Worker assignment and production planning with learning and forgetting in manufacturing cells by hybrid bacteria foraging algorithm

Accepted Manuscript Worker assignment and production planning with learning and forgetting in manufacturing cells by hybrid bacteria foraging algorithm Chunfeng Liu, Jufeng Wang, Joseph Y.-T. Leung PII: DOI: Reference:

S0360-8352(16)30083-3 http://dx.doi.org/10.1016/j.cie.2016.03.020 CAIE 4293

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Computers & Industrial Engineering

Received Date: Revised Date: Accepted Date:

28 November 2015 19 March 2016 22 March 2016

Please cite this article as: Liu, C., Wang, J., Leung, J.Y., Worker assignment and production planning with learning and forgetting in manufacturing cells by hybrid bacteria foraging algorithm, Computers & Industrial Engineering (2016), doi: http://dx.doi.org/10.1016/j.cie.2016.03.020

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Worker assignment and production planning with learning and forgetting in manufacturing cells by hybrid bacteria foraging algorithm Chunfeng Liua , Jufeng Wangb , Joseph Y.-T. Leungc,d,∗ a

School of Management, Hangzhou Dianzi University, Hangzhou 310018, P. R. China Department of Mathematics, China Jiliang University, Hangzhou 310018, P. R. China c Department of Computer Science, New Jersey Institute of Technology, Newark, NJ 07012, USA d School of Management, Hefei University of Technology, Hefei 230009, P. R. China b

Acknowledgment This research was supported by the Zhejiang Provincial Natural Science Foundation of China (Grant No. LY14G020014), Humanities and Social Sciences Youth Foundation of the Ministry of Education (Grant No. 14YJC630089), China Scholarship Council, Zhejiang Provincial Key Research Base of Humanities and Social Sciences in Hangzhou Dianzi University(Grant No. ZD03-201501), and the Research Center of Information Technology & Economic and Social Development. The authors are grateful for the financial supports.

∗

Corresponding author. Email addresses: [email protected] (Chunfeng Liu), [email protected] (Jufeng Wang ), [email protected] (Joseph Y.-T. Leung ) Preprint submitted to Computers & Industrial Engineering

March 19, 2016

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Worker assignment and production planning with learning and forgetting in manufacturing cells by hybrid bacteria foraging algorithm

3

Abstract

1

We consider a joint decision model of worker assignment and production planning in a dynamic cellular manufacturing system of fiber connector manufacturing industry. On one hand, due to the learning and forgetting effects of workers, the production rate of each workstation will often change. Thus, the bottleneck workstation may transfer to another one in the next period. It is worthwhile to reassign multi-skilled workers such that the production rate of bottleneck workstation may increase. On the other hand, because of the limited production capacity and variety of orders, late delivery or production in advance often occurs at each period. The parts with operational sequence need to be dispatched to the desirable cells for processing. The objective is to minimize backorder cost and holding cost of inventory. To solve this complicated problem, we propose an efficient hybrid bacteria foraging algorithm (HBFA) with elaborately designed solution representation and bacteria evolution operators. A twophase based heuristic is embedded in the HBFA to generate a high quality initial solution for further search. We tested our algorithm using randomly generated instances by comparing with the original bacteria foraging algorithm (OBFA), discrete bacteria foraging algorithm (DBFA), hybrid genetic algorithm (HGA) and hybrid simulated annealing (HSA). Our results show that the proposed HBFA has better performance than the four compared algorithms with the same running time. 4

Keywords: Cellular manufacturing system; Worker assignment; Production planning; Bacteria

5

foraging algorithm; Learning and forgetting; Operation sequence

Preprint submitted to Computers & Industrial Engineering

March 28, 2016

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1. Introduction

7

Cellular Manufacturing System (CMS) has emerged to cope with the production environments with

8

demands for mid-volume and mid-variety product mixes. It is a hybrid system that links the advantages

9

of job shops (flexibility in producing a wide variety of products) and flow lines (efficient flow and high

10

production rate). The CMS in labor intensive industries has been implemented with favorable results,

11

including better utilization of workforce, production efficiency, reduction of inventory and delay. All

12

of these benefits give rise to a decrease in operational costs. In a dynamic environment, a multi-

13

period planning horizon should be considered where each period has different product mix and demand

14

requirements. Therefore, the worker configuration and product portfolio in a period may not be optimal

15

and efficient for the next period. There are two important issues for dynamic CMS in labor intensive

16

industries. One issue is the worker flexibility and assignment/reassignment, and the other issue is

17

production planning.

18

For the first issue, workers in the manufacturing environment must constantly learn new skills,

19

technology and processes in order to keep up with the move toward rapid innovations of products and

20

production. Skill levels of the workers improve through practice or deteriorate if out of practice in

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a multi-period analysis. Due to the learning and forgetting effects of workers, the production rate of

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each workstation will often change. Thus, the bottleneck workstation with the least production rate

23

may transfer to another one in the next period. Consequently, it is essential and worthwhile to reassign

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workers to increase the production rate of bottleneck workstation in each cell, and hence improve the

25

efficiency of the CMS.

26

For the second issue, with shorter product life cycles and increasing diverse demands of customers,

27

there has been a shift from static environment to dynamic environment. In static environment, the

28

system runs for a single time period with known and constant product mix and demand, while in

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dynamic environment, the system operates with a different product mix and demand requirements in

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each period. Due to the limited production capacity (related to worker assignment) and variety of

31

orders in dynamic CMS, late delivery or production in advance often occurs at each period. Shop

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floor managers should make appropriate planning decisions for all types of products to smooth the

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production loads, so that the backorder cost and holding cost of inventory can be reduced effectively. 2

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Due to the high complexity of dynamic CMS in labor intensive setting, the above two issues are

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normally studied independently or sequentially, in spite of the inter-relationship between them. Worker

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assignment determines the production capacity of the cells and affects production planning of all prod-

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ucts in all periods. In contrast, the production quantities in each planning period also affect the indi-

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vidual and number of workers to be assigned to all cells. The optimization domain will be restrained

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and the optimal benefits of the dynamic CMS may not be fully realized when making decision on one

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issue after another. In addition, to the best of our knowledge, there are few studies considering learning

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and forgetting effects during multi-skilled workforce reassignment in this kind of problem. Therefore,

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simultaneous optimization of worker assignment and production planning becomes a very important

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area of research in minimizing the operational costs including backorder cost and holding cost.

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Ever since Passino (2002) invented the bacteria foraging algorithm (BFA), it has shown a high level

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optimization capability in dealing with very complicated NP-hard problems without significantly in-

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creasing the computational time. These problems involve portfolio asset selection in financial field

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(Mishra et al., 2014), bidding strategy of a supplier (Jain et al., 2015), margin of loading in multima-

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chine power system (Tripathy & Mishra, 2015), design strategy of stacked patch resonator (Jain, 2015),

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workspace volume of a three-revolute manipulator (Panda et al., 2014), etc. One of the main challenges

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for the BFA is to broaden its application to diverse optimization areas, especially for discrete problems.

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The major purpose of this paper is to build an integrated model which can simultaneously as-

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sign/reassign workers and make production planning in multiple periods to minimize the operational

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costs. This paper also attempts to develop a hybrid bacteria foraging algorithm (HBFA) embedding a

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two-phase based heuristic (TPBH) for solving this intractable problem.

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The remainder of this paper is organized as follows. The literature review related to worker assign-

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ment and production planning in CMS is presented in Section 2. The mathematical model integrating

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worker assignment and production planning with learning and forgetting effects is formulated in Sec-

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tion 3. In Section 4, the hybrid bacteria foraging algorithm embedding the TPBH is proposed. The

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validity of TPBH and HBFA is illustrated by a typical case. The hybrid genetic algorithm and hybrid

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simulated annealing for solving this problem are described in Section 5. In Section 6, numerical exper-

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iments are conducted to evaluate the proposed HBFA by comparison with the original bacteria foraging

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algorithm (OBFA), discrete bacteria foraging algorithm (DBFA), hybrid genetic algorithm (HGA) and 3

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hybrid simulated annealing (HSA). Finally, the paper closes with a general discussion of the proposed

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approach as well as a few remarks on future research directions in Section 7.

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2. Literature Review

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In this section, we present related literature review of studies about worker assignment and produc-

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tion planning in designing the CMS.

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2.1. Worker assignment in CMS

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Many research articles are involved with worker assignment in a single-period CMS. Some re-

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searchers paid attention to models and/or model comparison. S¨uer et al. (2013) focused on manpower

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allocation problem in CMS, and examined three different sharing strategies (no operator sharing al-

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lowed, sharing allowed without restrictions, sharing allowed with restrictions). They found that the

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models with little or no restrictions yield a higher production rate than the model with no operator

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sharing allowed. S¨uer et al. (2009a) investigated the effects of different fuzzy operators on fuzzy bi-

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objective cell loading problem in labor intensive CMS. The objective is to minimize the number of

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tardy jobs and the total manpower needed. S¨uer et al. (2008) presented four different bi-objective

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mathematical models to solve the cell loading problem with setup times and alternative operator con-

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figurations. Each model is to minimize two conflicted objectives including the number of tardy jobs

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and the total manpower needed. S¨uer & Dagli (2005) introduced a sub-problem of product-sequencing

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with the objective of minimizing the total intra-cell manpower transfers. Later, S¨uer et al. (2009b)

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extended this problem by adding manpower allocation phase with the objective of minimizing the pro-

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duction rate. Norman et al. (2002) studied worker assignment in CMS considering both human and

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technical skills and their impact on system performance. The objective is to maximize system per-

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formance including the productivity, output quality and training costs. Leopairote (2003) focused on

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workgroup composition, worker assignment, and scheduling assuming that workers are heterogeneous

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in task learning-forgetting behaviors.

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Some researchers are concerned with solving worker assignment problems in a single-period CMS.

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Egilmez et al. (2014) developed a four-phased hierarchical methodology for stochastic skill-based man-

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power allocation problem, where operation times and customer demand are uncertain, and the objective 4

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is to maximize the CMS production rate. Liu et al. (2016) proposed a discrete bacteria foraging algo-

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rithm for the assignment of workers and machines in the cell formation and task scheduling problem,

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in order to minimize the material handling costs as well as the fixed and operating costs of workers

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and machines. Azadeh et al. (2013) presented an integrated fuzzy data envelopment analysis and fuzzy

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computer simulation approach for optimizing operator allocation in multi-product CMS with learning

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effects. Aalaei & Shavazipour (2013) developed data envelopment analysis method to assign workers

96

in order to minimize backorder costs and intercellular costs. Azadeh et al. (2011b) presented a decision

97

making approach based on Fuzzy AHP, TOPSIS and computer simulation to determine the most effi-

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cient number of operators and the efficient measurement of operator assignment in CMS. Azadeh et al.

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(2011a) also presented a decision making approach based on a hybrid genetic algorithm and a TOPSIS

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simulation to solve a similar problem.

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A few research articles are involved with worker assignment in multi-period CMS. Mathur & S¨uer

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(2013) studied a CMS problem of determining weekly complete schedules with the overtime decisions

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on each weekend and weekday on each shift and on each cell. They compared the math model and GA

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approaches through experimentation and concluded that, the math model either finds optimal solution

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very fast, or finds a feasible solution better than the GA in relatively short period of time when the

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math model can not find the optimal solution. Later, S¨uer & Mathur (2015) provided four extensional

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mathematical models for this problem, each of which reflects different overtime workforce hiring prac-

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tices. S¨uer & Tummaluri (2008) proposed a multi-phase hierarchical approach for loading labor inten-

109

sive cells and assigning operators to operations by considering operator skill levels, operator-operation

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times and learning and forgetting issues. Bagheri & Bashiri (2014) proposed a mathematical model

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to solve the cell formation, operator assignment and inter-cell layout problems, simultaneously. The

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objective is to minimize inter-intra cell part trips, machine relocation cost and operator related issues.

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McDonald et al. (2009) described a worker assignment model that ensures job rotation and determines

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the levels of skill and training necessary to meet customer demand. The objective is to minimize net

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present cost which includes training costs, inventory costs and cost of poor quality. Aryanezhad et al.

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(2009) developed a model of dynamic cell formation and worker assignment, which considers part

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routing flexibility, machine flexibility and promotion of workers from one skill level to another. The

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objective is to minimize the machine-based and human-based costs. 5

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120

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2.2. Production planning in CMS Production planning problems in CMS are often integrated with cell formation, dynamic system reconfiguration and other manufacturing design problems.

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For production planning integrated with cell formation, Raminfar et al. (2013) developed a model

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of dynamic production planning and cell formation in CMS. The objective is to minimize the inter-cell

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material handlings, machine operating cost, finished goods inventory, and machine set-up costs. Safaei

125

& Tavakkoli-Moghaddam (2009) proposed an integrated model of the multi-period cell formation and

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production planning in a dynamic CMS. The objective is to minimize the machine, inter/intra-cell

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movement, reconfiguration, partial subcontracting, and inventory carrying costs. Chen & Cao (2004)

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suggested an integrated model for production planning considering inter-cell material handling, fixed

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charge costs and cell construction. They proposed a Tabu search based procedure to provide production

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planning decisions such as times to start part processing and levels of finished part inventory.

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For production planning integrated with dynamic system reconfiguration, Kioon et al. (2009) devel-

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oped a CMS model that integrates production planning, dynamic system reconfiguration, and multiple

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routings. Some linearization techniques were proposed to transform the model into a mixed integer

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linear programming formulation. Ahkioon et al. (2009) studied a CMS design problem that integrates

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multi-period production planning and dynamic system reconfiguration. They provided an in-depth dis-

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cussion on the trade-off between the increased flexibility versus the additional cost incurred through

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contingency routings.

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For production planning integrated with other manufacturing design, Hassan Zadeh et al. (2014)

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proposed a comprehensive framework including process planning as well as production planning &

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control in CMS, and developed a model based on Integrated Definition Modeling Language. Malakooti

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et al. (2004) provided an integrated approach determining the machine-part cells as well as part pro-

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cessing and production plans, while the total inter-cell part flow is minimized. They demonstrated

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a hospital planning problem, in which time and resource efficiency is accomplished through group-

144

ing patients in terms of their needed medical procedures. Gajpal & Nourelfath (2015) considered a

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multi-period production planning problem where the failure rate of machine depends on the load. They

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proposed a three-phase heuristic and tabu search based metaheuristic to minimize the total production

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costs. Chu et al. (2015) proposed a bi-level model for formulating an integrated planning and schedul6

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ing problem under production uncertainties. They developed a hybrid method combining MILP solver

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and agent-based method to solve this problem.

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2.3. Integrated decision of worker assignment and production planning

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Recently, some researchers started to exploit the integrated decision of worker assignment and pro-

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duction planning problem. Sakhaii et al. (2016) studied a dynamic CMS considering production plan-

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ning, operator assignment and unreliable machines. They developed a robust optimization approach

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to minimize the costs of machine breakdown and relocation, operator training and hiring, inter-intra

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cell part trip, and shortage and inventory. Saidi-Mehrabad et al. (2013) presented a linear program-

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ming model for dynamic CMS in the presence of worker training and production planning. This model

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is capable of determining the system configurations, worker assignment and production plan for each

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part type at each period. Soolaki (2012) offered an integer linear programming model for dynamic

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CMS with production planning, worker assignment and dynamic system reconfiguration. They sug-

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gested a non-dominated sorting genetic algorithm II to minimize the total cell load variation and sum

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of the miscellaneous costs. Mahdavi et al. (2011) presented a model of dynamic virtual CMS con-

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sidering production planning and assignment of workers, machines and parts. They proposed a fuzzy

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goal programming-based approach to minimize holding cost, backorder cost and exceptional elements.

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Mahdavi et al. (2010) designed an integer programming model of dynamic CMS considering produc-

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tion planning and worker assignment. They employed LINGO package to minimize the holding and

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backorder costs, inter-cell material handling cost, machine and reconfiguration costs, and human re-

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source costs.

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From the above mentioned integrated models, we can see that some researchers have focused on

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multi-period or dynamic characteristic of the CMS because of short product life cycles and variation

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of demands. Moreover, the importance of a workforce has been widely recognized especially in labor

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intensive industries. However, it is implicitly assumed that the skill levels of workers are constant.

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Few studies have investigated the effectiveness of learning and forgetting of multi-skilled workers on

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production rate of operations, location of bottleneck operations, cell performance, and production plan-

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ning. Due to the high complexity of these integrated models, optimization softwares were often applied

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to solve them at the expense of long runtime consumption. Consequently, efficient and effective meta7

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heuristics for the models have attracted an ever growing attention both from science and practice in

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recent years. In this regard this paper fills an important gap in the literature, and provides a hybrid

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bacteria foraging algorithm to the dynamic worker assignment and production planning in CMS under

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the impact of learning and forgetting of workers.

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3. Problem Statement and Formulation

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3.1. Background of the problem

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The problem arises from the optical fiber connector manufacturing plants. Producing the optical

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fiber connectors includes a lot of processing procedures. The termination procedure plays a key role

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in the delivery of the products. Many plants form manufacturing cells in order to meet the dynamic

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demands and diversity of the products. To increase the utility of the cellular manufacturing system, each

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cell is generally designed to process multiple types of optical fiber connectors such as ‘FC’, ‘SC’, ‘MT–

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RJ’ and ‘LC’ types (see Table 1). Each independent cell is able to perform the termination procedure as

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displayed in Table 2. The termination procedure is made up of several sequential processing operations

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including mounting, polishing, assembly, surface inspection, optical inspection, shape inspection and

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packing.

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The operation of each type of products in certain workstation may be different. For example, in

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assembly workstation, the installation procedure of SC connector is as follows: (1) clean the connector

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and coupling, (2) hold the connector by the boot, and (3) align the connector chamfers with the coupling

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(see Fig. 1(a)) and push into place. The installation procedure of FC connector is as follows: (1) clean

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the connector and coupling, and (2) engage the key in the slot while holding the connector by the

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boot (see Fig. 1(b)), and make sure that the key remains engaged while tightening the threaded nut.

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The difference is reflected in the processing complexity coefficient of products in the following model.

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Consequently, The system of optical fiber connector manufacturing plant can be described as a special

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cellular manufacturing system.

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It is important for the shop floor managers to appropriately assign some workers to each operation,

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and to make optimal decision of production planning of each product in each period. An example dis-

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played in Fig. 2 illustrates a cellular manufacturing system and the assignment of workers to operations

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during certain period. 8

(a)

(b) Figure 1: Installation of SC and FC connectors

Table 1: Optical fiber connector types

Short name

Long form

Typical applications

FC

Ferrule Connector

Measurement equipment, single-mode lasers

SC

Subscriber Connector

Data communication, telecommunication

MT–RJ

Mechanical Transfer Registered Jack Duplex multimode connections

LC

Lucent Connector

High-density connections

Figure 2: Cellular manufacturing system in the optical fiber connector plant

9

Table 2: Operations of termination procedure

Operations

Description

Mounting

Inject epoxy into the ceramic ferrule and thread the connector onto the fiber

Polishing

Polish the surface of connector in the polishing machine to make the surface smooth

Assembly

Install the housing onto the body of connector

Surface inspection

Insert the connector into a good quality optical microscope to check for blemishes and scratches

Optical inspection

Insert the connector into a good quality optical microscope to check for optical loss

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Shape inspection

Check for the overall polished shape and confirm there is no shape deviation

Packing

Pack the connectors into package and carton box

3.2. Problem assumptions and notations

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The problem is formulated according to the following assumptions:

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• Planning horizon assumption: There are several production periods (measured in weeks) in the

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cellular manufacturing system. The working time in a week is set to 40 hours.

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• Product assumption: The number of product types are known in advance. Each product type has

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its own production demand in each period. The total demands of products are approximately

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equal to the total production capacity. Shop floor managers hope to fully utilize cells to produce

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products in advance so that the inventory products may be delivered in the following high demand

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periods.

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• Processing procedure assumption: All types of products require the same sequential operations,

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i.e., the operations of each product are sequentially and continually processed on the workstations

215

in a cell.

216

• Processing time assumption: The processing time of the same operation may be different due 10

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to the difference in complexity (or requirements) of the product types and skill levels of the

218

associated workers. The unload and setup times of products between different workstations are

219

ignored.

220

221

• Cell and workstation assumption: The number of cells to be formed is given and constant through all production periods. Each workstation in the cell has limited positions and tools for workers.

222

• Worker skill assumption: Each worker has skills to perform all operations. All workers have the

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characteristics of learning and forgetting. That is to say, a worker’s skill level improves when

224

he/she performs a specific operation for a period of time, and his/her skill level deteriorates when

225

he/she doesn’t engage in the operation for a period of time.

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• Worker assignment assumption: The total number of workers is given. All workers are assigned

227

to workstations to perform operations. At least one worker is assigned to each workstation, but

228

the number can not exceed an upper bound because of limited positions and tools. Only at the

229

end of each period can the workers be reassigned to different cells and workstations.

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• Cost assumption: Backorder and holding inventories are allowed between periods with known

231

costs. The holding cost of inventory is much less than the backorder cost. Consequently, the

232

demand for certain product type in a given period can be satisfied in the preceding or succeeding

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periods.

234

235

236

The main parameters and variables used in the model are summarized as follows. The notations with symbol ♣ will be further explained in Subsection 3.3. Input parameters:

11

W

Number of workers, w denotes index for worker (w = 1, 2, . . . , W).

Q

Number of product types, q denotes index for product (q = 1, 2, . . . , Q).

J

Number of operations, j denotes index for operation ( j = 1, 2, . . . , J).

C

Number of cells, c denotes index for cell (c = 1, 2, . . . , C).

H

Number of periods, h denotes index for period (h = 1, 2, . . . , H).

Uj

Upper bound for worker level for operation j.

Dqh

Demand of product type q (measured in packs) during period h.

θq

Unit backorder cost of product type q at the end of each period.

φq

Unit holding cost of product type q at the end of each period.

fq ♣

Processing complexity coefficient of product type q.

kw j ♣

Steady-state production rate (measured in packs per hour) of worker w processing

237

operation j. pw j ♣

(Measured in weeks) accumulated initial experience of worker w for operation j.

rw j ♣

Cumulative operation work required to attain a production rate level of kw j /2, and 1/rw j indicates the learning rate of worker w for processing operation j.

238

239

240

αw j ♣

Degree to which worker w forgets the skill of processing operation j.

Decision variables: Xwc jh 1 if worker w is assigned to cell c for operation j during period h, and 0 otherwise. ρqch

Production quantity of product type q (measured in hours) which is assigned to cell c for processing during period h. ρqch ∈ {0, 1, 2, . . . , i, . . . , 40}.

241

242

243

244

According to the input parameters and decision variables, The intermediate variables are defined as follows:

245

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Gqch

1 if product type q is assigned to cell c for processing during period h, and 0 otherwise. Gqch = Min{ρqch , 1}.

yw jx ♣

Production rate (measured in packs per hour) of worker w, corresponding to x periods of accumulated operation work j, x denotes index of count for worker to process the operation in each period.

Rw jx ♣

Recency of experiential learning of worker w to perform x periods of accumulated operation work j.

246

Ywq jh ♣

Production rate (measured in packs per hour) of worker w who perform operation j of product type q during period h.

βqc jh ♣

Production rate (measured in packs per hour) of operation j of product type q assigned to cell c during period h.

Bqch ♣

Production rate (measured in packs per hour) of product type q assigned to cell c during period h.

247

248

249

3.3. Learning and forgetting effects and production rate

250

Mazur & Hastie (1978) introduced an individual learning model, and this model was modified to

251

include forgetting effects by Nembhard & Uzumeri (2000). We extend these models to represent the

252

learning and forgetting behavior of heterogeneous workers who process different operations.

253

First the extended learning model is denoted by Eq. (1). ( yw jx = kw j

) x + pw j , x + pw j + rw j

yw jx , kw j , pw j , x ≥ 0 and pw j + rw j > 0

(1)

254

where yw jx is a measure of the production rate of worker w, corresponding to x periods of accumulated

255

operation work j, and x denotes index of count for worker to process operation in each period. Parame-

256

ter kw j estimates the asymptotic steady-state production rate of worker w processing operation j, which

257

can be expected when all learning has been completed. Parameter pw j represents the accumulated initial

258

experience of worker w for operation j (see Fig. 3, Case 2). Parameter rw j is the cumulative operation

259

work required to attain a production rate level of kw j /2, starting from the point where yw jx is equal to 13

260

zero (see Fig. 3, Case 1). Hence, a smaller rw j corresponds to more rapid approach to steady-state

261

performance (i.e., more rapid learning).

Production Rate (packs/hour)

ywjx kwj Case 2 Initial Expertise

Case 1 No Initial Expertise

kwj 2

pwj

x

rwj

Cumulative Work (units)

Figure 3: Learning curve of individual worker

262

Forgetting is based on a measure of recency of experiential learning, Rw jx , which provides a relative

263

measure of how recently the practice of operation j was obtained by worker w. For each unit of

264

cumulative production x, Rw jx is determined by computing the ratio of the average elapsed time to the

265

elapsed time of the most recent unit produced, as in Eq. (2). The elapsed time for unit x is given by

266

(~w jx − ~w j1 ), which is the difference between the time stamps of the period of the current unit, ~w jx , and

267

the period of the first unit, ~w j1 .

Rw jx

    if x = 1   1 = ∑ x    i=1 (~w ji −~w j1 )  x(~ if x > 2 w jx −~w j1 )

(2)

268

The recency variable, Rw jx , ranges from 0 to 1. The value approaching 1 indicates that all experience

269

was obtained exactly in the current unit, whereas the value approaching 0 indicates that all experience

270

was obtained infinitely long ago.

271

In order to incorporate the impact of recency of experience on every worker, the cumulative work 14

α

272

x is discounted by the factor Rwwjxj , as in Eq. (3), where αw j represents the degree to which worker w

273

forgets operation j (see Fig. 4). Parameter αw j is restricted to be greater than or equal to zero. There is

274

no forgetting after a break when αw j = 0.

yw jx

  = kw j 

α

xRwwjxj + pw j α

xRwwjxj + pw j + rw j

  

(3)

275

Eq. (3) gives the general production rate of operation when worker performs standard product. In fact,

276

parameter x is a function of the variable h when worker w and operation j are given. We might as well

277

assume a threshhold value λ = 10. If the total production quantity (in hours) in cell c during period h

278

is greater than or equal to the value, it is thought that worker w assigned to operation j in cell c will

279

increase one unit of corresponding experience of operation j. Eq. (4) shows the relationship of x and

280

h. The variable ~w jx of Eq. (2) can be calculated by the inverse function of Eq. (4). The coefficient

281

fq denotes the processing complexity of product type q. Consequently, the intermediate variable of

282

production rate Ywq jh can be calculated in Eq. (5). The intermediate variable of production rate βqc jh

283

can be calculated in Eq. (6). The intermediate variable of production rate Bqch can be calculated in Eq.

284

(7), because production rate of cell is determined by its bottleneck operation.

Production Rate (packs/hour)

ywjx kwj Case 2 Initial Expertise

Case 1 No Initial Expertise

kwj 2

Forgetting after process interruption (

pwj

wj

! 0)

x

rwj

Cumulative Work (units)

Figure 4: Learning-forgetting curve of individual worker

15

x(h) =

C ∑ h ∑ c=1 t=1

 ∑Q     q=1 ρqct    Xwc jt · Min  1, x y ,      10

∀w, j

yw· j·x(h) , ∀w, q, j, h fq W ∑ = Ywq jh · Xwc jh , ∀q, c, j, h

(4)

Ywq jh =

(5)

βqc jh

(6)

w=1

{ } Bqch = Min βqc jh | j = 1, . . . , J ,

285

286

∀q, c, h

(7)

3.4. Mathematical model The problem can be formulated as a non-linear 0-1 integer programming model as follows:

Min

 h  h ∑ C   ∑ ∑     θq · Max  0, D − B · ρ qi qci qci      i=1 i=1 c=1 h=1 q=1  h C  Q H ∑ h   ∑ ∑    ∑∑  + φq · Max  0, B · ρ − D  qci qci qi     Q H ∑ ∑

h=1 q=1

s.t.

C ∑ J ∑

i=1 c=1

Xwc jh = 1,

(8)

i=1

∀w, h

(9)

c=1 j=1

1≤

W ∑

Xwc jh ≤ U j ,

∀c, j, h

(10)

∀q, h

(11)

w=1 C ∑

Min{1, ρqch } ≤ 1,

c=1 Q ∑

ρqch ≤ 40,

∀c, h

(12)

q=1

ρqch ∈ {0, 1, 2, . . . , 40}, Xwc jh ∈ {0, 1},

∀q, c, h

(13)

∀w, c, j, h

(14)

287

The objective function (8) consists of two cost items as follows:

288

(The first term) Backorder cost: The cost of delay in the delivery of products over all periods in the

289

290

planning horizon. (The second term) Holding cost: The holding cost of inventories of all products over all the periods 16

291

in the planning horizon.

292

The decision variable ρqch represents the production quantity of product type q (measured in hours)

293

which is assigned to cell c for processing during period h. It also serves as a system state variable which

294

identifies if this product type is assigned to the associate cell. The continuous integer version of ρqch

295

has considerable computational advantage over the 0-1 version.

296

Constraint (9) ensures that each worker is assigned to one operation in one cell during each period.

297

Constraint (10) ensures that at least one worker is assigned to each workstation, but the number can not

298

exceed an upper limit. Constraint (11) ensures that each product type can be assigned to at most one

299

cell for processing during each period, where Min{1, ρqch } is the expression of intermediate variable

300

Gqch . Constraint (12) ensures that the sum of time spent on processing the products can not exceed

301

the available time during each period. Constraint (13) ensures that each decision variable ρqch is in

302

the given integer interval. Constraint (14) provides the logical binary necessity for decision variable

303

Xwc jh . Eqs. (1) ∼ (7) provide the calculating method of intermediate variable Bqch which is actually the

304

function of decision variables Xwc jh and ρqch .

305

4. Hybrid Bacteria Foraging Algorithm

306

The classical bacteria foraging algorithm was firstly invented based on the foraging strategy of Es-

307

cherichia coli bacteria in human intestines. A “virtual” bacterium represents a point in n-dimensional

308

search space where each point is a potential solution to the optimization problem. The BFA is mod-

309

eled as an evolution process where bacteria seek nutrients to maximize their health. The process in-

310

volves three main stages, namely chemotaxis (including tumble and swimming), reproduction, and

311

elimination-dispersal (Passino, 2002).

312

In BFA, through chemotaxis the bacteria try to search for places with better nutrient gradient al-

313

ternating between “tumbling” and “swimming”. Through reproduction process the unhealthy bacteria

314

die and each fitter bacterium splits into two bacteria. When the local environment where a population

315

of bacteria live changes either gradually (e.g., via consumption of nutrients) or suddenly (e.g., due to

316

medicine influence), the group of bacteria is dispersed to a new region or all bacteria in the area are

317

eliminated.

318

This algorithm is suitable for the continuous optimization problem, so a modified hybrid bacteria 17

319

foraging algorithm (HBFA) is suggested to solve the above discrete optimization problem. In this

320

HBFA, the solution representation and neighborhood generation operators are elaborately designed.

321

4.1. Solution representation

322

Since our problem is a discrete problem, we should move the bacterium discrete position to find a

323

better solution in terms of the relationship of worker, cell, operation, product, production volume and

324

period. Thus, we suggest a schema which consists of two ingredients as follows:

325

The first ingredient related to decision variables Xwc jh is a W × H matrix composed of genes (c, j).

326

The row number to which the gene belongs is associated with worker index, and the column number

327

to which the gene belongs is associated with period index. An example is demonstrated in Fig. 5(a)

328

where the gene (1,2) with dashed rectangular indicates worker 4 is assigned to operation 2 of cell 1

329

during period 2. While completing the matrix, constraints (9), (10) and (14) should be satisfied.

330

The second ingredient related to decision variables ρqch is a Q × H matrix composed of genes

331

(c, ρ). Sometimes ρqch is abbreviated to ρ for simplicity. The row number to which the gene belongs is

332

associated with product type index, and the column number to which the gene belongs is associated with

333

period index. An example is demonstrated in Fig. 5(b) where the gene (2,14) with dashed rectangular

334

indicates product type 4 with production quantity value 14 hours is assigned to cell 2 for processing

335

during period 3. While completing the matrix, constraints (11), (12) and (13) should be satisfied. h

w

(2,3) (1,1) (1,3) (2,3) ! "(2,3) (1,1) (1,3) (1,1) # " # "(2,2) (2,1) (2,3) (1,2) # " # "(2,1) (1,2) (1,1) (2,3) # "(1,2) (2,2) (2,1) (1,3) # " # "(1,3) (1,3) (1,2) (2,1) # "(1,1) (2,3) (2,2) (1,3) # " # "$(2,1) (2,2) (2,3) (2,2) #%

(a) The first ingredient (c, j )W

h

q

(1, 21) " (1,10) " " (2,15) " " (2,23) "$ (1, 9)

(2,12) (2,14) (1,18) (1,22) (2,14)

(1, 7) (2,16) (2,10) (2,14) (1,30)

(2,25) ! (1,30) ## (1, 4) # # (2,10) # (1, 6) #%

(b) The second ingredient (c ! )Q! H H

Figure 5: Solution representation

Combining the two ingredients described above, the solution representation is as shown in (15). The ingredients can be referred as dimensions or directions, so it is very suitable for the bacterium of 18

HBFA to tumble in a randomly selected dimension and to swim in the same previous dimension. (c, j)W×H | (c, ρ)Q×H

336

(15)

4.2. Two-phase based heuristic for initial solution

337

Initial solution plays an important role in the exploration and exploitation process of the HBFA.

338

High quality initial solution is located near the optimal solution, and hence helps to improve the effi-

339

ciency and effectiveness of the search. Here, a heuristic is suggested to obtain an initial solution. For

340

the model proposed in Section 3.4, we just relax the learning and forgetting effects of workers and

341

suppose each worker always has steady-state production rate kw j during all periods. The original model

342

and relaxed model are denoted by P and P’, respectively. It is obvious that the solution of model P’ is

343

also the one of model P. So, a two-phase based heuristic (TPBH) for model P’ is proposed to obtain a

344

near-optimal solution of model P.

345

The first phase of TPBH mainly assigns the workers to the operations of cells so that the total

346

production rate is maximized:

347

1. 1: Compute the average steady-state production rate of each worker i (Ωi =

348

1. 2: Initialize: Let the production rate of each operation of each cell and the production rate of each

349

350

1 J

∑J

j=1 kw j ).

cell be equal to 0. 1. 3: For each period h = 1, · · · , H, execute the following steps:

351

(a) Assign an arbitrary unassigned worker to the bottleneck operation of cell with the least

352

production rate. If there are upper level of workers in the operation, preferentially select

353

the bottleneck operation of cell with less production rate. In case there are upper level of

354

workers in all bottleneck operations, randomly select a feasible operation.

355

(b) Update the production rate of each operation of each cell according to the sum of Ωi of

356

workers assigned to the operation. The production rate of each cell can also be derived

357

according to its bottleneck operation. If there are unassigned workers left, go to previous

358

sub-step (a). 19

359

The second phase of TPBH mainly assigns the quantity and mix of products to the cells by imposing

360

the production rate limit obtained in phase 1, so that the backorder and holding costs can be minimized:

361

2. 1: For each period h = 1, · · · , H, execute the followings:

362

(a) Randomly divide all product types into C groups, satisfying that each group is not empty.

363

Randomly determine the production quantity (measured in hours) of each product type,

364

satisfying that the total production quantity in each group is not greater than 40.

365

(b) Compute the priority index ξ of each product group (ξ equals to the average

θq fq

of the group).

366

Sort the product groups in order of descending ξ.

367

(c) Sort the cells in order of descending production rate.

368

(d) Assign the product group with greater ξ to the cell with greater production rate for process-

369

ing.

370

(e) Sort the product types in order of ascending

371

(f) For cell c = 1, . . . , C, execute the following:

φq fq

in each cell.

372

If there is production capacity left in cell c, assign product type with the least

373

for processing.

374

375

376

377

378

379

380

381

382

383

384

φq fq

to the cell

2. 2: For product type q = 1, · · · , Q, execute the followings: (a) For period h = 1, · · · , H, execute the followings: ∑ ∑ ∑ ∑H ∑C ∑H i. If( hi=1 Cc=1 ρqci Bqci > hi=1 Dqi and i=1 i=1 Dqi ),♠ c=1 ρqci Bqci > Reduce the following amount (measured in hours) of product q during period h: { ∑h ∑C } ∑h ∑H ∑C ∑H ∑C i=1 c=1 ρqci Bqci − i=1 Dqi i=1 c=1 ρqci Bqci − i=1 Dqi Min , , c=1 ρqch .♠ Bqch Bqch ♠ Bqch can be obtained by computing Eqs. (2)∼(7), since workers and products have bee assigned to cells. ∑ ∑ ♠ hi=1 Cc=1 ρqci Bqci represents the total production volume (measured in packs) of product type q during periods 1∼h. ∑ ♠ hi=1 Dqi represents the total demand (measured in packs) of product type q during periods 1∼h. ∑H ∑C ♠ i=1 c=1 ρqci Bqci represents the total production volume (measured in packs) of product type q during 20

385

386

periods 1∼H. ∑H ♠ i=1 Dqi represents the total demand (measured in packs) of product type q during periods 1∼H.

387

388

389

To better illustrate the proposed TPBH, let us consider a simple case. The parameters of the case are displayed in Table 3. The computational process is explained as follows. Table 3: Parameters of the case using TPBH

W = 6, Q = 6, J = 2, C = 2, H = 3 U j : U1 = 3, U2 = 3 Dqh : D11 = 57, D12 = 68, D13 = 98, D21 = 58, D22 = 55, D23 = 91, D31 = 80, D32 = 53, D33 = 61 D41 = 85, D42 = 67, D43 = 62, D51 = 85, D52 = 67, D53 = 86, D61 = 57, D62 = 85, D63 = 74 θq : θ1 = 9, θ2 = 6, θ3 = 10, θ4 = 9, θ5 = 9, θ6 = 9 φq : φ1 = 5, φ2 = 2, φ3 = 5, φ4 = 1, φ5 = 2, φ6 = 4 fq : f1 = 1.8, f2 = 1, f3 = 1.7, f4 = 1.5, f5 = 1.7, f6 = 1.6 kw j : k11 = 6, k12 = 5, k21 = 6, k22 = 10, k31 = 5, k32 = 8 k41 = 5, k42 = 6, k51 = 10, k52 = 8, k61 = 8, k62 = 6 pw j : p11 = 2, p12 = 2, p21 = 1, p22 = 1, p31 = 3, p32 = 3 p41 = 1, p42 = 2, p51 = 2, p52 = 1, p61 = 2, p62 = 2 rw j : r11 = 4, r12 = 5, r21 = 3, r22 = 6, r31 = 6, r32 = 3 r41 = 6, r42 = 6, r51 = 5, r52 = 4, r61 = 4, r62 = 3 αw j : α11 = 4, α12 = 3, α21 = 1, α22 = 2, α31 = 4, α32 = 1 α41 = 2, α42 = 3, α51 = 4, α52 = 3, α61 = 3, α62 = 1

390

In the first phase:

391

• In Step 1.1, we can obtain Ω1 = 5, Ω2 = 8, Ω3 = 6, Ω4 = 5, Ω5 = 9, Ω6 = 7.

392

• In Step 1.3 (e.g., h = 1), we can obtain “c1 : {5}, {2, 3}; c2 : {6}, {1, 4}”, i.e., worker 5 is assigned

393

to operation 1 of cell 1, workers 2 and 3 are assigned to operation 2 of cell 1, worker 6 is assigned

394

to operation 1 of cell 2, and workers 1 and 4 are assigned to operation 2 of cell 2. 21

395

• Through the first phase, we can obtain the first ingredient of solution shown in Fig. 6(a).

396

In the second phase:

397

• In Step 2.1, take h = 1 as example.

398

• In Step 2.1(a), we can obtain “g1 : {(1, 22), (5, 13)}; g2 : {(2, 14), (3, 7), (4, 12), (6, 5)}”, i.e., group

399

1 includes product type 1 with quantity 22 and product type 5 with quantity 13, as well as group

400

2 includes product type 2 with quantity 14, product type 3 with quantity 7, product type 4 with

401

quantity 12, and product type 6 with quantity 5. θq . fq

402

• In Step 2.1(b), group 2 has higher priority than group 1, since group 2 has greater average

403

• In Step 2.1(c), cell 1 has higher priority than cell 2, since cell 1 has greater production rate.

404

• In Step 2.1(d), product with greater backorder cost and less complexity coefficient should be

405

processed in cell with greater production rate. Therefore, we obtain

406

“c1 : {(2, 14), (3, 7), (4, 12), (6, 5)}; c2 : {(1, 22), (5, 13)}”, i.e., group 1 is assigned to cell 2 for

407

processing, and group 2 is assigned to cell 1 for processing. This step helps to increase production

408

volume and reduce backorder cost as much as possible.

409

• In Steps 2.1(e) and (f), we can obtain “c1 : {(4, 14), (2, 14), (6, 5), (3, 7)}; c2 : {(5, 18), (1, 22)}”. φq fq

410

As can be observed, product type 4 has the least

in cell 1, so the quantity of the product is

411

increased from 12 to 14, such that there is no production capacity left in cell 1. Similarly, the

412

quantity of the product 5 in cell 2 is increased from 13 to 18. This step helps to make full use of

413

production capacity and reduce holding cost as much as possible.

414

• Through Step 2.1, we can obtain the second ingredient of solution shown in Fig. 6(b).

415

• In Step 2.2,

416

i=1

c=1 ρqci Bqci −

Bqch

∑h i=1

Dqi

measures whether product type q has been produced too much

in period h according to the total production volume and demand during periods 1∼h. Similarly, ∑H ∑C

417

∑h ∑C

i=1

c=1 ρqci Bqci −

Bqch

∑H i=1

Dqi

measures whether product type q has been produced too much in period h

418

according to the total production volume and demand during the whole planning horizon. If the

419

product type has been produced too much, reduce the minimum of the two measurements. Of 22

∑C

ρqch .

420

course, the reduced amount can not be greater than the present production quantity

421

Through this step we can obtain an optimized second ingredient of solution shown in Fig. 6(c).

422

For example, product type 2 has been reduced from 38 to 29 in period 2. h

h

w

(2,2) "(1,2) " "(1,2) " "(2,2) "(1,1) " $(2,1)

(1,1) (1,2) ! (2,2) (1,2) ## (2,2) (1,1) # # (2,1) (2,2) # (1,2) (2,2) # # (1,2) (2,1) %

(a) The first ingredient (c, j )W after the first phase

q

H

(2,22) "(1,14) " "(1, 7) " "(1,14) "(2,18) " $(1, 5)

h

(2, 3) (2, 2) ! (1,38) (1,40) ## (1, 2) (2, 0) # # (2,11) (2, 4) # (2,19) (2,18) # # (2, 7) (2,16) %

(b) The second ingredient (c ! )Q! H after Step 2.1

q

424

4.3. Implementation of the proposed HBFA Some notations to be used in the HBFA are summarized as follows:

425

23

(2,22) "(1,14) " "(1, 7) " "(1,14) "(2,18) " $(1, 5)

(2, 3) (2, 2) ! (1,29) (1,39) ## (1, 2) (2, 0) # # (2,11) (2, 4) # (2,19) (2,18) # # (2, 7) (2,16) %

(c) The second ingredient (c ! )Q! H after the second phase

Figure 6: Solution of the case using TPBH

423

c=1

j

Index for the chemotactic step.

k

Index for the reproduction step.

l

Index of the elimination-dispersal event.

S

Number of bacteria in a population (the number is assumed to be a positive even integer).

Sr

Number of half population of bacteria.

θi ( j, k, l)

The ith bacterium position at the jth chemotactic step, kth reproduction step, and lth elimination-dispersal event, which corresponds to a feasible solution of the problem, θi ∈ R p , where p is the number of dimensions of the position (sometimes we drop the indices and refer to the ith bacterium position as θi ).

J(θi )

The ‘cost’ of being in the position θi (using terminology from optimization theory) or the nutrient surface (in reference to the biological connections), which

426

corresponds to the objective function value of the solution. Nc

Length of the lifetime of the bacteria as measured by the number of chemotactic steps they take during their life.

427

m

Counter for swimming steps.

Ns

Maximum number of swimming steps.

Nre

Number of reproduction steps.

Ned

Number of elimination-dispersal events.

ped

Probability of elimination-dispersal event for each bacterium.

i Jcur

The best cost of each bacterium i in the current generation.

i Jlast

The last cost of each bacterium i (a better cost may be found via a run).

i Jhealth

Accumulated cost of all the chemotactic steps of bacterium i in the current ∑ c +1 i generation; Jhealth = Nj=1 J(i, j, k, l)).

428

429

The HBFA simulates the foraging behavior of bacteria which tries to climb up the nutrient concen-

430

tration (finding lower and lower values of J(θi )), avoid the noxious substances, and search for ways out

431

of the neutral media through three nested loops of chemotaxis, reproduction, and elimination-dispersal. 24

432

For example, J(θi ) < 0, J(θi ) = 0, and J(θi ) > 0 represent that the bacterium i at position θi is in

433

nutrient-rich, neutral, and noxious environments, respectively. The bacterium tends to avoid being at

434

positions θi , where J(θi ) ≥ 0. The flowchart of the proposed HBFA algorithm is outlined in Fig. 7, and

435

its detailed procedure is described in Algorithm 1.

Algorithm 1: Hybrid Bacteria Foraging Algorithm 1. Initialize: S = 20, Nc = 5, N s = 8, Nre = 10, Ned = 8, ped = 0.3, j = k = l = 0; 2. Initial population: Generate one initial value for the θi by using the TPBH, and generate other initial feasible values for the θi randomly. 3. Elimination-dispersal loop: l := l + 1 3.1. Reproduction loop: k := k + 1 i 3.1.1. Let Jcur = J(θi ( j, k, l)) to save the best cost of each bacterium i in the current

generation. 3.1.2. Chemotaxis loop: j := j + 1 3.1.3. For i = 1, 2, . . . , S , take a chemotactic step for bacterium i as follows: i 3.1.3.1. Let Jlast = J(θi ( j, k, l)) to save this value since we may find a better cost via

a run. 3.1.3.2. Tumble: movement (mutation) of bacterium i in the dth dimensional direction (d is randomly selected), which results in new position θi ( j + 1, k, l) for bacterium i. 3.1.3.3. Compute J(θi ( j + 1, k, l)). 3.1.3.4. Swimming process of bacterium i if needed: Let m=0 (counter for swimming steps). While (m < N s ) i If (J(θi ( j + 1, k, l)) < Jcur )

Let J i = J(θi ( j + 1, k, l)). Swim in the above dth dimensional direction, and update θi ( j + 1, k, l). Compute J(θi ( j + 1, k, l)). Let m := m + 1. 25

Else Break i i 3.1.3.5. If(Jlast < Jcur ) i i Jcur = Jlast i 3.1.3.6. If(J(θi ( j + 1, k, l)) < Jcur ) i Jcur = J(θi ( j + 1, k, l))

3.1.4. If j < Nc , go to step 3.1.2. In this case, continue chemotaxis, since the life of the bacteria is not over. 3.2. Implement reproduction strategy: Sort the population in order of ascending historic ‘cost’ Jcur of each bacterium in the current generation. The S r bacteria with the highest Jcur values die. Each of the other S r bacteria with the lowest values moves to its best position in the current generation (associated with the value Jcur ). Randomly select two of them to cross over and put two offspring bacteria into the current generation until the generation size reaches S . 3.3. If k < Nre , go to step 3.1 (i.e., start the next generation in the chemotactic loop). 4. Elimination-dispersal strategy: For i = 1, 2, . . . , S , disperse a bacterium except the best one to a random position on the optimization domain with probability ped . This keeps the best bacterium remaining in the next generation. 5. If l < Ned , then go to step 3; otherwise end.

436

437

4.3.1. Chemotaxis

438

A chemotactic step is defined to be a tumble or a tumble followed by at most N s swimming steps.

439

Tumble refers to the movement (mutation) of a bacterium in a random direction displayed in Fig.

440

8. For example, Fig. 8(a) shows a bacterium runs a tumble in the first dimension, i.e., two rows of

441

matrix (c, j)W×H are selected randomly and their components are substituted, and then two columns of

442

the outcome are selected randomly and their components are substituted. Similarly, Fig. 8(b) shows a 26

Start

Initialize variables

Generate initial bacteria population using TPBH and random method

Terminated QR

Eliminationdispersal loop Counter, l:=l+1

\HV

l
Elimination-dispersal strategy

Reproduction loop Counter, k:=k+1 \HV Chemotaxis loop Counter, j:=j+1

k
Tumble for each bacterium i and compute its cost J( i(j+1,k,l))

Reproduction strategy \HV

Swimming counter m
QR

\HV

j
QR

Update Ji

J( i(j+1,k,l))
QR

\HV Swim m:=m+1 Figure 7: Flowchart of proposed HBFA algorithm

27

QR

443

bacterium run a tumble in the second dimension. Let Jlast and Jcur denote the last ‘cost’ of bacterium and

444

its best historic ‘cost’ in the current generation. The HBFA’s swimming strategy suggests that when the

445

‘cost’ of bacterium after a tumble is better than Jcur (instead of Jlast as in Passino’s swimming strategy

446

(Passino, 2002)), the bacterium will move (mutate) further in the same direction. If that movement

447

resulted in a better position according to the above comparison rules, another movement is taken. This

448

swim is continued as long as it continues to reduce the cost, but only up to a maximum number of

449

movements, N s . This represents that the bacterium will tend to keep moving if it is headed in the

450

direction of increasingly favorable environments.

(2,3) " (2,3) " " (2,2) " " (2,1) " (1,2) " " (1,3) " (1,1) " $" (2,1)

(1, 5) " (2,27) " " (1, 25) " " (1,10) "$ (2,12)

(1,3) (1,2) (2,3) ! (1,3) (2,2) (1,1) ## (2,3) (2,1) (1,2) # # (1,1) (1,3) (2,3) # (2,1) (2,3) (1,3) # # (1,2) (2,3) (2,1) # (2,2) (1,1) (1,3) # # (2,3) (2,1) (2,2) %#

(1,12) ! (2,22) ## (2,18) # # (1,22) # (2,14) (1,30) (1, 6) #%

(2,12) (2,14) (1,18) (1,22)

(1, 7) (2,16) (2,10) (2,14)

(1,3) (1,2) (2,3) ! (1,3) (2,2) (1,1) ## (1,2) (2,3) (2,1) # # (1,1) (1,3) (2,3) # (2,1) (2,3) (1,3) # # (2,3) (2,1) (1,2) # (2,2) (1,1) (1,3) # # (2,3) (2,1) (2,2) %# (a) Tumble of (c, j )W H

(2,3) " (2,3) " " (1,3) " " (2,1) " (1,2) " " (2,2) " (1,1) " $" (2,1)

(1,10) (1,22) (2,14) "(2,27) (2,14) (2,16) " "(1, 25) (1,18) (2,10) " "(1, 5) (2,12) (1, 7) "$(2,12) (2,14) (1,30) (b) Tumble of (c !

(1,22) ! (2,22) ## (2,18) # # (1,12) # (1, 6) #% ) Q! H

(1,2) "(2,2) " "(2,3) " "(1,3) "(2,3) " "(2,1) "(1,1) " $"(2,1)

(1,3) (2,3) (2,3) ! (1,3) (2,3) (1,1) ## (1,2) (1,3) (2,1) # # (1,1) (2,1) (2,3) # (2,1) (1,2) (1,3) # # (2,3) (2,2) (1,2) # (2,2) (1,1) (1,3) # # (2,3) (2,1) (2,2) %#

(1,10) "(2,27) " "(1, 25) " "(1, 5) "$(2,12)

(1,22) ! (2,22) ## (2,18) # # (1,12) # (1,30) (2,14) (1, 6) #%

(2,14) (2,16) (2,10) (1, 7)

(1,22) (2,14) (1,18) (2,12)

Figure 8: Tumble of bacterium

451

4.3.2. Reproduction

452

After Nc chemotactic steps, a reproduction step is taken. Different from Passino’s reproduction

453

i strategy (the population is sorted in order of ascending accumulated cost Jhealth of bacterium i (Passino,

454

2002)), the HBFA suggests that the population is sorted in order of ascending historic ‘cost’ Jcur of

455

each bacterium in the current generation. The S r bacteria with the highest Jcur values die. Each of the

456

other S r bacteria with the lowest values moves to its best position in the current generation (associated 28

457

with the value Jcur ). Randomly select two of them to cross over (cf. Algorithm 2 and Fig. 9) and put

458

the offspring bacteria into the current generation until the population size reaches S .

Algorithm 2: Crossover Operator Step 1: (Cf. Fig. 9(a)) randomly select two dash lines as crossover points among the columns of (c, j)W×H in two bacteria. Swap the genes between two dash lines for the two bacteria to generate two new sub-offsprings (c, j)W×H . Step 2: (Cf. Fig. 9(b)) generate two new sub-offsprings (c, ρ)Q×H as in Step 1. Step 3: Randomly select one sub-offspring from A and one sub-offspring from B, and combine them to generate one offspring. The left two sub-offsprings from A and B are combined to generate another offspring.

459

460

4.3.3. Elimination-dispersal

461

After Nre reproduction steps, an elimination-dispersal event takes place. Each bacterium in the pop-

462

ulation is subjected to elimination-dispersal with probability ped . The elimination-dispersal events al-

463

low the bacteria to look into more regions to find good nutrient concentrations. Obviously, if ped is cho-

464

sen appropriately, the elimination-dispersal events can help the HBFA jump out of the local optima into

465

a global optimum. However, if ped is large, the HBFA will degrade to random exhaustive search. Dif-

466

ferent from Passino’s elimination-dispersal strategy (all bacteria are subjected to elimination-dispersal

467

(Passino, 2002)), the HBFA suggests that each bacterium, except the best one, is to be dispersed to a

468

random position on the optimization domain with probability ped . This keeps the best bacterium remain

469

in the next generation, which may speed up the convergence.

470

471

4.4. Case study

472

To better illustrate the proposed problem and show its result using the HBFA, we further discuss

473

the case which has been presented in Section 4.2. Fig. 10 displays the final solution of the case using 29

(2,3) " (2,3) " " (2,2) " " (2,1) " (1,2) " " (1,3) " (1,1) " $" (2,1)

(1,1) (1,3) (2,3) ! (1,1) (1,3) (1,1) ## (2,1) (2,3) (1,2) # # (1,2) (1,1) (2,3) # (2,2) (2,1) (1,3) # # (1,3) (1,2) (2,1) # (2,3) (2,2) (1,3) # # (2,2) (2,3) (2,2) %#

(2,3) " (2,3) " " (2,2) " " (2,1) " (1,2) " " (1,3) " (1,1) " $" (2,1)

(1,3) (1,2) (2,3) ! (1,3) (2,2) (1,1) ## (2,3) (2,1) (1,2) # # (1,1) (1,3) (2,3) # (2,1) (2,3) (1,3) # # (1,2) (2,3) (2,1) # (2,2) (1,1) (1,3) # # (2,3) (2,1) (2,2) %#

(1,1) "(1,1) " "(2,1) " "(1,2) "(2,2) " "(1,3) "(2,3) " "$(2,2)

(1,3) (1,2) (2,1) ! (1,3) (2,2) (1,1) ## (2,3) (2,1) (1,3) # # (1,1) (1,3) (1,3) # (2,1) (2,3) (2,2) # # (1,2) (2,3) (2,3) # (2,2) (1,1) (1,2) # # (2,3) (2,1) (2,3) #%

(1,1) " (1,1) " " (2,1) " " (1,2) " (2,2) " " (1,3) " (2,3) " "$ (2,2)

(1,1) (1,3) (2,1) ! (1,1) (1,3) (1,1) ## (2,1) (2,3) (1,3) # # (1,2) (1,1) (1,3) # (2,2) (2,1) (2,2) # # (1,3) (1,2) (2,3) # (2,3) (2,2) (1,2) # # (2,2) (2,3) (2,3) #%

(a) Crossover of (c, j )W

H

(1, 21) " (1,10) " " (2,15) " " (2,23) "$ (1, 9)

(2,12) (2,14) (1,18) (1,22) (2,14)

(1, 7) (2,16) (2,10) (2,14) (1,30)

(2,25) ! (1,30) ## (1, 4) # # (2,10) # (1, 6) #%

(1, 21) " (1,10) " " (2,15) " " (2,23) "$ (1, 9)

(2,21) (1,13) (2,12) (1,27) (2, 7)

(1, 5) " (2,27) " " (1, 25) " " (1,10) "$ (2,12)

(2,21) (1,13) (2,12) (1,27) (2, 7)

(2,12) (1,12) ! (2, 23) (2,22) ## (1,12) (2,18) # # (2, 5) (1,22) # (1,23) (1, 6) #%

(1, 5) " (2,27) " " (1, 25) " " (1,10) "$ (2,12)

(2,12) (2,14) (1,18) (1,22)

(b) Crossover of (c ! )Q! H Figure 9: Crossover

30

(2,12) (2, 23) (1,12) (2, 5) (1,23)

(2,25) ! (1,30) ## (1, 4) # # (2,10) # (1, 6) #%

(1,12) ! (2,22) ## (2,18) # # (1,22) # (2,14) (1,30) (1, 6) #% (1, 7) (2,16) (2,10) (2,14)

474

the HBFA. The final objective function value (585) has decreased greatly through the stochastic search

475

of HBFA from the initial objective function value (14885) using the TPBH.

476

The intuitive diagram of the final solution is showed in Fig. 11. For example, During the 1st period,

477

w1 , w2 and w5 are assigned to the first operation in cell 1, and product type 1 is assigned to cell 2 for

478

processing with 7 hours of production quantity (equivalent to 78 packs). Because of the learning and

479

forgetting effects of workers, the production rate of each operation will change. Thus, the bottleneck

480

operation may transfer to another one in the next period. we can see that the workers are reassigned

481

during the 2nd and 3rd periods to smooth the operations.

w

h

h

(1,1) (1,1) (2,1) ! " (1,1) (2,1) (1,1) # " # " (1,2) (1,1) (1,2) # " # " (2,2) (2,1) (1,2) # " (1,1) (2,2) (1,1) # " # $ (2,1) (1,2) (2,2) %

(2, 7) (1,29) (1,29) ! " (2, 3) (2,13) (2,13) # " # " (2, 2) (2, 2) (2, 3) # " # " (2, 4) (2, 4) (2, 4) # " (2,13) (2, 7) (2, 7) # " # $ (1,29) (2, 3) (2, 2) %

q

(a) The first ingredient (c, j )W

(b) The second ingredient (c ! )Q! H

H

Figure 10: Final solution of the case using HBFA

VWSHULRG

QG SHULRG

q2

q3

q4

q5

         

q6

 

 

q1

q2

q3

q4

q5

q6

         

q1

q2

w4

w2,w4

q3

q4

q5

q6

          &HOO

&HOO

&HOO

w6

w3,w4

w2,w5

w6

w1,w3

w3

 

q1

&HOO

&HOO

&HOO

w1,w2,w5

UG SHULRG

w5

w1

w6

Figure 11: Diagram of the final solution

482

Fig. 12 shows detailed production planing of all types of products. Firstly, let us explain some 31

483

parameters with product type q5 as example. The demands for q5 in the 1st and 2nd periods are 85

484

packs and 67 packs, respectively. So the accumulated demand for q5 in the 2nd period is 152 packs.

485

The production volume of q5 in the 1st and 2nd periods are 93 packs and 62 packs, respectively. So the

486

accumulated production volume of q5 in the 2nd period is 155 packs. The holding volume of q5 in the

487

2nd period is the difference between accumulated production volume and accumulated demand (i.e.,

488

155-152=3 packs), and the corresponding holding cost is 6.

489

We can also see that the production volume of q5 in the 3rd period is less than its demand by 4 packs

490

(i.e., 86-82=4). Due to the holding volume (3 packs) of q5 in the 2nd period, its backorder volume in the

491

3rd period is only 1 pack. That is to say, production in advance may respond to the increasing customer

492

demand in the later period. In contrast, the backorder volume of q2 in the 1st period is 2 packs, and its

493

holding volume in the 2nd period is 10 packs. It implies that shortage of products can be solved in the

494

later period. In a word, the proposed HBFA aims to help the shop floor managers to make appropriate

495

worker assignment/reassignment and production planning so that the backorder and holding costs are

496

minimized.

497

5. Hybrid Genetic Algorithm and Hybrid Simulated Annealing

498

Genetic algorithm (GA) and simulated annealing (SA) are powerful and broadly applicable stochas-

499

tic search and optimization technique. GA is inspired by the natural evolution of the living organisms.

500

It simultaneously works on a population of solutions (i.e., chromosomes) which evolve through the

501

genetic operators, so that the chromosomes would approach the optimal solution generation by gen-

502

eration. SA is inspired by the physical annealing process studied in statistical mechanics. It repeats

503

an iterative neighbour generation procedure to improve the objective function value. While exploring

504

solution space, the SA also offers the possibility to accept worse neighbour solutions in order to escape

505

from local optima. Consequently, the GA and SA can be referred to as the classic methods of popula-

506

tion search and neighbour search, respectively. They have been benchmarks for comparison with other

507

proposed algorithms. Here, the hybrid genetic algorithm (HGA) and hybrid simulated annealing (HSA)

508

are suggested for solving the problem, so that we can follow this paradigm to compare the proposed

509

HBFA with HGA and HSA in the following experiment.

510

32

VWSHULRG

QGSHULRG

UGSHULRG

q1

q2

q3

q4

q5

q6

q1

q2

q3

q4

q5

q6

q1

q2

q3

q4

q5

q6

'HPDQGSDFNV





































$FFXPXODWHG GHPDQGSDFNV











            

SURGXFWLRQ YROXPHSDFNV













$FFXPXODWHG SURGXFWLRQ YROXPHSDFNV











            

%DFNRUGHU YROXPHSDFNV





































%DFNRUGHUFRVW





































+ROGLQJYROXPH SDFNV





































+ROGLQJFRVW









  



































Figure 12: Production planing of the case

33













511

Hybrid genetic algorithm:

512

• Chromosome representation: The chromosome representation employs the same solution struc-

513

ture of the HBFA. This helps to exclude the influence of different solution structures when com-

514

paring the HBFA with the HGA.

515

• Fitness function: Let Fgk denote the fitness value of the kth chromosome in generation g before

516

selection. It is computed as Fgk = ξ + max ψig − ψkg , where ψig is the objective function value i∈{1,...,E}

517

of the ith chromosome in generation g, E is the number of chromosomes in generation g before

518

selection, and ξ is a small constant (say 3). Obviously, the smaller the objective function value of

519

chromosome, the greater its fitness value.

520

521

• Initial population: Generate E = 20 initial chromosomes according to the method similar to Step 2 in the HBFA.

522

• Crossover: Randomly select two chromosomes in the parent generation to cross over until 0.9 · E

523

offsprings are generated. The crossover mechanism is similar to Algorithm 2. If the fitness value

524

of offspring is greater than the average fitness value of its parent generation, the offspring will be

525

accepted for the new generation; otherwise, it will be thrown out.

526

• Mutation: Each chromosome will mutate according to a given mutation probability Pm = 0.1.

527

The mutation mechanism is to randomly regenerate two feasible ingredients of the chromosome.

528

All offsprings from the mutation are accepted for the new generation.

529

• Selection: The most common method “roulette wheel” sampling is applied in the selection. Each

530

chromosome is assigned a slice of the circular roulette wheel and the size of the slice is propor-

531

tional to the selection value (i.e., fitness value) of the chromosome. The wheel is spun E times.

532

On each spin, the chromosome under the wheel’s marker is selected to be in the pool of parents

533

for the next generation.

534

• Stopping rule: The HGA is stopped when its runtime reaches a given CPU time.

535

Hybrid simulated annealing: 34

536

• Solution representation: The solution representation employs the same solution structure of the

537

HBFA. This helps to exclude the influence of different solution structures when comparing the

538

HBFA with the HSA.

539

• Initial solution: Generate an initial solution using the TPBH.

540

• Initial temperature: The initial temperature T 0 is set in such a way that the nonimproving solutions

541

are accepted with a probability of 95% in the primary iterations by using the equation T 0 =

542

−|OFV(X j )−OFV(Xi )| , ln(0.95)

543

X j , respectively. Based on this equation, at the initialization stage two solutions Xi and X j are

544

randomly generated and the initial temperature T 0 is determined (Kia et al., 2012).

545

546

547

548

where OFV(Xi ) and OFV(X j ) are objective function values of solutions Xi and

• Markov chain length: It is chosen in such a way that for each temperature level a thermal equilibrium can be attained. In the HSA, Markov chain length (ℓ) is set to 150. • Cooling rate: The temperature is decreased by using the common equation T r = λ · T r−1 , where λ is the cooling rate and it is set to 0.7.

549

• Stopping rule: The HSA is stopped when its runtime reaches a given CPU time.

550

• Neighborhood generation methods: One method generates neighborhood solution by using tum-

551

ble strategy (i.e., move in the first or second dimensional direction) which is shown in Fig. 8. The

552

other method randomly generates a feasible solution as neighborhood solution. The two methods

553

are employed with the same rate.

554

555

The pseudo code of the HSA is given in Algorithm 3.

6. Computational Experiments

556

We refer to the BFA with modified operators (including swimming strategy, reproduction strategy

557

and elimination-dispersal strategy) as DBFA, which employs randomly generated feasible solutions

558

as initial population. We refer to the BFA with original operators (including swimming strategy, re-

559

production strategy and elimination-dispersal strategy) as OBFA, which employs randomly generated

560

feasible solutions as initial population. Let HGA denote the GA which employs one solution generated 35

Algorithm 3: Hybrid Simulated Annealing 1. Initialize: Counter r = 0, n = 0 Generate initial solution as current solution Xc Set Xbest = Xc and compute objective function value OFV(Xc ) 2. While (1) Do 2.1. While(n < ℓ) Do 2.1.1. Generate neighborhood solution Xk and compute OFV(Xk ) 2.1.2. If(OFV(Xk ) 6 OFV(Xc )), then Xc = Xk If(OFV(Xc ) 6 OFV(Xb est)), then Xb est = Xc Else Generate a random number r in the interval [0-1], and set ∆ = OFV(Xk ) − OFV(Xc ) If(e−∆/Tr > r), then Xc = Xk 2.1.3. n := n + 1; 2.1.4. If the runtime reaches a given CPU time, then terminate the procedure 2.2. r := r + 1 and T r = λ · T r−1

36

561

by the TPBH and other randomly generated feasible solutions as initial population. Let HSA denote the

562

SA which generates an initial solution by the TPBH. We compare the performance of HBFA, DBFA,

563

OBFA, HGA and HSA through the following numerical experiments. To compare the solution quality

564

of these algorithms within the same CPU time, we modify the stopping rule of the DBFA and OBFA.

565

Let Ned of DBFA and OBFA be sufficiently large integers. The DBFA and OBFA are terminated if

566

their runtime reaches the HBFA’s runtime after reproduction. Moreover, the HGA is terminated if its

567

runtime reaches the HBFA’s runtime after selection, and the HSA is terminated if its runtime reaches

568

the HBFA’s runtime after the decrease of temperature.

569

The experiments are performed on a Pentium-based Dell-compatible personal computer with 2.30

570

GHz clock-pulse and 4.00 GB RAM. The HBFA, DBFA, OBFA, HGA and HSA algorithms are coded

571

in C++, compiled with the Microsoft Visual C++ 6 compiler, and tested under Microsoft Windows 7

572

operating system.

573

The performance of the five algorithms is to be evaluated by the use of five impact parameters,

574

including the number of workers (W), the number of product types (Q), the number of operations (J),

575

the number of cells (C), and the number of periods (H). Two groups of experiments are conducted. The

576

first group (displayed in Table 5) compares the performance between the HBFA, DBFA and OBFA, and

577

the second one (displayed in Table 6) compares the performance between the HBFA, HGA and HSA.

578

The instance (including small, medium and large size) have been randomly regenerated to verify the

579

proposed algorithm. Each group of experiment has five sets. For example, In the first set of Table 5, W

580

is allowed to vary to test its impact effect, given Q = 10, J = 3, C = 3 and H = 40. The other four sets

581

of Table 5 test the effects of varying Q, J, C and H, respectively. The other parameters for the randomly

582

generated instances are listed in Table 4. These parameters are given corresponding random integers

583

between the minimum and the maximum. Given a typical instance with W = 50, Q = 10, J = 15, C = 3

584

and H = 40, Fig. 13 shows the respective convergence of HBFA, DBFA, OBFA, HGA and HSA within

585

the same runtime.

586

Each entry of Tables 5 and 6 represents the average of its associated 10 randomly generated in-

587

stances. For example, 10 random instances are generated when C = 3, and another 10 random in-

588

stances are generated when C = 15. Let OFVHBFA , OFVDBFA , OFVOBFA , OFVHGA and OFVHS A denote

589

the average objective function values (OFV) using the HBFA, DBFA, OBFA, HGA and HSA, respec37

590

DBFA HBFA tively. Let ∆OFVOBFA denote the declining percentage of OFVDBFA over OFVOBFA , let ∆OFVDBFA

591

HBFA denote the declining percentage of OFVHBFA over OFVDBFA , let ∆OFVHGA denote the declining per-

592

HBFA centage of OFVHBFA over OFVHGA , and let ∆OFVHS A denote the declining percentage of OFVHBFA

593

over OFVHS A . 6

x 10

6

HBFA DBFA OBFA HGA HSA

5

OFV

4 3 2 1 0 0

100

200

300 CPU time (s)

400

500

600

Figure 13: Typical convergence of HBFA, DBFA, OBFA, HGA and HSA within the same runtime

594

In order to evaluate the effectiveness of modified operators (including swimming strategy, repro-

595

duction strategy and elimination-dispersal strategy) in the HBFA or DBFA and exclude the interference

596

of TPBH, we compare the performance of DBFA and OBFA instead of HBFA and OBFA. As can be

597

DBFA seen from Table 5, ∆OFVOBFA reaches 32%∼85% regardless of the variation of the five impact pa-

598

rameters W, Q, J, C and H. There exists the following condition: Through one or more chemotactic

599

steps, a bacterium may find a worse position than its best position in the current generation. So in the

600

swimming strategy of OBFA, a bacterium probably wastes many swimming steps but still results in

601

a worse position than its best position in the current generation. In the swimming strategy of DBFA,

602

i i Jcur , instead of Jlast , is used as a criterion for further swimming. Consequently, many invalid swimming

603

steps can be avoided.

604

In addition, for the accumulated cost scheme of reproduction of OBFA, it may not retain the fittest

605

bacterium for subsequent generation. In the reproduction strategy of DBFA, the bacterium with the best 38

Table 4: Parameters for randomly generated instances

Parameter

Min

Max

U j : Upper bound for worker level for operation j

x 2W y JC

x 3W y JC

Dqh : Demand of product types q during period h

50

100

θq : Unit backorder cost of product type q at the end of each period

5

10

φq : Unit holding cost of product type q at the end of each period

1

5

1.0

1.9

kw j : Steady-state production rate of worker w processing operation j

5

10

pw j : Accumulated initial experience of worker w for operation j

1

3

rw j : Cumulative operation work required to attain a level of kw j /2

3

6

αw j : Degree to which worker w forgets operation j

1

4

fq : Processing complexity coefficient of product type q

606

historic position in the current generation will be moved to the above position, and produce its offspring

607

replacing inferior bacterium. So the best bacterium can be passed on to the next generation. This

608

speeds up the convergence. Similarly, in the elimination-dispersal strategy of OBFA, the best bacterium

609

may be dispersed to an inferior position, whereas in the elimination-dispersal strategy of DBFA, the

610

best bacterium is kept unchanged and transferred to the subsequent stage. This also speeds up the

611

convergence and will not trap the solution into the local optima, because the tumble of chemotactic can

612

modify the position of each bacterium in a random dimension and helps to jump out of the local optima.

613

From Fig. 13 we can see that the original swimming, reproduction and elimination-dispersal strategies

614

of the OBFA cause frequent fluctuation of convergence curve, whereas the improved strategies of the

615

DBFA result in rapid and stable convergence.

616

In order to evaluate the effectiveness of TPBH in the HBFA, we compare the performance of HBFA

617

HBFA and DBFA. It can be observed from Table 5 that, ∆OFVDBFA reaches 5%∼97% regardless of the varia-

618

tion of the five impact parameters W, Q, J, C and H. Consequently, the TPBH plays an important role in

619

generating good initial solution of the HBFA to avoid the blind search of DBFA at the beginning while

620

exploring the solution space. In Fig. 13, the initial objective function values of the HBFA and DBFA

621

are 3.6 × 106 and 5.3 × 106 , respectively. The superior initial objective function value of the HBFA 39

Table 5: Performance comparison between the HBFA, DBFA and OBFA for impact parameters

Q = 10, J = 3 C = 3, H = 40

W

50 150 250 350 W = 50, J = 3 C = 3, H = 40

Q

10 20 30 40 W = 50, Q = 10 C = 3, H = 40

J

3 7 11 15 W = 50, Q = 15 J = 3, H = 40

C

3 7 11 15 W = 50, Q = 10 Q = 3, C = 3

H

40 80 120 160

OFVHBFA

OFVDBFA

OFVOBFA

DBFA △OFVOBFA (%)

HBFA △OFVDBFA (%)

CPU (s)

139,842 159,222 174,662 225,188

166,703 1,766,600 5,001,518 8,626,877

1,028,680 7,998,590 16,348,833 24,808,268

84 78 69 65

16 91 97 97

192 351 570 917

OFVHBFA

OFVDBFA

OFVOBFA

DBFA △OFVOBFA (%)

HBFA △OFVDBFA (%)

CPU (s)

136,804 245,743 361,624 852,902

163,921 257,742 414,966 1,478,485

1,038,430 1,062,984 1,726,800 4,610,533

84 76 76 68

17 5 13 42

146 306 551 694

OFVHBFA

OFVDBFA

OFVOBFA

DBFA △OFVOBFA (%)

HBFA △OFVDBFA (%)

CPU (s)

137,045 90,341 709,479 1,309,159

150,585 177,834 1,341,876 1,847,652

919,226 1,210,500 2,486,755 2,723,741

84 85 46 32

9 49 47 29

164 304 512 712

OFVHBFA

OFVDBFA

OFVOBFA

DBFA △OFVOBFA (%)

HBFA △OFVDBFA (%)

CPU (s)

206,338 248,127 289,874 296,077

221,085 263,313 306,955 347,926

1,036,135 1,247,965 1,417,453 1,253,237

79 79 78 72

7 6 6 15

227 411 506 694

OFVHBFA

OFVDBFA

OFVOBFA

DBFA △OFVOBFA (%)

HBFA △OFVDBFA (%)

CPU (s)

152,106 502,399 953,086 1,538,408

168,456 550,674 1,447,493 2,939,676

1,153,410 3,564,260 9,396,234 15,001,992

85 85 85 80

10 9 34 48

172 312 607 761

40

Table 6: Performance comparison between the HBFA, HGA and HSA for impact parameters

Q = 10, J = 3 C = 3, H = 40

W

50 150 250 350 W = 50, J = 3 C = 3, H = 40

Q

10 20 30 40 W = 50, Q = 10 C = 3, H = 40

J

3 7 11 15 W = 50, Q = 15 J = 3, H = 40

C

3 7 11 15 W = 50, Q = 10 Q = 3, C = 3

H

40 80 120 160

OFVHBFA

OFVHGA

OFVHS A

HBFA △OFVHGA (%)

HBFA △OFVHS A (%)

CPU (s)

139,842 159,222 174,662 225,188

273,452 263,650 763,022 977,968

330,190 287,613 2,303,369 4,327,560

49 40 77 77

58 45 92 95

192 351 570 917

OFVHBFA

OFVHGA

OFVHS A

HBFA △OFVHGA (%)

HBFA △OFVHS A (%)

CPU (s)

136,804 245,743 361,624 852,902

261,772 862,166 2,901,138 6,799,613

253,144 352,748 386,692 1,268,906

48 71 88 87

46 30 6 33

146 306 551 694

OFVHBFA

OFVHGA

OFVHS A

HBFA △OFVHGA (%)

HBFA △OFVHS A (%)

CPU (s)

137,045 90,341 709,479 1,309,159

257,610 1,369,820 2,966,801 3,162,651

296,654 228,666 1,234,307 1,643,250

47 93 76 59

54 60 43 20

164 304 512 712

OFVHBFA

OFVHGA

OFVHS A

HBFA △OFVHGA (%)

HBFA △OFVHS A (%)

CPU (s)

206,338 248,127 289,874 296,077

512,636 563,117 635,157 627,558

385,052 922,123 901,547 895,723

60 56 54 53

46 73 68 67

227 411 506 694

OFVHBFA

OFVHGA

OFVHS A

HBFA △OFVHGA (%)

HBFA △OFVHS A (%)

CPU (s)

152,106 502,399 953,086 1,538,408

259,593 837,169 1,938,859 2,993,194

270,556 1,050,900 2,639,394 3,688,998

41 40 51 49

44 52 64 58

172 312 607 761

41

622

gives rise to its better final result.

623

In order to evaluate the advantage of HBFA over other metaheuristics, we compare the performance

624

HBFA of HBFA, HGA and HSA. The results from Table 6 show that, ∆OFVHGA reaches 40%∼93%, despite

625

the variation of the five impact parameters W, Q, J, C and H. The reason can be demonstrated as follows:

626

There are some similarities and differences between the HBFA and HGA. The reproduction strategy

627

of HBFA is similar to the selection plus crossover of HGA, and the elimination-dispersal strategy of

628

HBFA is similar to the mutation of HGA. The HBFA, however, has its unique chemotactic strategy. It

629

is the tumble and the following swimming steps that lead to a deeper exploration of HBFA. In Fig. 13,

630

the HBFA obtains much better result than the HGA although they have similar initial objective function

631

HBFA values. We can also observe from Table 6 that, ∆OFVHS A reaches 6%∼95% in spite of the variation

632

of the five impact parameters. There is great difference of optimization mechanism between the HBFA

633

and HSA. For their comparison, the tumble step is employed in the neighborhood generation method

634

of HSA, but the HSA is often premature because of no swimming step for deep exploitation. Fig. 13

635

shows that the objective function value of the HSA can not decrease after CPU time reaches 160s.

636

7. Conclusions

637

In this paper a new optimization model of dynamic CMS in fiber connector manufacturing industry

638

is introduced along with a hybrid bacteria foraging algorithm (HBFA) embedding two-phase based

639

heuristic (TPBH). The advantage of the proposed model is simultaneously considering dynamic worker

640

assignment/reassignment and production planning by assuming multi-skilled workers, learning and

641

forgetting effects and operation sequence. Main constraints are workstation capacity for workers and

642

cell production capacity. The objective is to minimize the sum of backorder cost and holding cost of

643

inventory. The TPBH helps to generate a high quality initial solution for further search.

644

The main difference between the OBFA and DBFA is that the former applies Passino’s swimming

645

strategy, reproduction strategy and elimination-dispersal strategy, and the latter uses these modified

646

strategies. The HBFA tries to improve the search quality of DBFA by employing the TPBH to generate

647

initial solution. The performance of HBFA is evaluated and compared with the performance of DBFA,

648

OBFA, HGA and HSA in terms of objective function values within the same runtime.

649

It is observed that the quality of results obtained by HBFA is better than DBFA, OBFA, HGA and 42

650

HSA regardless of the variation of some important parameters. The TPBH plays an important role in

651

generating good initial solution of the HBFA to avoid the blind search of DBFA at the beginning while

652

exploring the solution space. The superiority of HBFA over OBFA lies in that the former may avoid

653

many invalid swimming steps and transfer the fittest bacterium to subsequent generation besides the

654

function of TPBH. So the HBFA pays more attention to the efficiency of exploitation and convergence

655

than the OBFA. The advantage of HBFA over HGA can be explained in that the HBFA has unique

656

chemotactic strategy besides the characteristics of crossover, mutation and selection which the HGA

657

possesses. So the HBFA does well in balancing the depth of exploitation and the width of exploration,

658

and pays more attention to the depth of exploitation than the HGA and even most efficient GA. The

659

HBFA also shows merit of deep exploitation when compared with the HSA.

660

An important research direction that may be pursued in the future is to extend single-level flexi-

661

bility of workers to multi-level flexibility, i.e., each worker has a different number of skills. The other

662

potential interest would consider some worker-related cost terms such as worker hiring cost and worker

663

firing cost when hiring and firing are allowed to adjust cell production capacity.

664

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46

Highlights     

Worker assignment and production planning decisions are made simultaneously. Bottleneck workstation may transfer due to learning and forgetting effects. Late delivery and production in advance result in backorder and holding costs. The hybrid bacteria foraging algorithm embeds a heuristic and evolution operators. The superiority of proposed algorithm over other metaheuristics is illustrated.