Undominated Valid Inequalities for a Stochastic Capacitated Discrete

Lot-sizing Problem with Lead Times, Cancellation and Postponement

Carlos E. Testuri

, H

ector Cancela

and V

ıctor M. Albornoz

Depto. de Investigaci

on Operativa, Instituto de Computaci

on, Facultad de Ingenier

ıa, Universidad de la Rep

ublica,

J. Herrera y Reissig 565, 11300 Montevideo, Uruguay

Depto. de Industrias, Campus Santiago Vitacura, Universidad T

ecnica Federico Santa Mar

ıa,

Avda. Santa Mar

ıa 6400, Vitacura, Santiago, Chile

Keywords:

Stochastic Lot-sizing, Multi-stage Stochastic Integer Programming, Valid Inequalities.

Abstract:

The problem addresses the expected cost minimization of meeting the uncertain demand of a product during

a discrete time planning horizon. The product is supplied by selecting ﬁxed quantity shipments that have lead

times. Due to the uncertainty of demand, corrective actions, such as shipment cancellations and postpone-

ments, must be taken with associated costs and delays. The problem is modeled as an extension of the discrete

lot-sizing problem with different capacities and uncertain demand, which belongs to the N P-hard class. To

improve the resolution of the problem by tightening its formulation, valid inequalities based on the (`,S) in-

equalities approach are used. Given that the inequalities are highly dominated for most experimental instances,

a scheme is established to determine undominated ones. Computational experiments are performed on the res-

olution of the model and variants that include subsets of undominated and representative valid inequalities for

instances of several information structures of uncertainty. The experimental results allow to conclude that the

inclusion of undominated and representative derived (`,S) valid inequalities enable a more efﬁcient resolution

of the model.

1 INTRODUCTION

The studied problem is the minimization of the ex-

pectation of the costs incurred in decisions taken to

meet the uncertain demand of a product over a ﬁnite

discrete time planning horizon. To meet the demand,

there are certain optional distinguishable shipments

(denominated as cargoes) with a non-fractional quan-

tity of the product that can be acquired at most once

with an associated cost. The cargoes have meaning-

ful delivery lead times within the planning horizon; so

that a signiﬁcant amount of time elapses between the

purchase decision and the moment when the cargo is

received. After meeting the demand in a given period,

the remaining quantity of product is stored, keeping

an inventory up to a certain capacity, to ﬂexibly sat-

isfy the future demand in subsequent periods. Due to

the passage of time, while the uncertainty of the de-

mand is revealed and changes, it could happen that at

a given time a cargo, that was already acquired (or-

dered) and has not yet been received, is no longer

necessary. In this case it could be decided to can-

cel its acquisition order or postpone its delivery; deci-

sions, which, in turn, have minimum execution times

in relation to the time of delivery and associated costs.

These decisions hedge against the risk of excess in-

ventory.

The problem can be modeled as an extension of

the lot-sizing formulation (Wagner and Whitin, 1958)

and particularly of the variant with variable capacity

and discrete dimensioning (Nemhauser and Wolsey,

1988). For the case where the parameters are known

with certainty (deterministic case), the dimension-

ing is continuous, and without capacity constraints or

with constant capacity, the problem has efﬁcient res-

olution through dynamic programming (Wagner and

Whitin, 1958; Wagelmans et al., 1992). In addition,

there are known formulations which determine the

convex hull of the feasible region: the extended facil-

ity location formulation (Krarup and Bilde, 1977) and

the (`,S) valid inequalities formulation (Barany et al.,

1984). The deterministic variant with discrete sizing

is a generalization of the binary knapsack problem,

and belongs to the N P -hard complexity class (Bitran

and Yanasse, 1982).

In the case that the parameters are random vari-

ables (stochastic variant) the problem can be for-

mulated by stochastic programming (Birge and Lou-

390

Testuri, C., Cancela, H. and Albornoz, V.

Undominated Valid Inequalities for a Stochastic Capacitated Discrete Lot-sizing Problem with Lead Times, Cancellation and Postponement.

DOI: 10.5220/0007395203900397

In Proceedings of the 8th International Conference on Operations Research and Enterprise Systems (ICORES 2019), pages 390-397

ISBN: 978-989-758-352-0

veaux, 1997). An adjusted extended formulation

of the stochastic continuos non-capacitated problem

with Wagner-Whitin conditions are not satisﬁed for

the stochastic variant (Ahmed et al., 2003). The (`,S)

inequalities are also valid for the stochastic continu-

ous variant, and they were extended to a general class

that allow to deﬁne facets of the feasible set (Guan

et al., 2006).

Other variants of the deterministic continuous

non-capacitated lot-sizing problem model delivery

time of the lots (e.g. due to production time). A

variant in which demands have a compliance inter-

val has efﬁcient resolution by dynamic programming

(Lee et al., 2001). There are two variants accord-

ing to whether the lots are or are not distinguish-

able with respect to delivery times (Brahimi et al.,

2006). For there are efﬁcient algorithms based on

dynamic programming for the distinguishable case

and for the undistinguishable case when the order-

delivery windows are not inclusive. For these vari-

ants, there are tight extended formulations (Wolsey,

2006). For the stochastic case, the problem can be

efﬁciently solved when delivery windows do not in-

tersect in time (Huang and K

uc¸

ukyavuz, 2008). The

distinctive features of the problem under study in the

present work are cancellation and postponement cor-

rective decisions with time delays in a stochastic set-

ting; these aspects are novel and were not found in the

literature review.

The present work is organized as follows. In Sec-

tion 2 an algebraic model of the problem is presented.

Valid inequalities for the model are presented in Sec-

tion 3. In Section 4 experiments are established to

determine utility of the valid inequalities formulation.

Work is completed with Section 5, where conclusions

and future work are discussed.

2 STOCHASTIC MODEL

FORMULATION

Basic index sets are established according to Table 1.

The planning time is represented by the set T of dis-

crete time periods. The set C of cargoes is partitioned

in two sets: the set A of already acquired cargoes –

cargoes ordered in past resolutions of the model– that

are pending reception, and the set P of possible car-

goes to be acquired from now on. Acquisition de-

cision could be made on possible cargoes to be ac-

quired, and cancellation and postponement decisions

could be made on cargoes that were acquired before

the actual planning horizon.

The uncertain demand is represented by a

discrete-time stochastic process indexed in the plan-

ning periods. The process is deﬁned in a ﬁnite prob-

ability space. It is assumed that the demand of the

ﬁrst period is deterministic, and that the demands of

the remaining periods are random with known dis-

tribution functions. The decisions made in a period

can not anticipate the realization of the uncertainty of

the next period. These decisions must simultaneously

take into account all possible revelations of the de-

mand uncertainty of the following periods. This infor-

mation structure can be represented by a tree structure

called tree of scenarios (R

omisch and Schultz, 2001).

This is a perfect directed tree, with the root node rep-

resenting the present time at period t = 1, and with

leaf nodes identifying the future scenarios at period

t = H.

Each node of the scenario tree describes the state

of the process at a given period, and it is identiﬁed

by a period and a scenario. An useful abbreviated

notation is to identify the nodes by a single index n

in a numerable set of nodes, N. For the ﬁrst period,

t = 1, there is a unique node, denoted by 1, that rep-

resents the root of the tree. Each node n ∈ N has an

immediate time predecessor node, p(n); the auxiliary

node 0 is deﬁned as the predecessor of the root node,

0 := p(1), such that 0 /∈ N. The period correspond-

ing to each node n is denoted as t(n). The proba-

bility of the state of each node n is denoted as π

such that

∑

n∈N|t(n)=t

= 1, for all t = 1,...,H. The

t-th time predecessor of node n is deﬁned as p(n,t) :=

p(p(n,t −1)), such that p(n,1) := p(n). The nodes of

the path from the root node to node n are denoted as

the ordered set P(n) := {p(n,t(n) −1),..., p(n,1),n}.

The set of successors of node n is deﬁned as S(n) :=

∈ N,k = 1,...,H −t(n)|n = p(n

,k)}. The set of

leaf nodes is L := {n ∈ N|t(n) = H}.

Table 1: Basic index sets.

T periods, {1,...,H}

A already acquired cargoes

P possible cargoes to be acquired

C cargoes, A ∪ P

N nodes of the scenario tree

L leaf nodes of the scenario tree

Parameters are described in Table 2. The demand

volume of the product at each node n is known and

denoted as d

. The demand distribution for period

t = 1,...,H is represented by (d

,π

) such that n ∈ N

and t(n) = t. Due to storage constraints, the inventory

of the product at the end of each period is restricted

between a minimum volume, s, and a maximum vol-

ume, s, and there is an initial storage volume, s

, at

the beginning of the planning horizon.

The period at which an already acquired cargo

Undominated Valid Inequalities for a Stochastic Capacitated Discrete Lot-sizing Problem with Lead Times, Cancellation and Postponement

391

Table 2: Parameters.

demand volume at node n ∈ N

probability of node n ∈ N

initial inventory volume

s,s min. and max. storage capacities by period

period in which already acquired cargo c ∈ A

is received

volume of cargo c ∈ C

delivery time of cargo c ∈ P, such that

0 ≤ γ

≤ H − 1

cancellation minimum time of already

acquired cargo c ∈ A,

such that 0 ≤ δ

≤ τ

− 1

postponement minimum time of already

acquired cargo c ∈ A,

such that 0 ≤ ε

≤ H − τ

acquisition unit cost of cargo c ∈ C

cancellation unit cost of cargo c ∈ A

postponement unit cost of cargo c ∈ A

storage unit cost in period t ∈ T

already acquired volume that is received in

period t ∈ T (auxiliary deducted parameter)

c ∈ A is received is ﬁxed, τ

, and it is decided in pre-

vious acquisitions (i.e. previous model resolutions).

Each cargo c ∈ C has a given volume, q

. The achieve-

ment of decisions on cargoes have latency times mea-

sured in periods. The delivery time of a cargo c ∈ P,

, establishes the length of the wait time (measured

in periods) between the acquisition decision and the

actual arrival of the cargo. The minimum time for

cancellation of a cargo c ∈ A, δ

, establishes the min-

imum number of periods prior to the delivery period

at which the cargo may be cancelled. The minimum

postponement time for a cargo c ∈ A, ε

, establishes

the minimum number of periods after the initial de-

livery period in which the posponed cargo can be re-

ceived. The achievement period of decisions on ac-

quisition, cancellation and postponement must take

place within the planning horizon.

For each cargo c ∈ C there are unit costs per vol-

ume associated with the decisions to acquire, ca

cancel, cc

, and postpone, cp

. In addition, there is a

unit cost associated with storage at each period t ∈ T ,

The already acquired volume that is scheduled to

be received at each period is determined by the sum

of the cargoes that are received in that period,

∑

{c∈A|τ

=t}

, ∀t ∈ T ;

this is an auxiliary parameter that summarize deci-

sions of previous model resolutions on a rolling hori-

zon scheme.

Table 3: Derived index sets.

nodes where it is possible to acquire

cargo c ∈ P, {n ∈ N|t(n) ≤ H − γ

}

nodes where it is possible to cancel and

postpone cargo c ∈ A, {n ∈ N|t(n) ≤ τ

− δ

}

periods to where it is possible to

postpone cargo c ∈ A, {t ∈ T |t ≥ τ

+ ε

}

Table 4: Functions and mappings of nodes.

t(n) period of node n ∈ N

p(n) immediate predecessor node of node

n ∈ N in the tree

p(n,t) t-th time predecessor node of node

n ∈ N in the tree

P(n) nodes in the path from root node to node

n ∈ N in the tree

S(n) successor nodes of node n ∈ N in the tree

Table 5: Variables.

inventory volume at the end of the period of

node n ∈ N

acquired volume incoming at node n ∈ N

if cargo c ∈ P is acquired at node n ∈ N

(binary)

cancelled volume outgoing of node n ∈ N

if already acquired cargo c ∈ A is cancelled

in node n ∈ N

(binary)

postponed volume incoming at node n ∈ N

if already acquired cargo c ∈ A is postponed

in node n ∈ N

to period t ∈ T

(binary)

Derived subsets of the sets of nodes and periods

that are indexed in the parameters are established in

Table 3 in order to facilitate the formulation. There

are subsets to abbreviate the denomination of nodes

where it is possible to acquire each cargo c ∈ P, N

and where it is possible to cancel and postpone each

cargo c ∈ A, N

. In addition, subsets of periods to

where it is possible to postpone each cargo c ∈ A are

established, T

. The subset subscripts are part of their

denomination.

Functions and mappings on the nodes of the tree

are summarized in Table 4.

In the stochastic model, all decisions depend on

the nodes of the tree according to Table 5. An ac-

quisition decision on a cargo c ∈ P at node n ∈ N

is represented by binary variable v

. For each of

these decisions, the delivery of cargo c will be on the

nodes of the subtree rooted on n with period t(n)+ γ

∈ S(n)|t(n

) = t(n) + γ

}. A cancellation decision

of an already acquired cargo c ∈ A at node n ∈ N

represented by binary variable x

. For each of these

decisions, the already acquired volume of cargo c that

ICORES 2019 - 8th International Conference on Operations Research and Enterprise Systems

392

was budgeted to be delivered on the nodes of the sub-

tree rooted at n with period τ

, {n

∈ S(n)|t(n

) = τ

is cancelled. A postponement decision of an already

acquired cargo c ∈ A is modeled by a cancellation de-

cision of the cargo in conjunction with a decision to

postpone it towards period t ∈ T

, that is represented

by binary variable z

. For each of these decisions, the

already acquired volume of cargo c that was budgeted

to be delivered on the nodes of the subtree rooted on

n at period τ

, {n

∈ S(n)|t(n

) = τ

}, is postponed to

the nodes of the subtree rooted on n at period t ∈ T

∈ S(n)|t(n

) ∈ T

}. The amounts of inventory,

acquisition, cancellation and postponement at node n

are summarized and represented by the variables, s

, w

and y

, respectively.

The indexes of periods in deterministic parame-

ters or variables are reduced to the temporary realiza-

tion of a node n by t(n). This is the case for parame-

ters corresponding to the already acquired volume and

storage unit cost.

From the previous deﬁnitions the formulation of

the multi-stage stochastic optimization model (SCS)

min

∑

n∈N

∑

{c∈P|n∈N

}

(1)

∑

{c∈A|n∈N

}

(cc

− ca

(2)

∑

{c∈A,t∈T

|n∈N

}

(cp

+ ca

− cc

(3)

+ h

t(n)

, (4)

s.t.

p(n)

+ a

t(n)

+ u

+ y

= d

+ w

+ s

, ∀n ∈ N,

(5)

s ≤ s

≤ s, ∀n ∈ N, (6)

∑

{c∈P|γ

+1≤t(n)}

p(n,γ

)

, ∀n ∈ N, (7)

∑

∈P(n)

≤ 1, ∀c ∈ P,∀n ∈ N|t(n) = H −γ

, (8)

∑

{c∈A|t(n)=τ

}

∑

∈P(n)|t(n

)≤τ

−δ

}

∀n ∈ N, (9)

∑

∈P(n)

≤ 1, ∀c ∈ A, ∀n ∈ N|t(n) = τ

− δ

(10)

≥ z

, ∀c ∈ A, ∀n ∈ N

,∀t ∈ T

, (11)

∑

{c∈A|t(n)≥τ

+ε

}





∑

∈P(n)∩N

}

,t(n)





∀n ∈ N, (12)

∑

∈P(n),t∈T

}

≤ 1, ∀c ∈ A, ∀n ∈ N

, (13)

≥ 0, ∀n ∈ N,

∈ {0,1}, ∀c ∈ P,∀n ∈ N

∈ {0,1}, ∀c ∈ A,∀n ∈ N

,∀t ∈ T

This formulation takes into account the information

structure of the scenario tree. It minimizes the expec-

tation of acquisition costs (1), cancellation costs less

acquisition costs in case of cancellation (2), postpone-

ment costs plus acquisition costs minus postponement

costs (3) (a postponement is modeled in conjunction

with a cancellation) and storage costs (4).

Constraints (5) set the volume balance for each

node. The lower and upper storage bounds at each

node are determined by inequalities (6). The amount

of product acquired that is received at each node is

determined by acquisitions of cargoes in the possi-

ble range of the corresponding acquisition periods ac-

cording to equalities (7). The constraints (8) state

that each cargo is acquired at a single node at most

in each path from the root node to a node whose pe-

riod coincides with the latest acquisition period of the

cargo. The product previously acquired that is can-

celled at each node is determined by the cancellations

of the nodes in the path from the root node to the

node, whose cancellation periods are less than the de-

livery period less the cancellation time, according to

(9). Constraints (10) state that each cargo to be can-

celled is at a single node at most in each path from

the root node to a node whose period coincides with

the receiving period minus the cancellation time of the

cargo. The postponement of the cargoes is modeled in

conjunction with the cancellation, i.e. only cancelled

cargoes can be postponed, (11). The already acquired

volume that is postponed in a node is determined by

the postponements of the cargoes in the nodes in the

path from the root to the node for all periods supe-

rior to the period of reception plus the delay time of

the node, according to (12). Constraint (13) state that

each cargo to be postponed is at a single node at most

in each path from the root node to a node in some pe-

riod greater than the receiving period plus the time of

postponement of the node.

Undominated Valid Inequalities for a Stochastic Capacitated Discrete Lot-sizing Problem with Lead Times, Cancellation and Postponement

393

3 VALID INEQUALITIES FOR

THE STOCHASTIC MODEL

Since (SCS) belongs to the time complexity class

N P -hard, there is no known polyhedral description

of the convex hull of its feasible solutions. It is nev-

ertheless interesting to derive valid inequalities which

can be used to strengthen the original formulation. In

some cases adding these inequalities can directly im-

prove the capacity of the solver to ﬁnd solutions for

larger instances in shorter times; even when this is

not the case, they may be used within a more sophis-

ticated solving strategy, such as branch and cut meth-

ods relying on constraint separation.

In this section, we discuss a variation of classic

(`,S) valid inequalities for the stochastic capacitated

discrete lot-sizing problem with lead times, cancella-

tion and postponement, and a scheme to obtain un-

dominated valid inequalities of the variation.

3.1 Derived (`,S) Valid Inequalities

A set of valid inequalities are derived por (SCS) based

on the (`,S) valid inequalities formulation for the de-

terministic uncapacitated lot-sizing problem (Barany

et al., 1984) while considering the extension for the

stochastic case (Guan et al., 2006). The derived in-

equalities establish bounds on decision variables for

the nodes of possible paths in the scenario tree (Tes-

turi et al., 2018).

Proposition 1 . Let ` ∈ N and S ⊆ P(`) then the de-

rived (`,S) inequality

∑

n∈S

≤

∑

n∈S

β(n) +

∑

n∈S

+ s

, (14)

where

β(n) :=

∑

{c∈P|γ

+1≤t(n)}

p(n,γ

)

∑

∈P(`)\P(p(n))

and

∑

∈P(`)\P(p(n))

is valid for the feasible region of (SCS).

As shown by the authors the derived (`, S) valid

inequalities has alternative and equivalent inequalities

without inventory variable, s

∑

n∈P(`)\S

∑

n∈S

β(n) +

∑

n∈S

− w

∑

n∈P(`)

≥

−

∑

n∈P(`)

t(n)

− s

, for all ` ∈ N, S ⊆ P(`).

(15)

3.2 Undominanted (`,S) Valid

Inequalities

Depending on the instance values of the parameters

and d

, some of the derived (`,S) inequalities (15)

of a given subset S ⊆ P(`), ` ∈ N, may be dominated

by other inequalities of a different subset. Therefore,

a procedure was established to determine undomi-

nanted inequalities on the power set of P(`).

Let χ

χ := [(v

)

c∈P,n∈N

,(x

)

c∈A,n∈N

,(z

)

c∈A,n∈N

,t∈T

]

be the composite variable. Let b := d

−

∑

n∈P(`)

t(n)

− s

, for each ` ∈ N, be the indepen-

dent term of (15). Given the power set of P(`),

:= {S

,...,S

}, let α

be the coefﬁcient vector of

variable χ

χ on the inequality (15) for subset S

,k ∈

{1,...,K

}. Therefore, the inequalities (15) can be es-

tablished as

χ ≥ b, k ∈ {1, ...,K

},` ∈ N.

Given i, j ∈ {1,...,K

} and that χ

χ is nonnegative, it

is said that α

χ ≥ b dominantes α

χ ≥ b, if the

componentwise comparison of α

and α

is such that

iλ

≤ α

jλ

for each component λ ∈ Λ, and at least for

one component the inequality is strict. Let S

be the

subset of dominant inequalities on S

The procedure to obtain S

by pairwise compari-

son of inequalities has and upper bound of O(K

|Λ|)

operations. If S

has few elements, an efﬁcient heuris-

tic to obtain a promising undominanted inequality

candidate,

∗

:= argmin

i∈K

∑

λ∈Λ

iλ

, (16)

takes Θ(K

|Λ|) operations. Lets denote S

d∗

:= {i

∗

Furthermore, lets denote S

the case where the

power set, S

, is approximated by a representative

subset of S

that contains only the root node, n = 1.

Tree variants of the original formulation, (SCS),

are generated by including to it the inequalities of the

sets S

, S

d∗

and S

, for each ` ∈ N, establishing formu-

lations denoted as (SCS-S

), (SCS-S

d∗

) and (SCS-S

respectively.

4 COMPUTATIONAL

EXPERIMENTS

This section explores the computational impact of

adding three families of inequalities introduced in the

previous section to the original formulation. These

are the power set inequalities (S

), the dominance re-

duction inequalities (S

d∗

), and the root representative

inequalities (S

). The original formulation and the

three modiﬁed formulations are tested over a set of

test instances, checking the quality of the obtained so-

lutions and the computational effort invested by the

solver.

In order to generate a number of diverse test in-

stances, six scenario tree structures were considered.

ICORES 2019 - 8th International Conference on Operations Research and Enterprise Systems

394

Each structure, depicted in Table 6, is determined by

the number of direct descendants of each node (tree

arity) and the number of periods of the planning hori-

zon. For each tree structure with arity g and H periods

there are g

H−1

escenarios and (g

−1)/(g−1) nodes.

Table 6: Size of scenario tree structures.

Arity(g) Periods(H) Scenarios Nodes

2 5 16 31

2 6 32 63

2 7 64 127

3 5 81 121

3 6 243 364

3 7 729 1093

The size of each tree structure model instance (num-

ber of equations and variables) for a given distribution

of cargos (C) is shown in Table 7.

Table 7: Instance size of scenario tree structures by cargo

distribution.

g H |C|(|A| + |P|) Eqs. Vars. (binary)

2 5 10 (2+ 8) 225 249 (124)

2 6 12 (3+ 9) 480 549 (296)

2 7 14 (3+11) 1.012 1.223 (714)

3 5 10 (3+ 7)

827 809 (324)

3 6 12 (4+ 8) 2.542 2.485 (1.028)

3 7 14 (4+10) 7.987 8.091 (3.718)

Thirty data instances were randomly generated for

each tree structure and cargo distribution, totaling 180

instances. Each instance has an initial storage, s

20, and a lower and an upper bound storage, s = 0

and s = 80, respectively. For each cargo c ∈ C there

is an uniformly distributed volume, q

∼ U[10, 50],

and there are costs evenly distributed according to

the operations of acquisition, ca

∼ U[150,250], can-

cellation, cc

∼ U[30, 50], and postponement, cp

∼

U[5, 12]. Each already acquired cargo c ∈ A has de-

livery period τ

= 1 or 2 with equal probability. Each

cargo c ∈ C has delivery time γ

= 1, cancellation time

= 1 and delay time ε

= 1. The unit storage cost at

each period t is h

= 1. For each scenario n ∈ L (leaf

node), a probability of state π

is established from a

distribution Beta(α = 2,β = 2); the probability of the

remaining nodes is obtained from the sum of the prob-

abilities of their corresponding immediate successor

nodes. Finally, the demand for each node is evenly

distributed, d

∼ U[10,50].

The computational implementation was per-

formed using AMPL (Fourer et al., 2002) for the al-

gebraic coding of the stochastic model, and GUROBI

7.5 (Gurobi Optimization, LLC, 2018) for the resolu-

tion of the instances through its branch and cut solver.

The calculations were carried out on an Intel Core

i7 5960X 3.5GHz computer with 20MB cache and

64GB RAM, operating with CentOS-7 Linux system.

For each instance, the original model and the vari-

ants were solved. A summary of the results of the

original model and each variant is presented in Ta-

ble 8, Table 9, Table 10 and Table 11, respectively for

(SCS), (SCS-S

), (SCS-S

d∗

) and (SCS-S

Table 8: Average results of formulation (SCS) by tree struc-

ture and cargoes.

g-H-C Time(s) MIP Nodes Cuts LP

2-5-10 0.68 - 6,449 125 10.31

2-6-12 13.07 - 30,706 189 18.85

2-7-14

†

493.45 0.26 1,093,185 713 9.84

3-5-10 13.39 - 31,260 300 12.40

3-6-12 ‡758.80 1.90 733,244 1,319 19.64

3-7-14

900.25 5.02 27,291 1,264 23.42

(†) 12 of 30 instances reach the time limit of 900 s.

(‡) 24 of 30 instances reach the time limit of 900 s.

(#) All instances reach the time limit of 900 s.

Table 9: Average results of formulation (SCS-S

) by tree

structure and cargoes.

g-H-C Time(s) MIP Nodes Cuts LP

2-5-10 1.11 - 2,132 88 7.44

2-6-12 17.48 - 24,885 253 11.25

2-7-14

†

430.79 0.25 431,172 1,184 6.97

3-5-10 7.51 - 3,189 206 9.72

3-6-12

‡

692.13 1.36 174,553 1,539 17.21

3-7-14

900.83 12.85 5,754 ? 36.60

(†) 12 of 30 instances reach the time limit of 900 s.

(‡) 19 of 30 instances reach the time limit of 900 s.

(#) All instances reach the time limit of 900 s.

Table 10: Average results of formulation (SCS-S

d∗

) by tree

structure and cargoes.

g-H-C Time(s) MIP Nodes Cuts LP

2-5-10 0.48 - 2,693 98 7.46

2-6-12 13.22 - 37,603 249 11.25

2-7-14

†

435.25 0.28 1,005,992 1,024 6.97

3-5-10 5.87 - 12,281 239 9.72

3-6-12

‡

720.13 1.71 513,898 1,609 17.24

3-7-14

900.19 4.66 41,495 1,367 20.53

(†) 12 of 30 instances reach the time limit of 900 s.

(‡) 21 of 30 instances reach the time limit of 900 s.

(#) All instances reach the time limit of 900 s.

The summary shows, for each tree structure deﬁned

by arity, periods and number of cargoes, depicted at

column “g-H-C”, the average results of the 30 in-

stances of the model (SCS) and its variant with the

corresponding valid inequalities. The average results

depicted are solver elapsed time at column “Time”,

solver MIP gap percentage for instances that reach the

time limit of 900 s at column “MIP”, solver number

of nodes of solver branch and cut method at column

Undominated Valid Inequalities for a Stochastic Capacitated Discrete Lot-sizing Problem with Lead Times, Cancellation and Postponement

395

Table 11: Average results of formulation (SCS-S

) by tree

structure and cargoes.

g-H-C Time(s) MIP Nodes Cuts LP

2-5-10 0.45 - 3,361 94 7.48

2-6-12 11.90 - 26,885 237 11.25

2-7-14

†

443.32 0.27 1,145,992 978 6.99

3-5-10 4.87 - 11,584 264 9.74

3-6-12

‡

449.50 0.73 330,166 1,546 17.20

3-7-14

900.18 4.84 29,881 1,416 20.90

(†) 12 of 30 instances reach the time limit of 900 s.

(‡) 12 of 30 instances reach the time limit of 900 s.

(#) All instances reach the time limit of 900 s.

“Nodes”, number of cuts added by solver’s branch

and cut method at column “Cuts”, and the relative ra-

tio percentage of the objective value with respect of

the objective value of the linear programming relax-

ation of the model, at column “LP”.

In the case of formulation (SCS-S

), it can be seen

that the average Time results for the tree structures (2-

5-10) and (2-6-12) are worse than the corresponding

to formulation (SCS). On the other hand, the aver-

age Time and MIP-gap results of the formulation for

the tree structures (2-7-14), (3-5-10) and (3-6-12) are

better than the corresponding to formulation (SCS).

Also, the formulation reduces to 19 the number of in-

stances of structure (3-6-12) that reach the time limit

of 900 s, compared with 24 of the (SCS) formulation.

Finally, except for structure (3-7-14), the formulation

obtains a reduction of the LP-gap of the remaining

structures compared with formulation (SCS).

In the case of formulation (SCS-S

d∗

), only the

average Time results for the tree structures (2-6-12)

are slightly worse than the corresponding ones of

formulation (SCS). The formulation has lower LP-

gap for all tree structures compared to formulation

(SCS). With regards to its comparison with formu-

lation (SCS-S

), the formulation obtains better Time

results for the tree structures (2-5-10), (2-6-12) and

(3-5-10); and it obtains equal or slightly worse MIP-

gap results, except for formulation (3-7-14), where it

gets better result.

Formulation (SCS-S

) obtains better Time results

than formulations (SCS) and (SCS-S

) for all tree

structures. While it get better MIP-gap results for

all tree structures than the formulation (SCS), it

gets slightly worse MIP-gap results than formulation

(SCS-S

), except for structure (3-7-14), where it gets

better result. In comparison with formulation (SCS-

d∗

), it has slightly better Time results, and similar

MIP-gap results. It reduces to 12 the number of in-

stances of structure (3-6-12) that reach the time limit

of 900 s, compared with 21 of the (SCS-S

d∗

) formu-

lation.

5 CONCLUSIONS

A stochastic multi-stage capacitated discrete lot-

sizing model formulation of the provision with lead

time of the uncertain demand of a product has been

proposed. The decisions on product lots are modeled

with their delay time, aspect that for cancellation and

postponement decisions is not covered in the previ-

ous literature. A discrete time stochastic process with

ﬁnite probability, summarized in a scenario tree, is

used to model the information structure of the uncer-

tain demand. The model is formulated by stochas-

tic programming with entities indexed by nodes of

the scenario tree. The model incorporates the cancel-

lation and postponement decisions with delay time,

which implied the revision of the deﬁnitions of the

variables and the restrictions to take into account the

structure of the scenario tree. To tighten the formula-

tion valid inequalities based on the (`,S) inequalities

approach were used. Since the inequalities are highly

dominated for most experimental instances, a scheme

is established to determine undominated ones. Three

variants of the formulation are obtained from the in-

clusion of the power-set, undominated and represen-

tative valid inequations. The original formulation and

the three variants are tested over a set of test instances,

checking the quality of the obtained solutions and the

computational effort invested by the solver. Compu-

tational experiments where carried out for several in-

stances within a few tree structures of different sizes.

Most computational experiments could be solved to

optimality for the small and medium-size tree struc-

tures. The representative and undominated formula-

tions obtains a slightly better results than the original

and power set formulations for all tree structures.

ACKNOWLEDGMENTS

This work was partially supported by the Comisi

Sectorial de Investigaci

on Cient

ıﬁca, CSIC, and the

Programa de Desarrollo de las Ciencias B

asicas,

PEDECIBA.

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