Cell Model Assignment

Cell transmission mechanism

To solve the DAURTN based on the continuous transmission, we build the CTM for the urban rail network (V,A). For describing the CTM, the cell transmission network is constructed based on cell from the network as follows.

We define each section as a cell chain, and each station as a station cell. For any section , as travel time is fixed, we divide the section into several transmission cells by ΔT. The transmission cells of one section compose a cell chain, and passengers flows can be transmitted forward between the transmission cells. Note that travel time may not be exactly divided by ΔT, so the time length of the tail cell can be equal to or exceed ΔT. The number of cells divided is , where ⌊x⌋ is a function of the maximum integer no larger than x. Therefore, we denote Cell(Lld,i,j) as the jth cell in the cell chain of section . Denote m(Lld,i) as the number of cells in the ith cell chain, and for simplifying the notation, denote Cell(Lld,i,end) the last cell in the cell chain of section . Denote yh(Lld,i,j,tk) as the flow of Cell(Lld,i,j) traveling to destination sh in tk. We also denote Cell(su) as the station cell of su, and yh(su,tk) as the flow of Cell(su) traveling to destination sh in tk.

The transmission relationship between cells is illustrated in Fig 3, where a hollow node represents a station cell, a solid node represents a transmission cell, a hollow rectangle represents the corresponding cell chain of a section, and the arrows represent transmission directions. In each time interval, passengers follow the instantaneous dynamic route choice principle [22] and are transmitted between cells. Transmission mechanism between cells is designed and transmission processes of flows between the cells are classified into 4 groups: where ‘⟹’ means the transmission process of flow from the left cell to the right cell.

The flows of station cells and transmission cells at the initial interval t0 are 0. In tk(k ≥ 1), the O–D demand {quh(tk)|shS} inflows into station cell Cell(su). Therefore, the initial value of the variables are yh(su,t0) = 0, yh(su,tk) = quh(tk), yh(Lld,i,j,tk) = 0, ∀h,u,i,j,Lld,k ≥ 1.

Next, we analyze the transmission mechanism in 3 steps.

Step 1: The transmission processes Cell(Lld,i,end) ⟹ Cell(Lld,i + 1,1), and

As the length of time in the tail cell Cell(Lld,i,end) is equals to or greater than ΔT, only a certain proportion of flow can outflow, and the proportion is . Thus, the outflow of Cell(Lld,i,end) in interval tk is (9)

Then we can obtain the detained flow of the tail cell (10) where ‘←’ denotes that the value of the right variable is assigned to the left variable.

According to the space priority principle, the outflow fh(Lld,i,end,tk) of tail cell Cell(Lld,i,end) has only two choices, i.e., Cell(Lld,i + 1,1) or , and the transmission choice is determined by the instantaneous dynamic route choice principle [22], i.e., the shortest path from platform to destination station sh at the current time interval in network (V,A). If the shortest path passes through , then flow fh(Lld,i,end,tk) is transmitted into cell Cell(Lld,i + 1,1); otherwise, it is transmitted to station cell . The above transmission choice is similar to the all-or-nothing assignment, i.e., the flows follow the shortest path.

We denote the set of destinations to which the shortest path from platform passes through as (11)

When , the flow fh(Lld,i,end,tk) is transmitted from Cell(Lld,i,end) to Cell(Lld,i + 1,1). Thus, (12)

When , the flow fh(Lld,i,end,tk) is transmitted from Cell(Lld,i,end) to . Note that the average transfer time at station is , so (13)

In order to realize the FCFS in transmission mechanism, we introduce a variable xh(su,tv), 1 ≤ vN, which represents the flows arriving at station su in tv and detained at the station in tk.

(14)

Step 2: The transmission process Cell(Lld,i,j) ⟹ Cell(Lld,i,j + 1)

After Step 1, in the tail cell of the cell chain, there may be some detained flows, and then the flow of the tail cell equals to the detained flows plus the flows from Cell(Lld,i,end − 1), so (15)

For other cells in the chain, it is only need to move flows from the forward cell to the backward cell in the chain, that is, (16)

Step 3: The transmission process

According to the principle of the space priority, flows from Cell(Lld,i − 1,end) are transmitted to Cell(Lld,i,1) and occupy the capacity of Cell(Lld,i,1) with priority. Thus, the surplus capacity of Cell(Lld,i,1) in tk is (17) The flows in tk, which are queuing at station su and head to Cell(Lld,i,1), have to compete for the surplus capacity with the FSFC principle.

In order to determine the queuing flow, passengers at station cell Cell(su) first determine which platform to queue. Similar to the method in Step 1, passengers determined the platform by the shortest path from station su to destination station sh at the current time interval in network (V,A). If the shortest path passes through , then flows traveling to destination station sh queue on platform . We denote the set of destinations to which the shortest path from station su passes through as (18) Thus, the flow competing for the surplus capacity is .

If , then (19)

If , which means the surplus capacity is insufficient, then there exists , and it makes that According to the FCFS principle, the flow traveling to each destination station can be transmitted, i.e., (20) and a portion of can also be transmitted. According to the equal proportion principle, the proportion of flows transmitted can be calculated by (21) Then (22)

After the above processes, the flow of each cell make a choice by the shortest paths and are all transmitted to the next cell. But the cost of each arc will be changed with the variable flow, so the method of successive average (MSA) is adopted to reach the instantaneous dynamic user optimal state in each time interval. The variables in the above model are updated in MSA.

An efficient method for solving the shortest path

In the CTM, it is needed to solve the shortest path in tk from s ∈ SSΩ to suS in network (V,A). We design a fast method for solving the shortest path as follows.

If the shortest path from sSSΩ to suS passes through several transfer stations, then the shortest path can be divided into three segments at most. The first segment of the shortest path is from origin s to the first transfer station , and its length is denoted as . The last segment of the shortest path is from the last transfer station to destination su, and its length is . As long as we solve the length of the shortest path between any two transfer stations , we can obtain the cost of the shortest paths in three cases as follows: (23)

If the shortest path from sSSΩ to suS does not pass through any transfer station, then it will only use one line and can be solved easily.

The above analysis indicates that the solving method for the shortest path from sSSΩ to suS can be decomposed into 3 steps.

Step 1: calculate the shortest path from sSSΩ to suS in each network G(LlU,LlD), which composed of a pair of opposite directional lines LlU,LlD shown in Fig 1.

Step 2: calculate the shortest path between each two transfer stations in the network.

Step 3: calculate all the shortest paths from sSSΩ to suS.

In step 1, for any destination , we can structure two subsets of nodes bounded by node , i.e. and , which can form two generated sub-networks of G(LlU,LlD). Obviously, the shortest paths from other nodes to su in the two sub-networks are equal to solving the shortest paths in G(LlU,LlD). In the former sub-network, there are three cases.

Case 1: solve the shortest paths from nodes to su along the directional line LlU;

Case 2: solve the shortest paths from

  • 1.

    Fry T, Breen M, Wilson M (1987) A successful implementation of group technology and cell manufacturing. Prod Inv Manag J 28(3):4–6Google Scholar

  • 2.

    Collet S, Spicer R (1995) Improving productivity through cellular manufacturing. Prod Inv Manag J 36(1):71–75Google Scholar

  • 3.

    Levasseur G, Helms M, Zink A (1995) Conversion from a functional to the cellular manufacturing layout at Steward Inc. Prod Inv Manag J 36(3):37–42Google Scholar

  • 4.

    Singh N, Rajamaani D (1996) Cellular manufacturing systems: planning and control. Chapman and Hall, New YorkGoogle Scholar

  • 5.

    Mungwatanna A (2000) Design of cellular manufacturing systems for dynamic and uncertain production requirement with presence of routing flexibility. Ph.D. thesis, Blacksburg State University VirginiaGoogle Scholar

  • 6.

    Seifoddini H (1990) A probabilistic model for machine cell formation. J Manu Sys 9(1):69–75CrossRefGoogle Scholar

  • 7.

    Harahalakis G, Nagi R, Proth J (1990) An efficient heuristic in manufacturing cell formation to group technology applications. Int J Prod Res 28(1):185–198 DOI 10.1080/00207549008942692CrossRefGoogle Scholar

  • 8.

    Song S, Hitomi K (1996) Integrating the production planning and cellular, layout for flexible cell formation. Prod Plan Con 7(6):585–593 DOI 10.1080/09537289608930392CrossRefGoogle Scholar

  • 9.

    Chen M (1998) A mathematical programming model for systems reconfiguration in a dynamic cell formation condition. Ann Oper Res 77(1):109–128 DOI 10.1023/A:1018917109580MATHCrossRefGoogle Scholar

  • 10.

    Wicks E (1995) Designing cellular manufacturing systems with time varying product mixed and resource availability. PhD thesis, Virginia Polytechnic Institute and State University, Blacksburg, VAGoogle Scholar

  • 11.

    Tavakkoli R, Aryanezhad M, Safaei N, Azaron A (2005) Solving a dynamic cell formation problem using met heuristics. App Math Comput 170:761–780 DOI 10.1016/j.amc.2004.12.021MATHCrossRefGoogle Scholar

  • 12.

    Safaei N, Saidi-Mehrabad M, Jabal-Ameli M-S (2006) A hybrid simulated annealing for solving an extended model of dynamic cellular manufacturing system. Eur J Oper Res, DOI 10.1016/j.ejor.2006.12.058Google Scholar

  • 13.

    Safaei N et al (2007) A fuzzy programming approach for a cell formation problem with dynamic and uncertain conditions: Fuzzy Set Syst, DOI 10.1016/j.fss.2007.06.014

  • 14.

    Bidanda B et al (2005) Human-related issues in manufacturing cell design, implementation, and operation: a review and survey. Comput Ind Eng 48:507–523 DOI 10.1016/j.cie.2003.03.002CrossRefGoogle Scholar

  • 15.

    Balakrishnan J, Cheng C-H (2007) Multi-period planning and uncertainty issues in cellular manufacturing: a review and future directions. Eur J Oper Res 177:281–309 DOI 10.1016/j.ejor.2005.08.027MATHCrossRefGoogle Scholar

  • 16.

    Suer G, Bera I (1998) Optimal operator assignment and cell loading when lot-splitting is allowed. Com & Ind Eng 35(3–4):431–434 DOI 10.1016/S0360-8352(98)00126-0CrossRefGoogle Scholar

  • 17.

    Askin R, Huang Y (2001) Forming effective worker teams for cellular manufacturing. Int J Prod Res 39(11):2431–2451 DOI 10.1080/00207540110040466MATHCrossRefGoogle Scholar

  • 18.

    Norman B et al (2002) Worker assignment in cellular manufacturing considering technical and human skills. Int J Prod Res 40(6):1479–1492 DOI 10.1080/00207540110118082MATHCrossRefMathSciNetGoogle Scholar

  • 19.

    Min H, Shin D (1993) Simultaneous formation of machine and human cells in group technology: a multiple objective approach. Int J Prod Res 31(10):2307–2318 DOI:10.1080/00207549308956859CrossRefGoogle Scholar

  • 20.

    Park P (1991) The examination of worker cross-training in a dual resource constrained job shop. Eur J Oper Res 51:291–299 DOI 10.1016/0377-2217(91)90164-QCrossRefGoogle Scholar

  • 0 comments

    Leave a Reply

    Your email address will not be published. Required fields are marked *