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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

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

GM - 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

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

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

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

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

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

assumptions - 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,

obsolescence - 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

time - 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

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

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

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.

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

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

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

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)

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

units,

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

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.

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.

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

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

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

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)

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

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

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

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

x - For this to hold we require only that co and cs ?

0 - 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

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, ?,

?)

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

vice-versa - 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

R - 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

distribution. - 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

decisions - 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

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

(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

(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

(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 - ?

(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

(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

equation

(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

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.

(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

year)

(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

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

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)

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

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

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)

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

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

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

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)

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

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

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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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

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

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)

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

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

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

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

disruptions

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

stations

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

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

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

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

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

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

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

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

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

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)

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