Crainic, Teodor

Perboli, Guido

Rei, W.

Watson J-P

Woodruff, D.L.

English

PORTO Publications Open Repository TOrino

5th International Workshop on Freight Transportation and Logistics
ODYSSEUS 2012, May 21-25, Mykonos, Greece
A Stochastic Bin Paking Model for Logistical
Capacity Planning
Teodor Gabriel Crainic
Department of Management and Technology,
School of Management, UQAM and CIRRELT, Montreal, Canada
Guido Perboli
Department of Control and Computer Engineering,
Politecnico di Torino, Turin, Italy
Jean-Paul Watson
Sandia National Laboratories,
Albuquerque, New Mexico, USA
David L. Woodruff
Graduate School of Management,
University of California, Davis, California, USA
Walter Rei
Department of Management and Technology,
School of Management, UQAM and CIRRELT, Montreal, Canada
Email: walter.rei@cirrelt.ca
1 Introduction
Most modern companies are part of international economic networks, where goods are
produced under different strategies (made-to-stock or made-to-order), then transported
over long distances, and stored for variable periods of time at different locations along the
considered network. In this context, having enough capacity to properly perform crucial
activities such as the supply, storage and distribution of goods is paramount if a company
is to be competitive. It should be noted that these activities are often performed by first
consolidating goods in appropriate bins, which are then either stored at warehouses or
shipped using multiple vehicles through various transportation modes. Companies thus
face the problem of having to plan for enough capacity, expressed here in terms of bins,
to be available at different locations throughout their network and for different periods
of time. Given that logistical activities are often subcontracted, capacity planning entails
351
5th International Workshop on Freight Transportation and Logistics
ODYSSEUS 2012, May 21-25, Mykonos, Greece
negotiations with third party logistics firms (3PL’s) to book the needed capacity. The
results of these negotiations often take the form of medium term contracts that specify
both the capacity to be used (the quantity and type of the bins) and the logistical services
to be performed (storage, transportation, bin operations, etc.). Therefore, bin packing
models represent important decision support tools for logistics managers, who bear the
responsibility for these tactical planning decisions.
Although bin packing optimization problems have been extensively studied, see [2], this
research has mainly been conducted under the assumption that all necessary information
concerning the different parameters used to model the problems is known and readily
available (i.e., deterministic parameters). Given the time lag that usually exists between
the capacity planning decisions made and the operational decisions that are taken and that
define how the planned capacity is used, such an hypothesis in unlikely to be observed.
Therefore, we propose a new stochastic bin packing model, referred to as the stochastic
variable size and cost bin packing (SVSCBP) model, that explicitly takes into account the
different sources of uncertainty that appear in capacity planning problems. The model
is based on a two-stage stochastic programming formulation that separates the tactical
capacity planning decisions (i.e., the a priori plan), that are taken when certain parameters
are unknown and dependent on a random event, from the operational decisions (i.e., the
recourse decisions), that define the adjustments made whenever the capacity is used and
all the information becomes known. To efficiently solve the proposed model, we develop a
progressive hedging (PH) based heuristic, which uses as the scenario subproblem solver the
algorithms proposed in [1]. The remainder of this abstract is organized in two sections. In
section 2, a detailed formulation of the considered problem is provided. We then conclude
with section 3 by providing a general description of the solution approach proposed and
of the experimental plan that will be conducted.
2 The stochastic variable size and cost bin packing model
Let us consider the problem where a given company must produce a tactical capacity
plan by choosing a set of appropriate bins (i.e., first stage decisions), which are defined
according to their specific volume and fixed cost. Fixed costs are used here to represent
the specific rates offered by the different 3PL’s for bins of different sizes. Selected bins are
to be available to the company under the terms of a contract specifying the parameters
of usage. The capacity plan produced is then repeatedly applied to perform specific
logistical operations for a given period of time (e.g., organize a maritime shipment of
goods every week for the next semester). In this context, the bins that are included
in the capacity plan are chosen in advance and not knowing exactly what items will
eventually be packed. Therefore, whenever packing operations are performed (i.e., second
352
5th International Workshop on Freight Transportation and Logistics
ODYSSEUS 2012, May 21-25, Mykonos, Greece
stage decisions), it may occur that the planned capacity is insufficient. When such a case
occurs, extra capacity must be purchased to properly perform the necessary operations.
We thus consider that extra bins may be added during the operational phase (i.e., second
stage decisions), provided at a premium by the 3PL’s (i.e., fixed costs defined according
to the spot market).
This problem can now be formulated as the SVSCBP model. Let J define the set of
available bins in the first stage of the problem where, ∀j ∈ J , fj and Vj are respectively
the fixed cost and volume associated with bin j. Let set Ω be the sample space of the
random event, where ω ∈ Ω defines a particular realization. Let vector ξ contain all
stochastic parameters defined in the model and ξ(ω) be a given realization of this random
vector. If we define the first stage variables as follows: yj = 1 if bin j ∈ J is selected, 0
otherwise; then the SVSCBP model is formulated as:
min
∑
j∈J
fjyj + Eξ[Q(y, ξ(ω))] (1)
s.t. yj ∈ {0, 1}, ∀j ∈ J , (2)
where Q(y, ξ(ω)) equals the extra cost that is paid for the capacity that is added in the
second stage of the problem, given the tactical capacity plan y and the realized vector
ξ(ω). The objective function (1) then minimizes the sum of the total fixed cost incurred
for the tactical capacity plan and the expected cost associated with the extra capacity
added during operations, while constraints (2) impose the integrality requirements on y.
To formulate Q(y, ξ(ω)), we consider the following stochastic parameters in ξ(ω): I(ω),
the set of items to be packed; vi(ω), i ∈ I(ω), the item volumes; K(ω), the set of available
bins in the second stage; and fk(ω), k ∈ K(ω), the cost of bins available in the second
stage. The second stage variables are defined as follows: zk = 1 if bin k ∈ K(ω) is selected,
0 otherwise; and xij = 1 if item i ∈ I(ω) is packed in bin j ∈ J , 0 otherwise; and xik = 1
if item i ∈ I(ω) is packed in bin k ∈ K(ω), 0 otherwise. We now define function Q(y, ξ(ω))
as follows:
Q(y, ξ(ω)) = min
∑
k∈K
fk(ω)zk (3)
s.t.
∑
j∈J
xij +
∑
k∈K(ω)
xik = 1, ∀i ∈ I(ω) (4)
∑
i∈I(ω)
vi(ω)xij ≤ Vjyj , ∀j ∈ J (5)
∑
i∈I(ω)
vi(ω)xik ≤ Vkzk, ∀k ∈ K(ω) (6)
xil ∈ {0, 1}, ∀i ∈ I(ω), ∀l ∈ J
⋃
K(ω) (7)
zk ∈ {0, 1}, ∀k ∈ K(ω). (8)
353
5th International Workshop on Freight Transportation and Logistics
ODYSSEUS 2012, May 21-25, Mykonos, Greece
The objective function (3) minimizes the cost associated with the extra bins selected.
Constraints (4) ensure that each item is packed in a single bin. Constraints (5) and (6)
make sure that the total volume of items packed in each bin does not exceed the bin
volume. Finally, (7) and (8) impose integrality requirements on all second stage variables.
3 Solution approach and experimental plan
The first step towards solving model (1)-(2), is to obtain a manageable approximation of
function Eξ[Q(y, ξ(ω))]. Therefore, sampling is applied to obtain a set of representative
scenarios (i.e, set S), which are used to approximate the expected cost associated with
the second stage. The PH solution approach, as recalled in [3], is then used to solve the
approximated problem. Following this approach, scenario decomposition is applied by re-
laxing the non-anticipativity constraints, which impose that a single tactical capacity plan
y is used for all considered scenarios. By doing so, the SVSCBP model can be decomposed
by scenarios thus obtaining, ∀s ∈ S, a deterministic packing problem. This enables the
use of the efficient heuristics proposed in [1] to solve these scenario subproblems.
A solution to the approximated problem is then constructed by iteratively applying
the following steps: 1) aggregate the scenario solutions obtained to produce a reference
point (the average over all scenario solutions is traditionally used as the reference point),
2) using an augmented Lagrangean strategy, penalize the differences observed between the
scenario solutions and the reference point (i.e., the objective function of each subproblem
is updated in such a way as to induce a single tactical capacity plan for all scenarios),
3) solve anew the updated scenario subproblems. In the presence of integer variables,
the PH algorithm may not converge given that the average over all scenario solutions
is not necessarily integer. Furthermore, considering that the algorithm solves at each
iteration a series of deterministic packing problems (i.e., one for each scenario), the overall
solution process may become very time-consuming as the number of iterations increases.
To address these issues, a series of algorithmic innovations have recently been proposed
in [3]. These innovations include different strategies to update the objective functions of
the subproblems, improve the convergence of the algorithm, provide termination criteria
or detect cyclic behavior in the solution process. These features are used to design an
efficient solution approach for the present problem. It should also be noted that the PH
algorithm proposed is implemented using the PySP library, which provides an effective
platform on which to develop such procedures, see [4].
Finally, the experimental plan is used to analyze both the impact that diversity (re-
garding 3PL’s) has on capacity planning and the effect of considering different levels of
variability and correlation on the stochastic parameters related to items, as well as to
measure the consequences of volatility in the spot market.
354
5th International Workshop on Freight Transportation and Logistics
ODYSSEUS 2012, May 21-25, Mykonos, Greece
References
[1] T. G. Crainic, G. Perboli, W. Rei and R. Tadei, “Efficient lower bounds and heuristics
for the variable cost and size bin packing problem”, Computers & Operations Research
38(11), 1474-1482, 2011.
[2] G. Wa¨scher, H. Haußner and H. Schumann, “An improved typology of cutting and
packing problems”, European Journal of Operational Research 183, 1109-1130, 2007.
[3] J.-P. Watson and D. L. Woodruff, “Progressive hedging innovations for a class of
stochastic mixed-integer resource allocation problems”, Computational Management
Science 8(4), 355-370, 2010.
[4] J.-P. Watson, D. L. Woodruff and W. E. Hart, “PySP: Modeling and Solving Stochas-
tic Programs in Python”, Mathematical Programming Computation, (to appear) 2012.
355


An improved typology of cutting and packing problems”,

Eﬃcient lower bounds and heuristics for the variable cost and size bin packing problem”,

Progressive hedging innovations for a class of stochastic mixed-integer resource allocation problems”,

PySP: Modeling and Solving Stochastic Programs

A Stochastic Bin Paking Model for Logistical Capacity Planning

A Stochastic Bin Paking Model for Logistical Capacity Planning

Abstract

Similar works

Full text

Available Versions

PORTO Publications Open Repository TOrino