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Abstract

To unify the merits of traditional in-plant parts logistics alternatives such as line stocking and kitting, the concept of line-integrated supermarkets is introduced to improve the part feeding in mixed-model assembly lines. First, the highly interdependent optimization problems of assigning stations and scheduling logistics operators are described, and mathematical models are established with the aim to minimize the fleet size of logistics operators and unit part delivery time as well. Together with particular theorems and lemmas, a nested dynamic programming is presented to obtain global optimum for small-sized instances while a modified harmony search algorithm is constructed for medium- or large-sized instances. Benefit from repeatedly dividing and reconstructing the harmony memory, the computation speed is significantly enhanced. Meanwhile, crossover and mutation operations effectively improve the diversity of solutions to overcome deficiencies such as limited search depth and tendencies to trapping into local optimum. Finally, experimental results validate that the proposed algorithm is of competitive performance in effectiveness and efficiency compared to some other basic or modified meta-heuristics.

Keywords

Part feeding line-integrated supermarkets mixed-model assembly lines dynamic programming harmony search algorithm

Introduction

In-plant logistics, which starts with parts reception from upstream suppliers and takes over all steps including parts storage, sequencing, delivery to assembly lines and line-side presentation,¹ is a complicated and indispensable process in automotive industry.^2,3 In today’s competitive environment, mixed-model assembly lines (MMALs) are widely adopted to tailor to the customization tendency, which increases the complexity of parts logistics. The planning process becomes a critical challenge on account of the enormous quantity and diversity of parts required in final assembly, ranging from color, texture to shape, size, weight and so on.

In the context of in-plant logistics, the principles of line stocking and kitting are extensively discussed and utilized to feed assembly lines in automotive industries.^4–6 Line stocking, also noted as continuous supply or bulk feeding,⁷ means that homogeneous parts stored in large containers are directly positioned at the border of the line (BoL). With this principle, additional double-handling operations are scarcely required. Moreover, it provides flexibility in case of unexpected events that may cause great loss such as line stoppages. However, these full packing units usually occupy the bulk of the limited space at the line,⁸ which obstructs the assembly process. Instead of taking the right part directly from the containers, assembly workers have to walk around and search the required one from the multiple instances of the same part.⁷ For these reasons, the traditional line stocking is increasingly displaced by kitting, where all required parts for a future unit of end item are selected and packed in a kit. Despite the fact that non-value adding activities^9,10 such as preparing kits in advance are inevitably increasing, it is more predictable to transport required kits to specific stations since further demand of new kits is synchronized with the production plan of the assembly line. According to the position the kits are transported to, the kitting policy can be further divided into traveling and stationary kitting.^11,12 Considered as a particular form of stationary kit, sequencing introduced by Johansson et al.¹³ is nowadays widely applied. Successful cases of kitting are especially practiced in lean production systems such as Toyota Motor Corporation,¹⁴ which is termed as Set Parts Supply (SPS) where quality, cognitive and flexible requirements are addressed as well.¹⁵ However, it is still impossible for practitioners and researchers to reach an agreement on what the best strategy is Hanson,¹⁶ Limère et al.⁷ and Caputo et al.¹¹

In recent years, additional hybrid part feeding methods noted as downsizing, Kanban-based continuous supply^12,17,18 and novel strategies such as part supply with line-integrated supermarkets¹⁹ are gradually proposed to actualize reliable and efficient parts feeding. Different from line stocking in the size of unit loads, downsizing⁸ or Kanban-based continuous supply^20–22 applies smaller containers with more frequent replenishments and deliveries. Especially with the concept of just-in-time (JIT) production system, Kanban-based policy is nowadays extensively adopted.²³ A detailed comparison of line stocking and Kanban-based policy can be found in Hanson and Finnsgård,²⁴ where interrelationship between size of unit loads and materials handling efficiency was discussed as well. Despite the similarities with line stocking, these hybrid strategies also possess some advantages and disadvantages of kitting such as the decrease in the walking distances and inventory at BoL, but obviously more additional handling activities. In general, parts are prepared in supermarkets^18,25 and transported to specific stations by means of automatic guided vehicles (AGVs), tow-trains or milk-run systems.^26–28 In this aspect, complex issues involving location planning, route optimization, tow-trains scheduling and loading^29–31 interrelate with each other and have aroused great attention.

To draw the merits of line stocking, kitting and hybrid part feeding method, Boysen and Emde¹⁹ introduced a novel concept known as line-integrated supermarkets. The idea has been successfully applied at a North-American plant of a popular German auto manufacturer. Compared with traditional supermarkets, line-integrated supermarkets are closer to the assembly line, and instead of AGVs or tow-trains, logistics operators are employed to prepare small kits just in sequence (JIS) and deliver to stations. Therefore, in contrast to kitting, double-handling activities are to some extent reduced and thus equipment costs are saved. However, required parts is generally available even though something unforeseen occurs. The JIS concept further facilitates the parts retrieval and alleviates ergonomic burdens on assembly workers. Battini et al.³² also mentioned it as fish-bone supermarkets in their research about in-plant logistics. They pointed out that in addition to logistics operators, monorail shuttle systems could be established to realize automated part feeding with higher efficiency and lower cost. Nevertheless, as far as we concerned, in spite of great theoretical and practical significance, there is no more literature about the novel line-integrated supermarkets. Accordingly, it is of great significance to implement a deep investigation into in-plant part feeding with the concept of line-integrated supermarkets.

To follow the new trend of in-plant logistics and provide a different alternative for automotive producers and even other manufacturing systems, a novel cyclic part feeding strategy for assembly lines is investigated in this article combining the line-integrated supermarkets. With the aim to clarify and balance the responsibilities of each logistics operator, all stations are first divided into several independent subsets and each is assigned to an operator. Moreover, for a predetermined subset of stations the scheduling problem is to be treated, which shares some similarities to scheduling a single vehicle^33,34 such as determining the right parts with proper quantity at the right time. It has been proved to be a non-deterministic polynomial-time (NP)-hard problem with high complexity. To reduce decision variables, a cyclic part feeding strategy is considered to determine an optimal time interval for each subset. Hence, the number of tours can be deduced and the departure time for each tour can be specified as well. In each tour, instead of satisfying a particular station, a logistics operator is required to satisfy all the assigned stations before the next tour. With the aim to minimize total cost, both fixed cost for employing logistics operators and parts delivery cost should be decreased. Consequently, for small-sized instances, a nested dynamic programming (NDP) is adopted in this article, and for comparatively medium or large-sized cases, a modified meta-heuristics is introduced.

While dynamic programming (DP) is widely used to obtain optimal solutions for small-sized optimization problems,^29,30 the exponential growth in search space makes it impossible to solve large-sized problems within desirable time. Therefore, meta-heuristics and related hybrid algorithms^33–35 are applied to gain satisfactory solutions. Harmony search algorithm (HSA), whose global convergence is proved in solving discrete optimization problems,³⁶ is simple in theory, easy to accomplish and has been widely used in various fields.^37–40 Meanwhile, there is still possibility that HSA will trap into local optimum under complex conditions. Consequently, a modified harmony search algorithm (MHSA), with an attempt to gain global optimum, is constructed in this article to tackle large-sized instances.

The remainder of this article is organized as follows. In section “Problem description,” a problem description is provided for the part feeding problem using line-integrated supermarkets with detailed assumptions, notations and mathematical models. Section “Algorithm construction” presents lemmas and theorems of the proposed model. The NDP and MHSA, respectively, for small-sized and medium or large-sized instances are introduced as well. Computational experiments are designed and conducted to validate the proposed algorithm in section “Computational study.” Finally, section “Conclusion” draws conclusions and highlights areas for further research.

Problem description

The novel part feeding system for a given assembly line segment using line-integrated supermarkets is schematically presented in Figure 1. In this context, the decentralized logistics areas to store the required parts are directly located next to the respective stations. At each station, one or multiple JIS bins are settled to temporarily store the parts required by assembly. It should be noted that each JIS bin is available to a certain kind of parts. Logistics operators are employed to select and pack the right parts into the right JIS bins in accordance with the production plan. This article treats two main optimization problems.

Figure 1.

Part feeding with line-integrated supermarkets.

Stations assignment problem

The assignment problem (or P1, for short) is to determine the number of logistics operators to be hired for parts delivery and the partition of the given stations. Each logistics operator is responsible for one or several consecutive stations. As shown in Figure 1, there are three operators to serve the five stations, each responsible for one, two and two stations, respectively. It is obvious that the probability of parts shortage will reduce as the number of logistics operators increases. However, the increasing number of operators results in higher labor cost and more idle time, making it reasonable to minimize the number on the contrary.

Operators scheduling problem

The scheduling problem (or P2, for short) is to make cyclic delivery schedules that each operator starts a delivery tour every a certain time interval. When the time interval is determined, the departure time of each delivery tour can be deduced with ease. For each logistics operator, required parts of all assigned stations should be taken into account in each tour. Thus, there is no need to specify the traditional decision variables like the destination station and the part quantity for each delivery tour. The decision of the optimal interval aims at avoiding stock-outs at the stations while minimizing the delivery time per part (the expression unit delivery time will be used in the following sections) at the same time. However, cyclic deliveries facilitate the control of part feeding and make it easier to employ the automated handling with a monorail shuttle system to displace logistics operators.

To precisely describe the part feeding system with line-integrated supermarkets, the following premises are introduced:

Parts shortage is not allowed, which is always the first priority of in-plant logistics.

All time units are normalized to the equidistant length of production cycle.

Given the operator capacity, the number of parts delivered by each operator within a tour is limited.

The delivery process cannot be interrupted. Once departing, a logistics operator must go through all the assigned stations and return back to the logistics area.

Each JIS bin is of certain size, which stores limited parts of a single type.

Time for picking parts, loading and unloading parts, and walking from logistics area to station or between stations are predetermined.

The production sequence and correspondingBill of Material (BOM) are known with certainty.

The manufacturing system is a relatively stable system, and uncertain factors are not considered.

Given the parameters and variables summarized in Table 1, and combined with the problem description and premises mentioned above, the optimization problem in this article can be formulated by the following objective functions and constraints

P 1 : Minimize Z 1 = \min | W |

(1)

s . t . δ_{1} = 1

(2)

δ_{w + 1} - 1 ⩾ δ_{w} \forall w \in W

(3)

δ_{| W | + 1} = | S | + 1

(4)

Table 1.

Notations for the mathematical model.

$S$	Set of stations (index $s \in S$ )
$W$	Set of logistics operators (index $w \in W$ )
$T$	Number of total production cycles(index $t = 1, 2, \dots, T$ )
$ct$	Duration of a production cycle
$B_{s}$	Capacity of the JIS bin at station $s$
$d_{s}^{t}$	Part demand at station $s$ in production cycle $t$
$wt$	Walking time from logistics area to station
$st$	Walking time between stations
$a t_{w}$	Time for picking, loading and unloading parts
$C_{w}$	Capacity of logistics operator $w$
$δ_{w}$	The left border of the stations assignedto logistics operator $w$
$T_{w}$	The delivery interval of logistics operator $w$ , and $T_{w} ⩾ 1$
$D_{w}$	The duration the logistics operator $w$ needsto finish a delivery tour, and $D_{w} = a t_{w} + 2^{*} [wt + st \cdot (δ_{w + 1} - 1 - δ_{w})]$
$R_{w}$	Number of delivery tours for logistics operator $w$ (index $r = 1, 2, \dots, R_{w}$ )
$Q_{r}^{w}$	The number of parts delivered by logistics operator $w$ in tour $r$
$F_{w}$	The fixed cost for employing a logistics operator
$u$	Parts delivery cost per part per cycle time

JIS: just in sequence.

The assignment problem aims to minimize the crew of logistics operators as shown by equation (1), where $| W |$ denotes the number of logistics operators and “||” means to get the number of elements in a set. Equations (2) and (3) ensure that each logistics operator serves at least one station and no overlap is allowed. Equation (4) confines the range of number of logistics operators and assigned stations. $| S | + 1$ and $| W | + 1$ are, respectively, the “dummy station” and “dummy operator,” which are not included in the objective. Moreover, with the common purpose to minimize the system cost, great efforts should be taken to decrease the cost for non-value adding activities like parts logistics. Therefore, objective function equation (5) of the scheduling problem for each logistics operator aims at minimizing the unit delivery cost. Equations (6) and (7) separately ensure that the quantity of parts delivered to stations in a single or overall tours meets the required demand. Equations (8) and (9) are, respectively, the operator and JIS bin capacity constraints. Equation (10) guarantees that a logistics operator has sufficient time to finish a tour before the next one

P 2 : Minimize Z 2 (T_{w}) = \max \frac{1}{R_{w}} \sum_{r = 1}^{R_{w}} \frac{D_{w}}{Q_{r}^{w}}

(5)

\begin{array}{l} s . t . Q_{r}^{w} = \sum_{s = δ_{w}}^{δ_{w + 1} - 1} \sum_{t = r \cdot T_{w} / c t}^{(r + 1) \cdot T_{w} / c t} d_{s}^{t}, \forall w \in W, s \in S, \\ r = 1, 2, …, R_{w} \end{array}

(6)

\begin{array}{l} \sum_{r = 1}^{R_{w}} Q_{r}^{w} = \sum_{s = δ_{w}}^{δ_{w + 1} - 1} \sum_{t = 1}^{T} d_{s}^{t}, \forall w \in W, s \in S, \\ r = 1, 2, …, R_{w} \end{array}

(7)

Q_{r}^{w} ⩽ C_{w}, \forall w \in W, r = 1, 2, \dots, R_{w}

(8)

\sum_{t = r \cdot T_{w} / ct}^{(r + 1) \cdot T_{w} / ct} d_{s}^{t} ⩽ B_{s}, \forall w \in W, s \in S, r = 1, 2, \dots, R_{w}

(9)

D_{w} ⩽ T_{w}, \forall w \in W

(10)

There is obviously correlation between the two sub-problems. The stations assignment problem determines the number of logistics operators and the subset of stations for each operator, which are adopted as given parameters to solve the scheduling problem. Accordingly, only when the scheduling problem for each operator is performed, it can be confirmed whether the solution for the assignment problem is feasible. Therefore, it is advisable to consider the two sub-problems simultaneously. Together with equations (1) and (5), the main objective function is constructed as equation (11), which aims to minimize the total cost consisting of logistics operators and part feeding costs, and “*” means the optimal solution to the scheduling problem for operator $w$ . Obviously, equation (11) subjects to constraint equations (2)–(4) and (6)–(10)

Minimize Z = Z 1 \cdot F_{w} + u \cdot \sum_{w \in W} [\sum_{r = 1}^{R_{w}} Q_{r}^{w} \cdot Z 2^{*} (T_{w})]

(11)

Algorithm construction

In this section, lemmas and theorems are first given to facilitate the problems. Then, an exact method for small-sized instances is described, which is based on DP and heuristic rules. However, with the increase in production cycles and stations, it is impossible to get a global optimum. Consequently, a modified meta-heuristics named MHSA is presented to obtain satisfactory solutions.

Lemmas and theorems

Theorem 1

If parameters $S, T, d_{s}^{t} and C_{w}$ are given, then when $\forall w \in W, T_{w} = 1$ and $\forall t = 1, 2, \dots, T$ , $Φ = \max \sum_{s \in S} d_{s}^{t}$ , the lower bound of $| W |$ can be defined as ${| W |}_{\min} = ⌈ Φ / C_{w} ⌉$ .

Proof

According to constraint equation (7), $\sum_{w \in W} \sum_{r = 1}^{R_{w}} Q_{r}^{w} = \sum_{s \in S} \sum_{t = 1}^{T} d_{s}^{t}$ , which means the total delivery amount by logistics operators must satisfy the total demand in the whole planning horizon. Then, $\forall w \in W,$ $T_{w} = 1$ , it is obvious that $R_{w} = ⌈ T / T_{w} ⌉ = T$ . That is to say, $\forall r = t = 1, 2, \dots, T$ , $\sum_{s \in S} d_{s}^{t} = \sum_{w \in W} Q_{r}^{w}$ . Together with constraints equations (2) and (8), the logistics operators must deliver all required parts to all assigned stations, so ${| W |}_{\min} = ⌈ \max \sum_{s \in S} d_{s}^{t} / C_{w} ⌉$ , and theorem 1 is proved.

Lemma 1

If logistics operator $w$ is responsible for stations $m, \dots, n$ , and the optimal interval and number of tours are, respectively, $T_{w}$ and $R_{w}$ , then when $Q_{1}^{w} = \dots = Q_{r}^{w} = \dots = Q_{R_{w}}^{w}$ , $Z 2 (T_{w})$ reaches the minimum.

Proof

If stations $m, \dots, n$ are assigned to operator $w$ , then $D_{w} = a t_{w} + 2^{*} [wt + st \cdot (n - m)]$ . Besides, $Z 2 (T_{w}) =$ $(1 / R_{w}) \sum_{r = 1}^{R_{w}} D_{w} / Q_{r}^{w} = (D_{w} / Q_{1}^{w} + … + D_{w} / Q_{r}^{w} + … D_{w} / Q_{R_{w}}^{w}) / R_{w}$ . Therefore, $Z 2 (T_{w})$ reaches the minimum when $Δ = D_{w} / Q_{1}^{w} + \dots + D_{w} / Q_{r}^{w} + \dots D_{w} / Q_{R_{w}}^{w}$ is minimized. In accordance with the theory of mathematical inequalities, then

\begin{matrix} (\frac{1}{Q_{1}^{w}} + \dots + \frac{1}{Q_{r}^{w}} + \dots \frac{1}{Q_{R_{w}}^{w}}) \\ (Q_{1}^{w} + \dots + Q_{r}^{w} + \dots + Q_{R_{w}}^{w}) \\ ⩾ (1 + \frac{Q_{2}^{w}}{Q_{1}^{w}} + \dots + \frac{Q_{r}^{w}}{Q_{1}^{w}} + \dots + \frac{Q_{R_{w}}^{w}}{Q_{1}^{w}}) \\ + (\frac{Q_{1}^{w}}{Q_{2}^{w}} + 1 + \dots + \frac{Q_{r}^{w}}{Q_{2}^{w}} + \dots + \frac{Q_{R_{w}}^{w}}{Q_{2}^{w}}) \\ + \dots + (\frac{Q_{1}^{w}}{Q_{R_{w}}^{w}} + \dots + \frac{Q_{r}^{w}}{Q_{R_{w}}^{w}} + \dots + 1) \end{matrix}

When and only when $1 / Q_{1}^{w} = \dots = 1 / Q_{r}^{w} = \dots = 1 / Q_{R_{w}}^{w}$ , or $Q_{1}^{w} = \dots = Q_{r}^{w} = \dots = Q_{R_{w}}^{w}$ , “=” can be gained, and Lemma 1 is proved.

Theorem 2

Logistics operator $w$ is responsible for stations $m, \dots, n$ . If there are two feasible time intervals $T_{w}^{X}, T_{w}^{Y}$ with corresponding numbers of tours $X, Y$ , and $a_{i}, b_{j} (i = 1, 2, \dots, X; j = 1, 2, \dots, Y)$ , respectively, means the delivery amount for each time, then when $T_{w}^{X} > T_{w}^{Y}$ or $X < Y$ , $Z 2 (T_{w}^{X}) < Z 2 (T_{w}^{Y})$ , which means $T_{w}^{X}$ is preferred.

Proof

According to Lemma 1, then $Z 2 (T_{w}^{X}) = [X \cdot D_{w} / (\sum_{i = 1}^{X} a_{i} / X)] / X = X \cdot D_{w} / \sum_{i = 1}^{X} a_{i}$ , and $Z 2 (T_{w}^{Y}) = [Y \cdot D_{w} / (\sum_{j = 1}^{Y} b_{j} / Y)] / Y = Y \cdot D_{w} / \sum_{j = 1}^{Y} b_{j}$ . Together with equation (7), $\sum_{i = 1}^{X} a_{i} = \sum_{j = 1}^{Y} b_{j} = \sum_{s = m}^{n} \sum_{t = 1}^{T} d_{s}^{t}$ . Hence, $Z 2 (T_{w}^{X}) < Z 2 (T_{w}^{Y})$ , and Theorem 2 is proved.

On the basis of Theorem 2, when the partition of stations is predetermined, and if the feasible number of tours of logistics operator $w (R_{w})$ is minimized, the value of $Z 2 (T_{w})$ is optimized. If $\forall w \in W, \exists w = w^{*}$ , the minimal is $Z 2 (T_{w *})$ , then the lower bound of $Z$ is calculated as $\underline{Z} = | W | \cdot F_{w} + u \cdot \sum_{s \in S} \sum_{t = 1}^{T} d_{s}^{t} \cdot Z 2 (T_{w *})$ .

Theorem 3

If the total number of logistics operators $| W |$ is constant, and a feasible alternative for stations assignment problem is $Ω = {1, δ_{2}, \dots, δ_{w}, \dots, S + 1}$ . To construct a neighborhood $Ω' = {1, δ_{2}, \dots, δ'_{w}, \dots, S + 1}$ , making $δ'_{w} \leftarrow δ_{w} \pm 1$ , accordingly $D'_{w} \leftarrow D_{w}$ , $T'_{w} \leftarrow T_{w}$ and $R'_{w} \leftarrow R_{w}$ . If $Z 2 (T_{w}^{'}) < Z 2 (T_{w})$ , the neighborhood is better than the original alternative.

Proof

Theorem 3 is obviously right. When $Z 2 (T'_{w}) < Z 2 (T_{w})$ and the number of operators keeps constant, which means the fixed cost remains, then in accordance with constraint equation (11), $Z (Ω') < Z (Ω)$ and the alternative is optimized and should be updated.

NDP

Actually, the solution to the two sub-problems is a multi-stage decision-making process. Let $G = (V, E)$ be a directed graph with a set of vertices $V = S \cup {| S | + 1}$ and a set of edges $E = {(i, j) | i, j \in V}$ . Each vertex represents a station, and $| S | + 1$ is the dummy station. The edges connecting two vertices are characterized by a non-negative cost $μ (i, j)$ , which stands for the employment cost and delivery cost concerning a logistics operator and assigned stations. Therefore, it is similar to the shortest path problem (SPP) that aims to find a route through all the vertices at the lowest cost. Consequently, to achieve the optimum to the optimization problem, the NDP together with lemmas and theorems above is described as follows.

Based on the mathematical models, the stations assignment problem aims to determine the leftmost station for each logistics operator, then the rightmost station and the responsible range are consequently determined since no missing or overlapping is allowed. Therefore, the decision process is divided into $| S | + 1$ stages, where each stage $i (i = 1, 2, \dots, | S | + 1)$ represents a station, and $| S | + 1$ is the termination stage. For each stage, the state is also denoted as $i$ , which means the leftmost station is assigned to a logistics operator. Transition from state $i$ to state $j$ indicates that stations $i, i + 1, \dots, (j - 1)$ are assigned to a specific operator. Starting from $i = 1$ with a forward recursion, a change of objective function value is given as

μ (i, j) = F_{w} + u \cdot \sum_{r = 1}^{R_{w}^{*}} {Q_{r}^{w}}^{*} \cdot Z 2 (T_{w}^{*})

(12)

where * denotes the optimal value of the corresponding operator scheduling problem with the stations $i, i + 1, \dots, (j - 1)$ . If $H (i)$ stands for the optimal objective function value obtained by equation (11) for stations from 1 to $i$ , then

H (j) = \min_{1 ⩽ i ⩽ j - 1} {H (i) + μ (i, j)}

(13)

The optimal value $H (| S | + 1)$ can be acquired from the combination of equations (12) and (13), then the optimal alternative for the stations partition can be achieved by a simple backtracking.

While solving the stations assignment problem, $D_{w}, T_{w}^{*}, R_{w}^{*}$ and $Z 2 (T_{w}^{*})$ should be calculated for each stage, which means that the corresponding operator scheduling problem must be simultaneously considered. If stations $i, i + 1, \dots, (j - 1)$ are assigned to operator $w$ , together with the mathematical models and assumptions, the steps for the scheduling problem are summarized as follows:

Step 1. Calculate the least number of replenishment tours for each station $s (s = i, i + 1, . . ., j - 1)$ : ${\underline{N}}_{s} = ⌈ \sum_{t = 1}^{T} d_{s}^{t} / B_{s} ⌉$ and the maximum of all stations: $\underline{N'} = \max {{\underline{N}}_{i}, \dots, {\underline{N}}_{j - 1}}$ .

Step 2. Calculate the least number of replenishment tours for the subset of $i, i + 1, \dots, (j - 1)$ : ${\underline{N}}_{w} = ⌈ \sum_{s = i}^{j - 1} \sum_{t = 1}^{T} d_{s}^{t} / C_{w} ⌉$ .

Step 3. Determine the least number of tours for logistics operator $w$ : ${\underline{N}}_{w}^{\min} = \max (\underline{N}', {\underline{N}}_{w})$ .

Step 4. Calculate the feasible number of tours for logistics operator $w$ : $R_{w} \in [{\underline{N}}_{w}^{\min}, T],$ and accordingly the delivery time interval: $T_{w} \in [1, ⌊ T /_{w}^{\min} ⌋]$ .

Step 5. Determine $T_{w}$ and $Z 2 (T_{w}^{*})$ as follows:

(a) Select $T_{w}$ according to Theorem 2, then calculate $R_{w}, Q_{r}^{w}$ and $\sum_{t = r \cdot T_{w} / ct}^{(r + 1) \cdot T_{w} / ct} d_{s}^{t} (s = i, \dots, j - 1)$ .

(b) Examine the constraint equations (8) and (9), if both satisfied, then go to (d), else go to (c).

(d) $Z 2 (T_{w}^{*}) \leftarrow Z 2 (T_{w)}$ , and turn to equation (12).

(e) If no feasible $T_{w}$ , then $Z 2 (T_{w}) \leftarrow \infty$ , which means the stations assignment is unreasonable and should be reconsidered $(j \leftarrow j - 1)$ .

A small-sized instance is provided here to illustrate the proposed NDP. There are five stations assembling three different models. The assembly sequence is predetermined as $M 1, M 2, M 2, M 3, M 3$ , related information such as BOM and JIS bin capacity $B_{s} (s \in S)$ are given in Table 2. More detailed parameters include $P = 5, T = P + | S | - 1 = 9, ct = 100 s,$ $a t_{w} = wt = st = 10 s, C_{w} = 5, F_{w} = 300, u = 1$ .

Table 2.

Parameters of the example.

$s$	$M 1$	$M 2$	$M 3$	$B_{s}$
1	1	0	1	5
2	2	1	0	4
3	0	2	2	5
4	2	1	1	4
5	1	0	1	5

Consequently, together with Theorem 2, the optimal solution to operator scheduling problem can be easily obtained by NDP and gives feedback to the stations assignment problem, which efficiently reduces the search space and generates the global optimum within feasible time. As depicted in Figure 2, the vertices represent stations, and the arcs denote the assigned range for a logistics operator. The value inside the bracket is the optimal time interval. For example, the arc connecting vertex 1 and vertex 4 denotes that stations 1, 2 and 3 are assigned to a logistics operator and replenished every two cycles. The optimal solution $Ω = {1, 4, 6}$ in bold is comprised two arcs, which means two logistics operators are required to serve the line. The first one is responsible for stations 1, 2 and 3, while the second for stations 4 and 5, and the optimal time intervals are, respectively, 2 and 3 cycles. This leads to a total objective function value of $Z = 1091.5$ .

Figure 2.

The results of the example with NDP.

Together with Figure 2, it can be obviously concluded that when there are $| S |$ stations, there are totally $| S | + 1$ vertices, and each vertex can be connected to at most $| S |$ other vertices. Therefore, there are possibly $O ({| S |}^{2})$ number of arcs, and for each arc, the weight $μ (i, j)$ should be calculated. If $T_{w}^{\min} = 1$ and accordingly $R_{w}^{\max} = T$ , the value $Z 2 (T_{w})$ can be evaluated in $| S | \cdot T$ combined with constraint equation (6). Therefore, the time complexity is bounded by $O ({| S |}^{3} \cdot T^{2})$ . Especially, as the number of stations and cycles to be considered by NDP increases, the search space grows explosively as well, and instances of real-world size are impossible to solve in reasonable time. Consequently, a modified meta-heuristic MHSA is presented as follows to tackle the medium or large-sized instances.

MHSA

As one of the most widely used meta-heuristics, HSA outperforms other traditional meta-heuristics and has been successfully applied in various non-linear optimization problems.^36,37 Nevertheless, there are still some deficiencies such as the stagnation phenomenon in the later phase, poor ability in local exploitation and trapping into local optimum. Therefore, a modified HSA is presented here to tackle such tough issues. The detailed iterative process of MHSA is summarized as follows:

Step 1. Initialize parameters: accuracy $ε$ , maximum iterations $SNI$ and $HMCR, PAR$ .

Step 2. Initialize Harmony Memory Size (HMS) and Harmony Memory (HM): randomly generate $HMS$ harmonies and construct HM with consideration of constraints equations (2)–(4).

Step 3. Generate sub-HMs: divide the whole HM into several small-sized sub-HMs with different lengths of harmonies.

Step 4. Improvise new harmony: in each sub-HM, improvise a new harmony with parameters $HMCR,$ $PAR$ .

Step 5. Update sub-HMs: update the sub-HM if the new harmony is better than the worst one.

Step 6. Judge the termination criteria: if the maximum iterations $SNI$ is reached, recombine all the sub-HMs to construct the HM, else repeat Step 4 and Step 5.

Step 7. Crossover and mutation: choose the worst harmony $X^{w 1}, X^{w 2}$ , implement crossover and mutation to form new harmonies $X', X ″$ and update the HM.

Step 8. Judge the termination criteria: if the difference of any two objective function values is smaller than $ε$ , then terminate the MHSA algorithm, else repeat Steps 3–7.

Presentation of harmony

The proposed MHSA employs a two-level encoding to define harmonies or, namely, solutions. For each harmony $X^{i} (i = 1, 2, \dots, HMS)$ , the length $| X^{i} |$ reveals the required number of logistics operators, which is informed as $(| X^{i} | - 1)$ . Moreover, the first level shows the detailed partition of stations. Each value represents the leftmost station $(δ_{w})$ assigned to a logistics operator $w$ . Together with constraints equations (2)–(4), it is obvious that the first level starts with 1 and ends with $| S | + 1$ in an ascending order. The second level holds the corresponding optimal time interval $T_{w}$ for each logistics operator. Figure 3 presents two harmonies for the instance introduced in section “NDP,” where the first harmony $X^{1}$ shows the optimal solution, and the second one $X^{2}$ means that each station is served by a separate logistics operator, and the optimal replenishment intervals are 5, 5, 2, 2, 5 cycles, respectively.

Figure 3.

The schematic diagram of harmonies.

Initialization of harmony memory

For a given set of stations $S$ , the length (dimension) of a harmony is bounded by $Le n_{\max} = | S | + 1$ . Meanwhile, combined with Theorem 1, the minimum can be calculated as $Le n_{\min} = {| W |}_{\min} + 1$ . Let ¹ $X_{j}^{i}$ and ² $X_{j}^{i}$ , respectively, be the jth values of the first and second level of a harmony $X^{i}$ and the length is noted as $Le n_{i} = | X^{i} |$ . To initialize the HM, HMS-feasible harmonies are randomly generated with consideration of constraints equations (2)–(4). For each harmony $X^{i} (i = 1, 2, . . ., HMS)$ , it should satisfy the following condition

{\begin{matrix} {{}^{1}X}_{j}^{i} < {{}^{1}X}_{j + 1}^{i}, {{}^{1}X}_{1}^{i} = 1, {{}^{1}X}_{Le n_{i}}^{i} = | S | + 1 \\ {{}^{2}X}_{1}^{i} = \dots = {{}^{2}X}_{j}^{i} = \dots = {{}^{2}X}_{Le n_{i} - 1}^{i} = 1, {{}^{2}X}_{Le n_{i}}^{i} = 0 \end{matrix}

(14)

Thus, the objective function value $f (X^{i})$ of the harmony $X^{i}$ is calculated and kept in HM simultaneously. Therefore, the initial HM matrix is composed of $HMS$ two-level harmonies, each of which is an integral-valued vector bounded by $| X^{i} |$ dimensions.

Updating sub-HMs

It is commonly accepted that the HSA employing comparatively smaller-sized HM outperforms larger one for medium-sized problems.^41,42 Therefore, the whole HM is divided into several sub-HMs with various smaller sizes, which depends on the length of harmonies $Le n_{i} (i = 1, 2, \dots, HMS)$ . With the aim to avoid premature convergence or trapping into local optimum, for each sub-HM, new harmonies are improvised by the rules with parameters harmony memory considering rate $(HMCR)$ and pitch adjusting rate $(PAR)$ , and the sub-HM is updated repeatedly. The pseudocode demonstrating the process is presented in Figure 4 as follows.

Figure 4.

Pseudocode for updating sub-HMs.

As shown in Figure 4, the improvisation of a new harmony in this article is different from the traditional HSA. Since all the harmonies in a specific sub-HM are of equivalent length, combined with Theorem 3 and the specific characteristics of encoding, the improvisation of new harmonies employs a neighborhood search. A roulette selection is applied to randomly select two harmonies, and a weight factor $α$ between 0 and 1 is introduced to generate a new harmony $X^{new}$ . Whether updating the sub-HM depends on the objective function values of $X^{new}$ and the current worst harmony $X^{worst}$ .

Updating the whole HM

Although small-sized sub-HMs can efficiently speed up convergence, they probably lead to premature convergence and local optimum. Therefore, all the sub-HMs are recombined as a whole HM every certain iterations. Moreover, crossover and mutation are implemented for better diversity and solutions. The main process is described as follows:

Crossover: with the predefined crossover rate $P_{c}$ , the crossover operation is implemented when a random number generated between 0 and 1 is smaller than $P_{c}$ . The two harmonies $X^{w 1}, X^{w 2}$ are selected by

X^{w} \leftarrow \arg \max_{i = 1, 2, . . . HMS} {f (X^{i})}

(15)

It is obvious that the two worst harmonies are selected, and the crossover can be achieved by following steps:

Step 1. Randomly select two crossover points, $C 1$ for $X^{w 1}$ and $C 2$ for $X^{w 2}$ , and accordingly $L 1 = | X^{w 1} |, L 2 = | X^{w 2} |$ , then the length of new harmonies $X'$ and $X ″$ is, respectively, $L' = C 1 + L 2 - C 2 and L ″ = C 2 + L 1 - C 1$ .

Step 2. If ${{}^{1}X}_{C 1 + 1}^{w 1} > {{}^{1}X}_{C 2}^{w 2}$ and ${{}^{1}X}_{C 2 + 1}^{w 2} > {{}^{1}X}_{C 1}^{w 1}$ and $L' ⩽ | S | + 1$ and $L ″ ⩽ | S | + 1$ , then go to Step 3, else go to Step 4.

Step 3. Exchange the first-level elements of $X^{w 1}$ and $X^{w 2}$ after $C 1$ and $C 2$ , and calculate the second-level elements ${}^{2}X'_{j}, {{}^{2}X ″}_{j}$ as the five steps in section “NDP” to form new harmonies $X'$ and $X ″$ , then calculate $f (X'), f (X ″)$ .

Step 4. Regenerate a different crossover point $C 2$ , and go to Step 3.

Step 5. The crossover of $X^{w 1}$ and $X^{w 2}$ is completed.

As shown in Figure 5, there are two harmonies $X^{1}, X^{2}$ , and the two crossover points are $C 1 = 2$ and $C 2 = 4$ . According to Step 2 and Step 3, exchange the latter elements and recalculate the optimal intervals, then the two new harmonies $X^{1'}$ and $X^{2'}$ are constructed, which are partly different from the original ones. It is apparent that the length of harmonies is altered ( $L 1$ changed from 3 to 4, $L 2$ changed from 6 to 5), which is important to take account of all possible number of logistics operators and improve the diversity.

2. Mutation: after the crossover operation, harmonies $X^{w 1}$ and $X^{w 2}$ turn into $X'$ and $X ″$ . If $f (X') < f (X^{w 1})$ and $f (X ″) < f (X^{w 2})$ , then $X^{w 1} \leftarrow X'$ and $X^{w 2} \leftarrow X ″$ , otherwise denote the harmony to be mutated as $X^{chng}$ . With the predefined mutation rate $P_{m}$ , the mutation operation is implemented if a random number generated between 0 and 1 is not larger than $P_{m}$ .

Figure 5.

The schematic diagram of crossover operation.

The detailed pseudocode demonstrating the crossover and mutation is presented in Figure 6 as follows.

Figure 6.

Pseudocode about crossover and mutation.

Computational study

To effectively evaluate the novel cyclic part feeding system with line-integrated supermarkets and the proposed MHSA, all related algorithms were programmed in MATLAB (2014b) platform and run on an x64 PC with a 1.4-GHz Intel^® Core™ i5-4260U CPU and 4 GB of RAM.

Since there is very few scientific research documenting the novel concept, established test data are not available. Consequently, together with the instance generation described by Boysen and Emde,¹⁹ necessary parameters and data were generated as follows.

A MMAL is considered, and the demand in the production cycle $t (t = 1, 2, \dots, T)$ at a station $s (s \in S)$ is randomly generated as $d_{s}^{t} = ⌊ rn d^{uni} (0, 3) ⌉$ , where $rn d^{uni} (0, 3)$ indicates a random value following uniform distribution within the given interval $(0, 3)$ and ⌊·⌉ means a roundness to the nearest integer. Similarly, the capacities of JIS bins and logistics operators are, respectively, drawn as $B_{s} = ⌊ rn d^{uni} (5, 10) ⌉$ and $C_{w} = ⌊ rn d^{uni} (5, 8) ⌉$ . Moreover, $ct = 100 s$ , $st = 0.1 ct$ , $wt = 0.2 ct$ and $a t_{w} = 0.3 ct$ . Experiments with various combinations of $| S |$ and $T$ were conducted and analyzed, where each combination was repeated 20 times and the averages were considered here.

Algorithm validation with small-sized instances

To validate the proposed MHSA, experiments were first conducted to deal with small-sized instances, and the results of MHSA were compared with that of the NDP. The related parameters ( $| S |$ , $T$ ) and averaged results are listed in Table 3, where the values given in brackets were the corresponding optimal number of logistics operators. The first two columns, respectively, denote the number of production cycles and stations. The experimental results of NDP and MHSA are, respectively, listed in the next two columns, and the gaps between the optimal function values $Z$ are presented in the last column.

Table 3.

Parameters and results of small-sized instances.

Parameters		NDP	MHSA
$T$	$\| S \|$	$Z$	$Z$	Gap (%)
10	5	2340.42 (2)	2431.93 (2.6)	3.91
10	10	5412.04 (5)	5577.48 (6.0)	3.06
10	20	11,782.12 (13)	12,058.58 (14.2)	2.35
20	5	3811.30 (3)	3932.33 (3.5)	3.18
20	10	8242.32 (7)	8393.56 (7.5)	1.83
20	20	−	18,258.78(15.2)	−

NDP: nested dynamic programming; MHSA: modified harmony search algorithm.

All the instances could be solved to optimality by NDP, which validated the effectiveness of the NDP. However, when $T = 20$ and $| S | = 10$ , there are totally 256 $(1 + C_{| S | - 2}^{1} + C_{| S | - 2}^{2} + \dots + C_{| S | - 2}^{| S | - 2})$ different forms of stations assignment. Thus, even though the heuristic described in section “NDP” for the scheduling problem can easily find an optimal delivery interval $T_{w}$ , it inevitably takes several hours to combine and filter the possible solutions and finally get the optima. For example, when $T = 20$ and $| S | = 10$ , it took 1587 s to get the optimum and when $T = 20$ and $| S | = 20$ , NDP could no longer get the optimum within the time limit (3600 s). Therefore, it is no longer advisable to employ the NDP. However, only a few seconds were required to run the MHSA. The results of MHSA averaged over 20 times are close to the optima. As shown in Table 3, the gaps between the function value of MHSA and NDP are lower than 5% without exception, which indicate that satisfactory solutions can be obtained by the proposed MHSA. Moreover, with the increase in $| S |$ and $T$ , the deviations from the optimal solutions are decreasing since the HM is enlarged and simultaneously a better diversity of harmonies is formed to avoid local optima. Therefore, the proposed MHSA is applicable to tackle the medium- or large-sized instances with a guarantee of the quality of the solutions.

Comparison and analysis of algorithms

With the increase in $T$ and $| S |$ , there is a sharp increase in the search space. Therefore, the exact algorithm NDP is inapplicable or impossible to obtain optimal solutions within a given time limit. On the contrary, the MHSA is able to solve the medium- or large-sized instances in a short period, and yields approximate optimal solutions. Consequently, a further validation of the MHSA is provided in this part.

Comparison of MHSA, ant colony optimization and simulated annealing

MHSA or the original HSA is inspired by the music improvisation process where musicians constantly polish their pitches with the aim of better harmony.⁴³ First, the MHSA was compared with two other meta-heuristics motivated by natural phenomena, where the ant colony optimization (ACO)^44,45 is motivated by animals’ behavior and simulated annealing (SA)⁴⁶ from the physical annealing process. The detailed parameters and statistical results are summarized in Table 4.

Table 4.

Comparison of experimental results of MHSA, ACO and SA.

Parameters		MHSA (×10⁴)	ACO (×10⁴)	Gap (%)	SA (×10⁴)	Gap (%)
$T$	$\| S \|$
20	10	0.85	0.86	1.90	0.86	1.77
20	50	3.37	3.48	3.18	3.55	5.32
20	100	6.66	6.91	3.77	7.10	6.72
100	10	4.05	4.17	2.94	4.18	3.28
100	50	15.78	16.61	5.28	16.76	6.24
100	100	26.26	28.24	7.56	28.70	9.32
200	10	5.88	6.34	7.83	6.48	10.37
200	50	27.36	29.96	9.50	30.40	11.14
200	100	43.75	48.24	10.24	50.42	15.22

MHSA: modified harmony search algorithm; ACO: ant colony optimization; SA: simulated annealing.

As shown in Table 4, the objective function values of MHSA are obviously smaller than that of ACO and SA, so that the optimization capacity of MHSA is better. For the comparatively small-sized instances, the average relative gaps generated by ACO and SA are lower than 5%. However, when it comes to the large-sized instances $(T ⩾ 100, | S | ⩾ 50)$ , the average gaps are dramatically increasing and even over 15%. Therefore, the experimental results show the better capacity of deep searching of MHSA and to some extent verify its effectiveness in solving large-sized instances.

Moreover, Figure 7(a) and (b), respectively, depicts the converging conditions of MHSA, ACO and SA when $T = 20, | S | = 10$ and $T = 100, | S | = 50$ . It can be informed that for the small-sized instances, the MHSA and ACO are able to converge rapidly to the satisfactory solutions (see Figure 7(a)). However, when $T$ and $| S |$ increase, the MHSA shows a great improvement in convergence speed with comparison to ACO and SA, which validates the effectiveness of the employment of sub-HM operations, crossover and mutation. Therefore, the MHSA significantly outperforms ACO and SA (see Figure 7(b)).

Figure 7.

Convergence performance of MHSA, ACO and SA: (a) $T = 20, | S | = 10$ and (b) $T = 100, | S | = 50$ .

Comparison of MHSA, improved harmony search, global-best harmony search and novel global harmony search

For further clarification of the superiority of the proposed MHSA, in this part, it is compared with three other modified HSA that have been widely discussed and cited. First, the improved harmony search (IHS)⁴⁷ refers to a strategy for tuning necessary parameters such as PAR and bandwidth (bw). The second denoted as global-best harmony search (GHS)⁴¹ is inspired by the concept from swarm intelligence. The last is a novel global harmony search (NGHS),⁴⁸ which includes position updating and genetic mutation, and has been successfully applied to solve the 0-1 knapsack problems. The experimental results are vividly presented in Figure 8.

Figure 8.

Experimental results of MHSA, IHS, GHS and NGHS: (a) S = 40, T ∈ [40,240] and (b) T = 120, $| S |$ ∈ [20,120].

Figure 8(a) shows the results with the constant number of stations $| S | = 40$ and various cycles $T$ . The polylines reflect the variation of objective function values of different algorithms while the histogram presents the average deviation of IHS, GHS and NGHS from MHSA. It is obvious that the averaged objective function values of MHSA are much smaller than that of the other three algorithms, and correspondingly, the average deviations range from 5% to 20%. Similar conclusions can also be drawn from Figure 8(b), where the number of cycles is fixed at $T = 120$ and the number of stations $| S |$ is assigned from 20 to 120 with an ascending interval of 20. Therefore, it validates that the proposed crossover and mutation as well as repetitive partitions and recombination of HM in this article improve the search depth to avoid local optima, which make the MHSA superior to IHS, GHS and NGHS.

Parameter analysis

With the aim to minimize the total operating cost of part feeding, it is necessary to find a trade-off between the number of logistics operators $| W |$ and optimal time interval $T_{w}$ . With accordance to equation (11), the parameters $F_{w}$ and $u$ directly make a difference on the overall objective function. The effect of $F_{w}$ (the fixed cost for employing a logistics operator) and $u$ (material handling cost per part per unit time) is investigated by experiments, in which $T = 20, | S | = 10$ , $u$ is set as 1, and $F_{w}$ varies from 0 to 1000 with a step of 100. The effect of $F_{w} / u$ is presented in Figure 9, where “ $▲$ ” is the objective function value of different $F_{w} / u$ and the dotted line shows the linear fitting. It is clear that with the linear increase in $F_{w} / u$ , the corresponding objective function values are mainly a linear distribution with a slope of 7.54 and the intercept of 6028.54, which, respectively, mean the average number of logistics operators and the basic material handling cost.

Figure 9.

The effect of $F_{w} / u$ on objective function values.

According to the analysis above, the variation of $F_{w} / u$ mainly affects the number of logistics operators. The value of $F_{w} / u$ is set between the interval [0,3000], and $T = 20$ while $| S |$ is taken from the interval [20,120] with a step of 20. Thus, the relation between the number of logistics operators $| W |$ and $F_{w} / u$ is depicted in Figure 10(a) and (b) ((b) is the projection of (a) on the plane YOZ). When the production cycles $T$ is predetermined, the number of logistics operators $| W |$ increases with the number of stations $| S |$ . However, for any given $| S |$ , with the increase in $F_{w} / u$ , $| W |$ conversely decreases and keeps constant when it decreases to roughly $| S | / 2$ , which means that a logistics operator should be responsible for two stations. Actually, the acquired minimum satisfying all the constraints with certain $T$ and $| S |$ is a bit larger than the value calculated by Theorem 1. When $| W |$ decreases to the minimum, it will no longer be affected by the increase in $F_{w} / u$ , which only leads to the increase in the objective function value $Z$ .

Figure 10.

(a) The effect of $F_{w} / u$ on $| W |$ and (b) the projection of (a) on plane YOZ.

Conclusion

A novel cyclic part feeding system with line-integrated supermarkets is introduced in this article to draw advantages of both line stocking and kitting to facilitate the part supply in MMALs. Two highly dependent sub-problems, the stations assignment and operators scheduling problem, are investigated simultaneously. On account of the complexity of the problems, an exact algorithm NDP is presented to find the optima for small-sized instances on the basis of the particular theorems and lemmas of the problem. Moreover, a meta-heuristic MHSA is proposed for medium- or large-sized instances. While inheriting the advantages of the original HSA, the MHSA has improved the searching speed by repeatedly dividing the HM into several small-sized sub-HMs and combining all sub-HMs to rebuild the HM, which is rarely mentioned or employed in other literature as far as we know. Meanwhile, neighborhood search together with crossover and mutation is introduced to improve the diversity of the solutions and avoid premature convergence. The experimental results validated that the proposed MHSA shows great competition in solving the referred optimization problem.

However, the stations assignment in this article may sacrifice the utilization of operators to some extent since operators are not supposed to deliver parts out of their predetermined assignment even though there is spare time and opportunity. Therefore, the future work may lay emphasis on the condition that a team of logistics operators is jointly assigned to a segment of the assembly line.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was supported by the National Natural Science Foundation of China under Grant No. 71471135.

ORCID iD

Binghai Zhou

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A novel optimized cyclic part feeding system with line-integrated supermarkets

Abstract

Keywords

Introduction

Problem description

Stations assignment problem

Operators scheduling problem

Algorithm construction

Lemmas and theorems

Theorem 1

Proof

Lemma 1

Proof

Theorem 2

Proof

Theorem 3

Proof

NDP

MHSA

Presentation of harmony

Initialization of harmony memory

Updating sub-HMs

Updating the whole HM

Computational study

Algorithm validation with small-sized instances

Comparison and analysis of algorithms

Comparison of MHSA, ant colony optimization and simulated annealing

Comparison of MHSA, improved harmony search, global-best harmony search and novel global harmony search

Parameter analysis

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References