004 thomas priceg

7/23/2019 004 Thomas Priceg

1/5

M A N A G E M E m S C I E N C E

VoL

It No. It

July,

1970

nn itnUM.A.

PRICE-PRODUCTION DECISIONS WITH

DETERMINISTIC DEMAND*

JOSEPH THOMAS

Cornell University

The problem

of

simultaneously making price

and

production decisions

for a

single product with

a

known deterministic demand function

is

considered.

The ob-

jectiveis

to

maximise profit,andbackordersare notallowed. Rraults which reduce

the com putations necessaryfor thepricing decisionaregiven, and planning horizons

are identifiedas

in

previous inventory research. These resultsarecombined into

an

efficient forward algorithm.

Introductioii

The investigationofdynam ic inventory

models has

largely assumed priceto be con-

stantandconcerned itself withtheproduction decision only. Notable exceptionsare

[3]and[4].It would seem worthwhileinmany situationstoconsider

the

total effect

on profitsof price-production decisionsandmakethe

two

decisions simultaneously.

This paper considers

the

problem

of

amonopolist who m ust

set

both decisions

in

each

of

T

periods. The inven tory work

of

Wagner and W hitin

[5]and

apaper based

on

their

workbyEppen, GouldandPashigian

[1]

are extendedto theprice-production case.

The major results are

in

obtaining planning horizons similar

to

those

in

[1],[5],

and [6]

for the price-production atua iio n

and in

limiting

the

calculations required

for

the pric-

ing decision.

F o n n u l a t i o n

Conffldera deterministic demand function, nonincreasinginpriceanddifferentin

each

of

Tperiods. No backorders are allowed,andprofit m aximization is the objective.

We assume the re exist setsP,D,

and

X where:

P

=

{p:p

is

anadmissble price}

I jd:

d

is

anadmissible demand}

X

=

jz:

z

is

an

admissible production level}

PandX contain

ail

decisions t ha tarebeing considered

pi

andXt,i =

1,

,

T),

and

X

is such that no lap ad ty constraint

exists.

There

isa

demand function as specified

above whichisdenoted:

d= * &> whereweassume *(?)issuch tha tpr4>t pi) takeson itsmaximum

valueforafinite valueofpiin allperiods.

The imm ediate costsineach period, L , ( ),aregivenbyequation1.

(1) L, I, X,, p,)

=

it-il +

h{Xi)Kt+ F,z, -

pMVt)

here:


2/5

7 4 8 JOSEPH THOMAS

Reference[5j givesaforward algorithm based on four theorem s with no price decisio

The price production analogsof these theoremsare stated below without proof sinc

the proofs follow directly from thosein[5]and [6].

L E M M A 1.

There exists

an

optimal program such that Ii-Xi

= 0/ or aU t.

L E M M A

2. There exists

an

optimal program such thai

for all t:

Xi = {0 or22y_ ^ i p , ) } for some k , t ^ k ^ T , and for some prices Pi, ,Pt ^ P

L E M M A

3. There exists

an

optimal program such Ihat

if

dt*

is

satisfied

hy

some

t**


3/5

PBICE PBODtTCnO N DECISIONB W ITH DBTERMINIBTIC DEMAND 749

PBOOF. We know (3) takes on its rnftinmiim for

a

finite

pi

since

pf


4/5

7 5 0 JOSEPH THOMAS

problem using (2) and Theorem 2; record optimal prices in conjunetion witii eaeh pos-

sible last t-up .

Step 4.1{j t)

=

m t),j t) 1

is a plamung

horizon,

production dm s io ns to i (0

1 and pricing de dao ns to j t) are optimal in the T-period pn ^r am , and they m ay be

recorded as such. (Step 1 guarantees th at they are not reconsidered.)

Step 5.

Record

F t)

andi(0. (Prices are recorded in 3.)

Step6. Repeat 1 to 6 for + 1 , , T.F{t), periods where set-ups occur, and o

timal prices are avdlable, and production quantities can be computed.

R e f e r e n c e s

1 . E P F B N , G . D . , G o tru ), F . J . , AN D FA SHiaiAN , B . P . , E xtemion s of the Planning Horison

Tteorem in the Djrnsmio Economic Lot Siie Model,

Management Science

Vol. 1 5 , N o . 5

(January 19%), pp. 2^-^277.

2. THOMAa, L. J. , Simultaneous Price-Produ ction De cisions with Determ inistic and Random

Dem and, impublished P h. D . dissertation, Yale Un iversity, 1968.

3 . W A Q N B K , H . M . ,

A Postscript to Dynamic Problems of the F irm ,'''

Naval Research offisticg

Quarterly Vol. 7 , N o . 1 (1960), pp. 7 -12 .

4.

, ANDW H IT IN , T . M . , Dynamic Problems in the Theory of the F irm,

Naval Research

Logistics QuarUrly

Vol. 5 , N o . 1 (1958), pp. 53-74 .

6. , and , Dynam ic Version of the Econom ic Lot Size F orm ula,

Management Science

Vol. 5 , N o. 1 (October 1 9ffi), pp. 89-96.

6. ZABEL, E . , Some Generalizations of an Inventory Planning H orizon Th eorem, Management

Science

Vol. 1 0 , N o. 3 (April 1 964), pp. 465-471 .


5/5

004 thomas priceg

Documents