minimization of off-grade production in multi-site multi-product plants by solving multiple...

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Minimization of off-grade production in multi-site

multi-product plants by solving multiple travelingsalesman problem

Andras Kiralya,∗, Maria Christidoub, Tibor Chovana, Evangelos Karlopoulosb,Janos Abonyia

a University of Pannonia, Department of Process Engineering, P.O. Box 158. Veszprem

H-8200, Hungary bCentre for Research and Technology Hellas, Chemical Process & Energy Resources

Institute, P.O. Box 95, 502 00 Ptolemaida, Greece

Abstract

Continuous multi-product plants allow the production of several products (prod-

uct grades). During grade transitions off-spec products are produced. The eco-

nomic losses and the environmental impact of these transitions are sequence

dependent, so the amount of off-grade products can be minimized by scheduling

the sequence of the production of different products. Applying parallel produc-

tion sites (m) increases the flexibility of multi-product plants. Since market de-

mands are changing, the production cycles of these sites should be re-scheduled

in certain intervals. Therefore, our task is to design m production cycles that

contains all required products by minimizing the total length of grade transi-

tions. Most production scheduling problems such as the one considered in this

paper are NP-hard. Our goal is to solve realistic problem instances in no more

than a couple of minutes. We show that this problem can be considered as

a multiple traveling salesmen problem (mTSP), where the distances between

the products are based on the time or costs of the grade transitions. The re-

sulted mTSP has been solved by multi-chromosome based genetic algorithm.

The proposed algorithm was implemented in MATLAB and is available at thewebsite of the authors (www.abonyilab.com). For demonstration purposes, we

∗Corresponding authorEmail address: [email protected] (Andras Kiraly)

Prep rin t submitted to Journal of Clean er Productio n January 3, 2 015



present an illustrative example. The results show that multi-product multi-site

scheduling problems can be effectively handled as mTSPs, and the proposedproblem-specific representation based genetic algorithm can be used in wide

range of optimization problems.

Keywords: optimizatio, off-grade production, mTSP

1. Introduction

Thanks to the increasing need for flexible processing facilities to produce

more than one product, the planning problem of multiproduct plants is becom-

ing more and more important. The planning of process systems involves theprocedures and processes of allocating the available resources and equipment5

over a period of time to perform a series of tasks required to manufacture one

or more products. Typical example for such problem is the optimization of

the transformation of biomass to energy. Biomass is usually locally available,

which defines it as a distributed resource, and requires extensive infrastructure

networks for harvesting, transportation, storage, and processing. The design10

and management of biomass supply chains should account for the local condi-

tions and constrains, such as the existing infrastructure, geographical features

of the studied region and the competition among several consumers. Biomass

are wood and forestry residues, energy crops, various kinds of straw, as well

as biowaste from food production, wood processing and use. Primary biomass15

resources are distributed over the area in a region and often available in remote

locations. Building the infrastructure to transfer biomass energy over longer

distances would tend to increase its cost. On the other hand, biomass offers

the potential to reduce the environmental impact of energy supply and poten-

tially saving costs for reacting to natural disasters in the future. An important20

factor to be considered is the security of energy supply, which has significantimportance [1, 2] . Energy generation from domestic sources helps reducing the

dependence on foreign imports of crude oil and natural gas. It increases the

economic stability and can improve significantly the foreign trade balances of

2



the respective regions and countries. The relatively low energy density (energy25

per unit volume) of most raw biomass feedstocks tends to increase the cost,emissions and complexity of supply chain. Therefore, the developments of com-

plete procedure for regional energy supply optimization become an important

task [3].

The goal of this paper is the development of a production planning and30

scheduling algorithm for parallel (multi-site) continuous processes in the pres-

ence of sequence-dependent switchover times. A similar problem is already

studied in [4]. TSP is already a widely applied technique for scheduling of parts

in a flowshop, for lot of streaming and scheduling problems, and for optimiza-

tion of robot movements in automated production cells. According to [5], most35

of real-life problems can be defined as a flowshop, where each machine per-

forms a single operation, and the sequence of the procedures is fixed. Bagchi

et al. give a novel classification of these problems, the most important for us is

the so-called ”group-scheduling”, where products belong to different groups and

optimization is performed in a two-stage method. The approach is similar to40

the ”manufacturing cell” problem, where similarity coefficients to group similar

parts into families. The problem is almost identical to the one we will discuss

in the next sections, however, our method is capable to optimize the productionprocess in one step and considers not only the characteristics of manufacturing.

Similar problems appear also in shop-floor logistics systems, which influences45

not only the production control’s performance, but the order management and

production system also [6]. A TSP-based model for medium-term planning

of a single-stage plant with a single continuous processing unit producing sev-

eral products with sequence-dependent changeovers has been already studied in

[7]. There a MILP model is defined for a single-stage plant with a single con-50

tinuous processing unit producing several products with sequence-dependent

changeovers. Though we deal with a multi-site, multi-product processing pro-

cedure, our objective is very similar, to maximize the profit, i.e. to minimize

product changeover cost which occur when switching from one product to an-

other. We will give a linear programming formulation also in the next sections,55

3



and our representative example is also derived from the work of Liu et al. Based

on this classic formulation of travelling salesman problem (TSP), an improvedmodel is developed, where the objective function considers the profit, inventory

deviations from the desired trajectories and price changes simultaneously. As

Liu et al. discussed in [8], if the production changeovers are dependent on the60

production sequence, and product groups are involved, the production sequence

may affect the capacity of the factory, as well as the efficiency of the whole

supply chain. Therefore, handling the sequence-dependent changeovers is an

important issue. The problem becomes more complex when a multisite produc-

tion system is needed to be optimized. Such productiondistribution network is65

made up of several production sites distributing to different markets. The plan-

ning and scheduling model has to include spatial scales that go from a single

production unit within a site to a geographically distributed network [9].

Since economic competition is growing rapidly, companies are greatly inter-

ested in reducing overall costs, including manufacturing, inventory, changeover70

expenses, as well as minimizing ecological footprint and waste production. There-

fore, the Enterprise-wide Optimization (OWE) has become a major objective

not only in the chemical industry. As Grossmann describes in [10], EWO is

concerned with the coordinated of the operations in supply chain, and the mainobjective is to maximize profits, responsiveness and asset utilization and to75

minimize costs and ecological footprint. Therefore, a complex cost function is

needed, while complexity should remain as low as possible. We will discuss

a compound objective function derived from the utility theory. Recently, very

good reviews has been published, dealing with the environmental impacts of em-

placement and allocation as well as the optimization according to the available80

resources and raw materials [11]. Lin et al. in [12] discuss the increasing and

close attention of green logistics, since the recent production and distribution

strategies are not sustainable in the long term. These processes are sensitive

environmentally, ecologically and socially, and therefore, it is particularly im-

portant to optimize them cautiously.85

In this paper we propose multiple-Traveling Salesman Problem based repre-

4



sentation for the optimization of multiproduct and multisite production systems,

where the distances between the products are based on the time or costs of thegrade transitions. The resulted mTSP has been solved by multi-chromosome

based genetic algorithm. The chromosome representation and especially the90

applied operators make our modified genetic algorithm especially effective in

the optimization of mTSP problems [13]. Furthermore, taking additional con-

straints into consideration, like the maximum number of salesmen or the maxi-

mal time a salesperson can travel makes it capable to solve complex structures

and in production planning to prevent a single site from overloading. The pro-95

posed algorithm was implemented in MATLAB and available at the website

of the authors (www.abonyilab.com). For demonstration purposes, we present

an illustrative example from the literature, which is from a real world poly-

mer processing plant and discussed by [7]. In this simple example we present

the optimization of the production of 10 products by 3 sites, using our genetic100

algorithm based mTSP solver.

2. Problem formulation

As we discussed in the introduction, production scheduling problem usually

handled by integer programming methods, like MIP or MILP. For mathematical

formulation we can’t find any better approaches than these articles present,105

therefore we follow the conventions of these formulations. The problem can be

illustrated by a schematic diagram, like the one in Fig. 1.

As the figure demonstrates, we have several (m p = 3) parallel production

sites in the plant which produce several (n = 15) products to fulfill the market’s

demand. Raw materials come from various deposits (there are mr = 4 deposits),

and each market (mm = 4) has a unique combination of products they require.

Within a site, during the transition of production from one product to another,many waste or off-grade products are producing. Our main task is to minimize

the amount of these waste items and the ecological impact of their production,

fulfilling the restrictions of raw materials needed for certain products, and the

5



Figure 1: Concept

distribution of products to the different markets. The problem can be defined

by a simple MIP formulation as follows.

minmp

i,j∈J m

C ij , ∀i, j ∈ I , m p ≥ 1 (1)

where I is the set of items, and J m is the mth production site. C i j represents

the compound cost for the production of product j after i. This cost incorpo-

rates expenses of off-grade products and ecological impacts (carbon footprint)110

as well as the total cost of changeover from product i to j. In our case, in-

stead of incorporating tons of additional parameters and costs into our MIP

model, we decided to handle uncertainties by the help of utility theory. Our

approach is similar to the fuzzy model presented in [14], however, we define a

6



compound Taguchi-type loss function to handle off-grade production and car-115

bon footprint simultaneously. The Taguchi method [15] is a simple but robusttechnique for process parameter optimization, involving the reduction of pro-

cess variation. Kim et al. in [16] used the Taguchi method to simultaneously

handle productivity and environmental impact in eco-friendly manufacturing.

Figure 2 represents our loss function defining cost related to the production120

process. Using classical nomenclature, this problem is a multi-objective opti-

mization problem, however, we apply the same methodology as Lim et al. in

[17] to convert it to a single-objective problem.

Figure 2: Compound Taguchi-type loss function. The final cost function is the sum of off-grade

production and carbon footprint expenses.

In our approach, we define a composite similarity between each pair of prod-

ucts, which includes the changeover time between the two products, and con-siders deposits and markets, from where row materials come from and to where

final products are transported. The concept for this similarity is derived from

the coefficient defined in [18]. We define the similarity coefficient between prod-

ucts in the following way:

pij = wt ∗ tchangeover + w pd ∗ pdepositij + w pm ∗ pmarket

ij (2)

where tchangeiver is the changeover time from product i to j, wt, w pd and w pm

are the weights, and the other two coefficients are defined by

pdepositij =

mijr

xi + yj + mijr

(3)

where

7



• pdepositij : Similarity coefficient between products i and j according to de-125

posits

• mijr : Number of raw materials come from deposits serving the manufacture

of products i and j

• xi: Number of raw materials come from deposits serving the manufacture

of products i but not of j130

• xj : Number of raw materials come from deposits serving the manufacture

of products j but not of i.

pmarketkl =

mklm

xk + yl + mklm

(4)

where

• pmarketkl : Similarity coefficient between products i and j according to mar-

kets135

• mklm: Number of products transported to markets, which turn products i

and j to account

• xk: Number of products transported to markets, which turn products i

but not j to account

• xl: Number of products transported to markets, which turn products j140

but not i to account.

Thus, our compound cost metric is defined by the following formula, where

T ij is the Taguchi function showed in Fig. 2 for the production of product j

after i:

C ij = T ij + pij (5)

Finally, to prevent the sites from overloading, we need to define additional

constraints about the minimum number of product types:

|J m| > r, ∀m ≥ 1 (6)

8



Using this formulation, we can define a distance matrix containing distances

between each pair of products. In this way, the problem presented in Fig. 1 canbe transformed into a special case of multiple traveling salesmen problem, where

the start and end locations are not fixed. We are explaining this approach in145

the next section.

2.1. mTSP-based formulation

Since we use a compound objective function, and a complex distance metric,

our production allocation problem including the minimization of waste items

and changeover times can be handled as a multiple traveling salesmen problem150

or mTSP. With the following reconciliation we can handle and solve the problem

as a route planning task and use an mTSP solver as we will see in the next

section:

• compound distance metric ←→ traveling distances

• products ←→ locations155

• production sites ←→ salesmen

• compound utility function −→ fitness function

In this way, the problem is transformed into an mTSP, thus, the following

formulation can be used. Let us define the following binary variable xijk :

xijk =

1 if arc (i, j) is used on the tour of the kth salesman

0 otherwise(7)

9



Using this binary variable, our objective function can be expressed as follows:

minimizen

i=0

nj=0

ctij ·mk=1

xijk (8)

so thatn

j=1

mk=1

x1jk = m, (9)

nj=1

mk=1

xj1k = m, (10)

ni=0

mk=1

xijk = 1, i = 2, . . . , n , (11)

nj=0

mk=1

xijk = 1, j = 2, . . . , n , (12)

ni=0

nj=0

ctij · xijk ≥ r, k = 1, . . . m , (13)

xijk ∈ {0, 1}, ∀(i, j) ∈ A, 1 ≤ k ≤ m, 1 ≤ m ≤ M

3. The proposed GA-based solution

As we saw in the previous section, using compound distance coefficients and160

loss functions, the problem presented in Fig. 1 can be interpreted as a multiple

traveling salesmen problem, where the start and end locations are not fixed, it

is a so-called open-mTSP. Applying this transformation, we are capable to use

a slightly modified version of our previously published multi-chromosome repre-

sentation based genetic algorithm for solving mTPS problems. Here, we discuss165

only the main aspects of the method, interested readers can find more details in

[13]. We only needed to remove the restrictions for start end end locations from

the algorithm, i.e. to remove the depot. The concept of our approach is pre-

sented in Fig. 3, together with the corresponding route system. In the bottom

part of the figure, the correspondence between the parallel multi-production site170

plant and the route system is demonstrated. As it can be seen, the whole prob-

lem is decoded into a single individual, where different chromosomes represent

the different salesmen or sites.

10



Figure 3: Concept of the multi-chromosome representation based genetic algorithm for the

production site scheduling problem. On the left side, a single individual is presented, con-

taining three chromosomes, corresponding to the three salesmen in the upper right corner

depicting a route plan, or to the three production sites of the production scheduling in the

bottom right corner.

As all GA, our algorithm starts from a random population, where each prod-

uct is produced by a random site, in a random order. The algorithm uses the175

evolutionary strategy to select, recombine and mutate the individuals with best

fitness value to produce the new population. The fitness function is defined

in equation 8 and in equations 2-5, as a compound measure including distance

metrics and loss functions, therefore the smallest the fitness the better the in-

dividual. After predefined iteration steps, the procedure stops and the best180

individual will represent the optimal solution of the problem.

Our method uses the special chromosome representation discussed above,

and therefore requires special operators to recombine the individuals inside a

11



population. These special and complex operators ensure the strong convergence

of our GA as well as the aforementioned constraints. We present one complex185

operator here, other ones can be found in [13]. The operator depicted in Fig. 4 is

a complex one, which combines two mutation operations applying them sequen-

tially. The first one is the so-called ’slide’ which moves the last gene (location,

product) from each chromosome (route, production site) to the beginning of an-

other one. The second operator, ”swap” chooses two random sequences of genes190

from two chromosomes and transposes them. Of course, these randomly chosen

sequences can be empty, thus, the operator will be realized as the insertion of

the nonempty sequence to a randomly chosen place in the other chromosome.

Figure 4: Complex genetic operator used by the algorithm for mutation. From left to right:

applying a ”slide” and a ”swap” sequentially.

The concept above can be realized as a special clustering, where not only

the identified clusters matter, but an internal order of the items also. Ng and195

Lam in [19] optimize a supply network in two phase, where one is clustering.

Our approach combines the two method and solve the problem in one step.

4. Results and discussion

In this paper we present a representative example to illustrate our GA based

mTSP solver to schedule multiproduct multisite problems. The example derived200

from a real world polymer processing plant is the same which is discussed by

Liu et al. in [7] as an extension of the instance presented in [20]. Here, 10 types

12



From/To A B C D E F G H I J

A 45 45 45 60 80 30 25 70 55

B 55 55 40 60 80 80 30 30 55

C 60 100 100 75 60 80 80 75 75

D 60 100 30 45 45 45 60 80 100

E 60 60 55 30 35 30 35 60 90

F 75 75 60 100 75 100 75 100 60

G 80 100 30 60 100 85 60 100 65

H 60 60 60 60 60 60 60 60 60

I 80 80 30 30 60 70 55 85 100

J 100 100 60 80 80 30 45 100 100

Table 1: Representative example: Product grade transition times (minutes) [7].

of products (A-J) are manufactured, and we use a multi-site plant to satisfy

customer demands. Table 1 shows the changeover times as it is in [7]. We also

present an illustrative example for the allocation of suppliers and markets to205

products in Table 2.

According to the compound distance metric defined in equation 5, we mod-

ified the distance table which can be used in the optimization algorithm. Since

our GA necessitates additional parameters, we need to define e.g. an upper

bound for distance for each route, or the maximal number of salesmen. Using210

the following parameters, our algorithm was capable to solve the production

schedule problem in the form of an mTSP:

• Maximum number of salesmen: 3

• Minimum number of cities per tour: 3

• Maximum tour length: 300215

• Population size: 160

• Iteration number: 200

13



SuppliersProducts

A B C D E F G H I J

1 1 0 1 1 1 0 0 0 0 1

2 0 1 0 0 1 0 0 1 1 0

3 0 0 0 1 1 0 1 0 1 0

4 0 1 0 1 0 1 0 0 0 1

Distribution centers

1 1 0 1 0 0 0 1 0 0 1

2 0 1 0 1 1 0 0 1 0 0

3 0 1 0 0 1 0 0 1 1 04 0 0 0 1 0 1 0 0 0 1

Table 2: Allocation of suppliers (raw materials) to products (upper part) and distribution

centers (markets) to products (lower part). Representative example.

From the updated distance table we can construct two-dimensional coordinates

for the products using a two-dimensional scaling technique, and the resulted

optimal plan can be visualized as it is in Fig. 5.220

As the figure shows, the processing of products is balances, and Fig. 6 rep-

resents the realized production schedule for the 10 products using three sites

for a one week time horizon. During the optimization, the algorithm takes into

consideration the source of raw materials, as well as the acquiring markets, and

minimizing the changeover times, the cost of waste items and the economical225

impact.

5. Conclusions

Multiproduct multisite scheduling problems can be effectively handled as

mTSPs. We proposed a problem-specific representation based genetic algo-

rithm that can be used in wide range of optimization problems. The proposed230

algorithm was implemented in MATLAB and available at the website of the

authors (www.abonyilab.com). For demonstration purposes, we presented an

14



Figure 5: Optimal solution of the mTSP problem. Coordinates are representative, they were

determined by MDS from the compound distance table.

illustrative example from the literature, which is from a real world polymer

processing plant and discussed in [7]. In this simple example we presented the

optimization of the production of 10 products by 3 sites, using our genetic al-235

gorithm based mTSP solver. Further research will focus on the extension of

more general vehicle rooting problems and more detailed application study on

the optimization of a complex bioenergy production supply chain, where the

sequence of the processing of different waste materials is scheduled.

Figure 6: Optimal production schedule of the representative example using the GA-based

mTSP solver. Corresponds to the results in Fig. 5, visualizing only changeover times.

15



Acknowledgements240

This work was supported by the Greek-Hungarian Bilateral Program under

project Greek General Secretariat for Research and Technology (GSRT) / Eu-

ropean Regional Development Fund (ERDF) no.HUN88 / TET 10-1-2011-0539.

The contribution of Janos Abonyi was supported by the European Union and

the State of Hungary, co-financed by the European Social Fund in the framework245

of TAMOP 4.2.4. A/2-11-1-2012-0001, ’National Excellence Program’.

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