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EIN 6392 Manufacturing Management


Interchangeable parts (and workers) Economies of Scale. Manufacturing History ... of mass markets. Mass Retailers (Sears) Horizontal and Vertical Integration ... – PowerPoint PPT presentation

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Title: EIN 6392 Manufacturing Management

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

(No Transcript)
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

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

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

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

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

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
  • Sloan devised a central corporate structure to
    oversee independent operating divisions of GM

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

Modern Manufacturing Corporation
  • Sloans collection of scientific management,
    organizational structuring, and market emphasis
    created a model for the modern U.S. manufacturing
  • 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

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

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

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

  • 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
  • Lack of focus on core competencies
  • Mergers and acquisitions led to overall

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

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

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

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

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
  • It is, however, very robust to situations not
    fitting the assumptions

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

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,
  • Q Order quantity/Lot size

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
  • Inventory falls at constant linear rate
  • Leads to optimality of a fixed order quantity/lot
    size with each order/setup

EOQ Analysis
  • We would like to determine the order quantity, Q,
    that minimizes the average cost incurred per unit
  • 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

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

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

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

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

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)

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

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

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

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.

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.

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
  • If P D, we increase our inventory at a rate P
  • If P D we cannot produce indefinitely (why
  • During the time we are not producing, inventory
    decreases at a rate equal to D.

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

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)

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.

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
  • 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

Dynamic Deterministic Demand
  • The parameters of a dynamic lot sizing problem
  • 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
  • Qt, Lot size (production quantity in period t (a
    decision variable)

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?

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

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.

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
  • 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.

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
  • 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.

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)
    Z2100(1)(50) Z3100 270 j44
  • 5) Z5 min100(1)(50)(2)(10)(3)(50)(4)(50)
    50)(2)(50)Z3100 (1)(50)Z4100 320

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

Dynamic Lot Sizing Comments
  • What are the pros and cons of this modeling
  • 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

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 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
  • 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

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

Single-Period Model
  • Notation
  • X A random variable denoting single-period
  • 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
  • Q Production/order quantity the decision

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

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

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
  • For this to hold we require only that co and cs ?
  • The ratio cs/(cs co) is sometimes called the
    critical fractile

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)

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.

Single-Period Example (contd)
  • What is the overage cost?
  • The price he paid less the salvage value, co
  • 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, ?,

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

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

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
  • Backorders and on-hand inventory cannot
    simultaneously be positive

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
  • The item we just ordered will be able to satisfy
    the Rth future demand if it is received before
    this demand occurs

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
  • If the desired fill-rate equals ?, then we need
    only set G(R) ?

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
  • Ignoring outstanding orders could resulting in
    high accumulation of inventory
  • Ignoring backorders would overstate our ability
    to fill future demand with outstanding orders

Continuous Review Model
  • We make the following assumptions in our model of
    a continuous review system under stochastic
  • 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

(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

(Q, r) Model for Continuous Review
  • The figure shows the average behavior of this
  • Given this average behavior, we can characterize
    the expected costs using methods similar to the
    ones we used in our EOQ analysis

(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
  • b cost per backorder (/stockout)
  • s safety stock, s r - ?

(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
  • 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
  • We have a bit of work to do to determine the
    expected backorder cost per year

(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

(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

(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

(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
  • Step 2 If Qt Qt-1 stop with Q Qt and r rt. Otherwise, let
    t t 1 and go to Step 1.

(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

(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

(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

(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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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

MRP Systems
  • The inventory control mechanisms we studied to
    this point are much better for single-item
  • 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
  • MRP systems encode the interdependence among
    various end-items and components

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
  • 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)

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)

MRP Overview
  • MRP Procedure (contd)
  • Time Phasing Offset due dates of required items
    based on lead times to determine order release
  • 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
  • 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

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
  • 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)

MRP Example
  • Consider the following BOM
  • And the table of reqts

MRP Example
  • We first see how far our on-hand can take us, and
    whether well have to adjust the scheduled
  • 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

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

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

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

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

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.

Lot-Sizing Rules
  • Part-Period Balancing Example
  • Suppose our requirements for the next 9 periods
  • (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)

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

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.

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.

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
  • 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
  • If yield equals y, multiple requirements by 1/y

MRP Problems
  • MRP does not account for production capacity
    limits (or their effects on lead times)
  • Inflated safety lead times lead to high WIP
  • 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
  • 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

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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

MRP II Hierarchy
  • 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

  • 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)

  • 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

  • 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
  • We will cover shop floor control in more detail
    towards the end of this course

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
  • 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

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

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

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)

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

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
  • 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

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
  • 100 inspection (if possible, usually with
  • N 2 approach Inspect first and last job on
  • Continuous improvement
  • Always strive to the zero-defect goal

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
  • Kanban is loosely translated from Japanese as
  • Kanban systems attach Kanban cards to jobs in the
    system these cards are used to authorize
  • Systems control WIP through number of cards No
    working ahead of downstream stations

Kanban Schematic
  • Onecard system (Fig. 4.5 in Text)

Problems of the Past
  • We have now covered standard, traditional
    inventory control methods, MRP and MRP II, and
  • 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

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

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
  • 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

JIT Issues
  • Pros
  • Emphasis on quality control and improvement,
    setup reduction, WIP reduction, tight supplier
  • 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

Enterprise Resources Planning
  • ERP system sales and related consulting
    expenditures have skyrocketed over the past five
  • 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

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
  • 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)

ERP Systems
  • ERP systems integrate all enterprise-wide
  • 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
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