Computers and Electronics in Agriculture 75 (2011) 139–146
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Computers and Electronics in Agriculture
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riginal paper
ood traceability systems: Performance evaluation and optimization
abrizio Dabbenea, Paolo Gayb,∗
IEIIT-CNR – Politecnico di Torino, 24 Corso Duca degli Abruzzi, 10129 Torino, ItalyD.E.I.A.F.A. – Università degli Studi di Torino, 44 Via Leonardo da Vinci, 10095 Grugliasco (TO), Italy
r t i c l e i n f o
rticle history:eceived 30 June 2010eceived in revised form 7 October 2010ccepted 8 October 2010
eywords:raceabilityptimization
a b s t r a c t
The aim of a traceability system is to collect in a rigorous way all the information related to the displace-ment of the different products along the supply chain. This information proves essential when facingfood safety crisis, and allows efficiently managing the consequent product recall action. Although a recallaction could be absolutely critical for a company, both in terms of incurred costs and of media impact,at present most companies do not posses reliable methods to precisely estimate the amount of productthat would be discarded in the case of recall.
The skill of limiting the quantity of recalled products to the minimum can be assumed as a measure of
upply chain managementatch dispersionILP
the performance and of the efficiency of the traceability system adopted by the company. Motivated bythis consideration, this paper introduces novel criteria and methodologies for measuring and optimizingthe performance of a traceability system. As opposed to previous introduced methods, which optimizeindirect measures, the proposed approach takes into direct account the worst-case (or the average)quantity of product that should be recalled in the case of a crisis. Numerical examples concerning themixing of batches in a sausage production process are reported to show the effectiveness of the proposed
approach.
. Introduction
Traceability in the agricultural/food chain is nowadays a fun-amental requirement, which is becoming mandatory in almostll developed countries. As discussed in Ràbade and Alfaro (2006),raceability represents a mechanism for reinforcing the level ofoordination between producers and firms, and between firms andetailers.
Primal goal of a traceability system is to precisely log the his-ory and the location of the different products along the supplyhain. Recently, the technological advances in this direction haveed to the design of ICT instruments, such as e.g. bar codes andF-ID devices, aimed at facilitating data acquisition and reducinghe traceability management costs (Gandino et al., 2009; Regattierit al., 2007; Sahin et al., 2002), and to the development of data basesnd web-based systems for data processing (Alfaro and Ràbade,009; Ruiz-Garcia et al., 2010). The information collected by the
raceability systems becomes strategic in the unfortunate casehen a batch of product has to be recalled (Bechini et al., 2008).
ndeed, besides the media impact of this action, the firm has toncur costs related to the recall and the destruction of all the prod-
ucts that are, in some way, connected with the incriminated batch(Jacobs, 1996). Since this occurrence could be absolutely critical fora company, some studies have been carried on for modeling andforecasting the effects of recall actions (e.g. see Kumar and Budin,2006; Randrup et al., 2008). However, at present most companiesdo not have reliable methods to precisely estimate the amount ofproduct that has to be discarded in the case of a recall. Indeed, thisquantity, to which we associate a recall cost (RC), depends on manyfactors:
– the size of the batches that have been individually tracked andmanaged by the traceability system;
– the way the batches of different components have been mixed toobtain the final product;
– the skill of the firm to manage and maintain segregated dif-ferent batches of product, especially in the case of continuousprocesses (e.g. milk processing in a dairy, grain or soya, see forinstance Thakur and Hurburg, 2009; Thakur et al., 2010; Thakurand Donnelly, 2010; Skoglund and Dejmek, 2007).
From the analysis of these factors, it is clear that a simple
reduction in the size of the batches, and their consequent increasein number, leading to a finer granularity of the traceability sys-tem (Bertolini et al., 2006), may be not sufficient to minimize theamount of product to be discarded. For a discussion on the roleof different levels of granularity the interested reader is referred
k,i,l indexesQk quantity of product contained in the node kQ̄k maximum capacity of the node kQ 0
kinitial amount of product in the node k
N+ set of natural positive numbersR set of real numbersm number of linksL link matrixLi,1 starting node of the link iLi,2 destination node of the link i˛i amount of material transferred through the link i¯̨ i binary variable equal to one when ˛i > 0I set of indexes of all nodesIin set of indexes of input nodesIout set of indexes of output nodesnnodes number of nodesnin number of input nodesnout number of output nodesink set of links entering the node koutk set of links leaving the node kSl
kbinary variable equal to one when a product frombatch node l is present at node k
∨ OR – operator∧ AND – operatora,b,r binary variablesM sufficiently large numberntype number of product typesp product typeTp set of nodes containing the type of product pD matrix of recipe disassembling coefficientsA matrix of recipe assembling coefficientsRC(l) recall cost of the product lwi weights of the weighted recall costWCRC worst-case recall costARC average recall costWRC weighted recall cost
tr
agpt
bricewBrctfaope
BCD batch dispersion costD DISP(l) downward dispersion from node l
o Karlsen et al. (2011) and references therein, where an exampleelated to farmed salmon is also presented.
The previous considerations suggest selecting the recall cost asnatural measure of the performance for a traceability system. Thisives raise to two fundamental problems: (i) the evaluation of theerformance of a given traceability system, (ii) the optimization ofhe supply-chain design in order to minimize its performance.
To better formalize these problems, some nomenclature has toe introduced. Moe (1998) has introduced the concept of traceableesource unit (TRU) for batch processes as “unique unit, mean-ng that no other unit can have exactly the same, or comparable,haracteristics from the point of view of traceability”. In mod-rn agricultural supply system, units must be uniquely identifiableithin each system in which they are processed. To this extent,ollen et al. (2007) introduced the identifiable unit (IU), whose sizeeflects the granularity of the traceability system. In many supplyhains granularity is the consequence of a combination of tradi-ion, short-term convenience and use of available facilities. In very
ew cases granularity depends on the results of a formal analysisnd optimization in the supply chain. The simple implementationf a finer granularity by itself has no value unless it provides morerecise traceability. The precision of a traceability system can bevaluated, as discussed in Bollen et al. (2007), as the ratio between
ics in Agriculture 75 (2011) 139–146
IUs at two points in the supply chain and it is the consequence ofthe number and the nature of the transformations of IUs and of theextent, nature and accuracy of data recorded. If a IU is split up, theseparated parts keep the identification of the parent IU, while ifsome IUs are put together, the identification of the IU is differentfrom the identification of the parent IUs.
One possible solution to maintain the same level of traceabil-ity precision consists of breaking the processing into segments ofrelative homogeneity, both for processing conditions and productorigin, and recording all relevant information.
Finally, one has to take into account if the product is processedin completely separated runs, or if some mixing can occur betweenproducts of two succeeding batches. In the latter case, it is neces-sary to specify if tolerances can be accepted. This problem has beenaddressed by Skoglund and Dejmek (2007) for the case of continu-ous processing where it has been referred as fuzzy traceability, whileRiden and Bollen (2007) considered the case of discrete products,with an application to packhouse processing transformations.
The problem of the performance evaluation and optimizationof traceability systems was first introduced by Dupuy et al. (2005),and successively applied in different endeavours (see for instanceDonnelly et al., 2009). Tamayo et al. (2009) employ genetic algo-rithms to solve the optimization problem proposed by Dupuy et al.(2005). Finally, Wang et al. (2010) propose the joined optimiza-tion of traceability and manufacturing performances, acting bothon batch sizes and batch dispersion, by introducing risk functions.In all these works, the performance of a traceability system is asso-ciated with the number of active paths between raw-materials andfinished products, as formally detailed in Section 3. This measureis indeed related to the final quantity to be recalled, since it aims atreducing the mixing of different batches, and was proven effectivein the above-mentioned works. However, it should be remarkedthat, in general, the minimization of this index does not necessar-ily result in the minimization of the recall cost, when intended asthe quantity of products to be recalled in the worst-case.
In this paper, we introduce a modeling framework and optimiza-tion strategy to cope with this problem, directly adopting the recallcost as performance criterion. Similarly to Dupuy et al. (2005), theoptimization problem is expressed in the form of mixed-integerlinear programming (MILP), for which efficient numerical solversare available. To show its effectiveness, the proposed approach hasbeen first applied to the numerical example presented in Dupuyet al. (2005) and in Tamayo et al. (2009), and finally to a larger testcase.
2. Modeling
A complete food production process can be seen as a sequence ofstorage/carrying actions and of unit operations. Bulk products arestored and carried in containers (as for instance tanks, vats, binsetc.) depending on the nature of the products. Unit operations canbe conducted on a batch of product at a time (e.g. concentrationin a bull, cooking in a oven) or continuously, as the processes ofmilk pasteurization/sterilization, or of concentration in a continu-ous evaporator.
From the point of view of traceability, this second instance(continuous unit operations) can also be interpreted as a batchprocess situation, by either guaranteeing proper cleaning cyclesbetween two subsequent lots, or by allowing (and then neglecting)small percentages of contamination (Skoglund and Dejmek, 2007).
Each container/processing-unit that individually stores/processesa batch of product, at a certain time, can be modeled as a node in agraph.
Formally, at each node k one associates a variable Qk, whichaccounts for the quantity of product contained in the node. This
F. Dabbene, P. Gay / Computers and Electronics in Agriculture 75 (2011) 139–146 141
F exampi
vatbwtaa
afntwt˛tt
˛
cosdsn(t
l
i
ig. 1. An illustrating example showing the formalism introduced in the paper. Anntroduced in (4) is given for node 6 and source 1.
ariable can be bounded by the capacity of the container, or by themount of product that can be processed at a time. This correspondso imposing the constraint Qk ≤ Q̄k. In some cases, it is also possi-le to introduce the equality constraint Qk = Q̄k to reflect the caseshen one wants to fix the quantity of material in node k precisely
o the value Q̄k. This is the case, for instance, of final products thatre sold in fixed-weight packages, or of middle nodes where a fixedmount of product is processed at a time.
The flow of the batches inside the supply chain is modeled vianumber of oriented arrows (links of the graph). These links are
ormally described introducing a m × 2 matrix L ∈ Nm,2+ of positive
umbers, where m is the number of links. The entries of L havehe following structure: Li,1 indicates the starting node of the link i,hile Li,2 represents its destination node. The amount of material
ransferred through the i-th link is expressed by the variable ˛i ∈ R,i ≥ 0, i = 1, 2, . . ., m. Associated to the variable ˛i, one can define
he binary variable ¯̨ i, which is true whenever the i-th link is active,hat is
¯ i ={
1 if ˛i > 00 if ˛i = 0.
(1)
Nodes can be schematically grouped into three sets: input, pro-essing and output nodes. Letting I =
{1, 2, . . . , nnodes
}be the set
f the indexes of all nodes, one can define Iin ⊂ I and Iout ⊂ I as theets of indexes of the input and output nodes, respectively. The car-inality of these sets (i.e. the number of elements belonging to theet) is denoted as nin and nout, respectively. Then, with each inputode k ∈ Iin is associated the initial quantity of available productraw material) Q 0
kthat has to be transferred and/or processed by
he network.For any node k ∈ I, one defines the sets of links entering and
eaving the node as
nk ={
i ∈ I : Li,2 = k}
and outk ={
i ∈ I : Li,1 = k}
. (2)
le of Eqs. (2) and (3) is reported for node 4. Also, explicit construction of the states
These indices allow expressing the mass balances for each node kas follows∑i ∈ ink
˛i =∑
i ∈ outk
˛i = Qk. (3)
The modeling framework proposed in this paper relies on thedefinition of specific “state variables” Sl
k∈
{0, 1
}, k, l ∈ I. The binary
variable Slk
is true whenever the node k contains a product arisingfrom node l, that is whenever there exists a path in the graph con-necting node l to node k. Notice that the state Sl
kcan be recursively
calculated using the following relation
Slk = ∨
i ∈ ink
(SlLi,1
∧ ¯̨ i), (4)
where ∨ and ∧ represent the logical OR and AND operators, respec-tively. The initial conditions for recursion (4) are given by
Skk = 1 for k ∈ Iin and Sl
k = 0 for k ∈ Iin\{
l}
. (5)
We recall that both OR and AND operators may be rewritten aslinear operations on binary variables, see for instance Achterberget al. (2007). More precisely, for binary variables a, b, r ∈
{0, 1
},
one can write
r = a ∧ b ⇔{
r ≤ ar ≤ br ≥ a + b − 1
(6)
r = a ∨ b ⇔{
r ≥ ar ≥ br ≤ a + b
(7)
A simple example of the modeling framework introduced in(1)–(5) is presented in Fig. 1. To illustrate the meaning of the intro-duced variables, in this figure the explicit construction of the statesintroduced in (4) is provided for node 6 and source 1 as an example.
The recursive nature of the states is clearly evidenced: the state S1
6depends on S1
4 and S15, which in turn are functions of the state of
the sources, which are given by the initial conditions introduced in(5). Notice that, after the recursion is resolved, one obtains a rela-tionship whose interpretation is clear: node 6 contains material
1 ectron
fs
airiogfdstraTaihp
tp
ifcm(cilcfipstat
3
aDwToTotoud
d
42 F. Dabbene, P. Gay / Computers and El
rom node 1 whenever both links ˛1 and ˛6, or links ˛2 and ˛8, areimultaneously active.
In a generic supply chain, the lots of products are displacednd/or processed according to some rules that govern each mix-ng occurrence. In the proposed setup, these rules are genericallyeferred to as recipe rules, and are defined on sets of nodes contain-ng homogeneous products. In this way, a (possibly large) numberf nodes that are devoted to contain the same type of product can berouped into a single set. Recipes, which are related to sets, are validor each node belonging the involved set. More formally, one canefine ntype disjoint sets Tp ⊆ I, p = 1, 2, . . ., ntype, of indices, where theet Tp is formed by the indices of the nodes that contain a product ofype p. Then, recipe constraints can be generally expressed as linearelationships between product types. In particular, one can definessembling and disassembling recipes as in Dupuy et al. (2005).his is done by introducing the matrices of coefficients D ∈ Rntype ,ntype
nd A ∈ Rntype ,ntype . Then, a disassembling constraint allows describ-ng the situation when each product belonging to a given type pas to be destined to nodes belonging to the set Tj according to theercentage expressed by Dp,j, i.e.∑
i ∈ outk
Li,2 ∈ Tj
˛i = Dp,j Qk for ∀k ∈ Tp and p = 1, . . . , ntype. (8)
Analogously, assembly constraints impose that each product ofype p has to be composed by product of type j, according to theercentage Ap,j∑i ∈ ink
Li,1 ∈ Tj
˛i = Ap,j Qk for ∀k ∈ Tp and p = 1, . . . , ntype. (9)
It should be remarked that the modeling framework proposedn this section improves upon the one in Dupuy et al. (2005) in theollowing points: (i) the original approach of Dupuy et al. (2005)onsiders only three-stages of production systems, with a raw-aterials stage, a components stage and a finished products one;
ii) only fully interconnected networks (each node at one stage isonnected to each node in the successive stage) are consideredn Dupuy et al. (2005). On the contrary, the introduction of theink matrix L allows considering arbitrary networks, that is graphonfiguration consisting of multiple stages and arbitrary link con-gurations. In particular, this second feature allows excluding ariori undesired or logistically unfeasible links, thus leading to aignificant reduction in the complexity of the ensuing optimiza-ion problem. Also, the introduction of the state variables S allowsclear formalization of the desired performance measures related
o the recall cost, as shown in the next section.
. Performance evaluation
As discussed in Section 1, different measures can be defined forssessing the performance of a traceability system. In particular,upuy et al. (2005) defined three performance indices: the down-ard dispersion, the upward dispersion and the batch dispersion.
he downward dispersion of a raw material batch is the numberf final batches that contain parts of a specific raw material batch.he upward dispersion of a finished product batch is the numberf different raw material batches used to produce this batch, whilehe batch dispersion is defined in Dupuy et al. (2005) as the sum
f links between the raw material batches and the finished prod-ct batches. The analytical expression of these indices is formallyefined at the end of this section.
As previously discussed, the approach in Dupuy et al. (2005)oes not take into account quantities, but only the active paths that
ics in Agriculture 75 (2011) 139–146
are upward/downward involved. However, this setup has the greatadvantage of allowing a formalization of the optimization problemin terms of mixed-integer linear programming (MILP). Motivatedby the fact that the number of variables and constraints in the ensu-ing MILP problem may increase exponentially, Tamayo et al. (2009)proposed to solve this problem by means of genetic algorithms.
Proceeding along the same lines that in Dupuy et al. (2005), inthis paper novel performance indices are introduced, which betterquantify the cost of product recall as perceived by the industry. Tothis extent, first introduce the recall cost of product l, RC(l), as thetotal amount of (final) product that has to be recalled in the casewhen the batch of raw material contained in node l is recognizedas lacking the requirements. This corresponds to the mass of finalproduct that contains – owing to mixing operations – part of thematerial originally stored in the incriminated node l.
On the basis of the formalism presented in the previous section,the recall cost relative to node l can then be directly defined as
RC(l) =∑
k ∈ Iout
Slk Qk. (10)
The typical interest of a company is to know–and possibly toreduce–the worst-possible amount of product that could be neces-sary to recall. This corresponds to defining the worst-case recall cost(WCRC)
WCRC = maxl ∈ Iin
RC(l) = maxl ∈ Iin
∑k ∈ Iout
Slk Qk, (11)
as the largest amount of product that has to be recalled when abatch of raw material is found unsafe. Analogously, it is possible todefine the average recall cost (ARC) index as
ARC = 1nin
∑l ∈ Iin
RC(l) = 1nin
∑l ∈ Iin
∑k ∈ Iout
Slk Qk, (12)
which represents the average mass of product to be recalled whenone of the entering material is found inappropriate.
It should be remarked that the ARC cost defined in (12) canbe readily adapted to the case when suppliers of the inputbatches have a different level of reliability and/or one can asso-ciate to different input batches different probabilities of lackingthe requirements. This can be modeled introducing appropriateweights wi, i = 1, . . . , nin, (that can be interpreted also as prob-abilities). This leads to the following weighted recall cost (WRC)index
WRC =∑l ∈ Iin
wiRC (l) =∑l ∈ Iin
wi
∑k ∈ Iout
Slk Qk, (13)
Finally, it should be remarked that the introduced setup can eas-ily handle the batch dispersion cost (BDC) introduced in Dupuy et al.(2005), by introducing the downward dispersion from node l as
D DISP (l) =∑
k ∈ Iout
y (l, k) , (14)
where
y (l, k) ={
1 if Slk
Qk > 00 if Sl
kQk = 0
. (15)
Then, the BDC index can be written as
BDC =∑
D DISP(l) =∑ ∑
y (l, k) , (16)
l ∈ Iin l ∈ Iink ∈ Iout
and it represents the sum of links between the raw material batchesand the finished product batches. Notice that (16) is exactly theindex minimized in Dupuy et al. (2005).
F. Dabbene, P. Gay / Computers and Electronics in Agriculture 75 (2011) 139–146 143
ing th
4
m
ttdhho
(ftiutr
2bdoWSmimi
wimw
r
Fig. 2. Optimal solution of the example in Tamayo et al. (2009) us
. Optimization
Different approaches can be adopted to optimize the perfor-ance of the traceability system.The first possibility is to compare different scenarios. Even if this
echnique cannot properly be referred to as optimization, it permitso compare via simulation some selected configurations of the pro-uction process and/or the supply chain (Gay et al., 2009). It is aelpful approach when a decision among few possible alternativesas to be taken. However, clearly this methodology does not bringut optimal solutions that have not been already a priori selected.
A more rigorous approach is direct optimization, as in Dupuy et al.2005) and Tamayo et al. (2009). This methodology is to be pre-erred whenever one or more parameters of the supply chain haveo be designed according to an optimality criterion. In our case, asn Dupuy et al. (2005), the parameters to be designed are the prod-ct flows ˛i, i = 1, . . ., m and the considered optimality criteria arehe batch dispersion recalled in (16), and the worst-case/averageecall costs defined in (11) and (12).
It should be noticed that the framework introduced in Sectionallows formulating the problem of minimizing both the original
atch dispersion measure (Dupuy et al., 2005) and newly intro-uced more realistic performance measures WCRC and ARC in termsf mixed-integer linear programs. To this end, first notice that bothCRC and ARC objective functions contain the product of terms
lk
∈{
0, 1}
and Qk ∈ R. These quantities depend both on the opti-ization variables ˛i. However, this nonlinearity can be converted
n a set of linear inequalities. To see this, remark that an opti-ization problem of the type min Sl
kQk can be reformulated by
ntroducing an additional real variable r = Slk
Qk, as follows
min Slk
Qk ⇔ min rsubject to : r ≥ 0
r ≥ Qk − M(
1 − Slk
)r ≤ Qk
(17)
here M is a sufficiently large number. Notice also that the min-mization of the cost function WCRC in (11), which presents a
aximization term, can be reduced to a linear problem by simplyriting
min maxl ∈ Iin
RC (l) ⇔ min �subject to : RC (l) ≤ �, l ∈ Iin
(18)
Finally, also the binary version ¯̨ i of ˛i introduced in (1) can beeformulated in the integer programming paradigm by introducing
e ARC index. The average recall cost of this solution is ARC = 3333.
the following two linear constraints
¯̨ i ≤ M˛i, ¯̨ i ≥ ˛i
M(19)
where again M is a sufficiently large number. The same operationcan be made for introducing Eq. (14).
5. Numerical examples
In order to demonstrate the effectiveness of the proposedapproach, the same example proposed in Dupuy et al. (2005), andsubsequently elaborated in Tamayo et al. (2009), is here consid-ered. The problem concerns a sausages fabrication chain modeledas a three-level network, consisting of four batches of input (raw)material divided into two types of product (RM1 and RM2), sixprocessing batch units (“components”, according to the notationintroduced in Dupuy et al., 2005) divided into two types (SP1 andSP2), two batches of bought components (additional inputs), oneof each type (i.e. SP1 and SP2 again), and four batches of finishedproduct, also divided into two types (FP1 and FP2). Since the net-work is fully interconnected, all ˛i, i = 1, . . ., 56, coefficients have tobe determined.
To solve the problem, the batch dispersion minimization and thenewly introduced WCRC and ARC minimization problems have beenwritten as MILPs using the YALMIP software, that allows parsing ofoptimization problems under Matlab, see Löfberg (2004) for addi-tional details. The resulting MILP programs were then solved usingthe commercial solver CPLEX (ILOG-IBM), on a 2.53 GHz MacbookPro.
In the first numerical example, the exact same numerical setupused in Tamayo et al. (2009) was adopted, with initial values forthe first nodes equal to Q 0
1 = Q 03 = 1000 and Q 0
2 = Q 04 = 1200, and
with final desired quantities in the last four nodes set to the valuesQ̄13 = Q̄14 = Q̄15 = Q̄16 = 2000. The solution minimizing the aver-age cost criterion ARC was sought. The CPLEX solver returned, after2.7 s of elaboration, the solution reported in Fig. 2, which is guar-anteed to be optimal. The average recall cost of this configurationis ARC = 3333. The graph contains a total of 23 links, and ten directsource–destination paths, hence providing a batch dispersion mea-sure BDC = 10. These figures can be compared with the numericalsolution obtained in Tamayo et al. (2009), which has a recall cost
ARC = 3.667. This corresponds to a 10% improvement of our solu-tion. However, it should be noted that the solution in Tamayo et al.(2009) presents 13 direct source-destination paths (BDC = 13), andhence it is not optimal also for the batch dispersion cost introducedby Dupuy et al. (2005). This fact is not surprising, since the genetic
144 F. Dabbene, P. Gay / Computers and Electronics in Agriculture 75 (2011) 139–146
Fig. 3. Second numerical example. Optimal solution obtained minimizing the BDC cost.
al sol
agtptt
Fig. 4. Second numerical example. Optim
lgorithm approach in Tamayo et al. (2009) does not provide anyuarantee of returning an optimal solution. Hence, we computed
he optimal solution using the BDC cost, which provided the sameerformance as ours (ARC = 3333 and BDC = 10), with a computationime of 5.8 s. A similar behavior was observed when comparing withhe worst-case optimality criterion WCRC.
Fig. 5. Second numerical example. Optimal solu
ution obtained minimizing the ARC cost.
Based on this observation, a second numerical example was run.This second example considers the same node configuration of the
first one, but with the quantities in the first nodes, and the finaldesired quantities in the last four nodes, now unbalanced. Thatis, Q 0
1 = 450, Q 02 = 2350, Q 0
3 = 150, Q 04 = 1450, and Q̄13 = 1750,
Q̄14 = 3150, Q̄15 = 2250, Q̄16 = 850. The solution of the three dif-
tion obtained minimizing the WCRC cost.
F. Dabbene, P. Gay / Computers and Electronics in Agriculture 75 (2011) 139–146 145
Table 1Numerical results of the second example.
Optimization index Solution time # of links BDC ARC WCRC
Bold values signify the best achieved performances.
Fig. 6. Initial configuration for the four-stage example.
age ex
futar
Fiamwpi
Fig. 7. Optimal configuration for the four-st
erent optimality indices BDC, ARC and WCRC were again computedsing the CPLEX solver. Optimal solutions were returned respec-ively in 92.5, 47.8 and 0.6 s. The respective optimal configurationsre shown in Figs. 3–5, and the relative quantities of interest areeported in Table 1.
A few comments are at hand for the figures reported in Table 1.irst, it can be observed that the BDC solution, although present-ng fewer direct paths between input and final nodes, provides
n average recall cost that is around 23% worse than the opti-al ARC one, and a worst-case recall cost that is more than 55%orse that the optimal WCRC one. Second, this improvement inerformance is obtained together with an even more significative
mprovement in terms of computational cost: the WCRC optimiza-
Fig. 8. Optimal configuration for the four-stage exa
ample obtained minimizing the BDC index.
tion was about 156 times faster than BDC, and about 81 times fasterthan ARC.
Finally, the BDC and WCRC optimization criteria were comparedin a larger example, consisting of on four layers, with 8 batches ofinput (raw) material, 7 nodes in the second layer, 16 nodes in thethird layer and 13 batches of finished product was considered. Thisnetwork is only partially interconnected, according to the diagramreported in Fig. 6. As it can be seen, the initial configuration contains
78 feasible links and 44 nodes (a fully interconnected configurationwould involve 376 links). The maximum capacity of each node, theamount of raw material of the input batches and the desired quan-tity in the output nodes are also reported in Fig. 6. Remark that theparticular features in this example, i.e. a number of layers and the
mple obtained minimizing the WCRC index.
146 F. Dabbene, P. Gay / Computers and Electronics in Agriculture 75 (2011) 139–146
Table 2Results of the optimization of the four levels problem considering BDC and WCRC optimization criteria.
Optimization index Solution time # of links BDC ARC WCRC
B
dt
FrmopctbsrppBstBcotve
6
aioastp
A
fatT
R
A
A
Thakur, M., Wang, L., Hurburgh, C.R., 2010. A multi-objective optimization approach
old values signify the best achieved performances.
efinition a priori of the set of feasible links, cannot be managed byhe formulation in Dupuy et al. (2005).
The solutions of the two minimizations are reported inigs. 7 and 8, respectively for the BDC and the WCRC criteria. Theelative quantities of interest are reported in Table 2. To com-ents these results, it can be observed that the direct minimization
f the worst-recall cost allows a 25% improvement of the WCRCerformance. This can be interpreted as follows: by adopting theonfiguration in Fig. 8 one is guaranteed that, no matter what ishe initial product found inadequate, the quantity of material toe recalled will be less than 24,600. Contrary, adopting the BDColution of Dupuy et al. (2005) reported in Fig. 7, one can incur aecall cost as high as 30,750. Also, it can be noted that, as a sideroduct, the WCRC minimization allows to exclude from the sup-ly chain four nodes (nodes 23, 26, 27, and 31), while the optimalDC one excludes two nodes only (22, 31). In a typical industrialituation, this would easily correspond to a save in the produc-ion costs. Moreover, also in this case, the computation time for theDC criterion was dramatically higher than the one of WCRC (7 daysompared to half an hour). This behavior, which was observed in allur simulations, can be explained by the fact that the BDC cost func-ion formulation (15) requires the introduction of nin × nout binaryariables, which are not present in the WCRC one. This seems toxplain the large increase in the computational time.
. Conclusions and future research directions
In this paper novel criteria and methodologies for measuringnd optimizing the performance of a traceability system have beenntroduced. As opposed to the methods previously adopted, whichptimize indirect measures, the proposed approach takes in directccount the worst-case or the average quantity of product thathould be recalled in the case of a crisis. Numerical examples testifyo the effectiveness of the proposed methodology, both in terms oferformance and of computational cost.
cknowledgements
The authors would like to thank Constantino M. Lagoa for theruitful and inspiring discussions on the topics of this paper thatrose while he was visiting the first author. This work was par-ially supported by the grants of the project Namatech-Convergingechnologies (CIPE2007), Regione Piemonte, Italy.
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