orf 411 13 the budgeting problem - princeton university 411... · notational style variables »...

© 2013 W.B. Powell

Outline

Notational style The budgeting problem

» The basic budget problem» Budgeting with an S-curve effect» The dynamic budgeting problem

© 2013 W.B. Powell

Notational style

Our challenge is converting real problemsinto a form that can be studied on the computer.

0

1 1

min ( , ( ))

( )

, ,

T

t t tt

t t t t

Mt t t t

C S X S

x X S

S S S x W

X

© 2013 W.B. Powell

Notational style

The challenge of communicating:» “The elevator lift is being fixed for the next day.

During that time we regret that you will be unbearable.” - Sign in a Bucharest business.

» “We will take your bags and send them in all directions.” - Airline office in Copenhagen.

» “You are invited to take advantage of the chambermaid.” - A Japanese hotel.

» “Please leave your values at the front desk” - sign at a Paris hotel

» From a tailor shop: “Place orders early because in a big rush we will execute customers in strict rotation.”

© 2013 W.B. Powell

Notational style

The challenge of communicating:» In an Acapulco hotel: “The manager has personally

passed all the water served here.” » “Please do not feed the animals. If you have any

suitable food, give it to the guard on duty.” - Sign at a Budapest zoo.

» In a Swiss eatery: “Our wines leave you nothing to hope for.”

» “Ladies, leave your clothes here and spend the afternoon having a good time.” A laundry in Rome.

© 2013 W.B. Powell

Notational style Principles of good mathematical notation:

» Notation is a language - if the language is hard to learn, others will have difficulty speaking it.

» Minimize the number of variables you introduce (i.e. keep your vocabulary small).

» Make variables as mnemonic as possible.» Follow consistent, standard conventions.» Organize your variables into natural groupings:

© 2013 W.B. Powell

Notational style Variables

» A basic variable - lower case and script:» Use subscripts to identify elements of a vector:

» Use superscripts to create different flavors of a variable:

x

tij ti tx x x

cxxcxcxc

ccccxxxx

cx

cx

cx

hhsspp

hsphsp

hh

ss

pp

costs Total

:hold

:sell

purchase:Costs:Decisions

,,,,

:

Notational style

Variables» Never use variables with more than one letter:

» Using multiple letter variables is popular in the business community. If you are presenting an equation in a memo to a management audience, it is perfectly reasonable to write:

This is acceptable in this setting, because the variables define themselves, and you are never going to manipulate this equation.

© 2013 W.B. Powell

Purchasing cost (is it times ?)Battery charge at time t

PC P CBC t

0 1 2t t tFleet Const Const Tons Const speed

© 2013 W.B. Powell

Notational style

Sequencing subscripts:» When you have multiple subscripts:

• Roughly sequence them in the order that you would place them in a summation:

• Always model discrete time periods as a subscript, because it is an element of a vector over different time periods.

Dollars invested in asset in time period .

( )

( )

ti

ti tit i

t ti i

x i t

C x c x

x x

© 2013 W.B. Powell

Notational style Arguments, superscripts and hats:

» Use arguments only when there is a functional dependence:

» If we are estimating a variable iteratively, again use a superscript to identify different versions of the variable:

Use n for iterations. If you need outer and inner iterations, use n and m.

» Batch vs. recursive data

( ) or ( )F x a x

1 ( )n nx F x

1 2

In statistics, it is common to represent a set of data points( , ,..., ,..., ), where is a vector with element .i

n N n n

nx x x x x x

© 2013 W.B. Powell

Notational style

Choice of letters:» Use hats and bars (etc) to indicate different

estimates/observations of the same variable:

» Constants:• General rule: a, b, c, d and e

» Physical parameters (ratios, speeds, …):• Use Greek letters:

• Avoid unusual Greek letters, e.g.

(hard to pronounce/remember)

xx ˆ,

© 2013 W.B. Powell

Notational style

Time:» Always use t or its variants.

• A flight going from city i to city j might start at time t and arrive a time t’.

• You might purchase an option at time t which can be exercised at time t’.

» Use to indicate an interval of time, such as the time required to complete an action.

» Let t = 0 represent “here and now.”

© 2013 W.B. Powell

Notational style

Modeling new information:» It helps to identify variables that represent new

information arriving in a time period.» Suggest using “hats” to indicate new information:

» Use “bars” for statistics calculated from exogenous information:

ˆ Customer demand for a product in time .ˆ Market price of a stock at time .ˆ A baseball player's batting average in year .

t

t

t

D tp t

b t

'' 1

1

1

1 ˆ

ˆ(1 )1 ˆ

t

t tt

t t tN

N n

n

D Dt

p p p

b bN

© 2013 W.B. Powell

Notational style Sets:

» Sets - Calligraphic (or script) capital letters

• Subsets - suggest using:

» Use sets for summations:• Sets are especially useful when there is not a natural indexing:

• We can also write vectors:

x a

i b

1 is better than:

n

i ii i

x x

Let be a multiattribute vector, where:

It is then easy to write:

aa

aa

x

t t tit ix x x x

“Monotype Corsiva” in equation editorCalligraphic in Latex

© 2013 W.B. Powell

Notational style

Notational style» Functions:

» Use lower case letters for individual functions; use upper case for sums:

Argument notation (preferred in engineering):( )( , )

Mapping notation (preferred in mathematics):::

F xF x y

FF

( ) ( )t tt

C x c x

© 2013 W.B. Powell

Notational style

Notational style» Matrices:

• Square, capital letters:A, B

» Multiplication of matrices and vectors:

» When typing equations in MS Word…• I recommend purchasing the MathType add-in. Much better

than Word’s built in equation editor:http://www.dessci.com/en/products/mathtype/

Use when it is understood that we want an innerproduct. The "transpose" is implicit. There is noneed to write:

T

cx

c x

© 2013 W.B. Powell

Outline


» The basic budgeting problem» Budgeting with an S-curve effect» The dynamic budgeting problem

© 2013 W.B. Powell

The budgeting problem The budgeting problem

» How much of a single resource should be allocated to different activities (sometimes over time).

» This can be viewed as taking a single “budget” and then setting the budgets of a series of smaller tasks.

» Resources:• Money

– Your division budget is fixed at the beginning of the year. You have to decide how your budget should be divided among your subdivisions.

• Investment– You have to allocate your funds over different investments.

• Time– How much time should you allocate to different assignments

before Monday?– How much time should be given to individual tasks?

© 2013 W.B. Powell

The budgeting problem

Basic model:Amount of a resource to be allocated over different activities

(this is our "budget")=A set of tasks (or projects, or time periods) to which the resource

should be allocated.=(1,2,..., )

Amount t

R

Tx

1 2

of the resource that should be allocated to task ., ,...,

An ( ) Total contribution from an allocation .

T

tx x x x

allocation.C x x

© 2013 W.B. Powell


To start, we are going to assume that ( ) is .That is, we can write:

( ) ( )

This means that the marginal value of increasing the allocationfor task is independent of the rest

t tt

C x separable

C x c x

t

of the allocation vector.

© 2013 W.B. Powell


The contribution function is typically concave (when we are maximizing):

tx

( )t tc x

© 2013 W.B. Powell


“The” resource allocation problem

We wish to solve:

max ( )

subject to:

"The resource constraint"

0

t tx t

tt

t

c x

x R

x

© 2013 W.B. Powell

The budgeting problemIt is sometimes convenient to turn the contribution function into a piecewise linear approximation:

tx

( )t tc x

1tu 1 2t tu u 1 2 3t t tu u u

1ty

3ty2ty

tx( )

where

,

t

t t ti tii

t tii

ti ti

c x c y

x y

y u

© 2013 W.B. Powell


Solving the budget allocation problem:

We can convert our original nonlinear program into alinear program using a standard modification:

max

subject to:

0

t

t

ti tix t i

tit i

ti ti

ti

c y

y R

y uy

© 2013 W.B. Powell

The budgeting problem The budget allocation problem as a (linear) network:

R

1x

2x

3x

4x

5x

© 2013 W.B. Powell

The budgeting problem Variations:

» Contribution function may be nonconcave:• Volume discounts• Fixed cost for investing in a project

– Time required to investigate the project– Fixed costs for allocating resources to the project

» Contribution function may be nonseparable• You are drilling for oil. Different oil wells tap the same basic

oil field. The output of one well reflects all the wells that are drilled (and their location).

» Contribution may depend on the history of investments• Expenditures in HIV treatment; impact in time t depends on

pattern of investment over time.» Evolving information

• We may learn new information as the project unfolds

© 2013 W.B. Powell

Outline



© 2013 W.B. Powell

The S-curve

Resource allocation with an S-curve contribution function

Resources allocated to an activity

Con

tribu

tion

© 2013 W.B. Powell

The S-curve

Examples:» Airlines

• How many flying hours of their aircraft fleet should be dedicated to different markets?

» Universities• What areas of research should a university invest faculty

resources in?

» Auto manufacturers:• What product segments should a company focus in?• What geographical regions should the company focus on

selling to?

© 2013 W.B. Powell

The S-curve

Example: airlines» Mixture of markets of different sizes

What would you do if you were a small airline?

Medium sized?

Large?

© 2013 W.B. Powell

The S-curve If you are medium-sized:

» Put everything you have in the biggest markets that you can dominate, whether they be the mid-sized…

© 2013 W.B. Powell

The S-curve

If you are big enough:» Allocate your resources across all markets in proportion

to their size:

© 2013 W.B. Powell

The S-curve The Canadian coffee market

In November 2000, Starbucks, the world’s leading gourmet coffee retailer, announced that it would open around 50 to 75 coffee shops in the province of Quebec in the following four years. At the time, it had only five outlets compared with the 38 stores of Second Cup, the Canadian leader in the industry.

Perhaps foreseeing the threat, Second Cup had earlier abandoned its expansion plans in the United States and decided to concentrate its efforts on its Canadian business. Within 15 months of the announcement, it opened 18 new shops, increasing its number to 56. During the same period, Starbucks was able to open only three more shops and later opened another four, for a total of 12 by December 2003.

If the expansion observed in those three years is any indication, it is fair to conclude that Starbucks will obtain less than a quarter of the market it had hoped to capture initially. Although there may be other reasons why Starbucks’ plan did not materialize, it is quite plausible that Second Cup’s expansion played a role.

Dasci and Laporte, “A Continuous Model for Multistore Competitive Location,” Operations Research, Vol 53, 2005.

© 2013 W.B. Powell

The dynamic budgeting problem

1 2

3

Assume that represents a set of time periods, so that the budgetallocation decisions must be made sequentially: before before

... (we can retain the interpretation that is a set of tasks ax x

x

1

nd then arbitrarily order them).

Now let:=Our original budget (initial resource allocation)=The amount of our original resource remaining before we

make the decision for time .( ) T

t

t t

RR

tV R

1 1

he value of having remaining before solving time .

We suggest that it is "intuitively obvious" that the following equationholds:

( ) max ( ) ( , )t

t

t t t t t t t tx

R t

V R c x V R R x

© 2013 W.B. Powell


1 1

1 1

Often we will write:

( ) max ( )

where the functional dependence:( , )

is implicit.

We solve this using . Assume that(1, 2,..., ). We next h

tt t t t t tx

t t t t t t

V R c x V R

R R R x R x

backward dynamic programmingT

1 1

1 1

ave to assume that is known

(for example, we may simply assume that 0). We would thensolve:

( ) max ( )

for all possible values of . This is easiest if we consider only adis

T

T T

T T

T T T Tx

T

V R

V R

V R c x

R

crete set of values for . TR

© 2013 W.B. Powell


1 1 1

We can continue this process backward through time. Assume now that we know for all possible values of . For example, if we aredeciding how much of our financial budget we can spend in

t t tV R R

1 1 1

month , thenwe might restrict our attention to budgets in increments of $10,000,and consider only expenditures in the same units. In this case, if we know

for values of (0,10000,20000,.t t t

t

V R R

1

1

..), then we can solve the equation:

( ) max ( )

for values of (0,10000, 20000,...). For each value of , we would

evaluate the term ( ) for x (0,10000,20000,..., )

(a

tt t t t t t tx

t t

t t t t t t t

V R c x V R x

R R

c x V R x R

ssuming we cannot spend more than what is in the budget - of course, we might be able to borrow money).

© 2013 W.B. Powell


Analytical example:Assume that:

( ) lnThis means:

( ) max ln

lnT T

t t t t

T T x R T T

T T

c x c x

V R c x

c R

-4

-3

-2

-1

0

1

2

0.05

0.35

0.65

0.95

1.25

1.55

1.85

2.15

2.45

2.75

3.05

3.35

3.65

3.95

4.25

4.55

4.85

ln(x) function implies declining marginal returns for resources (which is common in practice). What do you think would be an optimal allocation with this type of function?

© 2013 W.B. Powell


Analytical example (cont’d):

1 1

1 1

1 1 1 1 1 1

1 -1 -1 -1

-1 -1 -1

-1 -1 -1 -1

-1 -1 -1 -1

-1 -1 1

( ) max ln ( )

= max ln ln( )

( , )differentiating: 0

( )

T T

T T

T T x R T t T T

x R T T T T T

T T T T

T T T T

T T T T T

T T T T

V R c x V R x

c x c R x

f x R c cx x R x

c R x c xc R c c x

-1

-1 11

1

-1 -1 -1 -11 -1 1 -1

1 1

-1 -1 -11

1 1

( ) ln ln( )

ln ln( )

T

T TT

T T

T T T TT T T T T

T T T T

T T T TT T

T T T T

c Rxc c

c R c RV R c c Rc c c c

c R c Rc cc c c c

-1 -1( , )T Tf x R

© 2013 W.B. Powell


Analytical example (cont’d):

' t,t''

't,t'

1

t,t

t,t

Using "proof by extrapolation," let:

( ) ln

where:

...This allows us to express:

Example: What if 1 (all time periods are equally important)?Then:

T

t t t tt t

t

t t T

t t

t

V R c R

cc c c

x R

c

11

This means: allocate the budget equally over the remaining time periods.T t

This is the fraction of the budget we expect to be allocated to time period t’ when we are in time period t. Note that when we are in time period t, we don’t actually do an allocation to time periods t’ > t.

orf 411 13 the budgeting problem - princeton university 411... · notational style variables »...

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