The document introduces the profit maximizing capacitated lot-size problem with pricing (PCLSP) as a generalization of the capacitated lot-sizing problem (CLSP) that allows for pricing decisions. It presents the formulation and characteristics of PCLSP and describes how it is computationally more tractable than CLSP by introducing pricing variables. The document outlines a Lagrange relaxation algorithm for solving PCLSP using lower and upper bound subproblems and presents results showing PCLSP solves faster with smaller optimality gaps than CLSP on example problems. It discusses the practical relevance of PCLSP under different market conditions.

1. The profit maximizing capacitated
lot-size (PCLSP) problem
by Kjetil K. Haugen
Asmund Olstad
and
B˚ I. Pettersen
ard
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

2. 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

3. Outline
1) Introduce LSP, CLSP and PCLSP.
2) Brief discussion of algorithmic properties
of LSP and CLSP.
3) Introduce a Lagrange Relaxation algorithm for PCLSP.
4) Judge algorithmic performance by examples and compare with CLSP
5) Discuss practical relevance of PCLSP.
3

4. The simple lot-size problem (LSP) (1)
T
Min Z =
stδt + htIt + ctxt
(1)
t=1
s.t.
xt + It−1 − It = dt ∀t
0 ≤ xt ≤ Mtδt ∀t
(2)
It ≥ 0, ∀t
(4)
δt ∈ {0, 1} ∀t
(5)
(3)
4

5. 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

6. The simple lot-size problem (LSP) (3)
– Problem characteristics
Basic trade off:
• Many set-ups ⇒
• Few set-ups ⇒
t st δt
t st δt
↑ and
↓ and
t ht It
t ht It
↓
↑
Slight generalization of EOQ-model:
Given dt = d, ct = c, ht = h∀t ⇒ LSP = EOQ
6

7. 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

8. The capacitated problem (CLSP) (1)
T
J
Min Z =
sjtδjt + hjtIjt + cjtxjt
(6)
t=1 j=1
s.t.
J
ajtxjt ≤ Rt
∀t
(7)
xjt + Ij,t−1 − Ijt = djt ∀jt
0 ≤ xjt ≤ Mjtδjt ∀jt
(8)
Ijt ≥ 0, ∀jt
(10)
δjt ∈ {0, 1} ∀jt
(11)
j=1
(9)
8

9. 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 t
0 otherwise
Parameters:
T
J
sjt
hjt
cjt
ajt
Rt
=
=
=
=
=
=
=
number of time periods
number of items
setup cost for item j in period t
storage cost, item j between t, t + 1
unit production cost, item j at t
resource used, item j at t
capacity resource available at t
T
Mjt =
djs
s=t
9

10. CLSP (2) – characteristics
• Much harder to solve compared to LSP.
• Reason:
Violation of capacity constraint ⇒
”moving production around” (combinatorial).
• Due to non existence of polynomial algorithms (NP-hardness) heuristical (Lagrange relaxation based) approaches common.
• A very ”popular” OR research problem.
• Still: typical problem sizes not much
larger than 100 x 100 – not satisfactory
given product variety today.
10

11. PCLSP (1)
T
J
djtpjt − sjtδjt − hjtIjt − cjtxjt
Max Z =
t=1 j=1
(12)
s.t.
αjt − βjt · pjt = djt ∀jt
(13)
J
ajtxjt ≤ Rt
∀t
(14)
xjt + Ij,t−1 − Ijt = djt
0 ≤ xjt ≤ Mjtδjt
Ijt ≥ 0,
δjt ∈ {0, 1}
αjt
≥ pjt ≥ 0
βjt
∀jt
∀jt
∀jt
∀jt
(15)
(16)
(17)
(18)
∀jt
(19)
j=1
11

12. 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

13. PCLSP characteristics
• Linear demand,
(unrealistic).
Monopoly assumption
• PCLSP is a generalization of CLSP. ∗
• Immediate feasible solutions are obtainable (as opposed to CLSP) by ”pricing
out”. †
• PLSP (uncapacitated single item version) is well known from OR-literature.
Thomas (1970) constructed a polynomial
DP-algorithm with complexity as of the
Wagner/Whitin algorithm.
∗ Easy
to see by the special case pjt = pjt where pjt are
ˆ
ˆ
assumed constant.
† That
is, any capacity constraint violation can be
”removed” as any demand may be forced to zero
jt
(pjt = αjt ).
β
13

14. Basic hypothesis
The added problem flexibility obtained by introduction of price variables should make the problem easier
to solve.
14

15. LUBP sub problems
By relaxing the capacity constraint (14), the
PCLSP-problem (12) – (19) may be expressed
as:
T
Max Z = Z +
t=1
s.t.

λt Rt −
J

ajtxjt
j=1
constraints (13) to (19)
It is ”straightforward” to adjust the DPalgorithm by Thomas (1970) to provide very
efficient solutions to the LUBP sub-problems.
Note: Solving LUBP yields upper bound on
PCLSP.
15

16. LLBP sub problems
If δij ’s are fixed (Set-up structure) in PCLSP,
a standard quadratic programming problem is
obtained:
T
J
Max Z =
djtpjt − hjtIjt − cjtxjt
t=1 j=1
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

17. Algorithmic structure
LUBP
Set-up
structure
Z
*
k
PCLSP
λt
LLBP
Algorithmic structure
17

18. 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 obtained in step 1.) Define λt from this
solution as λk , where k denotes iteration
t
count.
Stop if: (define reasonable stopping criteria)
k
3. Update λt by smoothing: λk+1 = θk ·
t
k−1
λt +(1−θk )λk , ∀t, where θk is a smootht
ing parameter, 0 ≤ θk ≤ 1
4. Go to step 1.
18

19. Some results (1)
Problem
CAP93%
CAP91%
CAP73%
CAP61%
CLSP
#It. G(%)
50
10
50
3
50
2
50
1
PCLSP
#It. G(%)
17
0.59
32
0.41
6
0.2
22
0.06
• Cases captured from Thizy and Wassenhove (1985). 8 products over 8 time periods.
• #It. means number of iterations performed in algorithm.
• G(%)
means
gap
ZLU BP −ZLLBP
· 100.
Z
in
percent
LU BP
19
or

20. Some results (2)
Problem
CAP93%
CAP91%
CAP73%
CAP61%
CLSP
#It. G(%)
50
10
50
3
50
2
50
1
PCLSP
#It. G(%)
1
3.68
1
2.82
1
1.5
1
0.65
The same cases with Gaps after one iteration in PCLSP compared to 50 iterations in
CLSP.
20

21. PCLSP – practical relevance
Demonstrated (by some examples) that
PCLSP solves significantly faster than CLSP.
What about practical relevance?
Obvious facts:
• If monopolistic market
PCLSP
CLSP
conditions
⇒
• If Free markets (price taking behavior)
CLSP is ”correct”
• Most markets are neither (oligopoly) –
what then?
21

22. Price constraints
• Given ”relatively small” price changes, underlying Nash equilibrium may be stable.
• Price constraints may serve the purpose
• Relatively simple to introduce (no radical
algorithmic changes)
22