ORF 411 14 Deterministic inventory problems · Basic inventory problems The lot sizing problem »...

Lecture outline

Basic inventory problems The economic order quantity An inventory gameMultiperiod lot sizing

» Math programming formulation» Heuristics» Wagner-Whitin algorithm

© 2013 W.B. Powell 1


Basic inventory problems

Examples:» Products:

• Customers consume products over time.• Store replenishes periodically.

» People with specialized training:• People randomly leave the company over time.• Company periodically hires new graduates.

» Water (management of dams).• Rainfall randomly replenishes reservoirs.• Release water from dam to maintain level.

» Oil being stored in storage tanks:• Oil is steadily consumed.• Periodically is replenished from tankers.



Examples:» Housing stock

• Houses are continually being purchased.• Developers produce new developments or apartment buildings.

» Financial resources (startup company)• Cash is used to build up the company.• Periodically fresh capital is raised from venture capitalists.

» Features in a software program:• Accumulating features in a software program in response to

user requests and the ideas of developers, or due to bug fixes.• Periodically ship a new version of the program.

» Purchasing stock:• Funds become available for investment.• Periodically purchase new shares of stock.



Mutual fund cash balance

Stock market InvestorCash

How much cash do we keep on hand to strike a balance between the deposits and withdrawals of investors, and the behavior of the market?



There are a number of ways to refer to storing resources for the future:» Physical resources

• Inventory• Stockpile • Stock

» Financial resources• Savings • Nest egg • Reserve



Reasons for holding inventories:» Economies of scale

• Batches of goods• Discounts (purchasing)• Transportation economies (e.g. shipping in bulk)

» Uncertainties• Demand• Order lead times• Supply/price of raw materials (OPEC)• Supply/price of components (strikes)• Quality control

» Speculation • commodities prices• currency fluctuations



Reasons for holding inventories» Transportation

• In-transit or pipeline inventories» Smoothing production

• Respond to seasonal patterns in demand• Seasonal production of some items

– Certain foods– Syrup– Snow– Students

» Control costs• Lower inventories requires more sophisticated control systems



The lot sizing problem» Often, there are economies of scale when ordering new

resources:• Raising operating capital

– There is a fixed cost to going to the capital markets– Just as much work to raise $1m as $5m

• Shipping the latest version of a software program– New features are added over time– There is a fixed cost of shipping a new version of the code– How many new features do you add before you ship the

code?• Ordering new product for a store shelf

– Fixed cost for placing and shipping an order

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An aging and replenishment process (negative drift):» State is inventory – drift is due to customer demand.

Time

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Indexing time:» Deterministic indexing – Index based on when

something happens.» Stochastic indexing – Index based on when something

becomes known.» Deterministic indexing:

0t 1t 2t 3t

0t 1t 2t 3t 4t

0D 1D 2D 3D

0 0,R x 1 1,R x 2 2,R x 3 3,R x 4 4,R x

Continuoustime

Discretetime

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Basic inventory equation» Deterministic indexing

» In deterministic indexing, everything is modeled at the beginning of a time period.

1 max 0,t t t t t t tR R x D R x D

Demand (random at time t)

New orders

State (e.g. inventory level)

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Stochastic indexing» Information arrives continuously over time» A variable indexed by t contains exogenous information

up through time t.

» In stochastic indexing, everything is indexed at the end of a period.

1t 2t 3t 4t

0t 1t 2t 3t 4t

1D 2D 3D 4D

0 0,R x 1 1,R x 2 2,R x 3 3,R x 4 4,R x

Continuoustime

Discretetime

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Basic inventory equation» Stochastic indexing

1 1 1max 0,t t t t t t tR R x D R x D

Demand (random at time t)

New orders

State (e.g. inventory level)

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Inventory with positive drift» Cash is deposited in an account. Periodically the cash

is invested in batch amounts to reduce transaction costs.

Time

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Basic inventory with positive drift» Blood donations, water reservoirs, …

1 1t t t tR R x D

Exogenous drift

Endogenously controlled shift

State (e.g. accumulation)

Outline



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The economic order quantity

Order costs» Fixed cost of placing an order

• Paperwork, forms, telephone• Sending a truck out• Setting up a machine• Price structure imposed by supplier

» Variable cost of ordering a certain amount of product• Variable cost may be fixed (a linear function).• Or there may be economies of placing larger orders.

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Holding cost» Storage (heat, electricity, supervision, etc.)» Taxes and insurance» Breakage, spoilage, deterioration and obsolescence

• Careful with these - these “costs” convert to reduction in quantity.

» Opportunity cost (interest)• Hurdle rate for a company is generally much higher than bank

interest rates.• Let I = “interest rate”• cp = purchase cost of item• ch = Icp = holding cost (be careful with units; if I is interest rate

per year, ch is holding cost per year).

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Stockout cost» Cost of lost customers» Cost of pushing orders to future time periods

Notes:» Stockout costs are not relevant in our simplest

inventory system, because they cannot happen. » Stockouts arise when:

• Demand is random• Demand varies over time with production capacities• Order costs may vary as a function of time, possibly exceeding

the “benefit” of covering demand.

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Some assumptions:» Demand is deterministic with rate per unit time.

» Rate is constant – stationary process.

» Costs are stationary.

» Orders arrive immediately.

» All orders must be filled.

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t 1 1

t

The basic inventory equation:

[ ]where: Inventory at start of time Amount ordered at time Demand during period tNota

t t t

t

t

R R x D

R tx tD

tion:

[ ] max ,0x x

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The cost function:( , ) Total costs during period given order quantity and initial

inventory

( ) ( , )where:

( ) Order costs

0

0 0

t t t

to h

t t t

ot

pt t

t

c x R t xR

c x c R x

c x

K c x xx

0

Unit purchasing costs( , ) Holding costs

Unit holding costs per time

p

ht t

tht t

h

cc R x

c R x z dz

c

K

Order quantity xt

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The cost function:We can transform the nonlinear cost function into a linear one:

( )

where: 0 M "big M" (0,1)

If 0 then we force 0.

o pt t t

t

t t

t

t t

c x Ky c x

xx Myy

y x

Now we have transformed a nonlinear costfunction into a linear one, but we have added an integer variable.

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Infinite horizon problem:

, 0

, 0

We would like to solve:

min , ,

This is a really big number! A more formal way to write it is as an average cost:

1lim min , ,

t t

t t

t t tx y t

T

t t tT x y t

c R x y

c R x yT

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The economic order quantityIntuition suggests that we let inventories drop to zero,

and then “order up to” an amount Q:

Inve

ntor

y, R

t

Q

= reorder interval

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Q

Qc

Qc

Q

p

o

pc

intervalorder per costs Purchase )(

Q

intervalorder ofLength interval).per order one make always we(sinceK

intervalorder per cost Order )(tic)determinis

andconstant (assumed unit timeper rate Demandquantityorder

:intervalorder per cost ofin termsproblem theeReformulat

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h

h

2h h

c ( ) Holding cost per order interval (holding cost)*(average inventory)*(length of interval)

c2

Qc c2 2

c( ) Total cost per order interval

Q

Q

Q Q

Q

o h

2h

c ( ) ( ) c ( )

c2

p

p

Q c Q Q

QK c Q

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*

We can try differentiating C(Q) with respect to Q:( ) 0 2 0

2Solving for Q gives us:

Q !!!!!

What went wrong?

p h

p

h

dC Q Qc cdQ

cc

How do we find the optimum value of Q?

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Q

Cost per cycle =K plus quantity proportional to green area

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Q

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Q

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Q

The smaller Q gets, the lower our costs per cycle.

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2

Need to minimize cost per unit time, not cost per order interval.So, we want to solve:

( ) ( ) ( )min ( )/

2

2Differentiating with respect to and setting

p h

p h

C Q C Q C QC QQ Q

QK c Q cQ

K Qc cQ

Q

2

to 0:d ( ) 0

2

hC Q K cdQ Q

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h

Finally, solving for Q gives us:

2K Q The Economic Order Quantity (EOQ)c

Also called the Economic Lot Size.Properties of the optimal solution: 1. Purchase costs do not ente

h

h

r the equation (why?) 2. Order costs per unit time holding costs per unit time:

Order costs per unit time 22K

c

2K Holding costs per unit time 2 2 2c

h

h hh

K K c KQ

Q c c Kc

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The average cost function:

Q

Cos

ts

Holding cost

Order costs

2Kch

hc2K

Total cost

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Sensitivity analysis

Actual order quantity

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

Actual order quantity

Rela

tive

incr

ease

in c

ost

Outline



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An inventory game

The basics:» Random demand (uniform between 0 and 10)» Demands are not revealed until an order is entered.» Cost parameters:

Order cost 25Purchase 10Holding 1Stockout 5

40

Time InventoOrder DemanEndingSold Lost Order PurchaHoldingStocko Total Cum0 20 0 2 18 2 0 0 0 18 0 18 181 18 0 1 17 1 0 0 0 17 0 17 352 17 0 8 9 8 0 0 0 9 0 9 443 9 0 3 6 3 0 0 0 6 0 6 504 6 0 2 4 2 0 0 0 4 0 4 545 4 4 6 2 6 0 25 40 2 0 67 1216 2 6 4 4 4 0 25 60 4 0 89 2107 4 4 5 3 5 0 25 40 3 0 68 2788 3 5 8 0 8 0 25 50 0 0 75 3539 0 8 6 2 6 0 25 80 2 0 107 460

10 2 6 2 6 2 0 25 60 6 0 91 55111 6 2 3 5 3 0 25 20 5 0 50 60112 5 3 5 3 5 0 25 30 3 0 58 65913 3 5 5 3 5 0 25 50 3 0 78 73714 3 5 0 8 0 0 25 50 8 0 83 82015 8 0 2 6 2 0 0 0 6 0 6 82616 6 2 9 0 8 1 25 20 0 5 50 87617 0 8 1 7 1 0 25 80 7 0 112 98818 7 1 6 2 6 0 25 10 2 0 37 102519 2 5 4 3 4 0 25 50 3 0 78 110320 3 4 8 0 7 1 25 40 0 5 70 1173

375 680 108 10 1173

An inventory game Player: Joe

41

Time InventoOrder DemanEndingSold Lost Order PurchaHoldingStocko Total Cum0 20 0 2 18 2 0 0 0 18 0 18 181 18 0 1 17 1 0 0 0 17 0 17 352 17 0 8 9 8 0 0 0 9 0 9 443 9 0 3 6 3 0 0 0 6 0 6 504 6 0 2 4 2 0 0 0 4 0 4 545 4 20 6 18 6 0 25 200 18 0 243 2976 18 0 4 14 4 0 0 0 14 0 14 3117 14 0 5 9 5 0 0 0 9 0 9 3208 9 0 8 1 8 0 0 0 1 0 1 3219 1 20 6 15 6 0 25 200 15 0 240 561

10 15 0 2 13 2 0 0 0 13 0 13 57411 13 0 3 10 3 0 0 0 10 0 10 58412 10 0 5 5 5 0 0 0 5 0 5 58913 5 20 5 20 5 0 25 200 20 0 245 83414 20 0 0 20 0 0 0 0 20 0 20 85415 20 0 2 18 2 0 0 0 18 0 18 87216 18 0 9 9 9 0 0 0 9 0 9 88117 9 0 1 8 1 0 0 0 8 0 8 88918 8 0 6 2 6 0 0 0 2 0 2 89119 2 10 4 8 4 0 25 100 8 0 133 102420 8 0 8 0 8 0 0 0 0 0 0 1024

100 700 224 0 1024

An inventory game Player Jimmie:

© 2013 W.B. Powell 42

Time InventoOrder DemanEndingSold Lost Order PurchaHoldingStocko Total Cum0 20 -1 0 20 -1 0 0 0 20 0 20 1 20 -1 0 20 -1 0 0 0 20 0 20 2 20 -1 0 20 -1 0 0 0 20 0 20 3 20 -1 0 20 -1 0 0 0 20 0 20 4 20 -1 0 20 -1 0 0 0 20 0 20 5 20 -1 0 20 -1 0 0 0 20 0 20 6 20 -1 0 20 -1 0 0 0 20 0 20 7 20 -1 0 20 -1 0 0 0 20 0 20 8 20 -1 0 20 -1 0 0 0 20 0 20 9 20 -1 0 20 -1 0 0 0 20 0 20

10 20 -1 0 20 -1 0 0 0 20 0 20 11 20 -1 0 20 -1 0 0 0 20 0 20 12 20 -1 0 20 -1 0 0 0 20 0 20 13 20 -1 0 20 -1 0 0 0 20 0 20 14 20 -1 0 20 -1 0 0 0 20 0 20 15 20 -1 0 20 -1 0 0 0 20 0 20 16 20 -1 0 20 -1 0 0 0 20 0 20 17 20 -1 0 20 -1 0 0 0 20 0 20 18 20 -1 0 20 -1 0 0 0 20 0 20 19 20 -1 0 20 -1 0 0 0 20 0 20 20 20 -1 0 20 -1 0 0 0 20 0 20

0 0 420 0 420

An inventory game Player:

© 2013 W.B. Powell 43

Time InventoOrder DemanEndingSold Lost Order PurchaHoldingStocko Total Cum0 20 -1 0 20 -1 0 0 0 20 0 20 1 20 -1 0 20 -1 0 0 0 20 0 20 2 20 -1 0 20 -1 0 0 0 20 0 20 3 20 -1 0 20 -1 0 0 0 20 0 20 4 20 -1 0 20 -1 0 0 0 20 0 20 5 20 -1 0 20 -1 0 0 0 20 0 20 6 20 -1 0 20 -1 0 0 0 20 0 20 7 20 -1 0 20 -1 0 0 0 20 0 20 8 20 -1 0 20 -1 0 0 0 20 0 20 9 20 -1 0 20 -1 0 0 0 20 0 20

10 20 -1 0 20 -1 0 0 0 20 0 20 11 20 -1 0 20 -1 0 0 0 20 0 20 12 20 -1 0 20 -1 0 0 0 20 0 20 13 20 -1 0 20 -1 0 0 0 20 0 20 14 20 -1 0 20 -1 0 0 0 20 0 20 15 20 -1 0 20 -1 0 0 0 20 0 20 16 20 -1 0 20 -1 0 0 0 20 0 20 17 20 -1 0 20 -1 0 0 0 20 0 20 18 20 -1 0 20 -1 0 0 0 20 0 20 19 20 -1 0 20 -1 0 0 0 20 0 20 20 20 -1 0 20 -1 0 0 0 20 0 20

0 0 420 0 420

An inventory game Player:

Outline



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Math programming formulation

What do we do when the demands are nonstationary?» Dt= the forecasted demand for time period t, 0 < t < T.

We are going to use a point estimate of the demand, which produces a deterministic model.

» T = the planning horizon.

Objective function:

1

, 0min , ,

t t

T

t t t tx y tc R x y

© 2013 W.B. Powell 46


1

0

T

ttD

Production arcs

Consumption

The optimization problem can be visualized as a network:

1

0

T

ttD

Inventory

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Math programming formulation Integer programming formulation:

Notation:Activity variables:

inventory at beginning of period t demand during period starting at

Parameters: Unit holding cost per time period Unit purchase cost Fixed order cost

Dec

t

t

h

p

R D t

ccK

ision variables:amount ordered in period t

1 00 0

t

tt

t

x x

y x

t 1t

, t tR x

tD

© 2013 W.B. Powell 48


1

0,

1

Objective function:

min

subject to:

(M big number)

0(0,1)

This is an integer programming problem, which can be solved usingcommerc

T p ht t t t t ttx y

t t t t

t t

t t t

t

t

Ky c x c R x D

R R x Dx Myx D Rxy

ial solvers such as Cplex and Gurobi.

Math programming formulation We implement our math programming formulation as a

rolling horizon procedure» Optimize over 0-4, implement time 0

» Roll to time 1, see new information, solve updated problem for time periods 1-5:

» Roll to time 2, see new information, solve updated problem for time periods 2-6:

49

0 1 2 3 4 5 6

0 1 2 3 4 5 6

0 1 2 3 4 5 6© 2013 W.B. Powell


Rolling horizon procedures» These are deterministic approximations of the problem

over a planning horizon H

© 2013 W.B. Powell 50

' ' ' ' ' ''

' '' ,..., ' ,...,

, ' 1 ' ' '

' '

' ' '

'

Objective function:

min

where = , y =

subject to:

(M big number)

0

t t

t H p htt t tt tt tt ttt tx y

t tt t ttt t t H t t t H

t t tt tt tt

tt tt

tt tt tt

tt

Ky c x c R x D

x x y

R R x Dx Myx D Rx

' (0,1)tty

© 2013 W.B. Powell 51

Heuristics

Silver-Meal heuristic (Least average cost)

» One of the best known and most widely used heuristics in supply chain management.

periods. timeTnext for theenough Order s.TSet C(s). 1)C(s when Stop C(s). ..., C(2), C(1), Calculate

s1

periods times nextover theorder weifunit timeper cost average C(s)Let

1

0

s

tt

h tDcK

© 2013 W.B. Powell 52

Heuristics

Least unit cost

not???).(why $/day of in terms measured is one-Nomeasured.actually are managers way thereflectscost unit Least

s.TSet C(s). 1)C(s when Stop C(s). ..., C(2), C(1), Calculate

periods times next over theorder weif producedunit per cost average C(s)Let

1

0

1

0

s

tt

s

tt

h

D

tDcK

© 2013 W.B. Powell 53

Wagner-Whitin algorithm

Example: Seasonal TV demand» Parameters:

» Demands (in 1000’s):

Time Demand0 11 32 53 2

Setup cost 8Order cost 1Holding cost 1.3

© 2013 W.B. Powell 54


A network representation:

» The set of decisions represents a shortest path problem over a specialized network:

» The cost on each arc is the cost of the decision, including order costs and all holding costs.

» What does the optimal solutionn look like?

0 1 2 3 4

© 2013 W.B. Powell 55

Wagner-Whitin algorithm Properties of optimal solution

» We only need to make decisions when the inventory is zero.

» This means our decision variable is not the quantity, but the number of time periods into the future that we need to cover.

0 1 2 3 4Order enoughjust for time period 0

Order enough justfor time periods 0 and 1

Order enough justfor time periods 2 and 3

The state that we are at time period2 with zero inventory.

© 2013 W.B. Powell 56

Wagner-Whitin algorithm Start with the final node:

$43.7

$43.7$10

$10

$17.6

$17.6

$29.7

$29.7 $0

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$43.7

$43.7$10

$10

$17.6

$17.6

$29.7

$0$13

$22.5

$33.9

Links into time period 2:

$33.5 $29.7$28.6 $11

$15.9

© 2013 W.B. Powell 59


$43.7

$43.7$10

$10

$17.6

$17.6

$29.7

$0$13

$22.5

$33.9

$33.5 $29.7$28.6 $11

$15.9

Links into time period 1: Links into time period 1:

$9

© 2013 W.B. Powell 60

Wagner-Whitin algorithm Finally, we walk forward in time:

0 1 2 3 4

We use the values computed in the backward pass to walk forward and compute decisions.

© 2013 W.B. Powell 61


Strengths:» Very fast» Handles very general cost functions

• You can use virtually any shape order cost function.

» Handles time-dependent data (e.g. seasonal data, day of week effects or hour of day patterns).

» Handles forecasts of the future (which is a form of time-dependency).

© 2013 W.B. Powell 62


Limitations of this model:» Assumes demands are deterministic!!!

• “Optimal” solution is not really optimal.• Drives inventories to zero, which will create stockouts.• Have to reoptimize as forecasted demands change.

» Limitations:• Computationally demanding when you have to solve 100,000

problems (Wal-Mart!).• Solutions are not “obviously” better than good heuristics under

realistic conditions.• Gets complicated if you have multiple items and joint capacity

constraints (need to use integer programming formulation)

» But:• Serves as a useful subproblem in the context of larger

applications.• Highlights behavior of the problem.

© 2013 W.B. Powell 63


Important generalizations:» Upper bounds on order quantities

• What if we cannot order more than ut in time period t?

» Upper bounds on production and multiple items:• This is the problem that actually arises in practice.• Called the “capacitated multi-item lot sizing problem.”

» Limit on total production time, in the presence of setup times.

• The literature on setup times is very sparse.

ORF 411 14 Deterministic inventory problems · Basic inventory problems The lot sizing problem »...

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