufacturers, basing their organizations on
  this model, prospered and dominated world markets
  for the first half of the 1900s and for much of
  the second half

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The Landscape Changed

• By 1969 the top 200 American firms accounted for
  61 of the worlds manufacturing assets
• Much of Europe and Japan spent the 50s and 60s
  rebuilding their infrastructures
• In the 1970s and 80s American firms lost
  significant market share to foreign competitors
• Today the highest selling automobile in the U.S.
  is the Toyota Camry, and the highest selling car
  in the world is the Corolla

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Decline of U.S. Manufacturing

• Analysts cite a variety of reasons for the
  decline of U.S. firms
• The lack of competition made manufacturing and
  quality an afterthought
• The primary emphases were marketing and finance
• Manufacturing was viewed as a dead-end career

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Marketing and Finance Outlook

• Marketing focus was to imitate, not innovate
• Primary focus was sales
• If only they didnt have to make the stuff
• Finance short term returns
• ROI emphasis combined with career movements
  favored short term gains
• Improve ROI in short-run by decreasing investment
• Little incentive for long-term investment

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Diversification

• Finance outlook emphasized diversifying risk
  through broad investment
• In 1949, 70 of top 500 U.S. firms earned 95
  from single business
• In 1969, 70 of firms did not have a dominant
  business
• Lack of focus on core competencies
• Mergers and acquisitions led to overall
  inefficiencies

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New Competition

• These problems did not surface until new
  competitive threats arose
• In the mid-20th Century, firms in Japan created
  new manufacturing management systems that
  ultimately led to great economic growth in the
  1970s and 80s
• We will later consider the principles of these
  management systems and why they proved to be so
  successful

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Evolution of Scientific Management

• We will first consider the basic manufacturing
  management principles developed between the
  1950s and 1970s that formed the basis for
  modern manufacturing management
• These principles are fundamental to both modern
  U.S. and Japanese manufacturing management
  systems and focus on managing inventory, supply,
  and production flow in factories

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Inventory Control

• What purpose does inventory serve?
• It provides capacity to instantaneously meet
  downstream demands and requirements
• It provides a buffer between successive
  operations stages
• It insulates against future uncertainties
• Uncertainties in supply
• Uncertainties in demand
• Uncertainties in capacity
• Uncertainties in material value

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Inventory Control

• What motivates firms to hold inventory?
• Economic Motive
• Economies of scale
• Speculative Motive
• Future values of raw materials
• Transaction Motive
• Precautionary Motive
• Uncertainty in supply, demand, and operations

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Inventory Control Decisions

• What decisions are involved in inventory control?
• How should I track inventory?
• Continuously versus periodically
• When do I place an order?
• Reorder point
• How much should I order when I place an order?
• Order quantity

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Economic Order Quantity

• The oldest known mathematical inventory model
• Illustrates insights regarding economic tradeoffs
  in production
• Addresses economic and transaction motives
• Uses extremely simple, often impractical modeling
  assumptions
• It is, however, very robust to situations not
  fitting the assumptions

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Economic Order Quantity

• Modeling Assumptions
• Instantaneous production
• Immediate delivery
• Deterministic demand
• Constant demand rate
• Constant setup cost for any production run or
  order placement
• Products can be analyzed separately
• Assumptions can be easily relaxed

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EOQ Parameters and Decisions

• D demand rate (units per unit time)
• c Unit production/procurement cost, over and
  above any fixed order cost
• A Fixed order/setup cost
• h holding cost per unit per unit time
• h ic, where i is an interest rate reflecting
  cost of capital, warehousing, insurance,
  obsolescence
• Q Order quantity/Lot size

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EOQ Properties

• Since demand is deterministic and occurs at a
  constant rate, we can always time orders so we
  have zero inventory when a replenishment arrives.
• This constant and deterministic demand rate leads
  to a system whose behavior does not vary with
  time
• Inventory falls at constant linear rate
• Leads to optimality of a fixed order quantity/lot
  size with each order/setup

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EOQ Analysis

• We would like to determine the order quantity, Q,
  that minimizes the average cost incurred per unit
  time.
• Suppose we order a quantity Q
• Since delivery of Q units occurs instantaneously,
  we begin with Q units
• Inventory is depleted at a constant rate of D
  units per unit time
• When Inventory hits zero, we again order Q

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EOQ Analysis

• Inventory level follows a cyclical pattern
• Minimizing the cost per unit time in any given
  cycle is sufficient
• We therefore consider the costs incurred in a
  single cycle and average them out over the cycle
  length, T Q/D, to get average cost per unit time

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EOQ Analysis

• What costs do we incur in a cycle?
• Fixed order cost, A
• Variable procurement cost, cQ
• Holding costs
• Since holding cost is applied per unit per unit
  time, we multiply h by the inventory level at
  each instant in time and integrate over the cycle
• Holding cost
• Since the integral of the inventory level over a
  cycle is just the area of a triangle, we can
  simply determine this area

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EOQ Analysis

• In terms of the variables, Q and D, what is the
  area of the triangle?
• Area (1/2)QQ/D Q2/2D
• Inventory cost in a cycle hQ2/2D
• Total Cost in a Cycle
• A CQ hQ2/2D
• To get the average cost per unit time, we divide
  by the cycle length, T Q/D
• Y(Q) AD/Q cD hQ/2
• Y(Q) Annual order/setup procurement holding
  cost if D is annual demand rate

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EOQ Analysis

• We would like to minimize Y(Q) over all Q ? 0.
• We can show that Y(Q) is a convex function
• This means that if we take the derivative and set
  it to zero, we will find the global minimum
• Showing convexity requires showing that the 2nd
  derivative is always nonnegative
• First derivative of Y(Q)
• Y(Q) -AD/Q2 h/2
• Second derivative of Y(Q)
• Y(Q) 2AD/Q3 ? 0 for any Q ? 0

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EOQ Analysis

• Setting Y(Q) 0
• -AD/Q2 h/2 0
• EOQ Q
• The above gives a simple formula for minimizing
  average cost per unit time
• The EOQ formula illustrates the tradeoff made
  between setup/order cost and holding cost
• As A increases, so does order quantity (we order
  less often to incur less fixed costs)
• As h increases, Q decreases (we order more often
  to reduce overall holding costs)

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EOQ Properties

• -AD/Q2 h/2 0 implies hQ/2 AD/Q
• Annual holding cost Annual order/setup cost
• Note that there are D/Q cycles per year if D is
  annual demand rate
• Note that Y(Q) is very flat around Q

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EOQ Cost and Sensitivity

• If we plug Q back into Y(Q) we find that
• Y(Q) cD
• To find how sensitive Y(Q) is to deviations from
  the optimal value of Q we look at the quantity
  Y(Q)/Y(Q), which is always ? 1 (why?)
• We can derive the following formula
• Note that Y(Q)/Y(Q) - 1100 gives the
  percentage deviation from optimal cost for an Q
  other than Q

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EOQ Sensitivity

• Suppose we have incorrectly estimated the setup
  cost A, for example, by 100, i.e., we used 2A,
  but the correct setup cost equals A.
• We used Q but Q
• (1/2) 1.0607
• Deviation of 100 in order cost results in only
  6 deviation from optimal cost
• This shows the robustness of the EOQ to
  deviations in parameter estimates
• Also robust to relaxed modeling assumptions

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Lead Times

• Our EOQ analysis made an assumption that the
  entire order quantity is delivered immediately
• We can easily extend this to incorporate a
  positive constant lead time,?.
• We need only ensure that our order is timed so
  that the inventory level hits zero exactly when
  the order arrives.
• If ? ? T, then we should set our reorder point, R
  D? Q? /T, which equals both demand during
  lead time and the fraction of the order quantity
  consumed during the lead time.

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Lead Times

• Suppose however, that the lead time, ?, exceeds
  the cycle length T.
• If we use R Q?/T, we set our reorder point
  higher than the order quantity, and we will never
  reach this reorder point.
• We can still time the order receipt to coincide
  with a zero inventory level in a future cycle.
  Suppose, for example, ?/T 1.5. Then if we
  order when inventory level equals 0.5Q, the order
  will arrive in 1-1/2 cycles, when inventory level
  equals zero.
• As a rule, we consider the fractional remainder
  of ?/T, i.e., and multiply this by Q
  to get R.

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Economic Production Lot

• The next assumption we relax in the EOQ model is
  that of the delivery of the entire lot at the
  same time
• In a production environment, the lot is typically
  delivered over a period of time at a rate equal
  to the production rate, P.
• We must have P ? D (Why?)
• If P D, what does the inventory level look
  like?
• If P D, we increase our inventory at a rate P
  D.
• If P D we cannot produce indefinitely (why
  not?)
• During the time we are not producing, inventory
  decreases at a rate equal to D.

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Economic Production Lot

• The Figure illustrates the inventory level over
  time.
• We analyze this in the same way we analyzed the
  EOQ.
• Note that when we order Q, our inventory never
  reaches Q
• We must determine H to find the maximum inventory
  level.
• Observe that
• H (P D)T1 H DT2 T1 T2 T Q/D

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Economic Production Lot

• From this we determine that H (1 D/P)Q.
• We still have average annual procurement cost
  equal to cD, and average annual setup cost equal
  to AD/Q.
• The average annual holding cost equals h
  multiplied by the area of the triangle, divided
  by the cycle length, T.
• This gives h(1/2)(1 D/P)Q.
• The average annual cost then equals
• Y(Q) cD AD/Q (hQ/2)(1 D/P)

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Economic Production Lot

• Suppose we let h h(1 D/P) and rewrite the
  equation as
• Y(Q) cD AD/Q hQ/2
• This is the exact same form of the cost equation
  we derived in the EOQ case, except h replaces h.
• This implies that the Economic Production Lot
  size equals Q
• Note that if P ?, h h and we have the EOQ as
  a special case of the EPL.

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Dynamic Deterministic Demand

• EOQ model assumes constant demand rate, often a
  dubious assumption
• If we want to deal with general dynamically
  changing demand, we must focus on discrete-time
  models
• We allow demand to vary in daily, weekly, or
  monthly buckets, or periods
• This approach minimizes total costs over a finite
  planning horizon consisting of a fixed number of
  periods

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Dynamic Deterministic Demand

• The parameters of a dynamic lot sizing problem
  are
• T, number of periods in the planning horizon
• Dt, demand in period t.
• ct, variable production cost in period t.
• At, setup cost in period t.
• ht, holding cost per unit remaining at the end of
  a period
• It, inventory at the end of period t (a decision
  variable).
• Qt, Lot size (production quantity in period t (a
  decision variable)

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Dynamic Deterministic Demand

• We wish to satisfy all demand until the end of
  the time horizon at minimum total production and
  holding cost.
• Some potential policies
• Lot-for-lot rule Setup in every period and
  produce the requirements for that period.
• Maximum setup costs, minimum (zero) holding costs
• Is this likely to be an optimal policy?
• Fixed Order Quantity Any time we produce we
  produce the same amount, Q.
• Is this likely to be an optimal policy?

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Wagner-Whitin Model

• Wagner and Whitin (1958) provided a method to
  determine an optimal solution for this problem
• Their method relies on the following
  zero-inventory production property
• An optimal solution exists in which either the
  inventory carried from period t 1 to t equals
  zero, or we produce nothing in period t, i.e.,
  It-1Qt 0 for all t.
• Try to provide an intuitive argument for the
  justification of this property
• This property allows us to consider only a subset
  of the possible production quantities in any
  period, i.e., when I setup in period 1, I either
  produce D1 units, D1 D2 units, D1 D2 D3
  units,

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Wagner-Whitin Approach

• How does this property help us?
• We use a dynamic programming approach, in which
  we consider only a subset of the time horizon at
  each step. (Note that if ct is the same for all
  periods, then the total production costs will be
  fixed and we need not consider these costs in
  making our decision.)
• Let Zi denote the minimum total cost of an
  i-period problem. Let ji denote the last period
  of production in an optimal solution to an
  i-period problem.

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Wagner-Whitin Approach

• Start with 1-period problem
• Z1 A1 j1 1
• Consider the 2-period problem
• Z2 minA1 h1D2 Z1 A2
• If the first gives min, j2 1 otherwise j2
  2.
• Consider the 3-period problem
• Z3 minA1h1D2(h1h2)D3 Z1A2h2D3Z2A3
• If 1st term gives min, j31 if 2nd, j32
  otherwise j3 3.
• We continue this out until we obtain ZT.

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Wagner-Whitin Approach

• When finished, we can trace our jt values
  backwards to determine the periods in which
  production occurred.
• For example, if jT i, we know the last setup
  was in period i
• We then check ji-1 to see when the previous
  setup occurred, etc.
• At step t, we are computing the minimum cost for
  a t-period problem as follows the minimum cost
  to reach the end of period t equals the minimum
  among
• Min. possible cost if the most recent setup was
  in period 1,
• Min. possible cost if the most recent setup was
  in period 2,
• ,
• Min. possible cost if the most recent setup was
  in period t 1,
• Min. possible cost if the most recent setup was
  in period t.

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Wagner-Whitin Example

• 1) Z1A1100 j11
• 2) Z2min100(1)(50) Z1100 150 j21
• 3) Z3min100(1)(50)(2)(10) Z1100(1)(10)
  Z2100 170 j31
• 4) Z4 min100(1)(50)(2)(10)(3)(50)
  Z1100(1)(10)(2)(50)
  Z2100(1)(50) Z3100 270 j44
• 5) Z5 min100(1)(50)(2)(10)(3)(50)(4)(50)
  Z1100(1)(10)(2)(50)(3)(50)Z2100(1)(
  50)(2)(50)Z3100 (1)(50)Z4100 320
  j54

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Wagner-Whitin Example

• Since j5 4, the last setup was in period 4
• In that setup we produce all demand for periods 4
  and 5, which implies Q4 100
• Next we need j4-1j31, the setup prior to
  period 4 occurs in period 1
• In that setup we produce all demand for periods
  1, 2, and 3, which implies Q1 80.
• Q2, Q3, and Q5 all equal zero
• The minimum total cost equals Z5 320

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Dynamic Lot Sizing Comments

• What are the pros and cons of this modeling
  approach?
• Pros
• Most production facilities plan in periodic
  fashion, i.e., daily, weekly monthly
• Has the ability to handle varying demand
• Computationally simple
• Cons
• Assumes infinite capacity
• Assumes all parameters are known with certainty
• Assumes products are independent
• Assumes zero-inventory production property is
  optimal

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Stochastic Inventory Models

• The real world does not behave deterministically
• Assumptions of deterministic models rarely hold
  in practice
• Deterministic models usually fit best with
  make-to-order systems
• Make-to-stock systems are typically found in
  environments in which demand is non-deterministic
  (stochastic)

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Stochastic Inventory Models

• Stochastic inventory models still must assume
  some knowledge of the nature of demand
• We typically assume that although demand is not
  known with certainty, we can effectively
  characterize a probability distribution for
  demand
• Models for these systems require use of the tools
  of probability and statistics
• We will next consider a few inventory models for
  handling stochastic demand

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Single-Period Stochastic Model

• Newsboy or Christmas Tree Problem
• Assumptions
• One planning period
• Inventory remaining at the end of the period will
  incur a disposal cost or retrieve a salvage value
• All costs are linear in volume
• No fixed order cost component (although the model
  extends easily to include this)
• Penalty cost for unsatisfied demand
• Known probability density function (pdf) or
  probability mass function (pmf) of demand

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Single-Period Model

• Notation
• X A random variable denoting single-period
  demand
• G(x) cumulative distribution function (cdf) of
  demand, i.e., G(x) ProbX ? x.
• g(x) pdf of demand (g(x) dG(x)/dx)
• co Cost () per unit left over after demand
  occurs (overage cost)
• cs Cost () per unit of shortage (shortage
  cost)
• Q Production/order quantity the decision
  variable

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Expected Single-Period Cost

• The expected cost in a period is a function of
  the expected number of units short and the
  expected number of units remaining at the end of
  the period
• Units short maxX Q, 0
• EmaxX Q, 0
• Units over maxQ X, 0
• EmaxQ X, 0

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Minimizing Expected Cost

• We wish to minimize Y(Q) over all Q ? 0
• Y(Q) coEmaxQ X, 0 csEmaxX Q, 0
• d(EmaxQ X, 0)/dQ G(Q)
• d(EmaxX Q, 0)/dQ G(Q) 1
• Y(Q) coG(Q) cs(G(Q) 1)
• Y(Q) (co cs)g(Q) 0 (?Y(Q) convex)
• Setting Y(Q) 0 gives
• G(Q) cs/(co cs), or
• Q G-1(cs/(co cs), where G-1(?) is the
  inverse cdf

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Minimizing Expected Cost

• The optimal order quantity is a function of the
  ratio of the shortage cost to the sum of the
  overage plus shortage cost
• Note that for any valid cdf, 0 ? G(x) ? 1 for any
  x
• For this to hold we require only that co and cs ?
  0
• The ratio cs/(cs co) is sometimes called the
  critical fractile

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Extending to Multiple Periods

• Consider a series of periods for which each
  periods demand is a random variable
• Inventory remaining at the end of a period can be
  used to satisfy demand in the following period
• If all demand is backordered and period demands
  are independent and identically distributed
  (iid), the critical fractile continues to give
  the optimal beginning target inventory level
• It is no longer the optimal order quantity
  since we may have inventory remaining from a
  prior period
• Expected cost in a period is a function of the
  starting inventory level
• With further analysis, we can develop similar
  equations for shortages that result in lost sales
  (under iid demands)

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Single-Period Example

• On consecutive Sundays, Mac, the owner of a local
  newsstand, purchases a number of copies of The
  Computer Journal, a weekly magazine. He pays 25
  cents for each copy and sells each for 75 cents.
  He can return unsold copies to his supplier for a
  10 cent recycling credit during each week.
• Mac has kept records of past demand and has found
  that weekly demand is normally distributed with
  mean ? 11.73 and standard deviation ? 4.74.

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Single-Period Example (contd)

• What is the overage cost?
• The price he paid less the salvage value, co
  0.15
• What is the shortage cost?
• The opportunity cost of lost profit, cs 0.75 -
  025 0.50
• cs/(cs co) 0.5/(0.5 0.15) 0.77
• G(Q) 0.77 Q G-1(0.77)
• We want ProbX ? Q 0.77.
• For a normal distribution we consult a normal
  table (or use Excel function normsinv(0.77)) to
  find that the corresponding z-value equals
  approx. 0.74
• This implies that Q ? z0.77?, or Q 11.73
  (0.74)(4.74) 15.24, or approx. 15.
• We could also use the function norminv(0.77, ?,
  ?)

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Base-Stock Model

• Assumptions
• One-for-one ordering (order one each time a sale
  is made)
• A fixed lead time exists for stock replenishment
• Demands occur one at a time
• Any unmet demand is backordered
• No fixed order cost (or negligible)
• Notation
• L replenishment lead time
• X random variable for demand during lead time

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Base-Stock Model

• Notation (contd)
• G(X) cdf of demand during lead time (in years)
• ? EX mean demand during lead time
• r reorder point (in units)
• R Base-stock level r 1 (in units)
• s r - ?, defined as the safety stock (expected
  amount on-hand when a replenishment arrives)
• Our decision is how to set the value of R, which
  uniquely determines r and s, in order to meet
  some service level

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Base-Stock Model

• Service level can be defined in several ways
• For now we focus on the fill rate, i.e., the
  proportion of demands met immediately from the
  shelf (equivalently the probability that a demand
  is met from stock)
• Note that R on-hand inventory inventory
  on-order Backorders
• If we have inventory on-hand, each time on-hand
  inventory decreases by one, on-order increases by
  one, and vice versa.
• If we have backorders, each time backorders
  increase by one, on-order increases by one and
  vice-versa
• Backorders and on-hand inventory cannot
  simultaneously be positive

63
Base-Stock Model

• Suppose we have just placed an order, immediately
  following a demand
• The item ordered will satisfy the Rth future
  demand, since we use either currently on-hand or
  on-order items to satisfy the current demand and
  all demand up to the Rth future demand
• Since our on-hand on-order backorders always
  R
• The item we just ordered will be able to satisfy
  the Rth future demand if it is received before
  this demand occurs

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Base-Stock Model

• This implies that the probability that this item
  can satisfy demand immediately equals the
  probability that the demand during lead time is
  less than R, i.e., ProbX distribution ProbX ? R continuous
  distribution.
• If the desired fill-rate equals ?, then we need
  only set G(R) ?

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Continuous Review Model

• It is becoming increasingly common through
  information technologies to be able to track
  inventory position continuously
• We define Inventory Position, IP, by the equation
  
• IP on-hand on-order backorders
• Note that in systems with lead times we need to
  keep track of IP to make correct replenishment
  decisions
• Ignoring outstanding orders could resulting in
  high accumulation of inventory
• Ignoring backorders would overstate our ability
  to fill future demand with outstanding orders

66
Continuous Review Model

• We make the following assumptions in our model of
  a continuous review system under stochastic
  demand
• The average demand rate is constant over time
• Placing an order incurs a fixed cost, A
• We have a fixed lead time, L
• We can characterize the distribution of demand
  during lead time
• We will use a (Q, r) policy if inventory
  position is at or below the reorder point, r, we
  order a fixed quantity of Q units

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(Q, r) Model for Continuous Review

• Q determines our average cycle stock, which is
  inventory that is required to keep total order
  costs low
• r determines our safety stock, which is the
  average amount on the shelf when a replenishment
  arrives

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(Q, r) Model for Continuous Review

• The figure shows the average behavior of this
  system
• Given this average behavior, we can characterize
  the expected costs using methods similar to the
  ones we used in our EOQ analysis

69
(Q, r) Model for Continuous Review

• We first define some additional notation
• D Expected annual demand
• X random variable for demand during lead time
• ? EX mean demand during lead time
• G(X) cdf of lead time demand
• g(x) pdf of lead time demand
• c production or purchase cost per item
• h annual holding cost per unit (/unit per
  year)
• b cost per backorder (/stockout)
• s safety stock, s r - ?

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(Q, r) Model for Continuous Review

• We would like to minimize the total expected
  ordering, holding, and backorder costs per year
• Note that, on average, the inventory level falls
  from Q s to s in a cycle which has length T
  Q/D
• This implies an average inventory level equal to
  Q/2 s (the area of the triangle of base Q/D
  plus a rectangle with sides s and Q/D, divided by
  the cycle length, Q/D)
• The expected annual holding cost then equals
  h(Q/2 s), or h(Q/2 r - ?)
• This cycle length implies an average of D/Q
  cycles per year
• This results in an expected annual order cost of
  AD/Q
• We have a bit of work to do to determine the
  expected backorder cost per year

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(Q, r) Model for Continuous Review

• Determining expected annual backorder costs
• We incur a cost of b for each backorder
• We know that there are, on average, D/Q cycles
  per year
• If we can determine the expected number of
  backorders in a cycle, then we need only multiply
  this by the average number of cycles per year to
  get the expected number of backorder per year
• We incur a backorder if the demand during lead
  time, X, exceeds the reorder point, r.
• The number of backorders in a cycle equals maxX
  r, 0

72
(Q, r) Model for Continuous Review

• The expected number of backorders in a cycle then
  equals EmaxX r, 0
• This is similar to what we did for the
  single-period model
• Let n(r) EmaxX r, 0
• Expected number of backorders per year then
  equals (D/Q)n(r)
• We can now formulate our expected annual cost
  equation

73
(Q, r) Model for Continuous Review

• Expected annual costs for a (Q, r) system
• Y(Q, r) (D/Q)A h(Q/2 r - ?) (D/Q)bn(r)
• We next take partial derivatives with respect to
  both Q and r and set them to zero
• This results in
• Q
• G(r) 1 hQ/bD
• Note that the equation for Q is a function of r,
  and the equation for r is a function of Q,
  implying that well need one to get the other

74
(Q, r) Model for Continuous Review

• The following simple algorithm converges to the
  optimal Q and r
• Step 0 Let r0 solve G(r0) 1 h(EOQ)/bD
• Step 1 Let Qt and let rt solve G(rt) 1
  hQt/bD
• Step 2 If Qt Qt-1 stop with Q Qt and r rt. Otherwise, let
  t t 1 and go to Step 1.

75
(Q, r) Model Insights

• The equations resulting from the (Q, r) model
  provide some interesting insights
• The equation for Q, Q is the same as the
  EOQ equation, with an added term in the numerator
• This means that the order quantity is always
  higher under uncertainty
• Increasing order quantity increases the time
  between replenishments, which results in less
  exposure to stockouts

76
(Q, r) Model With Service Levels

• It is typically very difficult for a manager to
  quantify with certainty the value of b, the cost
  per stockout, due to loss of customer goodwill
  and its effects on future sales
• Note that the higher the value of b, the higher
  Q, and the higher the average service level
• Rather than explicitly stating a value for b, we
  can specify a minimum required service level,
  which is much easier to quantify intuitively

77
(Q, r) Model With Service Levels

• We can reformulate our problem to minimize
  expected inventory investment subject to a
  minimum required service level constraint
• Our average inventory level was Q/2 r - ?,
  implying that our average investment in inventory
  equals c(Q/2 r - ?)
• We can characterize the fill rate as 1 n(r)/Q
• This is 1 minus the proportion of demands
  backordered in a cycle, which gives the
  proportion of demands filled from stock
• We also assume a maximum replenishment frequency,
  F (this implies a minimum time between successive
  orders, or a maximum number of replenishments per
  year)

78
(Q, r) Model With Service Levels

• Our problem now is
• Minimize c(Q/2 r - ?)
• subject to D/Q ? F 1 n(r)/Q ? S, where
  S is the minimum fill rate
• Note that reducing Q reduces inventory
  investment, so here we would like Q as small as
  possible while meeting the constraints
• This implies that we will set Q D/F
• Given this Q, we want the smallest r such that
  n(r) ? Q(1 S), which will result when n(r)
  Q(1 S).

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80
Material Requirements Planning

• First widely available software implementation of
  a manufacturing planning system (IBM 1960s)
• APICS MRP Crusade launched in 1972
• Quickly became the manufacturing planning
  paradigm in the U.S.
• The problems of production planning were all
  solved, right?
• By 1989 total sales and support for MRP systems
  exceeded 1 Billion

81
MRP Systems

• The inventory control mechanisms we studied to
  this point are much better for single-item
  planning
• Many products manufacturers produce have a
  complex bill-of-materials (recipe of components)
• Demand for components is dependent on end-product
  demand (which well call independent demand
  items)
• MRP systems encode the interdependence among
  various end-items and components

82
MRP Overview

• MRP is known as a push system, since it plans
  production according to forecasts of future
  demand and pushes out products accordingly
• MRP planning is based on time buckets (or
  periods)
• Orders (current demand) and forecasts (future
  demand) for end-items drive the system
• These requirements drive the need for
  subassemblies and components at lower levels of
  the bill-of-materials (BOM)

83
MRP Overview

• The end-item demands are translated into a Master
  Production Schedule (MPS)
• MPS contains
• Gross Requirements
• On-Hand Inventory
• Scheduled Receipts
• MRP Procedure
• Netting Subtract out on-hand and scheduled
  receipts from Gross Requirements
• Lot Sizing Given net requirements, determine
  periods in which production will occur, and the
  corresponding lot sizes (often uses Wagner-Whitin
  lot sizing procedure)

84
MRP Overview

• MRP Procedure (contd)
• Time Phasing Offset due dates of required items
  based on lead times to determine order release
  times
• BOM Explosion Go down to the next level in the
  BOM and use the lot sizes at the higher level to
  determine gross requirements
• Repeat for all levels in the BOM
• Notes on Netting
• We first use on-hand inventory to satisfy gross
  requirements
• If on-hand inventory is insufficient to meet some
  future demand and scheduled receipts are
  scheduled following this future demand, it
  doesnt make sense to plan a new order, since an
  outstanding order exists

85
MRP Overview

• Notes on Netting (contd)
• Instead of generating any new orders, we first
  attempt to expedite currently scheduled receipts
  so they arrive earlier (we assume this is
  possible, if not, the schedule will be infeasible
  and customers will require notification of a
  delay)
• When currently scheduled receipts are exhausted
  and netted out, we then have a set of net
  requirements that we use as requirements for the
  lot sizing procedure.
• For now well assume one of two very simple lot
  sizing rules
• Lot-for-lot
• Fixed order period (FOP)

86
MRP Example

• Consider the following BOM
• And the table of reqts

87
MRP Example

• We first see how far our on-hand can take us, and
  whether well have to adjust the scheduled
  receipts
• Since the first 3 periods demand equals 85, and
  the sum of the on-hand plus SRs until then is 40,
  we should adjust SRs by expediting the order
  receipt scheduled for period 4
• We can then project on-hand inventory

88
MRP Example

• From period 6 on we have no on-hand or scheduled
  receipts, so the deficit becomes net requirements

89
MRP Example

• Suppose our lot-sizing rule is an FOP 2
• Suppose producing Part A (given that all of its
  components are available takes 2 periods
• We then generate the planned order releases

90
MRP Example

• Next, we move down in the BOM to component 100
• Component 100 has 40 on-hand, no scheduled
  receipts and a 2 week lead time

91
Lot Sizing Rules for MRP

• We discussed three lot-sizing procedures
• Lot-for-lot, FOP, and Wagner-Whitin
• Here we consider additional heuristic rules
• Fixed Order Quantity and EOQ
• Each time we order, we order a set amount
• We cannot directly apply the EOQ formula, since
  we have no constant demand rate, D
• One strategy is to use the average demand per
  period in place of D in the EOQ formula and use
  the result as the fixed order quantity
• We schedule order receipts for periods in which
  we project negative on-hand inventory

92
Lot Sizing Rules for MRP

• Part-Period Balancing Heuristic
• Based on the observation that in the EOQ, the
  optimal solution has order/setup costs equal to
  holding costs
• Also satisfies Wagner-Whitin zero-inventory
  production property
• We begin with a setup in the first period with
  net requirements, call this period i. We then
  consider the total holding costs incurred if we
  satisfy demand for period i only, periods i and i
  1, periods i, i 1, i 2, etc., until holding
  costs exceed the setup cost.
• We next decide the number of periods for which we
  will produce based on which of these options
  results in holding costs closest to setup cost.
• We then repeat this process. For example, if the
  past decision was to produce for periods i, i
  1, , i k, we repeat, beginning with a setup in
  period k 1.

93
Lot-Sizing Rules

• Part-Period Balancing Example
• Suppose our requirements for the next 9 periods
  are
• (0, 15, 45, 0, 0, 25, 15, 20, 15)
• Let A 150, and let h 2 per unit per period
• Our first setup is in period 2
• Since 90 is closer to 150, we use the setup in
  period 2 to satisfy demand for periods 2 and 3
  (Q2 60)

94
Lot-Sizing Rules

• Part-Period Balancing Example (contd)
• Our next setup is in period 6
• The setup in period 6 covers demand until period 8

95
More Lot-Sizing Rules

• Least-Unit Cost Heuristic
• Do a setup in the first period necessary (call
  this period i), then
• Work forward, period by period (as with PPB) and
  calculate the average cost incurred per unit
• Stop at the first period in which the cost per
  unit increases, call this period i k.
• The setup in period i covers demand from period i
  to period i k 1.

96
More Lot-Sizing Rules

• Silver Meal Heuristic
• Do a setup in the first period necessary (call
  this period i), then
• Work forward, period by period (as with PPB) and
  calculate the average cost incurred per period
• Stop at the first period in which the cost per
  period increases, call this period i k.
• The setup in period i covers demand from period i
  to period i k 1.

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Safety Stock and Safety Lead Times

• MRP assumes data are deterministic
• Lead times are fixed
• Demand requirements are certain
• Lot size yields are 100
• This is clearly not the case in most production
  environments
• Safety stock inflates requirements to buffer
  against demand uncertainties
• Safety lead times inflate expected lead time to
  ensure supply availability at production stages
• Also inflate requirements based upon expected
  yield
• If yield equals y, multiple requirements by 1/y

98
MRP Problems

• MRP does not account for production capacity
  limits (or their effects on lead times)
• Inflated safety lead times lead to high WIP
  levels
• System nervousness MRP is not robust to changes
  in customer requirements
• Replanning the current schedule based on changes
  can lead to infeasible schedules
• Frozen zones specify a number of periods in which
  the schedule is fixed (cannot be changed)
• Can lead to problems with sales and marketing
  depts.
• Time fences are usually used, where the first X
  weeks are absolutely frozen, the next Y weeks can
  allow changes with a possible customer financial
  penalties, and beyond X Y weeks is open for any
  changes

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100
Manufacturing Resources Planning

• As MRP became known for problems in dealing with
  capacity and uncertainties, it became apparent
  that enhancements were necessary
• Manufacturing Resources Planning (MRP II) came
  about in response to these concerns
• MRP II embeds planning and control functions
  around the MRP functionality to make it more
  responsive to these problems

101
MRP II Hierarchy
102
MRP II

• Long-Range Planning
• Forecasting short-term and long-range
• Feeds Demand Management function (Intermediate)
• Resource planning
• Long-term capacity requirements
• Feeds into Aggregate Planning
• Aggregate planning
• Production, staffing, inventory, overtime levels
  over long term

103
MRP II

• Intermediate Planning
• Demand Management
• Actual and anticipated orders
• Available to promise (ATP)
• Compares committed production to available and
  planned production
• Master Production Scheduling (MPS)
• With help of rough-cut capacity planning to
  create a capacity-feasible MPS
• Rough-Cut Capacity Planning
• Quick check of critical (bottleneck) resources
  check capacity feasibility of potential MPS
• Uses Bill-of-Resources for each item on the MPS
  (hours required on critical resources)

104
MRP II

• MRP module performs the MRP functions we
  discussed earlier
• Feeds the job pool
• Job release function (short-term control) decides
  how to allocate parts to jobs
• Capacity requirements planning (CRP)
• More detailed check of production schedule output
  from MRP
• Does not generate a capacity-feasible plan
  rather it shows the required resource commitments
  given the MRP output
• Helps user identify problem sources
• Generates load profile for each processing center

105
MRP II

• Short-term Control
• Shop floor control
• job dispatching (sequencing jobs)
• input/output control (WIP level monitoring to
  determine whether job release rate is too fast or
  slow)
• We will cover shop floor control in more detail
  towards the end of this course

106
Just-in-Time (JIT) Manufacturing

• JIT in its broadest sense consists of a
  manufacturing paradigm quite different from the
  MRP paradigm
• JIT encompasses a variety of ideas and
  manufacturing principles and practices
• The origins of JIT are typically attributed to
  Toyotas manufacturing systems
• The success of Japanese auto manufacturers in the
  1970s and 80s is largely attributed to JIT
  practices
• Deteriorating performance of U.S. and European
  firms led to a desire to understand the source of
  the Japanese competitive advantage and eventually
  resulted in implementation of JIT (with varying
  success) at many U.S. firms

107
JIT Manufacturing

• JIT goals The seven zeroes
• Zero defects
• Zero excess lot size
• Zero setups
• Zero breakdowns
• Zero handling
• Zero lead time
• Zero surging
• Unachievable goals, but the point is clear

108
JIT Manufacturing Enablers

• JIT is often seen as synonymous with the Kanban
  pull production system (we will look at this in
  more detail later)
• Requires smooth production levels
• Translate monthly output requirements into an
  hourly production rate
• Mixed model lines necessitate low setups for this
  to work
• Dealing with variability capacity buffers
• Plan buffer capacity in each day for possible
  disruptions

109
JIT Manufacturing Enablers

• Setup Reduction
• U.S. manufacturers typically regarded setup times
  as given constraints
• Japanese (Toyota) continuously strived to create
  new way to reduce setup times
• Internal vs. external setups
• Make as much of setup external as possible
• Standardize product designs (commonality)

110
JIT Manufacturing Enablers

• Cross training
• U.S. traditionally held workers at one task
• Japanese focused on enabling workers to perform
  multiple tasks, which reduces boredom, increases
  flexibility, and gives workers broader view
• Plant layout
• U-shaped cells became common for labor-intensive
  lines to enable workers to move quickly between
  stations

111
JIT Manufacturing Enablers

• Total Quality Management (TQM)
• Probably the first elements of JIT to be adopted
  in the U.S.
• Although much of this was initially rhetoric and
  not practiced
• JIT requires a low amount of rework to be
  effective
• To keep production levels smooth
• Again, a case of U.S. manufacturers often taking
  rework as a necessary evil, while Japanese
  manufacturers took a more scientific root-cause
  approach

112
JIT Manufacturing Enablers

• TQM (contd)
• Seven principles essential to quality practice
• Statistical Process Control (SPC)
• Easy-to-see quality (charts, displays)
• Compliance to specifications at all stations
• Line stopping (empowers line workers)
• Correcting own errors (as opposed to U.S. rework
  lines)
• 100 inspection (if possible, usually with
  automation)
• N 2 approach Inspect first and last job on
  line
• Continuous improvement
• Always strive to the zero-defect goal

113
Kanban Production System

• Kanban is an alternative to MRP for controlling
  production flow
• MRP pushes out production according to forecasts
• Work releases are scheduled in advance
• Kanban pulls production through the system based
  on actual demand
• Work releases authorized as downstream demand
  occurs
• Kanban is loosely translated from Japanese as
  card
• Kanban systems attach Kanban cards to jobs in the
  system these cards are used to authorize
  production
• Systems control WIP through number of cards No
  working ahead of downstream stations

114
Kanban Schematic

• Onecard system (Fig. 4.5 in Text)

115
Problems of the Past

• We have now covered standard, traditional
  inventory control methods, MRP and MRP II, and
  JIT
• Each of these systems has improved productivity
  relative to past practices
• Each system has brought about a new set of
  problems and challenges
• We briefly consider why these different methods
  have been successful in some cases and failed
  miserably in other cases

116
Traditional Methods

• Pros
• Take a scientific approach to management
• Sharpen managerial insight by characterizing
  critical tradeoffs
• Cons
• Traditional inventory models and methods optimize
  costs under the model assumptions
• Model assumptions fail to hold in practice
• Constant demand rate, fixed and known setup cost,
  infinite capacity, complete shortage backlogging
• We often generate an optimal solution for the
  wrong problem
• Models typically assume a single-stage or product
• Real production systems are much more complex

117
MRP Paradigm

• Pros
• Perform extremely well in make-to-order systems
  with little uncertainty and ample capacity
• Deterministic demand is not a poor assumption
• Efficiently encodes the relationships and
  interdependencies of highly complex production
  systems
• Cons
• MRP systems have been hugely successful in terms
  of industry usage and sales
• Users have more often than not been less than
  pleased with the problems associated with MRP
• MRP is based on a flawed model
• Unlimited capacity, deterministic demand and lead
  times

118
JIT Issues

• Pros
• Emphasis on quality control and improvement,
  setup reduction, WIP reduction, tight supplier
  relationships
• Pull system responsive to state of the system
• Cons
• Implementation is long term investment
• Complete paradigm and attitude shift for workers
• Reliance on supplier relationships
• Requires continuous attention to detail
• Requires real management commitment, not rhetoric

119
Enterprise Resources Planning

• ERP system sales and related consulting
  expenditures have skyrocketed over the past five
  years
• SAP R/3
• Baan
• Peoplesoft
• What are ERP systems and how are they different
  from MRP and MRP II?
• ERP systems dont just target production
  operations, but all operations of the firm
• A control system for the entire enterprise

120
ERP Systems

• According to a 1997 Business Week article, SAPs
  R/3 software can
• act as a powerful network that can speed
  decision-making, slash costs, and give managers
  control over global empires at the click of a
  mouse
• Despite this hyperbole, ERP systems can at least
  do the first two things (speed decisions and
  slash costs) if used correctly
• This cost slashing, however, ironically comes at
  a steep price (SAP implementations typically cost
  in excess of 1 Million)

121
ERP Systems

• ERP systems integrate all enterprise-wide
  systems
• Finance
• Accounting
• Manufacturing
• Human Resources
• Marketing Sales
• It provides consistency across the enterprise in
  terms of user interfaces, data, and vendor and
  customer relations
• For example, when sales closes a deal and enters
  it in th