**Page maintained by M. Delorme (maxence.delorme2@unibo.it), M. Iori (manuel.iori@unimore.it), and S. Martello (silvano.martello@unibo.it).**

**The BPPLIB is a collection of codes, benchmarks, and links for the one-dimensional Bin Packing and Cutting Stock problem.**

Problem description

The bin packing problem (BPP) can be informally defined in a very simple way. We are given *n* items, each having an integer weight *w _{j} (j = 1, ..., n)*, and an unlimited number of identical bins of integer capacity

*c*. The objective is to pack all the items into the minimum number of bins so that the total weight packed in any bin does not exceed the capacity.

Similarly, in the cutting stock problem (CSP), we are given *m* item types, each having an integer weight *w _{j}*

_{ }and an integer demand

*d*

*, and an unlimited number of identical bins (frequently called rolls in the literature) of integer capacity*

*(j = 1, ..., m)*_{j}*c*. The objective is to produce

*d*

*copies of each item type*

_{j }*j*using the minimum number of bins so that the total weight packed in any bin does not exceed its capacity. A CSP instance can be easily transformed into an equivalent BPP instance and vice-versa.

For a recent **survey **on these problems, see M. Delorme, M. Iori, and S. Martello. Bin Packing and Cutting Stock Problems: Mathematical Models and Exact Algorithms. *European Journal of Operational Research*, 255(1):1–20, 2016.

Previous surveys on the BPP and the CSP

E.G. Coffman Jr., J. Csirik, G. Galambos, S. Martello, and D. Vigo. Bin packing approximation algorithms: Survey and classification. In P.M. Pardalos, D.-Z. Du, and R.L. Graham, editors, *Handbook of Combinatorial Optimization*. Springer New York, 2013.

G. Wäscher, H. Haußner, and H. Schumann. An improved typology of cutting and packing problems. *European Journal of Operational Research*, 183(3):1109–1130, 2007.

J.M. Valerio de Carvalho. LP models for bin packing and cutting stock problems. *European Journal of Operational Research*, 141(2):253–273, 2002.

E.G. Coffman Jr., M.R. Garey, and D.S. Johnson. Approximation algorithms for bin packing: a survey (file BPchapter.pdf). In D. S. Hochbaum, editor, Approximation algorithms for NP-hard problems, pages 46–93. PWS Publishing Co., Boston, MA, USA, 1996.

P.E. Sweeney and E.R. Paternoster. Cutting and packing problems: a categorized, application-orientated research bibliography. *Journal of the Operational Research Society*, pages 691–706, 1992.

R.W. Haessler and P.E. Sweeney. Cutting stock problems and solution procedures. *European Journal of Operational Research*, 54(2):141–150, 1991.

H. Dyckhoff. A typology of cutting and packing problems. *European Journal of Operational Research*, 44(2):145–159, 1990.

Codes (click the name of a code to download it, or to access it through a link).

**Branch-and-bound**

MTP is a branch-and-bound algorithm for the BPP proposed by S. Martello and P. Toth. in *Knapsack Problems: Algorithms and Computer Implementations*. John Wiley & Sons, Chichester, 1990.

BISON is a branch-and-bound algorithm for the BPP proposed by A. Scholl, R. Klein, and C. Jürgens in Bison: a fast hybrid procedure for exactly solving the one-dimensional bin packing problem. *Computers & Operations Research*, 24(7):627-645, 1997. The code is not available online, but can be obtained at request from the authors (armin.scholl@uni-jena.de).

The CVRPSEP package is a collection of routines dedicated to the Capacitated Vehicle Routing Problem. One of them is a branch-and-bound algorithm, similar to MTP, specifically designed to solve the BPP. The package relies on A new branch-and-cut algorithm for the capacitated vehicle routing problem. *Mathematical Programming*, 100(2):423–445, 2004, by J. Lysgaard, A.N. Letchford, and R.W. Eglese.

**Branch-and-price**

BELOV is a branch-and-cut-and-price algorithm for the CSP proposed by G. Belov and G. Scheithauer in A branch-and-cut-and-price algorithm for one dimensional stock cutting and two-dimensional two-stage cutting. *European Journal of Operational Research*, 171(1):85–106, 2006.

SCIP-BP is a branch-and-price algorithm for the BPP provided as an example when downloading the SCIP Otimization Suite.

**Pseudo-polynomial formulations**

For the codes implemented by ourself, we propose a version requiring the commercial ILP solver Cplex, and an alternative version requiring the free ILP solver SCIP.

ONECUT is an algorithm based on a pseudo-polynomial formulation of the CSP solved throught an ILP solver. It uses the model proposed by H. Dyckhoff in A new linear programming approach to the cutting stock problem. *Operations Research*, 29(6):1092–1104, 1981.

ARCFLOW is an algorithm based on a pseudo-polynomial formulation of the CSP solved throught an ILP solver. It uses the model proposed by J.M. Valério de Carvalho in Exact solution of bin-packing problems using column generation and branch-and-bound. *Annals of Operations Research*, 86:629–659, 1999.

DPLOW is an algorithm based on a pseudo-polynomial formulation of the BPP solved throught an ILP solver. It uses the model proposed by H. Cambazard and B. O’Sullivan in Propagating the bin packing constraint using linear programming. In *Principles and Practice of Constraint Programming–CP 2010*, pages 129–136. Springer, 2010.

VPSolver is a software than can solve vector packing problems using pseudo-polynomial formulation. Limited to 1 dimension, this problem becomes a CSP. It relies on Bin packing and related problems: General arc-flow formulation with graph compression. *Computers & Operations Research*, 69:56-67, 2016, by F. Brandão and J.P. Pedroso.

Benchmarks

All instances are given in the two following formats:

- In the BPP format:
- Number of items (
*n*) - Capacity of the bins (
*c*) - For each item
*j*(*j = 1,...,n*):- Weight (
*w*)_{j}

- Weight (

- Number of items (
- In the CSP format:
- Number of item types (
*m*) - Capacity of the bins (
*c*) - For each item type
*j*(*j = 1,...,m*):- Weight (
*w*) and demand (_{j}*d*)_{j}

- Weight (

- Number of item types (

The solution of all instances can be found **here**.

These are the instances used by E. Falkenauer in A hybrid grouping genetic algorithm for bin packing. *Journal of Heuristics*, 2(1):5–30, 1996. They were downloaded from the OR-library. They are divided into two classes of 80 instances each: the first class has uniformly distributed item sizes (‘Falkenauer U’) with *n *between 120 and 1000, and *c *= 150. The instances of the second class (‘Falkenauer T’) includes the so-called triplets, i.e., groups of three items (one large, two small) that need to be assigned to the same bin in any optimal packing, with *n *between 60 and 501, and *c *= 1000.

These are the instances used by A. Scholl, R. Klein, and C. Jürgens in Bison: a fast hybrid procedure for exactly solving the one-dimensional bin packing problem. *Computers & Operations Research*, 24(7):627–645, 1997. They were downloaded from the authors' web page. They are divided into three sets of 720, 480, and 10, respectively, uniformly distributed instances with *n *between 50 and 500. The capacity *c *is between 100 and 150 (set ‘Scholl 1’), equal to 1000 (set ‘Scholl 2’), and equal to 100 000 (set ‘Scholl 3’), respectively;

These are some of the instances (those regarded as the most difficult ones) used by G. Wäscher and T. Gau in Heuristics for the integer one-dimensional cutting stock problem: a computational study. *OR Spectrum*, 18(3):131–44, 1996. They were downloaded from the ESICUP website. These 17 hard instances have *n* between 27 and 239 and *c* = 10 000.

These are the instances used by P. Schwerin and G. Wäscher in The bin-packing problem: a problem generator and some numerical experiments with FFD packing and MTP. *International Transactions in Operational Research*, 4(5–6):377–389, 1997. They were downloaded from the ESICUP website. They are divided into two sets (‘Schwerin 1’ and ‘Schwerin 2’) of 100 instances each with *n *= 100 and *n *= 120, respectively, and *c *= 1000;

**Schoenfield_Hard28 (BPP - CSP) **

These are the instances used by J.E. Schoenfield in Fast, exact solution of open bin packing problems without linear programming. Technical report, US Army Space and Missile Defense Command, Huntsville, Alabama, USA, 2002. They were downloaded from the ESICUP website. These 28 instances have *n* between 160 and 200, and *c* = 1000.

**Randomly Generated Instances (BPP - CSP) **

These are the instances randomly generated by M. Delorme, M. Iori, and S. Martello in Bin Packing and Cutting Stock Problems: Mathematical Models and Exact Algorithms. *European Journal of Operational Research*, 255(1):1-20, 2016. They have different values of n (50, 100, 200, 300, 400, 500, 750, 1 000), c (50, 75, 100, 120, 125, 150, 200, 300, 400, 500, 750, 1 000), and of the minimum (0.1 c, 0.2 · c) and maximum (0.7 c, 0.8 · c) item weight. For each quadruplet (n, c, minimum weight, maximum weight), 10 instances were generated.

**Augmented Non IRUP and Augmented IRUP Instances (BPP - CSP) **

These are the difficult instances proposed by M. Delorme, M. Iori, and S. Martello in Bin Packing and Cutting Stock Problems: Mathematical Models and Exact Algorithms. *European Journal of Operational Research*, 255(1):1-20, 2016.

These are the instances proposed by T. Gschwind and S. Irnich in Dual Inequalities for Stabilized Column Generation Revisited. *INFORMS Journal on Computing*, 28(1):175-194, 2016.

BppGame: An interactive visual solver

An open source visual ScalaFX application to interactively solve BPP and CSP instances can be downloaded, as a compressed file, from http://www.or.deis.unibo.it/staff_pages/martello/Tools/T.html. Additional information on the whole architecture and (future) enhanced versions can be found at http://gianlucacosta.info/*Tech. rep. DEI, University of Bologna*, 2017.

The easiest way to test BppGame is to execute the following sequence of actions:

- Click on "Bin Packing Problem" and extract its contents
- Click on "BppGame-1.2"
- Click on "bin"
- Execute "BppGame.bat" (Windows) or "BppGame" (Linux)
- Click on "Sample problems".

The figure on top of the page shows the BppGame visualization of an instance.

Bibliography and useful links

This is a BibTex file of relevant references for the bin packing and the cutting stock problems.

Here are some interesting links related to the BPP and the CSP :

- ESICUP
- OR-library
- Bin Packing page of A. Scholl and R. Klein