The profit maximizing capacitated

lot-size (PCLSP) problem

by Kjetil K. Haugen

Asmund Olstad

and

B̊ard I. Pettersen

Molde University College

Servicebox 8, N-6405 Molde, Norway

E-mail: Kjetil.Haugen@hiMolde.no

European Journal of Operational Research, 176:165–176, 2006

1

Idea – Abstract

• Introduce a ”new” set of lot-sizing models

including pricing – PCLSP.

• PCLSP – practically at least as relevant

as CLSP.

• PCLSP – computationally more feasible

than CLSP.

2

Outline

1) Introduce LSP, CLSP and PCLSP.

2) Brief discussion of algorithmic properties

of LSP and CLSP.

3) Introduce a Lagrange Relaxation algo-

rithm for PCLSP.

4) Judge algorithmic performance by exam-

ples and compare with CLSP

5) Discuss practical relevance of PCLSP.

3

The simple lot-size problem (LSP) (1)

Min Z =T∑

t=1

stδt + htIt + ctxt (1)

s.t.

xt + It−1 − It = dt ∀t (2)

0 ≤ xt ≤ Mtδt ∀t (3)

It ≥ 0, ∀t (4)

δt ∈ {0,1} ∀t (5)

4

The simple lot-size problem (LSP) (2)

Decision variables:

xt = amount produced in period t

It = inventory between t, t + 1

δt = 1 if xt > 0 in period t ; 0 otherwise

Parameters:

T = number of time periods

st = setup cost in period t

ht = storage cost between t, t + 1

ct = unit production cost in period t

Mt = ”Big M” in period t

5

The simple lot-size problem (LSP) (3)

– Problem characteristics

Basic trade off:

• Many set-ups ⇒ ∑t stδt ↑ and

∑t htIt ↓

• Few set-ups ⇒ ∑t stδt ↓ and

∑t htIt ↑

Slight generalization of EOQ-model:

Given dt = d, ct = c, ht = h∀t ⇒ LSP = EOQ

6

The simple lot-size problem (LSP) (4)

– Algorithmic characteristics

• Polynomial DP-algorithm by Wagner and

Whitin (1950).

• Solves very fast.

• Planning horizon theorems; xt · It = 0.

• Interesting candidate as sub-problem

solver in more advanced lot-size problems.

7

The capacitated problem (CLSP) (1)

Min Z =T∑

t=1

J∑

j=1

sjtδjt + hjtIjt + cjtxjt (6)

s.t.

J∑

j=1

ajtxjt ≤ Rt ∀t (7)

xjt + Ij,t−1 − Ijt = djt ∀jt (8)

0 ≤ xjt ≤ Mjtδjt ∀jt (9)

Ijt ≥ 0, ∀jt (10)

δjt ∈ {0,1} ∀jt (11)

8

CLSP (2)

Decision variables:

xjt = amount of item j produced in t

Ijt = inventory of item j between t, t + 1

δjt =

{1 if item j is produced in period t0 otherwise

Parameters:

T = number of time periods

J = number of items

sjt = setup cost for item j in period t

hjt = storage cost, item j between t, t + 1

cjt = unit production cost, item j at t

ajt = resource used, item j at t

Rt = capacity resource available at t

Mjt =T∑

s=t

djs

9

CLSP (2) – characteristics

• Much harder to solve compared to LSP.

• Reason:

Violation of capacity constraint ⇒”moving production around” (combi-natorial).

• Due to non existence of polynomial al-gorithms (NP-hardness) heuristical (La-grange relaxation based) approaches com-mon.

• A very ”popular” OR research problem.

• Still: typical problem sizes not muchlarger than 100 x 100 – not satisfactorygiven product variety today.

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PCLSP (1)

Max Z =T∑

t=1

J∑

j=1

[djtpjt − sjtδjt − hjtIjt − cjtxjt

]

(12)

s.t.

αjt − βjt · pjt = djt ∀jt (13)J∑

j=1

ajtxjt ≤ Rt ∀t (14)

xjt + Ij,t−1 − Ijt = djt ∀jt (15)

0 ≤ xjt ≤ Mjtδjt ∀jt (16)

Ijt ≥ 0, ∀jt (17)

δjt ∈ {0,1} ∀jt (18)αjt

βjt≥ pjt ≥ 0 ∀jt (19)

11

PCLSP (2)

Decision variables added to CLSP:

pjt = price of item j in period t

Parameters added to CLSP:

αjt = constant in linear demand,

item j at t

βjt = slope in linear demand,

item j at t

12

PCLSP characteristics

• Linear demand, Monopoly assumption(unrealistic).

• PCLSP is a generalization of CLSP. ∗

• Immediate feasible solutions are obtain-able (as opposed to CLSP) by ”pricingout”. †

• PLSP (uncapacitated single item ver-sion) is well known from OR-literature.Thomas (1970) constructed a polynomialDP-algorithm with complexity as of theWagner/Whitin algorithm.

∗Easy to see by the special case pjt = p̂jt where p̂jt areassumed constant.†That is, any capacity constraint violation can be”removed” as any demand may be forced to zero(pjt = αjt

βjt).

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Basic hypothesis

The added problem flexibility ob-

tained by introduction of price vari-

ables should make the problem easier

to solve.

14

LUBP sub problems

By relaxing the capacity constraint (14), the

PCLSP-problem (12) – (19) may be expressed

as:

Max Z = Z +T∑

t=1

λt

Rt −

J∑

j=1

ajtxjt

s.t. constraints (13) to (19)

It is ”straightforward” to adjust the DP-

algorithm by Thomas (1970) to provide very

efficient solutions to the LUBP sub-problems.

Note: Solving LUBP yields upper bound on

PCLSP.

15

LLBP sub problems

If δij’s are fixed (Set-up structure) in PCLSP,

a standard quadratic programming problem is

obtained:

Max Z =T∑

t=1

J∑

j=1

[djtpjt − hjtIjt − cjtxjt

]

s.t. (13), (15), (17), (19).

Any solution to LLBP is feasible and typically

non-optimal. Hence, Any solution to LLBP

is a lower bound to PCLSP.

16

Algorithmic structure

LUBP

LLBP

Z*

PCLSP

Set-upstructure λλλλ

t

k

Algorithmic structure

17

Algorithm: Lagrange relaxation

0. Define λt = 0,∀t = 1,2, . . . , T

1. Solve LUBP (Obtain Set-up structure)

2. Solve LLBP (for Set-up structure ob-tained in step 1.) Define λt from thissolution as λk

t , where k denotes iterationcount.

Stop if: (define reasonable stopping crite-ria)

3. Update λkt by smoothing: λk+1

t = θk ·λk−1

t +(1−θk)λkt ,∀t, where θk is a smooth-

ing parameter, 0 ≤ θk ≤ 1

4. Go to step 1.

18

Some results (1)

CLSP PCLSPProblem #It. G(%) #It. G(%)CAP93% 50 10 17 0.59CAP91% 50 3 32 0.41CAP73% 50 2 6 0.2CAP61% 50 1 22 0.06

• Cases captured from Thizy and Wassen-

hove (1985). 8 products over 8 time pe-

riods.

• #It. means number of iterations per-

formed in algorithm.

• G(%) means gap in percent orZLUBP−ZLLBP

ZLUBP· 100.

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Some results (2)

CLSP PCLSPProblem #It. G(%) #It. G(%)CAP93% 50 10 1 3.68CAP91% 50 3 1 2.82CAP73% 50 2 1 1.5CAP61% 50 1 1 0.65

The same cases with Gaps after one itera-

tion in PCLSP compared to 50 iterations in

CLSP.

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PCLSP – practical relevance

Demonstrated (by some examples) that

PCLSP solves significantly faster than CLSP.

What about practical relevance?

Obvious facts:

• If monopolistic market conditions ⇒PCLSP Â CLSP

• If Free markets (price taking behavior)

CLSP is ”correct”

• Most markets are neither (oligopoly) –

what then?

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Price constraints

• Given ”relatively small” price changes, un-

derlying Nash equilibrium may be stable.

• Price constraints may serve the purpose

• Relatively simple to introduce (no radical

algorithmic changes)

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