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ARTICLE IN PRESS Expert Systems with Applications xxx (2009) xxx–xxx

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An algorithm portfolio based solution methodology to solve a supply chain optimization problem Salik R. Yadav a, Raja Ram M.R. Muddada b, M.K. Tiwari b,*, Ravi Shankar c a

Department of Engineering Management and System Engineering, Missouri University of Science and Technology, MO 65409, USA Department of Industrial Engineering and Management, Indian Institute of Technology, Kharagpur, West Bengal 721302, India c Department of Management Studies, Indian Institute of Technology, Delhi 110016, India b

a r t i c l e

i n f o

Keywords: Supply chain optimization Algorithm portfolio Platform and independent supply chain

a b s t r a c t This paper introduces the algorithm portfolio concept to solve a combinatorial optimization problem pertaining to a supply chain. The supply chain problem is modeled with capacity constraints and demand variations over different time periods to minimize the total supply chain conﬁguration cost. The algorithm portfolio is implemented over various problem instances to inspect and alleviate the computational expensiveness of a solution strategy. A bunch of ﬁve algorithms are utilized hereby viz. AIS, GA, Endosymbiotic Optimization, PSO and Psychoclonal algorithm. The observations reﬂect the appropriateness and effect of algorithm portfolios over the adopted supply chain, and viability over other optimization problems. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction The supply chain conﬁguration is comprehended differently with reference to different problem context. For instance, it can be deﬁned in terms of the selection of an optimal product family (Lamothe, Hadj-Hamou, & Aldanondo, 2006) or supplier option selection (Graves & Willems, 2005). A common issue in supply chain management is to minimize the supply chain conﬁguration cost (Ganeshan, Jack, Magazine, & Stephens, 1999). However, the globalization and enlarged product variety in a supply chain leads to product proliferation (Chong, Ho, & Tang, 1998) which complicates the supply chain structure and raises the cost. To compromise over price-premium and manufacturing cost, modular strategies (He & Kusiak, 1996) and delayed differentiation (Lee, Robertson, & Ulrich, 1998) are utilized. Meanwhile, component commonality and differentiation determining cost and revenue are to be pertinently realized by standardizing one/some of the components while differentiating others for different market products. A product platform is generally referred to the common components, parts, subassemblies or assets shared across a product family (Sawhney, 1998). Platform based product development (PPD) obviously alleviates the costs and time to market and in addition to that it is a vital strategy for manufacturing mass customized product (Salvador, Forza, & Rungtusanatham, 2000). Moreover, as evident from the supply chain and product development literature, the inventory levels of common modules, in case of PPD, are highly reduced what is known as ‘risk pooling’ effect and is an outcome of * Corresponding author. Tel.: +91 3222 283746. E-mail address: [email protected] (M.K. Tiwari).

component commonality. Various supply chain problems have been solved in the literature to obtain the minimum supply chain conﬁguration cost like product cycle time and product variety (Piramuthu, 2005), uncertainties in supply and demand (Kwon, Im, Kun, & Lee, 2007), integration of manufacturing/distribution planning decision (Liang & Cheng, 2008), scheduling problems (Chan, Chung, & Chan, 2005), logistical operations and reduction of bullwhip effect (Zarandi, Pourakbar, & Turksen, 2008). This paper researches a supply chain problem entangled with product development implications; however, it strives more to present a renovated solution methodology to promptly and economically resolve the supply and manufacturing decisions. Precisely, it addresses the inexorable quest to develop a versatile, prompt and efﬁcient solution device to boldly tackle the noise and variations in a given supply chain issue that may arise from time to time and situation to situation which has been barely attended in supply chain contexts. For this purpose, it adopts a combinatorial cum integer supply chain optimization problem pertaining to raw material vendor selection, manufacturing process selection and transportation mode selection in a supply chain. The undertaken problem is pretty similar to Graves and Willems (XXXX) who employed deterministic solution methodology viz. Dynamic programming for one instance of the problem but overlooked the promptness, versatility and computational cost issues for the problem particularly for large size variants of the problem. Huang, Zhang, and Liang (2005) considered the same problem; solved it by a random search algorithm viz. GA with appreciable promptness for one problem instance, however, ignored the establishment of its performance over diverse problem instances and nor did they adjudged other competing stochastic algorithms. Akcay

0957-4174/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2008.10.076

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and Xu (2004) employed a combination of two deterministic heuristics to resolve similar problem but pertaining to optimal base stock policy and component allocation. However, the computational time involved in solving the problem will be large at the cost of exact optimality of solution (as is the case with deterministic algorithms), particularly in case of large and complex variants of the problem. Moreover, these algorithms possess not much ﬂexibility when the problem complexity and nature is varied substantially. As reﬂected from the above discussion, most of the previous works have either utilized algorithms based on the performance over a few similar problem instances or lack a sophisticated performance evaluation strategy. At the same time, they do not allow a negotiated consideration of computational cost, solution quality and versatility. Particularly, the deterministic algorithms are associated with large computational cost and effort; while, on the other hand, the run time of the stochastic algorithms varies for the same problem instance (from run to run) as well as for different problem instances. Obviously, the performance is not similar with different problem instances of the same problem. Thus, the related literature on supply chain has sparse attention on overall performance evaluation, performance consistency maintenance and the algorithm selection issue. Though few attempts to optimize the overall performance of the algorithms such as Dean and Boddy (1988) are found in the literature but seldom have these or other overall performance based strategies been harnessed in supply chain optimization problems. However, recently, an audacious strategy viz. algorithm portfolio – ‘‘a collection of different algorithms and/or different copies of same algorithm running on different processor” Gomes and Selman (1997) – has been proposed by Gomes and Selman (2001) to analyze complexities of the results of computationally hard problems that may lead to better amelioration in terms of overall performance. This paper resolves and inspects a supply chain optimization problem through algorithm portfolio conceptualization with statistical and experimental evaluations over various problem instances with different problem complexities. The undertaken problem is mainly a combinatorial cum integer supply chain problem in which raw material vendor selection, manufacturing process selection and shipping mode selection issues juxtaposed with product platform considerations and service time decisions are addressed. The model is formulated to optimize the total supply chain conﬁguration cost for a product family with and without product platforms. In order to overcome the above-mentioned shortcomings of usual optimization algorithms and their performance evaluation strategies, this paper inspects several algorithms bunching them into portfolios based on cumulative performance over various problem instances of the underscored problem. Summarily, the paper focuses on following implications (i) algorithm performance improvement in terms of computational cost and a quality deviation from the expected solution (ii) exploitation of the performance diversity of multiple algorithms (iii) minimization of the overall risk associated with the use of an algorithm (iv) ranking the portfolios based on its overall performance (v) provide the supply chain decision maker with a tool to evaluate and ensure promptness and reliability of the solution methodology (vi) an efﬁcient strategy and algorithm kit for taking supply chain decisions in stipulated time frame.

conﬁguration of supply chain by making optimal supplier selection, manufacturing process and shipping decisions. For this purpose, a speciﬁc example from Graves and Willems (2005) or Huang et al. (2005) is taken for study. In this example, a laptop manufacturer company manufactures/assembles a laptop family comprising of laptop A and laptop B. The laptops are targeted at two markets US and Europe. In US (major market) both of the laptops A and B are sold whereas Europe (minor market) purchases only laptop A. The ﬁrm preferably uses PPD and not independent product development, IPD (IPD passes on zero commonality whereas PPD instills a non-zero commonality to the product family). Analogically, the supply chain also has two possible design alternatives viz. platform (PSC) and independent supply chains (ISCs). The optimization model minimizes the total supply chain cost which is considered exclusive of the product development cost, i.e, the supply chain cost is supposed to be governed by option selection for each of the above-mentioned decisions at different stages of the supply chain. 2.1. Generic bill of material and PPD A versatile representation alternative for a product family proposed by Hegge and Wortmann (1991) and later extended by Akcay and Xu (2004) is the concept of generic bill of material (GBOM). A GBOM is a tree containing ‘ex-OR’ (exclusive OR) nodes and ‘AND’ nodes (Jiao, Tseng, Duffy, & Lin, 1998). To realize one member of product family, Ex-OR nodes require one and only one of the children whereas ‘AND’ nodes require all of their children (commonality) (Huang et al., 2005). Also, there can be added a notion of real and pseudo items (Lamothe et al., 2006). In Fig. 1, the concerned laptop family is shown to bifurcate to subassembly (‘AND’ and ‘real’) and co-subassembly (‘Ex-or’ and pseudo) nodes. The laptop A is realized by selecting ‘CD Drive’ whereas DVD Drive is opted to procure the laptop B. Both the laptops share the common subassembly comprising of ﬁve components and the unique is CD or DVD drive. 2.2. Pictorial representation of the laptop supply chain For the undertaken example, a directed graph is used to symbolize the supply chain wherein each node captures exactly one item of the GBOM and one stage of the supply chain. In Fig. 2, the ﬂow of material from different stages of the supply chain is indicated through directed arcs and nodes are categorized in three kinds viz. procurement, assembly or demand stage (Huang et al., 2005). In case of platform supply chain (PSC), there is a node called differentiation point where common components are assembled with differentiating components. Resolving the supply chain about this point to design separate supply chains for laptops A and B leads to Independent supply chain designs ISCA and ISCB, respectively (Fig. 3). There are two decision variables, option selection and service time. The ﬁrst decision deals with selecting alternative raw material vendors, for procurement stages, alternative manufacturing/ assembling methods, for assembly stages, and to alternative endproduct delivery mode, for demand stages. The second decision variable viz. service time means the time taken by a stage to ﬁll an order after the order is received.

2. Platform product development and the supply chain: an example

3. Mathematical formulations

The product family must possess adequate product variety to address functional, physical or process demand diversity (Lamothe et al., 2006). This paper attends physical product diversity for product development and focuses on the determination of an optimal

A multi-period model formulated hereby possesses scope of studying of PSC, ISCs and other evaluation means such as AOH inventory, WIP inventory, time to market etc. The necessary assumptions made are as under (1) immediate processing of any

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Miscellaneous Component

Part w/8 Week LT

LCD Display

Part w/4 Week LT

Circuit Assembly

Subassem bly

Part w/2 Week LT

Metal Housing Laptop Family

Parts on Consignmen

Battery Pseudo Item Real Item

DVD Drive

AND Node Co-Sub assembly

Ex-OR CD Drive

Fig. 1. A GBOM representation of laptop product family.

1

2

5

Part w/4 Week LT

6

LCD Display

3

Part w/2 Week LT

7

Miscellane ous

4

Parts on consign

8

Metal Housing

9

11

Circuit Board

Part w/8 Week LT

CD Drive

13

Laptop A Assembly

15

US Laptop A

16

EUROPE Laptop A

17

US Laptop B

Subassem bly

10

14

Laptop B Assembly

Battery 12

CD Drive

Procurement stages (P)

Assembly stages (A) Demand stages (D) Fig. 2. Supply chain for laptop family manufactured through PSC.

arriving order; (2) small values of average back orders; (3) restricted capacity at each stage; (4) number of options, production cost and processing times vary over different time periods; (5) demand distribution of end-products varies with the time period but remain stationary over a given time period. The model is similar to that presented by Huang et al. (2005) but a signiﬁcant extension of their model is realized by generalizing the model through a multiperiod transformation along with added assumptions 3–5 to impart a realistic outlook. The total supply chain cost is given by

exists an indirect or dependent demand for stages ‘P’ and ‘A’ derived from the demands of end product using BOM requirement of items nij (number of items ﬁnished at stage ‘i’ required to produce on unit of item ‘j’). Let D(i) be the set of demand stages that require the items ﬁnished at stage ‘i’. Then, for i 2 P [ A the dependent demand is given by

Mit ¼

X d2DðiÞ

nid Mdt

and S2it ¼

X d2DðiÞ

n2id S2id þ

X X

ð1Þ

SCC Cost ¼ Inventory Cost þ Production Cost þ Procurement Cost þ Transportation Cost: The supply chain being a multi-stage system denotes the set of all the stages by ‘N’. The three kinds of stages are connoted by ‘P’, ‘A’ and ‘D’ representing, respectively, the sets of procurement, assembly and demand stages and are capable of holding AOH and WIP inventory as well. During a time period ‘t’, the demand dit follows a normal distribution with mean Mit and the standard deviation Sit and correlation between customers’ demand given by the correlation coefﬁcient Rijt (where i, j 2 D, i – j). For each stage, there also

nid nie Sdt Set Rdet

de 2DðiÞ d2DðiÞ e–d

Earlier discussion in the paper implies that a stage ‘i’ (i 2 P [ A) guarantees a service time sit to its immediate successors during period ‘t’ (t 2 T). Also, a stage cannot start its operation until each of the orders to its entire immediate predecessor stages are ﬁlled determining the input service time sv(i)t of a stage ‘i’. For i 2 P, the input service time is nil for there are no predecessor nodes and for i 2 P [ A, sv(i)t is given by the greatest of the service times of its predecessors

sv ðiÞt ¼ maxfsjt g for i 2 P [ A; j 2 v ðiÞ; t 2 T:

ð2Þ

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1

Part w/8 Week LT

2

Part w/8 Week LT

Circuit Board Assembly

5

CD Drive

11

LCD Display

6

3

Part w/8 Week LT

7

4

Part w/8 Week LT

8

Miscellaneou s C Metal Housing

13

Laptop A Assembly

Subassembly

10

15

US Laptop A

16

EUROPE Laptop A

Procurement stages (P) Assembly stages (A)

Battery

9

Demand stages (D)

(a): ISCA

1

Part w/8 Week LT

2

Part w/8 Week LT

5

Circuit Board Assembly

Procurement stages (P) Assembly stages (A)

6

3

Part w/8 Week LT

7

4

Part w/8 Week LT

8 9

LCD Display Demand stages (D) Miscellaneou s C Metal Housing

Subassembly

10

Laptop B Assembly

14

17

US Laptop B

Battery CD Drive

12

(b): ISCB Fig. 3. Supply chain for laptop family manufactured through Independent development of products.

The second decision variable viz. option selection determines the supplier for i 2 P, manufacturing/assembly process for i 2 A or transportation modes selected for i 2 D during period ‘t’. Meanwhile, it yields the corresponding cost (cit) incurred and processing lead time (Cit) at a stage ‘i’ accomplished by equating Cit to Citoit and cit to citoit where oit (oit 2 wit; wit being the set of options for stage ‘i’ during time period ‘t’) is the option selected for stage ‘i’. For deriving the inventory costs terms, the notions of replenishment lead times Lit and inventory coverage period Uit are utilized where former is the time span over which the stock point ‘i’ can be reﬁlled comprising cumulatively of waiting time for inputs and processing time of the stage ‘i’. Mathematically, it is given by

Lit ¼ sv ðiÞt ¼ Cit

8i 2 N:

ð3Þ

The inventory coverage time is the time span during which the average on hand inventory (AOHI) should cover and guarantee a demand following a reviewed base stock policy with review period (V). So, the inventory coverage cost Uit and base stock level Bit is expressed (Piramuthu, 2005) as

U it ¼

Lit sit

8i 2 P [ A 8i 2 D

Lit sit þ V

pﬃﬃﬃﬃﬃﬃ and Bit ¼ M it U it þ fi Sit U it

pﬃﬃﬃﬃﬃﬃ 1 VMit þ fi Sit U it 8i 2 D and 2 pﬃﬃﬃﬃﬃﬃ AOHIit ¼ fi Sit U it 8i 2 P [ A

AOHIit ¼

WIP inventory is calculated using mean dependent demand for the stage ‘i’ items and the corresponding processing time as

WIPIit ¼ Mit Cit

fi being the predetermined service factor for capturing service level for stage ‘i’. For demand stages, AOH inventory level can be determined as

8i 2 N

ð6Þ C AOH it

In maintaining the AOH inventory and WIP inventory, the costs and C WIP associated with the items ﬁnished or in process, respecit tively and stored at stage ‘i’ during a time period ‘t’ which are estimated as

C AOH ¼ cit it

8i 2 P;

cit þ ,

C WIP it

¼ cit

8i 2 P;

cit

X

C jt

8i 2 A [ D and

j2v ðiÞ

2þ

X

C jt

8i 2 A [ D

ð7Þ

j2v ðiÞ

Thus, the total supply chain conﬁguration cost can be evaluated as

SCC cos t ¼ ð4Þ

ð5Þ

X X rate AOH time ½hit fC it AOHIit þ C WIP cit M it it WIPIit g þ Ht i2N

t2T

ð8Þ Using this and the above-mentioned derivations the optimization model can be elaborately formulated as a combinatorial cum integer problem

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2 Minimize SCC cos t ¼

P

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rate hit cit fi Sit it it

P

( rate hit

!

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ fi Sit Cit þ sv ðiÞt sit þ

P

6 C s þ Mit Cit þ cit þ C jt X6 i2A j2v ðiÞ 6 i2P ( ! 6 6 P pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ P t2T 4 rate þ hit cit þ C jt VM2 it þ fi sit Cit þ sv ðiÞt sit þ V þ i2D

j2v ðiÞ

cit 2

P

þ

! C jt Mit Cit

j2v ðiÞ

cit 2

) þ

P

þ

! C jt Mit Cit

j2v ðiÞ

P i2N

Htime cit M it t

)3 7 7 7 7 7 5 ð9Þ

X

xitoit oit ¼ 1 8i 2 N; t 2 T

ð10Þ

Citoit xitoit ¼ 0 8i 2 N; t 2 T

ð11Þ

oit 2/it

X oit 2/it

X

C itoit xitoit cit ¼ 0 8i 2 N; t 2 T

ð12Þ

Table 1 Three problem instances dealt. Problem no.

Size

Time variation in demands

Correlation b/w customers’ demand

1 2 3

Small Intermediate Large

Not addressed Addressed Addressed

Not accounted Not accounted Accounted

oit 2/it

Cit sit P 0 8i 2 P; t 2 T

ð13Þ

sv ðiÞt þ Cit sit þ V P 0 8i 2 A [ D; t 2 T

ð14Þ

VMit =2 þ fi Sit

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Cit þ sv ðiÞt sit þ V þ Mit Cit þ Htime M it 6 Q i t

8i 2 D; t 2 T qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ fi Sit Cit þ sv ðiÞt þ V þ M it Cit þ Htime Mit 6 Q i t

ð15Þ 4.1. Algorithms deployed

8i 2 P [ A; t 2 T ð16Þ

xitoit ¼

1; if oit is selected 0;

otherwise

algorithm portfolio to enhance the decision ﬂexibility of the system and simultaneously to maintain a repository of elite portfolios for future investigations by availing multiple preferable alternatives and assessment tool for prospective portfolios in terms of their applicability and efﬁciency to this class of problem.

ð17Þ

Eq. (9) displays the objective function comprising mainly of four terms. The ﬁrst, second and third terms are the sum AOH and WIP inventory costs for three kinds of stages, respectively. The fourth term represents total production cost which includes the procurement cost, manufacturing/assembling cost, transportation cost as per the stages. Constraints (10) agree that any stage can opt one and only one option. Constraints (11) and (12) stipulate that the processing time and operations cost, respectively of a stage ‘i’ are in accordance with the options available for that stage. Constraints (13) and (14) deﬁned, respectively for procurement and assemblydemand stages; ensure that the inventory coverage time for a procurement stage is non-negative. Constraints (15) and (16) restrict the total capacity of any stage ‘i’ by the total capacity of the stage Qi. Constraints (17) draws the consensus that xitou is a binary decision variable for selecting as option for any stage ‘i’. 4. Designing the algorithm portfolio Addressing the performance variations and risk of various algorithms is a cumbersome job (Gomes & Selman, 1997) prominently for computationally hard problems. Moreover, the ‘‘winner takes all” strategy for the algorithm selection for a problem cannot maintain an efﬁcient performance over different problem instances (Rice, 1997) of a supply chain problem. Motivated by these adversities, this paper enquires for a viable and insightful perspective to an optimization problem via algorithm portfolios (Gomes & Selman, 2001). To investigate the portfolio approach for the model discussed in the paper, few conceptual and enumerative assumptions are varied to generate different problem instances (Table 1) over which the algorithm portfolios are tested. An ‘efﬁcient frontier’ can be deﬁned using

4.1.1. Representation schema Four nature inspired algorithms viz. genetic algorithm (GA), artiﬁcial immune system (AIS), particle swarm optimization (PSO), Psychoclonal algorithm and endosymbiotic optimization (ESO) are explored hereby for their performance. The representation schema for all these algorithms is same except the endosymbiotic (ESO). Integer encoding of a random solution requires a total of 2jTjjNj bits with 2jNj bits for each of jTj periods. For each pair of bits, ﬁrst one represents the option selected Oit and the second one captures the service time sit. A period wise sequential encoding is followed with a strict sequence of different stages to follow input–output and service time relationship decisions between stages. This paper follows a sequence indicated in Fig. 2. The constraints handling strategy in all of the algorithms is taken same as in (Piramuthu, 2005) except the Psychoclonal algorithm. In all the algorithms except ESO a population P the encoded solutions is initialized representing the speciﬁc entity of the algorithm. 4.1.2. Genetic algorithm (GA) The genetic algorithm used in this paper is pretty similar to that deployed by Piramuthu (2005). However, the differences lie in the utilization of Roulette wheel selection and elitistic approach in survival decisions, for the offsprings, adopted hereby. Each of the chromosomes (random solutions) is subjected to genetic operators viz. crossover and mutation (one cut point crossover and bit alter mutation). The ﬁtness function Fit is evaluated as, where k is a constant equal to 109 in this paper

FIT ¼ k=SCC cost

ð18Þ

4.1.3. Artiﬁcial immune system AIS, though, emerged recently has well established itself as a potential optimization algorithm backed by superior results it has yielded (de Castro & Timmis, 2002). It incorporates clonal selection and evolutionary principles altogether with a fair trade off between exploration and exploitation.

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4.1.4. Particle swarm optimization (PSO) PSO was initially developed as an optimization algorithm has great admirations form various areas such as single and multi objective optimization (Parsopoulos & Vrahatis, 2002). PSO is regarded as a behavioral algorithm based on swarm intelligence methods. The search space is explored by guiding a particle to its best position as well as to the globally best position found so far. 4.1.5. Psychoclonal algorithm The concept of AIS can be extended by incorporating Maslow’s need hierarchy theory in order to impart robustness for constrained optimization and ability to escape local optima (Tiwari, Prakash, Akhilesh, & Mileham, 2005). That is, evolution of antibodies is realized through fulﬁllment of ﬁve hierarchical needs viz. physiological, safety, social, growth and self actualization needs. Computationally, these needs, respectively, correspond to objective function evaluation, stochastic constraint satisfaction through penalty, cloning, hypermutation and recognition as best individuals.

of function evaluation required to meet a prespeciﬁed performance in this paper). 4.2.1. Statistical evaluation Let the set L = {L1, L2, . . . , Ln} encompasses all the algorithms entrapped in a portfolio with O processors algorithm and Li2L run over Lip processors so that O ¼ fL1p ; L2p ; . . . ; Lnp g. Also, Let R be the combined probability distribution for all Lis 2 L and P (Ri = r) be the probability that the algorithm Li requires rth outcome to meet the prespeciﬁed objective performance (Trivedi, 2003). Then, the experiments can be regarded as sequence of Bernoulli trials counting number of trials until ﬁrst success (attainment of prespeciﬁed performance) rather than counting number of success in ﬁxed number of trials. The sample space SS of the experiments (success and failure connoted, respectively by 1 and 0) can be represented by the set of all binary strings containing r number of 0s succeeded by a single 1 (r being an arbitrary number). That is

SS ¼ f0r1 1 : r ¼ 1; 2; 3; . . .g

ð19Þ r1

4.1.6. Endosymbiotic optimization (ESO) A symbiotic evolutionary algorithm imitates the biological coevolution phenomenon which is based on the serial reciprocal changes in two or more cooperative interacting species (Potter, 1997) solve problems containing interrelated sub-problems (Kim, Kim, & Kim, 2000). Hereby, an Endosymbiotic evolutionary algorithm Kim, Kim, and Kim (in press) is adopted which maintains two populations P–o and P–t (partial solutions) by encoding the option selection and service time variables, respectively along with a population P–ot of different endosymbionts (integration of individuals from the two populations) is maintained as well.

Supposing that the sample point 0 1 be assigned a value for the random variable Ri, the probability mass function (pmf) of Ri can be obtained for the event P[Ri] = r iff there exist a sequence of r 1 failure followed by one success, the probability of each success being p, as follows:

pRi ðrÞ ¼ pð1 pÞr

r ¼ 1; 2; 3; . . . ; e

ð20Þ

Therefore, probability distribution function (pdf) of Ri can be estimated as

Q Ri ðtÞ ¼

t X

pð1 pÞr ¼ p½1 ð1 pÞt =1 ð1 pÞ ¼ 1 ð1 pÞr

r¼1

4.2. Portfolio evaluation and performance measures

ð21Þ

The algorithm portfolios are meant hereby to derive computational proﬁt and quality maintenance to undertaken problem. So, they are evaluated by the risk (measured by the standard deviation) associated with them. For this purpose, a random variable (computational cost) and standard deviation (solution quality) are utilized. Let R = {R1, R2, . . . , Rn} with R1, R2, . . . , Rn as n corresponding random variables for n algorithms is so deﬁned that each of Ri belongs to the set {1, 2, 3, . . . , e}, set capturing different possibilities of expected outcome of ith trial, (considered the number

If for two algorithm Li and Lj running on different processor and the corresponding variables be Ri and Rj so that the random variable R indicating that at least one algorithm has met the prespeciﬁed success level, then

R ¼ minfRi ; Rj g

ð22Þ

Proposition 4.1. For all geometrically distributed and independent variable R is also geometrically distributed.

100 90 80

cumulative frequency

70 60 AIS ESO GA PSO Psycho

50 40 30 20 10 0

0

1000

2000

3000

4000

5000

6000

No. of fitness evaluation Fig. 4. Results with 5 algorithms 1 processor system for problem 1.

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Cumulative frequency

80 70 60 AIS ESO GA PSO Psycho

50 40 30 20 10 0

0

2000

4000

6000

8000

10000

12000

14000

No. of fitness evaluation Fig. 5. Results with 5 algorithms 1 processor system for problem 2.

100

90

80

Cumlulative frequncy

70

60

50 40

AIS ESO GA PSO Pyscho

30

20

10

0

0

0. 5

1

1. 5

2

2.5 x 10 4

No. of fitness evalulation Fig. 6. Results with 5 algorithms 1 processor system for problem 3.

Similarly

Proof

PðR > tÞ ¼ PðRi > t

and Rj > tÞ ¼ PðRi > t

and Rj > tÞ

ð23Þ

Q Rj ðtÞ ¼ 1 ð1 pÞr

ð28Þ

In terms of pdf

PðR > tÞ ¼ 1 Q R ðtÞ

ð24Þ

¼ ½1 Q Ri ðtÞ½1 Q Rj ðtÞ

ð25Þ

Table 2 Mean and variance of results form 5A1P. Algorithm

) Q R ðtÞ ¼ Q Ri ðtÞ þ Q Rj ðtÞ Q Ri ðtÞQ Rj ðtÞ ðfrom equation ð20ÞÞ ð26Þ Q Ri ðtÞ ¼

t X r¼0

pð1 pÞr ¼ 1 ð1 pÞr

ð27Þ

AIS ESO GA PSO Psycho

Problem 1

Problem 2

Problem 3

Mean

Variance

Mean

Variance

Mean

Variance

2621 2279 3117 2770 2763

17,620 19,981 17,620 17,400 17,400

4855 4881 5034 663 9 5210

96,330 80,073 94,696 80,634 89,080

7685 4881 8553 9824 7617

22,883 80,075 22,956 21,765 20,219

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Table 3 Best solution for ﬁrst problem instance. Items/stages in supply chain

PSC

ISC Laptop A

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Laptop B

Option selected

Service time

Option selected

Service time

Option selected

Service time

1 2 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2

10 10 0 0 0 0 0 0 0 5 5 5 6 6 0 0 0

1 3 2 1 1 1 1 1 1 1 1 – 1 – 2 2 –

0 0 0 0 0 0 0 0 0 5 5 – 6 – 0 0 –

3 2 1 1 1 1 1 1 1 1 – 1 – 1 – – 2

10 10 10 0 30 30 30 30 30 35 – 35 – 36 – – 0

Total SCC cost ($)

270,016,507

186,436,361

84,006,915 270,443,275

Total AOH inventory cost ($)

2,561,832

1,912,892

979,006 2,891,897

Total WIP inventory cost ($)

12,619,675

8,334,430

3,719,458 12,053,888

Total production cost ($)

254,835,000

176,189,040

79,308,450 255,497,490

Now, using Eq. (26), we have r

r

2r

Q R ðtÞ ¼ 2½1 ð1 pÞ ½1 2ð1 pÞ þ ð1 pÞ

ð29Þ

¼ 1 ð1 pÞ2ðrÞ ¼ ½1 ð1 pÞ2 r

ð30Þ

Hence, R is too geometrically distributed with the parameter 1 [(1 p)2]. h Proposition 4.2. The random variable R associated with the portfolio of ‘e’ processor is geometrically distributed with parameter 1 (1 p)e Proof. The Proposition 4.3 can be extended as

Q R ðrÞ ¼

e Y

½1 Q Ri ðtÞ ¼

i¼1

e e Y Y ½1 ð1 ð1 pÞt Þ ¼ ½ð1 pÞt i¼1

i¼1

ð31Þ Hence, the random variable R is geometrically distributed with the parameter 1 (1 p)e. h

Table 4 Various algorithm cases explored. No. of algorithms

Cases

2 Processor

3 Processor

4 Processor

2

1 2

GA–AIS ESO–Psycho

GA–AIS ESO–Psycho

3 4

GA–AIS ESO– Psycho ESO–GA AIS–Psycho

ESO–GA AIS–Psycho

ESO–GA AIS–Psycho

3

1

–

GA–ESO–Psycho

4

2 3 4 1

– – – –

GA–ESO– Psycho ESO–AIS–GA Psycho–AIS AIS–GA–Psycho –

ESO–AIS–GA Psycho–ESO–AIS AIS–GA–Psycho GA–AIS–ESO– Psycho

Proposition 4.3. The number of function evaluation for a search to end with desired quality solution has always more probability when used with ‘e’ processors than with single processors Table 5 Different portfolio cases considered. Cases

3P–3A

4P–2A

4P–3A

4P–4A

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

3/0/0 0/3/0 0/0/3 1/1/1 2/1/0 2/0/1 0/2/1 1/2/0 0/1/2 1/0/2

4/0 3/1 2/2 1/3 0/4

4/0/0 0/4/0 0/0/4 3/1/0 3/0/1 0/3/1 1/3/0 0/1/3 1/0/3 2/2/0 2/0/2 0/2/2 1/1/2 1/2/1 2/1/1

4/0/0/0 0/4/0/0 0/0/4/0 0/0/0/4 3/1//0/0 3/0/1/0 3/0/0/1 0/3/1/0 0/3/0/1 1/3/0/0 1/0/3/0 0/1/3/0 1/0/0/3 0/1/0/3 0/0/1/3 2/2/0/0 2/0/2/0 2/0/2/0 2/0/0/2 0/2/2/0 0/0/3/1 0/0/2/2 2/1/1/0 2/1/0/1 0/2/1/1 1/2/0/1 1/2/1/0 0/1/2/1 1/0/2/1 1/1/2/0 1/1/0/2 1/0/1/2 1/0/2/1 0/1/1/2 4/0/0/0

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i:e:; Q R ðtÞ > Q Ri ðtÞ

" # e1 Y t t f ¼ ð1 pÞ ð1 pÞ 1

ð32Þ

ð35Þ

i¼1

Proof. Let

Q t Again, since ð1 pÞt < e1 i¼1 ð1 pÞ , So, it can be concluded that Qe1 t ð1 pÞ < 1. i¼1 Therefore, the second term of the above product is negative. For any portfolio e > 1, hence the above-mentioned implies that a portfolio always works faster than the cases with the single algorithm. i.e. Q R ðtÞ > Q Ri ðtÞ. In case of single algorithm with geometrically distributed random variable Ri and pmf PRi ðrÞ ¼ pð1 pÞr r ¼ 1; 2; 3; . . . ; e, the expectation is given as

f ¼ Q Ri ðtÞ Q R ðtÞ ð33Þ ( ) e e Y Y ¼ 1 ½ð1 pÞt ¼ f1 ð1 pÞt g ¼ ð1 pÞt ð1 pÞt i¼1

i¼1

ð34Þ since p is the probability of success of a single trial, so p < 1 and t, e > 0. So, f can be written as

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D½Ri ¼

1 X

rpð1 pÞr1 ¼ p

r¼1

¼

p ð1 mÞ

2

¼

1 p

1 X dðmr Þ dm r¼1

m ¼ ð1 pÞ

From (39), it is clear that if probability of success and failure at any stage be constant (in case of random stochastic algorithms), the variance can be considered constant. From dianoetic application of Chebyshev’s inequality Trivedi (2003) the proposition gets proven.

ð36Þ ð37Þ

Similarly, for the case with e algorithms

1 D½Ri ¼ 1 ð1 pÞe

4.2.2. Experimental setup The experiments lay emphasis mainly on evaluating performance of the algorithms and the portfolios, none the less the superiority of platform supply chain (PSC) over independent supply chain (ISC) is established through an experiment in which cost optimization is performed for both the PSC and ISC (ISCA plus ISCB) and compared. The relative performance of all the ﬁve algorithms

ð38Þ

Proceeding similarly, we get the variance var as

var½Ri ¼ ð1 pÞ=p2

and var½R ¼

1 ½1 ð1 pÞe

ð39Þ

½1 ð1 pÞe 2

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are tested for using PSC strategy for all of the three problem instances and running them on a single processor for 100 runs, shown through Figs. 4–6, and Table 2. These ﬁgures show the plots of cumulative frequency versus the number of ﬁtness evaluation while Table 2 indicates the mean and variance of number of ﬁtness evaluations in each case. The stopping criterion for each experiment is speciﬁed as the attainment of the optimum solution within 0.05% of the best solution found out by rigorous preliminary experiments with all the algorithms. For instance, the best solution for the ﬁrst problem instance is shown in Table 3. As evident from the Table 2, the variance of algorithm varies over a very short interval which advocates the proposition proved in Equation 69. It can be inferred from the Figs. 4–6, and the Table 2 that most of the algorithms perform competitively. However, PSO could not perform well enough particularly in case of problem in-

stances 2 and 3. This might be due to local entrapment because of no variation of acceleration coefﬁcients with generations (Ratnaweera & Halgamuge, 2004). The integer coding can be replaced by real coding leading to further analysis of PSO over the problem and hence a future scope of research. Due to a relatively poor performance of PSO, rests of the experiments are carried out via four algorithms only. The experiments are performed on 2–4 processors systems for designed portfolios of 2–4 algorithms with various combinations shown in Tables 4 and 5. Utilizing the algorithmic combinations shown in Table 4, each algorithm is evaluated its performance for different processor-combinations. A ‘/’ symbol is used to denote these combinations, e.g., 2/0 means that, for the 2 processor 2 algorithm system, the ﬁrst of the selected algorithm is run over both of the processor while the second is run over none.

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5. Results and discussions: graphical and statistical insights The data for the ﬁrst problem instance is taken from Huang et al. (2005) and that for the rest of two are appropriately simulated which is available on the URL www.geocities.com/salik_nifft/ data_portfolio. This section presents results observed as the costs implications in the supply chain followed by themes and inferences drawn for the portfolios. 5.1. Supply chain cost implications As evident from Table 3, the supply chain cost, in case the manufacturing ﬁrm adopts PSC, is lesser than the cost in ISC by 0.16%. It is observed that PSC reduces AOH inventory cost by much more amount than any other cost. So, PSC can be proposed to be useful enough where AOH inventory holding is a crucial concern. The production cost too, in case of PSC, is found 0.26% less than that in case

Algorithm

GA-AISESOPsycho

Algorithm

GA- ESO Psycho

ESO -AIS-GA

AIS-GA-Psycho

4Processor 4Algorithm Portfolio Priority vector 4/0/0/0 0.01344 0/4/0/0 0.01704 0/0/4/0 0.00986 0/0/0/4 0.03479 3/1//0/0 0.07407 3/0/1/0 0.01402 3/0/0/1 0.01133 0/3/1/0 0.03440 0/3/0/1 0.04237 1/3/0/0 0.01118 1/0/3/0 0.00852 0/1/3/0 0.01416 0/0/3/1 0.03221 1/0/0/3 0.12906 0/1/0/3 0.29659 0/0/1/3 0.04293 2/2/0/0 0.01709 2/0/2/0 0.01092 2/0/0/2 0.04481 0/2/2/0 0.02201 0/2/0/2 0.02223 0/0/2/2 0.01943 2/1/1/0 0.03084 2/1/0/1 0.01002 0/2/1/1 0.09809 1/2/0/1 0.02221 1/2/1/0 0.02802 0/1/2/1 0.00846 1/0/2/1 0.02039 1/1/2/0 0.01753 1/1/0/2 0.02579 1/0/1/2 0.05134 1/0/2/ 0.01784 0/1/1/2 0.01115 1/1/1/1 0.01113

3Processor 3Algorithm Portfolio Priority vector 3/0/0 0.0208 0/3/0 0.0283 0/0/3 0.1076 1/1/1 0.0314 2/1/0 0.0576 2/0/1 0.0940 0/2/1 0.0180 1/2/0 0.0322 0/1/2 0.0266 1/0/2 0.0147 3/0/0 0.0373 0/3/0 0.0148 0/0/3 0.0307 1/1/1 0.0221 2/1/0 0.0342 2/0/1 0.0269 0/2/1 0.0162 1/2/0 0.0200 0/1/2 0.0318 1/0/2 0.0385 3/0/0 0.0156 0/3/0 0.0178 0/0/3 0.0241 1/1/1 0.0531 2/1/0 0.0314 2/0/1 0.0175 0/2/1 0.0372 1/2/0 0.0316 0/1/2 0.0311 1/0/2 0.0370

Rank 26 23 31 9 4 22 22 8 7 20 24 19 7 2 1 4 14 16 3 8 6 8 3 11 1 4 2 8 3 4 2 1 1 1 1

Rank 22 17 1 14 3 2 24 10 19 30 6 29 16 21 9 18 27 23 11 5 28 25 20 4 13 26 7 12 15 8

of ISC. However, the inventory cost incurred in for holding workin-process is less for ISC than PSC; thus indicating one of the pitfall of PSC. Similar data interpretations can be made for other two problem instances. 5.2. Experimental runs The whole experiment is branched into three parts based on the number of processors utilized. Each portfolio is executed 100 runs and the corresponding average number of ﬁtness evaluation is obtained. The experimental details of different portfolios corresponding to various algorithmic and processor cases utilized are available on www.geocities.com/salik_nifft/data_portfolio for each of 2 processor, 3 processor or 4 processor systems. The results obtained thereby for each of the three problem instances are shown by Figs. 7–12, respectively. In each of these ﬁgures, ordinates represents

Algorithm selected GA-AIS

ESO -Psycho

ESO -GA

AIS-PSYCHO

4Processor 2Algorithm Portfolio Priority vector 4/0 0.0679 3/1 0.0411 2/2 0.0601 1/3 0.0290 0/4 0.0511 4/0 0.0511 3/1 0.0381 2/2 0.0330 1/3 0.0445 0/4 0.0413 4/0 0.0606 3/1 0.0312 2/2 0.0320 1/3 0.0428 0/4 0.0703 4/0 0.0652 3/1 0.0311 2/2 0.1414 1/3 0.0351 0/4 0.0332

Rank 3 12 6 20 7 8 13 16 9 11 5 18 17 10 2 4 19 1 14 15

Rank1 Rank2 Rank3

2Processor 2Algorithm Algorithm selected Portfolio Priority vector 2/0 0.0714 GA-AIS 1/1 0.0773 0/2 0.0665 2/0 0.1950 ESO -Psycho 1/1 0.0494 0/2 0.0633 2/0 0.0484 ESO -GA 1/1 0.1330 0/2 0.0503 2/0 0.0647 AIS-PSYCHO 1/1 0.0681 0/2 0.1127

Algorithm GA-AIS

ESO -Psycho

ESO -GA

AIS-Psycho

3Processor 2Algorithm Portfolio Priority 3/0 0.0461 2/1 0.0564 1/2 0.0464 0/3 0.0584 3/0 0.0384 2/1 0.0620 1/2 0.0574 0/3 0.0649 3/0 0.1601 2/1 0.0342 1/2 0.0530 0/3 0.0686 3/0 0.0345 2/1 0.0443 1/2 0.1282 0/3 0.0471

Rank 5 4 7 1 11 9 12 2 10 8 6 3

Rank 12 8 11 6 14 5 7 4 1 16 9 3 15 13 2 10

Fig. 13. Priority and rank vectors for different portfolios with 2 processor, 3 processor and 4 processor systems.

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the no. of ﬁtness evaluation and the abscissas indicates the portfolio cases; three rows corresponds to three problem instances.

13

complexities, thus, enabling to plan the selection of an alternative within the efﬁcient frontier, imparting ﬂexibility to decision making process.

5.3. Observing the results 6. Conclusions and future research 5.3.1. Searching the best portfolio Owing to variation in performance of different portfolios, selecting best portfolios with minimum risk seems dilemmatic at ﬁrst glance has to be an effective procedure. This paper harnesses the analytical hierarchy process (AHP) as decision making tool for selecting best portfolio. It involves a hierarchical pairwise comparison to induce weights of alternatives. Employing AHP forces one to recognize each of the portfolio as an alternative for a particular processor system to which portfolio belongs to and the results over three different problem instances are considered as8of different attributes. For this purpose, a matrix 9 < a11 a12 a13 = , the columns of which connotes Z is built as Z ¼ ; : the attribute (3 problem instances) and the rows indicating the alternative or portfolio explored. The elements aij are the normalized objective value for the experiments characterized by set (i, P j), i.e, aij ¼ F ij j F ij (Fij being the objective which is made the basis for measuring the solution quality and corresponds hereby to the average number of ﬁtness evaluations involved by the particular test alternative). Thereafter, a priority matrix qiuxu for each of the attributes ‘i’, where ‘u’ is the number of alternatives for a particular experiment is evaluated as qijxk ¼ aij =aik . This is followed by the calculation of the associated weight vectors W i1xu for each attribute i by taking geometricq mean for the rows corresponding to matrix qi, ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Qu i i u i i.e., W 1xw ¼ fwj g ¼ k qjk . Now, the weight vectors are normalized to yield normalized priority vectors PVs computed as .P u i PV ij ¼ wij j¼1 wj . Finally, a relative rank vector RV is obtained as the weighted sum of the three priority vectors. Hereby, the priority weights are assumed as 1/3. Fig. 13 reveals the priority vectors and the ranks of different portfolios for each system. For instance, there are 12 possible alternatives in case of 2 processor system (4 and 3 algorithmic and processor combinations, respectively). Similarly, in case of 3 and 4 processor system there are 46 and 55 alternatives, respectively. As indicated from the ﬁgure, the analysis of different portfolios via AHP can yield a set of best alternatives, in case they exist. For instance, in case of two processor system, the portfolio with algorithm combination ESO–Psycho and processor combination 3/0 is the best. Likewise, for 3 processor system, two portfolios come out to be best viz. GA–ESO–Psycho with characteristic 0/ 0/3; and there are 6 best cases, 5 with four algorithm and 1 with 2 algorithm, for 4 processor system. These best portfolios found so far are the representatives of the efﬁcient frontier discussed in previous section. It can be observed from the results that the algorithms with multiple population and neighborhood search strategy (ESO) perform much better than others attributed to its neighborhood strategy capable of exploring larger search space. This can be attributed to the complex nature of combinatorial cum integer supply chain problems where critical exploitation and exploration of the random solutions are required. Moreover, Psychoclonal – an extension of AIS – is much more viable than others under considerations. This feature of Psychoclonal can be accredited to the introduction of better constraint handling and exploration mechanism. More or less, it has also been observed that considering the parallel runs of the same algorithm emerges out to be a better option than to use incoherent combination of various algorithms. The best portfolios found above can be regarded best strategies as they are accompanied by minimum risk, good performance and better capability to tackle the dimensional

This paper introduces the algorithm portfolio concept to obtain an optimal supply chain conﬁguration for a ﬁrm producing a platform based product. An extended optimization model is harnessed for this purpose making vendors, manufacturing process and product delivery mode selections at various stages of the supply chain. The problem is resolved through a platform and an independent supply chain (PSC and ISC, respectively) approaches where the lower cost is incurred in the PSC. As supply chain designer and decision makers have to make decisions within stipulated time limit, the reliability and promptness of the solution strategy are substantial issues. Algorithm portfolios are employed to induce promptness as well as reliability by juxtaposing several algorithms with superlative performances. In order to address the urge to select a best portfolio, analytical hierarchy process employing sophisticated quantitative methods is employed to select the best portfolios to generate efﬁcient frontier. Though the model adopted hereby encompasses various decisions as well as determination aspects, it can be made more comprehensive by introducing customization concepts, effect of excise rates and taxes, supply chain cycle time and so on. One of the important extensions of the problem is the incorporation of component allocation along with the option selection and service time decisions in order to impart the supply chain problem a more sophisticated look. The portfolio developed here can be tested over more complex supply chain models. References Akcay, Y., & Xu, S. H. (2004). Joint inventory replenishment and component allocation optimization in an assemble-to-order system. Management Science, 50(1), 99–116. Chan, F. T. S., Chung, S. H., & Chan, P. L. Y. (2005). An adaptive genetic algorithm with dominated genes for distributed scheduling problems. Expert Systems with Applications, 29, 364–371. Chong, J. K., Ho, T. H., & Tang, C. S. (1998). Product structure, brand width and brand share, product variety management. Research advances. USA: Kluwer Academic Publisher. Dean, T., & Boddy, M. (1988). An analysis of time-dependent planning. In Proceedings of the seventh conference on artiﬁcial intelligence (AAAI-88) (pp. 49– 54). St. Paul. de Castro, L. N., & Timmis, J. I. (2002). Artiﬁcial immune systems: A new computational intelligence approach. London: Springer-Verlag. Ganeshan, R., Jack, E., Magazine, M. J., & Stephens, P. (1999). A taxonomic review of supply chain management research. Quantitative models for SCM. Boston: Kluwer Academic. Gomes, C. P., & Selman, B. (1997). Algorithm portfolio design: Theory vs. practice. Proceedings of thirteenth conference on uncertainly in artiﬁcial intelligence (UAI97). Morgan Kaufmann Publishers. Gomes, C. P., & Selman, B. (2001). Algorithm portfolios. Artiﬁcial Intelligence, 126(1– 2), 4362. Graves, S.C., & Willems, S.P., (2005). Optimizing the supply chain conﬁguration for new products, Working paper, Leaders for management program and A.P. Sloan school of management: MIT. Hegge, H. M. H., & Wortmann, J. C. (1991). Generic bill of materials: A new product model. International Journal of Production Economics, 23, 117–128. He, D. W., & Kusiak, A. (1996). Performance analysis of modular products. International Journal of Product Research, 34(1), 253–272. Huang, G. Q., Zhang, X. Y., & Liang, L. (2005). Towards integrated optimal conﬁguration, manufacturing processes, and supply chains. Journal of Operations Management, 23, 267–290. Jiao, J. X., Tseng, M. M., Duffy, V. G., & Lin, F. H. (1998). Product family modeling for mass customization. Computers and Industrial Engineering, 35(34), 495–498. Kim, Y. K., Kim, J. Y., & Kim, Y. (in press). An endosymbiotic evolutionary algorithm for the integration of balancing and sequencing in mixed-model U-lines. European Journal of Operational Research. Kim, Y. K., Kim, S. J., & Kim, J. Y. (2000). Balancing and sequencing mixed model U lines with a coevolutionary algorithm. Production Planning and Control, 11(8), 754–764.

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An Improved Algorithm for the Solution of the ...