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Procedia - Social and Behavioral Sciences 108 ( 2014 ) 219 234

1877-0428 2013 The Authors. Published by Elsevier Ltd. Selection and peer-review under responsibility of AIRO.doi: 10.1016/j.sbspro.2013.12.833

ScienceDirect

Finding Pattern Congurations for Bank Cheque Printing

R. Cerullia,1, R. De Leoneb,2, M. Gentilia,1,3

aDepartment of Mathematics, University of Salerno, Italyb School of Science and Technology, University of Camerino, Italy

Abstract

The problem we address in this paper arises in large-scale manufacturing of bank cheques. Due to security reasons,the cheques must be printed on special (expensive) paper. The rst step in the printing process is to prepare the platesthat will be used by the composing machine. If the imprint (image) of a particular cheque is on a plate, each timethe composing machine uses this plate a new cheque of this type is produced. Each plate has a predened number ofpositions to be impressed. Due to delivery due dates, there is an additional constraint requiring each cheque not to bepresent in more than a predened number of dierent plates. There are two dierent production costs that have to beconsidered: overproduction costs and printing costs. Each overproduced cheque can be either destroyed or stored in aproper location under surveillance. Both these alternatives have a huge environmental impact, indeed, on the one hand,paper waste is produced, while, on the other hand there is a huge energy consumption. The problem consists in deningthe pattern (i.e. the conguration of cheque images) of each plate to be used and the corresponding frequency, such thattotal costs are minimized.

We study this real world problem that is strictly related to the cutting stock problem with pattern minimization.Such a problem is addressed actually by a large cheque manufacturer in Southern part of Italy. We dene a very ecientheuristic to solve it. The proposed solution methodology is currently used by the above mentioned manufacturer todene the cheque allocation of the plates.

c 2011 Published by Elsevier Ltd.Keywords: cutting problem, heuristic algorithm, cheque printing

1. Introduction

The problem we consider here arises in large-scale manufacturing of bank cheques. Due to security reasons,the cheques must be printed on special (expensive) paper and the number of ocial printing manufacturer isvery limited. Each cheque must contain information on the bank name and on the name of the branch of thebank where the cheque holder has the account. Moreover, all cheques have the same shape and dimensionand do not contain any information on account numbers. Each branch of the bank sends a request for a xedquantity of cheques to a master collection place that gathers the individual requests and, when a speciedlevel is reached, the individual requests are sent to the central printing manufacturer.

1Email: {raaele,mgentili }@unisa.it2Email: renato.deleone@unicam.it3Corresponding author

Available online at www.sciencedirect.com

2013 The Authors. Published by Elsevier Ltd. Selection and peer-review under responsibility of AIRO.

220 R. Cerulli et al. / Procedia - Social and Behavioral Sciences 108 ( 2014 ) 219 234

The rst step in the printing process is to prepare the plates that will be used by the composing machine.If the imprint (image) of a particular cheque is impressed in the plate, each time the composing machineuses this plate a new cheque of this type is produced. An image can be replicated more than once on a plategenerating multiple instances of the cheque any time the plate is used. Each plate has a predened numberof positions to be impressed. Moreover, due to delivery due dates, there is an additional constraint requiringeach cheque not to be present in more than a predened number of distinct plates. There are two dierentproduction costs that have to be minimized:

cost due to cheque overproduction: the cost of producing an extra cheque is very large due, not onlyto the cost of the paper, but also to the fact that unused cheques must be disposed of properly (forsecurity reasons);

cost due to the number of dierent plates: each time the printing scheme is changed, it is necessaryto set up the composing machine and this operation requires extra cost in terms of workerhours andmaterials.

Each overproduced cheque can be either destroyed or stored in a proper location under surveillance.Both these alternatives have a huge environmental impact, indeed, on the one hand, paper waste is produced,while, on the other hand there is a huge energy consumption. Solving the problem requires determining (i)the composition of each plate, (ii) the number of times each plate has to be printed such that all outstandingrequests are satised and total cost is minimized.

For instance, suppose there are 4 dierent cheques to be produced, i.e., C1, C2, C3, C4, whose requestsare, respectively, 10, 10, 7, 5. That is, 10 copies of cheque C1 are required, 10 of cheque C2, 7 of cheque C3and 5 of C4. Suppose that each plate contains 3 dierent positions that can be impressed and that the costof producing and extra cheque is equal to 10 units of money, while the cost of using a new plate is equalto 20 units of money. The optimal solution for this example has a cost equal to 60: we could use either 3plates with no overproduction or also 2 plates with an overproduction equal to 2. The optimal solution withthree plates uses: 7 times the plate with allocation {C1,C2,C3}, 3 times the plate with allocation {C1,C2,C4}and 1 time the plate with allocation {C4,C4}. On the other hand, the optimal solution with 2 plates and 2overproduced cheques uses: 10 times the plate with allocation {C1,C2} and 7 times the plate with allocation{C3,C4} (the overproduced cheque is C4). Notice that, if the cost of using a new plate increases, for exampleit is equal to 100, then the cost of the two above solutions is not the same: the former has a cost equal to 300while the latter (that is also the optimum one) has a cost equal to 220.

The problem considered in this paper is a real problem of a large cheque manufacturer in the South-ern part of Italy that manually solved its instances. As explained in the next section the problem can beformulated as a cutting stock problem with additional constraints whose exact solution would require com-putational times that are not acceptable from a practical point of view. In this paper we analyze the problem,provide a mathematical formulation and present a heuristic to solve it that is currently used by the chequemanufacturer above cited to dene the cheque allocation of the plates.

The paper is organized as follows. In Section 2 we give a complete mathematical formulation of theproblem that is useful to better understand the heuristic approach described in Section 3. Computationalresults are reported in Section 4. Conclusions are reported in section 5.

2. Problem Formulation and Related Literature

The problem considered here is related to the one dimensional cutting stock problem. To dene a genericinstance of such a problem, we are given a set of stock rolls with the same length L, set of m products of givenlengths li, i = 1, 2, . . . ,m, and the respective demands di, i = 1, 2, . . . ,m. A cutting pattern is a combinationof products on a given rolls. The problem consists in dening a set of patterns and their frequencies (i.e. thenumber of times such a pattern is used) such that a given cost function is minimized. The most common costfunction to be considered is the trim loss, and this problem has been extensively studied since the rst worksof Gilmore and Gomory [2], [3]. However, some other type of costs have been object of research, one ofthem is the cost associated with pattern changes that led to the one-dimensional cutting stock problem with

221 R. Cerulli et al. / Procedia - Social and Behavioral Sciences 108 ( 2014 ) 219 234

pattern minimization for which several solution approaches have been presented (greedy type heuristics [5],[6], pattern combination based heuristics [1], [4], local search heuristics [7], [8], and exact approaches [9]).

The problem we address in this paper is related to the one above described, indeed a cheque allocationon a plates can be viewed as a cutting pattern, and the generic cutting pattern is completely identied bythe number of times a particular cheque is present on the plate. However, unlike the general cutting stockwith pattern minimization, there are two main dierences in the problem due to the particular application weare dealing with: (i) our objective function minimizes not only the cost related to the used patterns but alsothe overproduction cost; (ii) we consider an additional type of constraints for which each product (cheque)cannot be placed on more than a certain number of dierent patterns.

The general linear formulation for our problem can be obtained by dening:

the pattern matrix with element ai j that counts the number of times cheque i is present on plate j; bi j that is a 0-1 parameter equal to 1 if cheque i is present on plate j and is equal to 0 otherwise; di the number of cheques of type i that must be produced; pi the maximum number of dierent plate where cheque i can be placed.

Variables of the problem are: y j an integer decision variable denoting the number of times plate j is used(i.e., the number of times a particular cutting pattern is used) and wj a binary variable that is equal to 1 ifpattern j is used and is equal to 0 otherwise. The formulation is then the following.

minCC

mi=1

nj=1

(ai jy j di) +CP

nj=1

wj

(1)

s.t.nj=1

ai jy j di i = 1, 2, . . . ,m (2)

wj y j i = 1, 2, . . . ,m j = 1, 2, . . . , n (3)nj=1

bi jw j pi i = 1, 2, . . . ,m (4)

y j 0 j = 1, 2, . . . , n integer (5)wj j = 1, 2, . . . , n binary (6)

where n is the number of possible dierent plates, m is the total number of cheques, CC and CP are,respectively, the unitary cost associated with cheque overproduction and the cost of using an additionalplate.

The objective function (1) requires the minimization of the total cost. Constraints (2) requires the de-mand for each cheque to be satised. Finally, constraints (3)-(4) ensure each cheque i cannot be placed onmore than pi dierent plates.

It is easy to observe that in the above formulation the number of possible dierent plates n is extremelylarge, the only requirement being that the sum of the entries in each column must be equal to the number ofdierent positions on a plate. However, for all the practical instances considered here, the total number ofdierent plates used is small. For the instances manually solved by the experienced schedulers this numberwas always less than 20, and in many cases smaller than 10.

Note that the cost we minimize has two components: the cost of producing the plates (CP) and the costassociated with the overproduction of cheques (CC). It is reasonable to assume that each of the individualcost is linear. The exact values of CC and CP are not relevant from the model point of view: only the ratioS F = CPCC is important. This quantity S F can be interpreted as the cost of producing a new plate normalizedwith respect to the cost of a single cheque. It will not be protable to produce a new plate if the waste is less

222 R. Cerulli et al. / Procedia - Social and Behavioral Sciences 108 ( 2014 ) 219 234

than or equal to C. This observation will be heavily used and will form the basis of our heuristic algorithmdescribed in the next section.

Moreover, note that the formulation presented in this section does not take into account the trim lossadditional cost due to the disposal of unused paper. Such a cost will be considered, however, in the proposedsolution heuristic procedure as additional criterion to choose among solutions having the same cost both interms of number of used plates and of overproduced cheques. In the next section, we will refer to such acriterion as Max Coverage criterion.

3. The heuristic procedure

In this section we describe in detail the proposed heuristic technique. Let us start with some denitionsthat will be used in the sequel.

S F the maximum allowed cheque overproduction. The proposed heuristic will generate a solution suchthat the number of overproduced cheques will not exceed S F (S F = CPCC ). That is, S F is the maximumnumber of cheques overproduction that is allowed since its cost is lower than the cost of using a newplate.

sk the cheque overproduction when the kth plate is used. Clearly, s1 S F and sk S F k1j=1

s j for

k > 1.

drki the outstanding demand for cheque i when the allocation of cheques on the kth plate is considered. We

have drki = max{di k1j=1

xi jy j, 0}.

prki number of distinct plates on which cheque i can be placed, once the allocation of cheques on the rst

k 1 plates is already done. From equation (??), prki = pi k1j=1

zi j.

STEP in our heuristic technique, we will restrict our attention to values of yk that are integer multiple ofSTEP. The use of this parameter drastically decreases the number of possible solutions to be exploredwithout aecting too much the quality of the nal solution as it will be clear in the sequel of thissection and from the computational results presented in section 4.

yink largest multiple of STEP such that it is possible to use the kth plate without any further cheque over-

production.

y f ink largest multiple of STEP such that it is possible to use the kth plate with a cheque overproduction

below S F.

The proposed heuristic technique minimizes the number of distinct plates used while maintaining thecheque overproduction below the allowed overproduction S F. The algorithm performs an incompletedepthrst search in the subspace of all the feasible solutions of the original problem. Each level of thetree corresponds to a dierent plate and with each node of the tree a cheque allocation on the plate is associ-ated as well as the number of times the plate has to be used. A nal feasible solution corresponds to a pathin the tree from the root node to a leaf.

The exploration of the tree, as well as the generation of the nodes of the tree, is leaded by two maincriteria strictly connected with the cost to be minimized and based on some practical considerations, asalready observed in the previous section:

Max Coverage Criterion we prefer patterns that cover as much position of a plate as possible in orderto reduce the trim loss cost due to the disposal of unused paper;

223 R. Cerulli et al. / Procedia - Social and Behavioral Sciences 108 ( 2014 ) 219 234

Max Usage Criterion: we prefer patterns that allow to use the plates as much as possible;The procedure consists in forward moves and backtracking steps. Suppose we are at level k of the tree. Firstof all, we determine yink and y

f ink and, for a xed value of y

k [yink , y f ink ] (i.e., for a xed number of times theplate k is used), various feasible allocations are considered. Once the cheque allocation on plate k has beenxed, we proceed to the next level k + 1...