reducing dispersion in food distribution
TRANSCRIPT
Reducing Dispersion in Food Distribution
Martin Grunow† 1, Aiying Rong2 and Renzo Akkerman3 Department of Management Engineering, Technical University of Denmark,
Produktionstorvet 425, 2800 Kgs. Lyngby (Copenhagen), Denmark Email: [email protected]†1
Abstract. After a number of food safety crises, the design and implementation of traceability systems became an important focus of the food industry. As a result, food product traceability ranks high on senior management agendas for supply chain activities. In the literature, numerous studies deal with traceability from the viewpoint of information system and technology development such as radio frequency identification (RFID) and DNA-based techniques. However, traceability and its implications for food safety are thus far not incorporated in the standard operations management literature on production and distribution planning. In this paper, we develop a production and distribution planning model for food supply chains which minimizes production and logistics costs and at the same time reduces food safety concerns, limits the size of potential recalls, and satisfies product quality requirements throughout the supply chain. We also present heuristics for solving the resulting mixed integer linear programming model and demonstrate the effectiveness of the developed methodology in a numerical investigation. Keywords: Traceability, food safety, mixed-integer linear programming, food industry, production and distribution planning.
1. INTRODUCTION Food safety issues have been at the forefront of
societal concerns in recent years. These concerns are caused by a sequence of food scandals and incidents during the last decade, such as mad cow disease, bird flu, and cases of salmonella. Consequently, customers call for food quality, integrity and safety; retailers are adding further safety issues to their supplier requirements; and governments are imposing new legislation that enforces traceability of food products during all stages of production, processing, and distribution (e.g. European Parliament and Council, 2002). In reaction to this, the food industry has implemented systems to improve product safety (including HACCP systems), while at the same time making information about the food products transparent at the level of supply chains (Beulens et al., 2005).
The introduction of traceability systems can be viewed as a strategic response by the food industry to the increase of the consumers’ overall risk perceptions of food products (Wang and Li, 2006). Traceability systems are developed to track products throughout the supply chain, and to provide the possibility to trace back products from anywhere in the chain. Moe (1998) classified food traceability into chain traceability and internal traceability. Chain traceability
tracks products through the production chain from harvest through transport, storage, processing, distribution and sales. Internal traceability traces product batches internally in one of the steps in the chain, for example the production process. Up to now, numerous studies deal with traceability from the viewpoint of technology development (Asensio et al., 2008; Thomas et al., 2007) and information systems (Bello et al., 2005; Cimio et al., 2005; Jansen-Vullers et al., 2003; Trienekens and Beulens, 2001). Schwägele (2005) gives an overview of the traceability legislation in Europe and its implications on the technological requirements in the meat sector.
Traceability systems limit the impact of potential food safety problems. With their help it is exactly known which products are affected, and which supply network paths are involved. Control measures can therefore easily be derived. However, traceability itself does not change the safety and quality of the products. To efficiently manage food safety risks, the implementation of traceability systems must be complemented with suitable production and distribution planning approaches. If, for example, a product batch is distributed through diverse channels to numerous customers, the implications of a product recall are severe if the product batch turns out to have safety problems. The amount of products, the logistics effort, and the damage to
________________________________________ † : Corresponding Author
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the reputation are large despite the implementation of a traceability system. Ideally, we would like to both trace products efficiently, and limit the impact of food safety problems. No previous research exists on this problem.
Dupuy et al. (2005) used the concept of batch dispersion to deal with internal traceability in the production stage. The authors define batch dispersion as the sum of downward dispersion of raw material batches and upward dispersion of finished product batches. Downward dispersion is the number of finished product batches that contain part of a raw material batch. Upward dispersion is the number of different raw material batches used to produce a finished product batch. Here, we are extending the concept of dispersion to the supply chain context.
Chain dispersion is a result of the decisions made in distribution. As food safety problems often originate from contamination of a certain production batch (e.g. with salmonella), limiting the number of retailers served by the same batch can reduce the impact of a recall.
Therefore, limiting batch sizes in the production stage seems a straightforward way of reducing safety concerns. This would, however, lead to an increase in production setups, cleaning efforts, etc., leading to increased production costs and decreased efficiency. In today’s food supply chains, this can be difficult to achieve without compromising competitiveness. Essentially, this means that there is a trade-off between reducing production costs of the food products and reducing the concerns for food safety.
In addition, batch sizing decisions are influenced by product quality degradation, as products are often perishable and cannot be stored indefinitely. This could be reflected in storage costs, but this is not sufficient for most food products, where product quality and its degradation are such essential issues that they need to be taken into account explicitly. Quality degradation of food products occurs during all storage and distribution stages in the chain, and it depends on the environmental conditions and the time spent in those environments. This means that we also need to take the quality degradation of the individual product batches into account when making the production and distribution decisions with the aim to enhance traceability.
We formulate the problem as a mixed-integer linear programming (MILP) model. Next to batch setup costs, we introduce dispersion costs to penalize batch dispersion in relation to the aforementioned safety concerns. To capture the dynamics of the problems, we pursue a multi-period modelling approach.
The remainder of this paper is organized as follows. In the next section, we discuss traceability and batch dispersion in more detail. This is followed by the formulation of the aforementioned MILP model in Section 3. As solving this model is quite difficult we then develop
in Section 4 two heuristic procedures. In Section 5, we test the model and the heuristics on two test cases. Finally, we present our conclusions and a discussion of the results.
2. TRACEABILIY
To provide an illustrative setting for our modelling
framework, we consider a simple production and distribution network consisting of one plant and multiple retailers as shown in Figure 1. The plant can perform both production and storage tasks. The retailers are the customers, which specify certain quality requirements in their demand. In practice, these requirements are often related to a remaining shelf life of the products.
2.1. Batch dispersion
Batch production is prevalent the food industry. This is
sometimes the result of a certain fixed capacity of the processing equipment, but can also be a decision based on efficiency requirements. In the literature, the former is known as batch processing and the latter as family scheduling (Webster and Baker, 1995). In food production systems, a certain minimum product quantity is often required for the process to run smoothly, and generally, large batch sizes are aimed for to increase capacity utilization and competitiveness. The batching opportunities of a specific food company depend on the position of the company on the continuum between batch production systems and continuous flow production systems, as introduced by Fransoo and Rutten (1994). Furthermore, the number of products also affects the frequency of product switches, the associated equipment cleaning operations, and hence capacity utilization.
We introduce the concept of batch dispersion, based on the number of retailers served by a production batch, to assess the impact of the decisions made in production and distribution. We define the batch dispersion measure Db as
Db = n(n − 1)/2, (1) where n is the number of retailers served by the batch. This concept is illustrated in Figure 2. According to our
R2R1
PP
… R..
Figur tion e 1: A simple distribution network with one producplant and several retailers.
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Product batch
Retailers
definition, batch dispersion only occurs when a batch serves more than one retailer. An increase in the number of retailers served from a single batch results in a quadratic increase in the dispersion measure. Defining batch dispersion in this way not only reflects our aim to prevent the distribution of batches to numerous retailers, but it also makes it easier to include it in the model formulation presented later. 2.2 Tracking quality information
A special focus in food production and distribution systems is maintaining product quality in the storage and transportation facilities. Quality degradation of food products is a complex process, due to the range and dynamics of product characteristics and storage conditions. In general, degradation of food products in storage (or during transport) is dependent on storage time, storage temperature, and various constants (e.g., activation energy, gas constant). For a more detailed discussion of food quality degradation, we refer to e.g. Labuza (1982). Numerous models have been developed for specific food products and some are even included in decision support software (Brul et al., 2007). Rong et al. (2008) presented a first approach that utilizes these models in logistical decision-making. We will use their approach in this paper. In short, this means that we are describing product quality on a scale of discrete quality levels. For every storage or transport activity, products will lose a certain amount of these quality levels. Rong et al. (2008) use this approach to determine optimal temperature levels in a supply network. Here we assume that the temperature is set properly so that the quality degradation in each link of the network is moderate.
3. PROBLEM FORMULATION
In managing safety risks in supply chains, we need to determine (i) the number of batches, (ii) the batch sizes, and (iii) which batches to use to fulfil each retailer’s demands in each period, so that the total costs can be minimized. Next to the setup and holding costs, which are considered in traditional lot-sizing literature, we also include production costs that depend on the quality of the produced product and dispersion costs that reflect safety concerns.
To minimize the total costs in the model, we need to obtain a trade-off between setup costs, dispersion costs and production costs. To reduce the setup costs, we need to increase batch size. This means that one batch must serve multiple retailers, possibly even over multiple periods. However, if the same product batch serves multiple retailers, this will cause batch dispersion as defined in Section 2 and will lead to dispersion costs. The primary trade-off of interest in this paper is thus between dispersion costs and setup costs. In addition, the quality level of the product batch must be as high as any of the quality requirement of the supplied retailers. This may imply an increase of production costs. If the same batch serves the retailers in multiple periods, we must further increase the initial quality levels of the product batch to compensate for quality degradation in the storage, which again means an increase of production costs. Overall, we have a trade-off between dispersion costs, production costs and setup costs.
Traceability requires tracking each batch as an entity. We therefore identify each product batch by a batch ID, which consists of the information about batch number, product type, production time, and production location. We also track the quality degradation of the products for the individual batches.
1 2 3 4
0 1 3
Number ofRetailers served
BatchDispersion (Db) 6
Figure 2: Illustration of the batch dispersion measure.
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3.1 Assumptions
To facilitate the modelling, we further make the following assumptions:
• Quality degradation follows a linear (or linearized) relationship with time for a given temperature (see Rong et al. (2008) for a more detailed description). Quality changes an integer number of quality levels per period.
• To fulfil demand from period 1 to period T over the planning horizon, a planned transport lead time ωmax is added to the planning interval, similar to Spitter et al. (2005). Hence, the planning interval covers the periods [1 - ωmax, T].
• Setup cost from batch to batch is considered constant, i.e., not sequence-dependent.
• Production capacity is assumed to be sufficient.
3.2 Notation The following notation is introduced for the model.
Indices: b batch ID, j retailer index, p product type index, q quality level index, t time index, referring to either a period or a point
in time. The period t is between point t - 1 and t, t ∈ {1 - ωmax,…,T},
Sets: P set of product types, R set of retailers, Q set of quality levels, St,p set of batch IDs for product p produced in the
plant in period t,
Sp set of batch IDs for product p produced in the
plant in all periods, , Ut
pmax1 ω−=
T
ptSS .=
UPp
pSS∈
=
S set of batch IDs for all products produced in the plant in all periods, ,
Parameters: M a large positive value, cdispersion penalty dispersion cost, csetup set up cost from batch to batch, cp,q cost for producing one unit product p with
quality level q, dj,p,t demand of product p (with minimum quality
requirement) by retailer j in period t, h holding cost for one unit product in the
plant storage, qp,max maximal quality level for product p, qp,j,min minimum quality level of product p specified
retailer j, minimum quality level of product p at the plant,
which can satisfy the quality requirements of at least one retailer,
pq
Δqp quality degradation of product p during one period in the plant storage facility,
Δqj,p quality degradation for product p during transport from the plant to retailer j,
uj number of periods it takes to transport the product from the plant to retailer j.
Decision variables: Ib,q,t inventory of batch ID b with quality level q at
the plant at the end of period t, xb,j,q,t flow quantity of batch ID b on arc (i,j) in period
t with starting quality level q, yb,q binary variable indicating whether batch ID b
with quality level q is produced, zb,q production volume of batch ID b with quality
level q, θb,j binary variable indicating whether batch ID b of
product p serves retailer j, ob,j,k binary variable indicating whether batch ID b
serves two different retailers j and k simultaneously.
3.3 Model formulation
The multi-period multi-product production and
distribution planning for improving traceability can now be formulated as follows.
Min ∑∑∑
∈ ∈∈≠Sb Rj
Rkjk
kjbdispersion oc 2/,,
∑ ∑ ∑ ∑−= ∈ ∈ ≥
+T
t Pp Sb qqqb
setup
pt p
ycmax ,1
,ω
(2)
∑ ∑ ∑ ∑ ∑∑ ∑−= ∈ ≤ ∈ Δ+≥∈ ≥ ⎟⎟
⎟
⎠
⎞
⎜⎜⎜
⎝
⎛++
T
t Pp t Sb qqqtqb
Sb qqqbqp
p ppt p
hIzcmax ,,1
,,,,ω τ τ
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Subject to
,,,
,
, ,,,1,,, p
qqqRj
tqjbtqqbtqb qqPpxII
pjp
p≥∀∈∀−= ∑
Δ+≥∈
−Δ+
,,1},,,2{ ,max pSbtTt ττω ∈∀−≤−∈∀ K (3)
,,,
,
,,,,,,, p
qqqRj
tqjibqbtqb qqPpxzI
pjp
≥∀∈∀−= ∑Δ+≥
∈
,},,,1{ ,max ptSbTt ∈∀−∈∀ Kω (4)
,,,,,,, min,,
, tpjRj ut Sb qq
utqqjb dxj p jp
jpj=∑ ∑ ∑ ∑
∈ −≤ ∈ ≥−Δ+
τ τ
},,,1{,, TtPpRj K∈∀∈∀∈∀ (5)
,1, ≤∑≥
pqq
qby ,, pSbPp ∈∀∈∀ (6)
,,,,,, ppqbqb qqSbPpMyz ≥∈∀∈∀≤ (7)
,,,,,,,,,
PpRjMx jbqqq
utqjbjbpjp
j∈∈∀≤≤ ∑
Δ+≥− θθ
,,},,...,1{ , pj SbutTt ττ ∈−≤∈∀ (8)
,,,,,1 ,,,, SbkjRkRjo kjbjbkb ∈≠∈∈≤−+θθ (9)
,,,0,, QqSbI tqb ∈∀∈∀≥ },,,1{ max Tt Kω−∈∀ (10)
,,,0, QqSbz qb ∈∀∈∀≥ (11)
,,},1,0{, QqSby pqb ∈∀∈∀∈ (12)
,,},1,0{, RjSbjb ∈∈∀∈θ (13)
,,,,},1,0{,, kjRkRjSbo kjb ≠∈∈∈∀∈ (14) In the above formulation, the objective function (2)
aims to minimize the total costs. Total costs consist of dispersion costs, batch setup costs, production costs (which vary according to the product quality produced) and storage costs. The first term in (2) reflects the dispersion costs. They are determined according to the formula (1). In order to calculate dispersion Db, binary variables ob,j,k are summed up, which indicates whether batches are used to supply multiple retailers. Constraints (3) and (4) model the inventory balance for individual product batches. In those equations also the quality degradation during storage is
accounted for. The products are either from the inventory in previous periods (3) or from the production in the current period (4). Constraints (5) enforce that demand and its quality requirements must be satisfied. Constraints (6) and (7) make sure that a product batch has only one quality level at the time of production. Constraints (8) check whether the specific batches are delivered to retailers. Constraints (9) force ob,i,k = 1 if product batch b serves the two different retailers i and k simultaneously. Constraints (10)-(14) define the domains of the decision variables. 4. HEURISTICS
Several test runs have been performed with the model presented in Section 3, which have shown that it is difficult to solve using the standard MIP solver, even for relatively small instances. We therefore present heuristics to solve the problem. 4.1 Heuristic principle
We start with individual small batches of products that serve each retailer in each period separately based on the latest production time of the product batch. If there is demand for retailer j in period t, this leads to a production batch in period t - uj, i.e. the transport time is taken into account. Subsequently, we merge these small batches into larger batches that can serve multiple retailers in multiple periods based on the trade-off between setup cost, dispersion cost and the production cost.
We propose three heuristics based on this idea. All heuristics work in a multi-stage merging process. But differ in the way merging possibilities are identified. The first heuristic starts by merging batches for the same retailer, but for different production periods, while the second heuristic starts by merging batches for different retailers produced in the same period. The third heuristic uses a hybrid approach, looking both at different retailers and different production periods.
For all three heuristics, it is possible that there exist some batches that can be further merged to reduce the number of setups cost-efficiently. For this reason, all heuristics have a final merging process where all combinations of batches are checked for further merging opportunities. 4.2 Merging conditions
Since the basic idea of Heuristic 1 and Heuristic 2 is to
form larger batches by merging small batches cost-efficiently, we introduce some merging conditions.
To facilitate the description, we assume that production cost of the product p is a linear function of
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product quality and that the marginal cost for an additional level of quality for product p is Δcp (Δcp > 0). For reasons of simplicity, we ignore the constant part of this linear relationship, as this would be the same for all products. To guarantee the existence of the feasible solution, we require that the initial product quality is high enough to satisfy the retailers’ requirements:
RjPpqqq jppjp ∈∈≥Δ− ,,min,,,max, (15) It is important to note that the difference between
these qualities, combined with the quality degradation, also determines the number of periods that a product can be stored at the production plant. The following additional notation is introduced. Ω, Ω1, Ω2 subsets of R (set of retailers) Bj,p,t batch size of batch for product p produced in
period t, which serves retailer j. BΩ,p,t batch size of batch for product p produced in
period t, which serves retailers Ω ⊆ R (|Ω| ≥ 1).
Nj,p,t quality level of batch for product p produced in period t, which serves retailer j.
NΩ,p,t quality level of batch for product p produced in period t, which serves retailers Ω ⊆ R (|Ω|≥1).
4.2.1 Merging batches in one production period
In each period t, we sort the individual batches
delivered to different retailers based on the decreasing order of Nj,p,t. (see also Figure 3). We define R(j) as the j-th retailer in the ordered set. If a batch serves retailers Ω ⊆ R simultaneously, then the dispersion costs are calculated by
2/)1|(||| dispersionc−ΩΩ
∑Ω∈
Ωi
tpiRtp BB ,),(,,
ptptp cNB ΔΩΩ ,,,,
tpN ,,Ω
K
1 j2
(16) The batch size is calculated by
= (17)
The quality level for the batch must accommodate the highest level for the batch serving the individual retailer, i.e
, therefore the production cost is tpN ,,Ω
(18) If the batch serving retailers Ω will be merged with an
additional batch serving the j-th retailer, then the total cost for serving the |Ω|+1 retailers is
ptptpjRtp
dispersionsetup
cNBBcc
Δ++Ω+Ω+
ΩΩ ,,,),(,, )(2/||)1|(| (19)
If the batch serving retailers Ω is not merged with the batch serving the j-th retailer, then the total cost for serving the |Ω|+1 retailers is
ptpjRtpjRsetup
ptptpdispersionsetup
cNBc
cNBcc
Δ+
+Δ+−ΩΩ+ ΩΩ
,),(,),(
,,,,2/)1|(||| (20)
Comparing (19) and (20), we find that merging is beneficial (i.e. cost-effective) and should therefore be done if
setup
ptpjRtptpjRdispersion
c
cNNBc
<
Δ−+Ω Ω )(|| ,),(,,,),( (21)
This procedure is performed iteratively until the addition of the next individual batch would not be cost-effective anymore, i.e., if (21) does not hold. 4.2.2 Merging batches from different periods
If we want to merge two batches from different
production periods t1 and t2 (t2 > t1), the merged batch must be produced in period t1. It means that the original batch from period t2 must be increased to a quality level that accommodates for the quality degradation between t1 and t2. The batches from periods t1 and t2 serve retailers Ω1 ⊆ R and retailers Ω2 ⊆ R respectively, where we use Ω = Ω1 ∪ Ω2. In contrast to Section 4.2.1, we here here present a more general formulation in which we allow both batches under consideration to be previously merged batches, which is why we use two sets. To fulfil all quality requirements, the quality level of the final merged batch should be
))12(,max( 2,,21,,1max
1,, ptptptp qttNNN Δ−+= ΩΩΩ (22)
tpRN ,),1(
tpRN ,),2(
tpjRN ,),(
Qualitylevel
K K Retailer
Figure 3: Required quality levels for the ordered set of retailers.
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where we require max,1,, ptp , to make sure that the maximum potential product quality is not exceeded. This also limits the time difference between t1 and t2. Only if this requirement is met, the merging possibility can be considered.
max qN ≤Ω
If two batches are not merged, then the total cost for those batches is
ptptp
dispersionsetup
ptptp
dispersionsetup
cNBcc
cNBcc
Δ+−ΩΩ+
+Δ+−ΩΩ+
ΩΩ
ΩΩ
2,,22,,2
1,,11,,1
2/)1|2(||2|
2/)1|1(||1|
(23)
If the two batches are merged, then the total cost is
2,,2max
1,,2,,21,,1 )12()(
2/)1|(|||
tpptptptp
dispersionsetup
BtthcNBB
cc
ΩΩΩΩ −+Δ+
+−ΩΩ+ (24)
Then, by comparing (23) and (24), merging is cost-effective if
setuptp
dispersion
dispersion
ptptptp
ptptptp
cBtth
c
c
cNNB
cNNB
<−
+−ΩΩ+−ΩΩ
−−ΩΩ
+Δ−
+Δ−
Ω
ΩΩΩ
ΩΩΩ
2,,2
2,,2max
1,,2,,2
1,,1max
1,,1,,1
)12(
2/))1|2(||2|)1|1(||1(|
2/)1|(||(|
)(
)(
(25)
The two batches should be merged if this condition holds. (25) is a general merging condition for merging any two batches starting either in the same period or different periods, and (21) is the special case of (25).
4.2.3 Heuristic procedures Using the merging conditions from the previous section, we propose two different heuristics to approach the problem described in Section 3. Heuristic 1 Step 0 For each product p, set up batch production for
each retailer j in period t - uj to satisfy its demand in period t, dj,p,t. The total number of small batches produced is |P||T||R|.
Step 1 Starting with the first batch for the first retailer, merge this batch with batches for this retailer for subsequent periods based on (25). Stop adding batches whenever the merging condition does not hold.
Step 2 Starting with the next period, repeat Step 2 until all periods have been considered. Continue this
process for all of the retailers. Step 3 Starting in the first period, for each product p,
rank the production batches in that period (generated in Step 1 and 2) based on decreasing order of quality level.
Step 4 Starting with the batch with the highest quality level requirement, merge this batch with subsequent batches based on (25). Stop adding batches whenever the merging condition does not hold. Repeat this until all batches in that period have been considered.
Step 5 Perform Step 3 and 4 for all of the production periods.
Step 6 Check all resulting batches from the previous steps for merging possibilities using (25). Note that this does also include batches produced in different time periods.
Heuristic 2 Step 0 As in Heuristic 1. Step 1 For each product p and each period t, rank the
retailers based on decreasing order of quality level of batches delivered to them generated in Step 0.
Step 2 Starting in the first period with the retailer with the highest quality level requirement, merge this batch with batches for subsequent retailers based on (21). Stop adding retailers whenever the merging condition does not hold.
Step 3 Starting with the next retailer, repeat Step 2 until all retailers have been considered. Continue this process for all of the production periods.
Step 4 As Step 6 in Heuristic 1. Heuristic 3 Compared to Heuristic 2, this heuristic has an additional stage between Step 2 and 3, which we label Step 2a. Step 2a After a merged batch is produced in Step 2,
check further whether the small batches of the same retailers starting in the next periods can be merged into the current batch based on (25). These batches are then excluded when merging batches for that specific time period.
5. COMPUTATIONAL RESULTS
As mentioned in Section 4, the motivation for
developing heuristics is that standard MIP solvers have difficulties solving even relatively small instances of the model, let alone large real-life applications. In this section,
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we illustrate this, using a small instance with one plant, four retailers, two products and a five-period planning horizon as an example; we compare the performance of the heuristics against a standard MIP solver, CPLEX 10.2. This experimental design results in test instances with more than ten thousand constraints and variables (of which around three thousand are binary). Moreover, we use a second, larger test instance with one plant, ten retailers, two products and fourteen-period planning horizon to analyze the trade-offs between setup costs, dispersion costs and production costs based on the heuristic solution. All test runs were performed on a 1.86 GHz Pentium 4 PC with 1GB RAM. The heuristics were coded in C++ and implemented in the Visual Studio 2005 C++ software environment.
01234567
0.1 0.2 0.4 0.6 0.8 1.05
Ratio of dispersion cost and setup cost
Retailers
Periods
Figure 4: Maximum number of periods and maximum number of retailers of a batch against α.
5.1 Performance of the heuristics
The cost scenarios selected for the small benchmark
instance to test the performance are given in Table 1. We are increasing the dispersion costs cdispersion from 50 to 550. To simplify the analysis, we are define the cost ratio α as cdispersion/csetup and are neglecting the holding costs.
Table 1: Cost scenarios for the test instance.
Scenarios
1 2 3 4 5 6
csetup 500 500 500 500 500 500
cdispersion 50 100 200 300 400 550
α 0.10 0.20 0.40 0.60 0.80 1.05
Table 2 shows the percentage gap between the objective function values of the heuristics and CPLEX as well as the solution times. CPLEX takes more than three hours to obtain a solution. It is important to note that CPLEX cannot guarantee to solve the problem optimally because of computer memory limitations. In fact, in some cases, the heuristics generate a better solution than CPLEX.
Table 2: Performance comparison between heuristics and
CPLEX (H1: Heuristic 1, H2: Heuristic 2) Sc. Gap (%) Solution time(s)
H1 H2 H3 CPLEX H1 H2 H3 1 9.2 8.6 4.5 12605 0.052 0.011 0.062
2 6.9 9.0 3.3 11988 0.055 0.019 0.063 3 2.3 -3.0 2.0 11700 0.062 0.130 0.078 4 0.6 1.4 1.0 11376 0.062 0.270 0.078 5 -0.4 0.3 0.5 11600 0.065 0.600 0.078 6 -0.4 17.4 -0.4 11520 0.052 0.900 0.062
Av. 1.8 5.6 1.8 11798 0.058 0.180 0.070
All heuristics can generate solutions very quickly.
Overall, Heuristic 1 and 3 perform better than Heuristic 2 in terms of both optimality performance (gap) and solution time. The maximum gap between the heuristic and CPLEX is the smallest (4.5%) for Heuristic 3. For Heuristic 1 and 2, this gap is 9.2% and 17.4% respectively.
5.2 Analysis of cost trade-offs
In this section, we analyze the trade-off between setup,
dispersion and production costs by solving a larger instance. The cost scenarios used here are the same as those shown in Table 1. For the results, we solved the scenarios with all three heuristics, and selected the best solution among them.
Figure 4 shows the maximum number of periods and the maximum number of retailers that a batch that can serve against α. The results show that the number of retailers served by a batch decreases as α increases. This means that α can be used to control dispersion. However, α has no influence on the number of periods served by a batch, as this does not affect dispersion.
At high levels for α, when the relation between dispersion cost and setup cost prevents batches to serve multiple retailers, we can only reduce the number of setups by allowing a batch to serve the same retailer for multiple periods. For our test instance, the number of periods served by the same batch assumes a maximum value of three. However, if a batch serves multiple periods, then the quality level of the product must be increased to compensate for the quality degradation in storage. If quality degradation would increase, the maximum number of periods covered by a batch would decrease. Hence, the number of setups must increase. This will be analysed in greater detail in the following section.
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Figure 5 shows the total dispersion, the total number of batches, and the total number of shared batches against α. Shared batches are batches that serve more than one retailer at the same time. The figure shows that, as α increases, dispersion decreases. This is the result of a decrease in the number of retailers served by the same batch. This does require an increase in batches, as can also be seen in the figure. The increasing gap between the number of batches and the number of shared batches is the direct result of a decrease in the average number of retailers that is served per batch. We can also see that dispersion and the number of shared batches are identical for α = 0.60 and higher, which for α = 0.60 and α = 0.80 means that all shared batches exactly serve two retailers. For α = 1.05 both are zero, meaning that no batches are shared and no dispersion occurs.
Finally, Figure 6 shows total setup costs, total dispersion costs and production costs against α. Total setup costs increases because the total number of setups (total number of batches) increases as was shown in Figure 5. Even though the dispersion is higher when α is lower, a low α also means a low cost factor cdispersion and thus total dispersion costs are lower. Initially, the dispersion costs increase due to the increase of cdispersion that is not completely compensated by a decrease of dispersion. But for higher values of α this is the case, up to the point at which dispersion is completely suppressed, and the total dispersion costs become zero.
Analysing the production costs is less straightforward. Initially, as α increases, the costs decrease because the decrease in dispersion makes it possible to reduce the batch quality requirements, hereby reducing production costs. Having to cover a smaller range of quality requirements with each batch means that there is less ‘quality wastage’. For higher values of α, production costs are again higher and fairly constant. This is likely caused by the fact that the
reduction in dispersion seen with an increase of α from 0.40 to 0.60 is the result of an increase in the number of batches that serve a retailer over multiple periods. This can also be seen in Figure 5, as the number of shared batches decreases significantly, without causing an increase in the total number of batches. Hence, this has to be compensated by more expensive, higher quality batches that can serve retailers over multiple periods. For higher values of α we see that this decrease in shared batches is compensated by an increase in the total number of batches, which in fact leads to minor reductions in production costs, as it leads to an improved match between production batch qualities and quality requirements of the retailers. 5.3 Effects of quality degradation
The quality degradation in storage affects the number
of periods that a batch can be used for. An increase in quality degradation (e.g. caused by higher environment temperatures) will lead to a lower maximum storage time, and will also affect the different cost factors. To illustrate this, we increased the quality degradation in the instance used in Section 5.2, so that the time span from the production of a maximum quality product the moment it reaches the minimum quality requirement of any retailer decreases from 6 and 7 time periods for the two products to 4 and 4 periods respectively.
Table 3 shows the resulting relative increases for the various cost factors after the increase in quality degradation. Overall, we can see that all costs, i.e. production, dispersion, setup, and total costs, increase as the quality degradation increases.
Number of batches Setup CostsShared batches Production Costs
Figure 5. Total dispersion, total number of batches and total number of shared batches against α.
Figure 6. Total setup costs, total dispersion costs and total production costs against α.
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Table 3: Relative cost increase at a higher quality degradation.
α Relative cost increase (%)
Production Dispersion Setup Total
0.10 25.0 18.0 12.3 14.1 0.20 27.0 30.7 16.5 19.0 0.40 26.1 15.3 18.0 19.1 0.60 31.9 117.1 9.5 18.5 0.80 22.1 240.9 9.8 19.9 1.05 57.8 0.0 7.9 20.4
Average 31.6 70.3 12.3 17.7
Concerning dispersion and the related costs, the effects are less straightforward. On the one hand, it is possible that dispersion increases because there are fewer possibilities to reduce setups by serving the same retailer for multiple periods. On the other hand, quality degradation may also prevent serving more retailers because the initial quality levels of the products are now increased and a merged batch must be up to the highest quality requirements of any of the supplied retailers.
Based on above analysis, we can clearly see that preserving product quality (i.e. reducing quality degradation) is important to both reduce safety concerns (by controlling dispersion) and to reduce costs related to production and setup. Temperature is known to be an important environment factor to affect the quality degradation (Labuza, 1982). Reducing quality degradation requires a reduction in, or tighter control of, storage or transportation temperatures. It is important to note that this might lead to an increase in storage costs and cooling costs (see also Rong et al., 2008).
6. CONCLUSIONS
Food safety is increasingly important for all actors in
the supply chain, which is reflected in the industry-wide implementation of traceability systems. Traceability by itself does however not change the quality and safety of the food products. We also need to address the quality and safety concerns in production and distribution planning. To this purpose we introduced batch dispersion as a measure for the extent in which production batches are spread among retailers, i.e. dispersed in the distribution network. Dispersion can be limited by reducing batch sizes. However, this is often not feasible due to the efficiency losses incurred by extensive setup times. The challenge is to find an efficient and effective way to reduce the number of production setups without causing safety concerns.
In this paper, we contributed to this challenge by presenting a multi-period, multi-product, production and distribution planning model that aims to improve food
safety and traceability based on the concept of batch dispersion.
Since the model cannot be solved for problem instances of realistic size, we developed a number of heuristics. We have used a simplified network that consists of one plant and multiple retailers to illustrate the effectiveness of the methodology as well the trade-off between dispersion, setup, and production costs.
Through selecting a proper level for the dispersion cost coefficient, a suitable level of dispersion can be achieved. Hereby, the food safety concerns can be managed proactively. How this penalty cost is to be determined is not part of this work. To do this, further research could focus on gaining insight into the risk attitude of the actors involved in food production and distribution.
Preserving food quality is also an important key to reduce safety concerns. If quality degradation is moderate, one product batch can serve the same retailer in multiple periods to avoid batch dispersion.
ACKNOWLEDGEMENTS
The authors would like to thank the FoodDTU
research centre (www.fooddtu.dk) at the Technical University of Denmark for partial funding of this research. The third author would also like to acknowledge support from a H.C. Ørsted postdoctoral fellowship from the Technical University of Denmark. REFERENCES
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AUTHOR BIOGRAPHIES
Martin Grunow is professor of Operations Management and leader of the theme "Catering and convenience, freshness and supply chains" in the FoodDTU Centre at the Technical University of Denmark. Earlier, he worked at the Technical University Berlin and at Degussa AG's R&D department. His research interests are in production and logistics management with a focus on supply chain management in the process industries, including food. He has co-authored more than 80 publications amongst others in International Journal of Production Economics, International Journal of Production Research, European Journal of Operational Research, CIRP Annals, and OR Spectrum. For the latter journal, he also acts as an editor (since 2001). His email address is [email protected].. Aiying Rong received her master degree in Industrial Engineering and Engineering Management at Hong Kong University of Science and Technology and her Ph.D. degree in Computer Science (Algorithmics) at the University of Turku, Finland. Currently, she is working as a postdoctoral researcher in the Department of Management Engineering at the Technical University of Denmark and is involved in research on supply chain management in the food industry. Her research interests include operations, planning and scheduling of production activities in different industrial sectors such as the energy industry, the food industry, and the iron & steel industry. She has published papers in European Journal of Operational Research, International Journal of Production Research, Journal of the Operational Research Society and Applied Energy. Her email address is [email protected]. Renzo Akkerman is a H.C. Ørsted postdoctoral research fellow at the Technical University of Denmark in the field of Operations Management. He obtained his Ph.D. in Operations Management from the University of Groningen in The Netherlands, where he previously received an M.Sc. in Econometrics & Operations Research. His research interests are in operations management and supply chain management, mainly related to the food industry, but also in the health care sector. His research has been published in, amongst others, International Journal of Production Economics, International Journal of Production Research, British Food Journal, and Health Care Management Science. His email address is [email protected].
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