The Single Item profit maximizing

capacitated lot-size (PCLSP) problem

with fixed prices and no set-up

by Kjetil K. Haugen1),∗]Asmund Olstad1)

Krystsina Bakhrankova1)

and

Erik Van Eikenhorst2)

1)Molde University College, Norway2)University of Edinburgh, UK

∗]E-mail: Kjetil.Haugen@hiMolde.no

16th International Scientific Conference onMathematical Methods in Economy and Industry

Ceske Budejovice, June 15-18, 2009

1

Idea – Abstract

• Even though modern LP-solvers (and

computers) are extremely efficient, fast

specialized sub-problem solvers may be of

interest.

• Here we focus on an LP arising as a typical

sub-problem in Dynamic Pricing problems.

• We demonstrate the algorithmic devel-

opment and conclude with some simple

speed tests, demonstrating computational

efficiency.

2

Background

Haugen, Olstad and Pettersen defined the

PCLSP problem in:

1) K. K. Haugen, A. Olstad, and B. I. Pet-

tersen. The profit maximizing capacitated

lot-size (pclsp) problem. European Jour-

nal of Operations Research, 176:165–176,

2007.

2) K. K Haugen, A. Olstad, and B. I. Pet-

tersen. Solving large-scale profit maxi-

mization capacitated lot-size problems by

heuristic methods. Journal of Mathemat-

ical Modelling and Algorithms, 6(1):135–

149, 2007.

3

PCLSP – formulation

Max Z =T∑

t=1

J∑

j=1

[djtpjt − sjtδjt − hjtIjt − cjtxjt

]

(1)

s.t.

αjt − βjt · pjt = djt ∀jt (2)J∑

j=1

ajtxjt ≤ Rt ∀t (3)

xjt + Ij,t−1 − Ijt = djt ∀jt (4)

0 ≤ xjt ≤ Mjtδjt ∀jt (5)

Ijt ≥ 0, ∀jt (6)

δjt ∈ {0,1} ∀jt (7)αjt

βjt≥ pjt ≥ 0 ∀jt (8)

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PCLSP – variables and constants

Variables:

djt = demand for item j in period t

pjt = price of item j in period t

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

Constants:

αjt = demand constant, for item j at t

βjt = demand slope, for item j at t

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

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Single item – negligible set-up costs

Many modern production settings (JIT) in-volve negligible set-up costs (and times). Inthe previous model we hence focus on a ver-sion with J = 1 (single item) and sjt ≈ 0(negligible set-up costs).

Hence, removal of demand variables (djt) bysubstitution gives:

Max Z =T∑

t=1

[(αt − βt · pt)pt − htIt − ctxt] (9)

s.t.

atxt ≤ Rt ∀t (10)

xt + It−1 − It = αt − βt · pt ∀t (11)

xt ≥ 0 ∀t (12)

It ≥ 0, ∀t (13)αt

βt≥ pt ≥ 0 ∀t (14)

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Simplifying assumptions

• Capacity constraint: Without loss of gen-erality, equation (10) can be substitutedwith xt ≤ Rt where Rt = Rt

at.

• Given prices: If we assume that all pricesp1, . . . , pT are given, let’s say by p1, . . . , pT ,the objective (9) can be rewritten as:

Max Z =T∑

t=1

(αt − βt · pt)pt −T∑

t=1

[htIt + ctxt]

= C −T∑

t=1

[htIt + ctxt] (15)

or

Min Z =T∑

t=1

[htIt + ctxt] (16)

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The reformulated LP

Additionally, defining;

Dt = αt − βt · pt (17)

problem (9) – (14) may be redefined as thefollwing LP-problem:

Min Z =T∑

t=1

[htIt + ctxt] (18)

s.t.

xt ≤ Rt ∀t (19)

xt + It−1 − It = Dt ∀t (20)

xt ≥ 0 ∀t (21)

It ≥ 0, ∀t (22)

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Assumptions on c and h

Logistics problems of this type (”Lot-sizing”)

will typically not have a very large time hori-

zon. Consequentually, making assumptions

on stability of production and storage costs

seems reasonable. We assume the following:

c1 = c2 = . . . , cT = c (23)

and

h1 = h2 = . . . hT = h (24)

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Minimization of total inventory

Utilizing assumptions (23), (24), the objec-tive (16) may be expressed:

T∑

t=1

[htIt + ctxt] = hT∑

t=1

It + cT∑

t=1

xt (25)

Next, it is straightforward to realize by sum-ming up the left and right side of equa-tion (20) that:

T∑

t=1

xt = IT − I0 +T∑

t=1

Dt (26)

The right hand side of equation (26) is a con-stant so is h and c, giving:

Min Z =T∑

t=1

It (27)

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The algorithmic logic

• Now, Suppose we relax the capacity con-

straints (19). Then, the optimal solution

to the LP (18) – (22) is obvious (a ”Chase

Demand” or ”JIT” strategy):

x∗t = Dt and I∗t = 0, ∀t (28)

• Taking the capacity constraints back into

consideration, it is likewise obvious that

any period where (19) binds must lead to

production as close to this period as pos-

sible in order to minimize total inventory.

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The algorithm

• Summing up: The algorithm could be de-scribed verbally as: Start out with theJIT solution. If it is feasible it is alsooptimal. If infeasible, run through all in-feasible points (ie all periods where x∗t >

Rt) and utilize ”closest” possible availableproduction capacity to remove infeasibili-ties.

• A formal version:

0. LET x∗t = Dt, ∀t

1. IF x∗t ≤ Rt, ∀t STOP (x∗t is optimal)

2. IF next period is T + 1 STOP

3. ELSE find next period, τ where x∗t > Rt andproduce a total of x∗t − Rt in previous periods τ −1, τ−2, . . . as close as possible to τ . (If impossible,problem is infeasible STOP)

4. SET x∗τ = Rτ and update x∗τ−1, x∗τ−1, . . . corre-

spondingly

5. GOTO 2.

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Relaxing cost assumptions further

i) c1 = c2 = . . . cT = c and h1 6= h2 6= . . . hT

Previous arguments hold – similar mathe-

matical reformulation. However, the final

objective changes from total inventory to

toal inventory costs:

Min Z =T∑

t=1

htIt (29)

Obviously, the algorithm will still hold. It

is no point moving production to an earlier

period than the closest possible as total

inventory costs must increase under such

a strategy.

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Relaxing cost assumptions further

ii) c1 > c2 . . . > cT and h1 6= h2 6= . . . hT

In this case, which should be quite natural

– productivity should increase over time,

the algorithm must also hold. Again, as

production costs are larger if we move

back in time, it must be optimal to pro-

duce as close to the capacity violation as

possible. Both total inventory and pro-

duction costs are then minimized.

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Relaxing cost assumptions further

iii) ctht

= Constant = c ⇒ ct = c · ht

In most reasonably competitive markets, the

value of a product is proportional to the pro-

duction costs. Of course, in a perfectly com-

petitive market, price equals marginal costs,

and the above assumption is ”correct” if the

main contribution to inventory costs are due

to storage value – as most inventory experts

assume.

Surely, such an assumption opens up also

for increasing prodction costs, which in cer-

tain situations may be predictable – wage in-

creases, economic growth etc.

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Algorithmic consequences if ct = c · ht

Rewriting (11) as:

xt = Dt + It − It−1 (30)

and substituting ct = c · ht into the objec-

tive (18) yields:

Z =T∑

t=1

[htIt + c · ht(Dt + It − It−1)

](31)

Now, assuming a given initial inventory I0 and

the elimination of a constant, the objective Z

above may be replaced by the following:

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Z =T∑

t=1

htIt (32)

where

ht = (c + 1)ht + cht+1 and hT+1 = 0 (33)

Finally, comparing the objectives Z of equa-

tion (32) and Z of equation (29), we observe

structural equality and our algorithm would

work also for the case with a constant ratio

between production and inventory costs.

Some simple numerical experiments

The previsouly defined algorithm was imple-

mented in Fortran 95 and executed and com-

pared with state of the art commercial LP-

software (CPLEX) on a modern PC. The ta-

ble below shows the results (CPU secs.).

T = 10k T = 100k T = 1m

CPLEX 0.219 1.766 31.156Algorithm 0.031 0.093 0.672Change (%) 700 % 1893 % 4637 %

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