A novel locust swarm algorithm for the joint replenishment problem considering multiple discounts simultaneously

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A novel locust swarm algorithm for the joint replenishment problem considering multiple discounts simultaneously Ligang Cui a, Jie Deng b,∗, Lin Wang c,∗, Maozeng Xu a, Yajun Zhang c a

School of Economics and Management, Chongqing Jiaotong University, No. 66, Xuefu Ave., Chongqing, 400074, China Intellectual Property Institute of Chongqing, Chongqing University of Technology, No.69 Hongguang Ave., Ba’nan District, Chongqing, 7F Room 15, 400054, China c School of Management, Huazhong University of Science and Technology, No. 1037, Luoyu Road, Wuhan, 430074, China b

a r t i c l e

i n f o

Article history: Received 10 December 2015 Revised 27 July 2016 Accepted 5 August 2016 Available online xxx Keywords: joint replenishment quantity discount PSO locust swarms

a b s t r a c t In B2C E-Commerce operations, multiple quantity discount offers are commonly practiced in the multiitem replenishment environment. In this paper, a novel joint replenishment model (JRP) is presented considering two quantity discounts, all-unit quantity discount, incremental quantity discount, simultaneously. A novel swarms search technique, locust swarms algorithm (LS) is introduced and redesigned to solve the novel formulated JRP model. Numerical experiments and parameter sensitivity analyses reveal that LS is an effective and efficient algorithm for solving the proposed model in terms of solution quality and searching stableness comparing to some other meta-heuristic algorithms, such as GA, DE and PSO. Moreover, management insights such as the mutual effects of multiple quantity discounts to the total cost, and the role of multiple quantity discounts to different stakeholders in replenishment are outlined. © 2016 Published by Elsevier B.V.

1. Introduction For most of the item replenishment operations, the motivations for buyers and suppliers to adopt the joint replenishment policy can attribute to two aspects: to pursue the savings in ordering cost [1] and to acquire the discounted order quantities [2]. For example, due to the fast development of B2C E-Commerce in China, many corporations of this kind, taking Jingdong, Amazon and Suning for instance, experience unprecedented prosperities in recent years and take up a large share of the market cake, which was once owned by traditional sectors. However, behind the great gains obtained, B2C companies often fall into the dilemmas on deciding a best way to group more items in one batch during the replenishment to save ordering cost, and decrease per unit purchasing cost through acquiring the ‘coupons’ in each order. B2C companies are always obsessed by this plight, which inspires them to explore/experiment better strategies to decrease risks in replenishment operations, which is also the original motive that pushes us to lead this research.

∗

Corresponding authors. E-mail addresses: [email protected] (L. Cui), [email protected] (J. Deng), [email protected] (L. Wang), [email protected] (M. Xu), [email protected] (Y. Zhang).

In the multi-item system, great benefits can be obtained through grouping multiple items in one order [3]. Traditionally, the joint replenishment problem (JRP) is just presented for coping with such kinds of problems, and generally defined as the multi-item inventory problem of coordinating the replenishment of a group of items that may be jointly ordered from a single supplier [4]. The original JRP assumes two types of ordering cost, the major ordering cost, which is independent of items in an order and can be taken as the fixed cost, e.g. the set-up cost of the supplier for manufacturing, and the minor ordering cost, e.g. the labor cost and insurance fees, which is closely correlated to the specific item in an order and assumed as the variable cost. However, the superiority of JRP is not fully reflected in classic JRP models, as group buying different items can not only realize ordering cost savings, but also can acquire discount benefits in replenishment operations. By investigating the state of the art of JRP research, the variations of JRP are mainly concentrated on the supplying end and the selling end of a specific supply chain [5]. For example, Hsu et al. [6] assumes satellite factories offering raw materials in the supplying end. Cha et al. [7] consolidates the JRP model with delivery process among multiple retailers considered at the selling end. Currently, the research emphases of JRP can be generalized to two classifications, JRP model-emphasized research and model solving approach-emphasized research [5]. For example, in the former classification, Nielsen and Larsen [8] introduces Q(s, S) inventory policy to JRP, Chen and Chen [9] embed channel coordination

http://dx.doi.org/10.1016/j.knosys.2016.08.007 0950-7051/© 2016 Published by Elsevier B.V.

Please cite this article as: L. Cui et al., A novel locust swarm algorithm for the joint replenishment problem considering multiple discounts simultaneously, Knowledge-Based Systems (2016), http://dx.doi.org/10.1016/j.knosys.2016.08.007

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procedures into JRP, Moon et al. [10] and Wang et al. [11] investigate JRP with delivery considerations. In the latter classification, Kaspi and Rosenblatt [12], Porras and Dekker [13], Zhang et al. [14] and other researchers provide heuristic algorithms for solving JRP. While as the NP-hard nature of JRP has been testified [15], some researchers, like Chan et al. [16], Wang et al. [11,17] and Cui et al. [18], try to present evolutionary algorithms to solve the classical JRP model and its extensions. Of all JRP research, one type of JRPs with discount consideration has drawn some researchers’ attention. However, comparing to various discounting schemes offered by the suppliers in practice and discussed by researchers [19], only a handful of researches have focused on JRPs with discount consideration. For example, Chakravarty [20] set up a single retailer JRP model total volume discount. Cha and Moon [21] present a JRP model with all-unit quantity discount and an efficient heuristic algorithm. Moon et al. [10] construct a JRP model with all-unit quantity discount under multiple-supplier environment. Sari et al. [22] consider the situation that the supplier offers time-based quantity discount and construct a joint economic JRP model considering the imperfect items and all-unit quantity discount, simultaneously. By addressing the characters of these research models with discounts, however, the corresponding assumptions of these papers are either on a single product replenishing/distribution among several different suppliers or retailers [10,23,24], or on multi-item problems with one type of discount [25–27], no one research takes into consideration of the multiple discounts within the multiple items environment. In reality, it is a common practice that providing multiplediscount offers considering the supplying ability of specific items in replenishment operations. The inherence of performing discount schemes may stem from the fact that it encourages buying larger batches of items [2], but the presence of different discount schemes often complicates the item purchasing operations [28]. Since different discount schemes, such as the all-unit quantity discount and incremental quantity discount, have different structures, it makes the model even more trivial by integrating discount parameters in one model [2,29]. Conventionally, strategies for grouping items are classified as the direct grouping strategy (DGS) and the indirect grouping strategy (IGS) [4], but Olsen [[30]] and Wang et al. [31] note that IGS outperforms DGS. Since JRP has been testified as a NP-hard problem [15], it is rather challenging for finding an efficient and effective algorithm to solve JRP and its variation models [11]. Current approaches for JRPs are categorized as heuristics and meta-heuristics, but shortages of these algorithms are reflected in searching universality inefficiently or searching in-depth ineffectively. Henceforth, it is essential to find a proper algorithm for solving the proposed JRP model. Locust Swarms (LS) algorithm is extended from particle swarm optimization (PSO) that simulates flocks of birds and schools of fish [32] but enhanced the fine searching ability of PSO by continuing to search the neighborhood of the current best-found result obtained from PSO. Like other probability based algorithms, such as PSO [33], artificial bee colony (ABC) algorithm [34] and biogeography-based optimization (BBO) algorithm [35], LS is also a new multi-swarm system and does not make any assumptions about the problem, but has been strengthened the fine searching ability comparing to PSO [36], ABC [37], BBO [38] and other swarm mimic algorithms [39,40]. LS is originally designed for multi-modal problems [41,42], the superior performance of which algorithm has been vigorously tested by common benchmark functions [43]. However, its searching abilities in discrete spaces are still unclear, and it has not been applied to solve some more specific problems with integer variables, like JRP. The decision variables of JRP are discretely distributed among the whole solution spaces, since LS featured with the balance of exploration and exploitation in

searching abilities, it might be one of the most promising algorithms for solving the JRP model with discount considerations. In the following contents, two discount schemes, all-unit discount quantity and incremental discount quantity, are considered simultaneously to construct a new JRP model, but the LS algorithm should be redesigned to solve the proposed JRP model. The main contributions addressed in this research are shown as below. (1) A novel JRP model is developed considering two quantity discounts, the all-unit quantity discount and the incremental quantity discount, simultaneously. In the proposed model, the decision variables are inherited from those in the conventional JRPs, namely, the basic cycle time of all items and ordering frequencies of each item. The objective of this new model is to investigate the mutual effects of two discount schemes on the minimized total cost, through which the basic cycle time and ordering frequencies of each item are obtained. (2) A locust swarms algorithm (LS) is introduced and redesigned to solve the proposed model. Considering the structure of the proposed JRP model, LS is redesigned in its encoding mechanism considering both the integer and real number decision variables, and the searching schemes for generating ‘smart’ locusts and velocities to update current best points. (3) Based on the results obtained through numerical experiments, superiorities of LS in searching solutions have been testified by a typical case comparing to other meta-heuristic algorithms, such as GA, DE and PSO. Parameter sensitivity analyses of the new JRP have been led and the mutual effects of different discounts to the total cost have been testified. Further experiments of LS on random parameters of JRP and five large scale JRP cases confirm the robustness, efficiency and effectiveness of LS comparing to PSO. The remainder of this paper is organized as follows. In Section 2, assumptions, notations and the JRP model formulation are presented. Section 3 introduces LS and its revisions for the proposed JRP. Computational results of numerical cases are presented in Section 4. Section 5, conclusions and directions for future research are provided. 2. JRP model with two quantity discounts 2.1. Problem description, assumptions and notations The assumptions of the new model are inherited from those of the conventional JRP model, for example, the demand is assumed to be deterministic and conforms to the uniform distribution, no shortages are allowed, no quantity discount, linear holding cost, and so on [4]. The assumptions considered throughout this paper are given below: The demand of each item is deterministic and constant. Shortages are not allowed. The items are replenished when the inventory level drops to zero. The inventory holding cost is known and constant. The discount structures are offered by the supplier and known by buyer. Each type of items is offered one and only one possible discount scheme

i j

The notations of the proposed model are given as follows: the index of items, and set I = {i|i = 1, 2, . . . , n} the index of discount intervals, and set J = { j| j = 1, 2, . . . , Ji }

Please cite this article as: L. Cui et al., A novel locust swarm algorithm for the joint replenishment problem considering multiple discounts simultaneously, Knowledge-Based Systems (2016), http://dx.doi.org/10.1016/j.knosys.2016.08.007

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n0

the number of items that are offered no discount (denoted by N) by the supplier, and set N0 means the items with N in N0 the number of items that are offered all-unit quantity discount (denoted by A) by the supplier, and set N1 means the items with A in N1 the number of items that are offered incremental discount (denoted by I) by the supplier, and set N2 means the items with I in N2 unit cost/price of item i that the buyer pays to the supplier with N discounted unit price of item i in the j-th interval under A scheme discounted unit price of item i in the j-th interval under I scheme binary variable: if and only if the order quantity of item i falls in the interval of j, xi j = 1, otherwise xi j = 0 threshold (breakpoint) of each discount interval, μi,0 = 0 and μij ∈ Ui total annual cost of all items major ordering cost of each order minor ordering cost of each item demand rate of item i annual holding cost of item i basic cycle time (decision variable) integer multiplier of item i (decision variable), ki ∈ K, K is the set of all ki

n1

n2

ci

α ij β ij xij

μij TC S si Di hi T ki

2.2. Model formulation 2.2.1. The general JRP model Based on IGS, Ti for each item i is an integer multiplier of ki and T. Thus, the replenishment cycle of item i is:

Ti = ki T

(1)

Thus, the optimal value TC0 is obtained after optimal T and ki s are fixed. In general, the purchasing cost of items is not included in the total cost of joint replenishment process. In practice, however, most of the practitioners prefer to perform the joint replenishment strategy not only for the sake of acquiring ordering cost decreasing benefits, but also want to save more cost through ordering different items in large batches with different of discount offers. Therefore, the total joint cost of JRP with N scheme is presented as

T C (T , K ) = Ch + Co + C p =

(2)

Ch =

i=1

n T  Qi hi /2 = ki Di hi 2

(3)

i=1

And the annual total ordering cost per unit time is:

Co = S/T +

n  





 1 S+ (si /ki ) T

si / ( ki T ) =

i=1

n



(4)

i=1

Accordingly, the annual total cost per unit time TC0 under N scheme is:

   n n T  1 T C0 (T , K ) = Ch + Co = ki Di hi + S+ (si /ki ) 2 T i=1

(5)

i=1

where ki ∈ K, i = 1, 2, . . . , n. The annual total cost is the summation of two terms: the (major and minor) ordering cost and the inventory holding cost. For simplicity, the annual total cost per unit time is defined as TC0 , and the objection is to find the minimized TC0 of JRP. Obviously, TC0 is a convex function of T for a given set of K = (k1 , . . . , kn ) ∈ Nn . Thus, the optimal basic cycle time T∗ can be easily obtained by taking the first-order derivative of TC0 and let it be zero, see Eq. (6).

   n n 1 dT C0 (T , K ) 1 = ki Di hi − 2 S + (si /ki ) = 0 dT 2 T i=1

(6)

i=1

Then, T∗ is given by Eq. (7) below.



T∗ =



2 S+

n  i=1

(si /ki )

  n

ki Di hi

i=1

(8)

i=1

where Cp is the total purchasing cost (it can be considered as the occupational cost or inventory carrying cost per unit time) of items in each order. 2.2.2. JRP with two quantity discounts The total purchasing cost of the buyer depends on the cost structure offered by the supplier. Two discount structures, the allunit quantity discount and the incremental quantity discount, are utilized to formulate the purchasing cost. Here below the cost function of each discount structure is given as (1) Purchasing cost with the all-unit quantity discount In the all-unit quantity discount scheme, the supplier offers price discount according to the possible order quantities of different items. The price is stepped down as the ordering quantity of an item increases progressively in different intervals, and the ordering quantity intervals are divided from the maximum and the minimum ordering data in the supplier’s supplying history. Thus, the total purchasing cost per unit time with all-unit discount is formulated as:

T CA =

n1  

αi j Qi j

(9)

i=1 j∈J

The annual total holding cost per unit time is: n 

    n n n T  1 ki Di hi + S+ (si /ki ) + ci Qi 2 T i=1

and the order quantity Qi of item i is:

Qi = Ti Di = Di ki T

3

(7)

Ji Ji where x Q = Qi and x = 1, which means that for j=1 i j i j j=1 i j item i and only one j ∈ J, Qi j = Qi if and only if μi, j−1 ≤ Qi j ≤ μi, j . It is also assumed that the unit price is stepped down as αi1 > αi2 > · · · > αiJi . Therefore, if j is fixed, the TCA can be simplified as

T CA =

n1  

αi j ki T Di

(10)

i=1 j∈J

(2) Purchasing cost with the incremental quantity discount For the incremental quantity discount scheme, the slightly difference comparing to the all-unit quantity discount lies in that the incremental discount applies only for quantities exceeding the price break quantities. The cost function TCI under incremental discount scheme is given as:

T CI =

n2   

j−1 

i=1 j∈J

k=1

βi j (Qi j − xi j μi, j−1 ) + xi j

βik (μi,k − μi,k−1 )



(11)

 where j∈J xi j = 1, which means that there only one xij equals to 1 and the others equal to 0 for j ∈ J. It is also assumed that the unit price is stepped down as βi1 > βi2 > · · · > βiJi . The comparisons of relations of total purchasing cost and a specific quantity discount are illustrated in Fig. 1. Here we should be noted that even in the same type of a quantity discount, the numbers and the lengths of intervals may be different, which increase the difficulties in processing multiple orders.

i=1

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Fig. 1. Structures of two quantity discounts.

After two purchasing cost functions have been constructed, the total joint cost of JRP with multiple discounts is given below:

T C  (T , K ) = Ch + Co + C p

(12)

where C p is the total purchasing cost, including the total cost of items purchased with no quantity discount, with the all-unit quantity discount, and with the incremental quantity discount. Accordingly, C p is modeled as

C p = C p (n=n0 ) + CA (n=n1 ) + CI (n=n2 ) =

n0 

ci Qi +

i=1

n1  

αi j Qi j +

i=1 j∈J



n2   i=1 j∈J

j−1

+ xi j

βi j (Qi j − xi j μi, j−1 )

βik (μi,k − μi,k−1 )



Algorithm 1 Pseudo codes for LS Sub-Swarms 1: 1: Generate R random points 2: Pick S¯ best points with fitness function 3: Assign random velocity to each point 4: Run particle swarm for n1 iterations 5: Find current best point with fitness function Sub-Swarms 2: 6: Generate R random points around the last best point 7: Pick S¯ (R > S¯) best points with fitness function 8: Assign outward velocity to each point 9: Run particle swarm for n2 iterations 10: Find the best optimal point with fitness function 11: Return the best-found results

(13)

k=1

Ji where x = 1. j=1 i j In the proposed JRP model, the most trivial processing cost sources from two aspects, one lies in that the different numbers and lengths of intervals for items with a single type of discount, the other lies in that discount offers are from two different types. Therefore, an effective and efficient LS algorithm is introduced in the following contents. 3. The revised LS algorithms 3.1. The basic LS algorithm The basic or standard LS algorithm is a novel multi-optima swarms search technique developed from PSO [41]. Differing from PSO reflects the behaviors of swarms of fish or birds, LS mimics the behavior of locust swarms searching the spaces for foods [41]. The searching process of LS is guided by a ‘devour’ sub-swarm and a ‘move on’ sub-swarm, which means that after a sub-swarm of locusts ‘devours’ a relatively small region of the searching space (to find a local optimum), new locust scouts are deployed to look for new promising regions to ‘move on’. Thus, this LS process is called the ‘devour and scout’ strategy, and this strategy presents a useful metaphor for designing of optimization algorithms. The two phases of locust swarms, known as migrating and devouring or exploring and exploiting, have been designed and coordinated using a ‘coarse search + greedy search’ framework. According to Chen and Vargas [43], the coarse search phase of locust swarms has been implemented using ‘smart’ start points. Hence, LS uses PSO as part of its coarse search to find the smart point to realize a paired greedy search process. The pseudo codes of the original LS are presented as Algorithm 1 below. In Algorithm 1, Sub-Swarms 1 called ‘devour’ helps LS coarsely search and find the local optimum, while Sub-Swarms 2 is designed to ‘move on’ from the last best-found point to the optimal

point. A small “-” is added over S to discriminate the major ordering cost and the number of best candidate locusts for Sub-Swarms 2 search, S¯ is initialized by Chen [41] as S¯ = 30. Specifically, SubSwarms 1 performs the standard PSO operations, which has been extensively studied by numerous scholars [32,44,45]. Thus, our focus comes to the operations of Sub-Swarms 2. The R (is initialized as R = 50 0 0 by Chen [41]) scouting locusts are generated randomly in line 6 to searching for the most promising S¯ point (‘smart point’) as shown in line 7. Then, initial particle velocities in line 8 is assigned to the last optimum point for moving away from current local best point. After the whole process is repeated for n2 times (line 9), the last optimal found point (line 10, 11) is the point we the desired to find. By referring to Kennedy and Eberhart [32] and Chen [41] on PSO, in the ‘scouting process’, Sub-Swarms 1 is performed in below Eqs. (14) - (17).

velocity0 = 0.5 ∗ (range/2 ) ∗ (2 ∗ rand − 1 )

(14)

locationi+1 = locationi+1 + 0.95 ∗ velocityi

(15)

gravit yvectori = best locat ion − locationi

(16)

vel ocityi+1 = M ∗ vel ocityi + G ∗ gravityvectori

(17)

where velocity0 is the initial velocity, rand is utilized to generate random numbers in (0, 1) uniformly, M is defined as the ‘momentum’ factor M in Eq. (16) to guide the particles to ‘hurtle past their target’ with the introduction of the ‘gravity’ factor G. After the Eqs. (14) - (16) are repeated n1 = 500 times, the local optimal point is obtained, which is then applied as the candidate point for scouting search. The devouring process is guided by Eqs. (18) and (19) below.

delta = ± range ∗ {gap + abs(randn ∗ spacing)}

(18)

velocity0 = v ∗ (locationi − previousoptimum ) + 0.5 ∗ (range/2 ) ∗ (2 ∗ rand − 1 )

(19)

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Sub-swarms 2 starts from line 6 to line 10. Eq. (17) is to create random points (line 6) around the last best-found locust, and the minimum delta comprises two main parts, the gap (=0.2) that denotes the search gaps of the ‘smart’ locusts and the last optimal, plus an additional variation of distance called spacing (=0.2). Eq. (18) is followed to generate initial velocity for launching the particles away from the previous optimum, and v is initially set to 0.8. Through n2 iterations, the best-found optimum is returned as the final answer of LS. Originally, LS is designed and intensively experimented to solve multi-modal and continuous functions [42], but its searching ability in searching discrete functions is not clear. Moreover, its searching ability is largely affected by the process of its Sub-swarms 2. Since LS and PSO are all swarm intelligence that being part of evolutionary computation [46], not only the superiorities, such as high swarms search abilities [47], being easy to implement (than Genetic Algorithm (GA)) [48], and taking into account of the communication topology influences in the convergence rate of the search [42], but also the weakness like being not possible to continuously ‘fly’ particles through a discrete-valued space [49] and compromising of exploration and exploitation in the whole searching process [42], have been inherited. Hence, how to develop the superiorities and avoid negatives of LS is essential to solve the new JRP model efficiently and effectively. 3.2. An improved LS The improvements to LS have already drawn researchers’ focus by generating the initial velocities of the last optimal point [42]. However, as the superiorities of conventional PSO are mainly on its searching abilities to the optimization problems with continuous variables, its searching performance for some problems with integer variables is rather skeptical [32], though the discrete version of PSO is developed to solve some problems with binary decision variables [33,50]. Thus, to solve the proposed JRP model, LS should be redesigned with integer decision variables. Specifically, the positions and velocities of locusts, as well as the updating mechanism of LS, should be redesigned according the specific structure of the proposed JRP model. 3.2.1. Initialization process The solution representation is the key issue when designing the swarms algorithm [51], the initialization is inherited from the basic LS but considering the nature of decision variables. Thus, the initialization process ascertains following contents: the representation of decision variables, including the integer variables ki s and the continuous variable T, the length of the operating individuals (particles or locusts), and initial positions and velocities of the searching locusts. Thus, the locust i and solution individual are represented as follows.

Xi¯ = {xi¯1 , xi¯2 , . . . , xi¯J¯}

(20)

where J¯ is the maximum number of particles of a solution. The last element xi¯J¯ denotes the decision variable T, which is initialized in (0, 1), and remaining elements represent decision variables ki s, which are initialized as integer numbers. If we define the scalar Pt as the population (swarm) contains Xi¯, then

Pt = {X1t , X2t , . . . , Xi¯t , . . . , X pt }

(21)

where t is the t-th iteration and p is the size of the swarm. After the (solution) particles have been initialized, the moving velocity is another vital factor that should be given for LS. According to Kennedy and Eberhart [32], the velocity of the original LS is applied to guide the current particle to move from current position to the present best position (pbest) and the global best position gbest, simultaneously. However, the velocity (in lines 4 and 9

5

of Algorithm 1) of the original LS is updated following the process of PSO, which has some risks at trapping particles in local optima rather than moving them toward the gbest (pbest) solution. By referring to Sha and Lin [52], the velocity V − i¯ of particle X − i¯ at t-th generation is repressed as

Vi¯t = {vti¯1 , vti¯2 , . . . , vti¯ j¯, . . . , vti¯J¯}, 1 ≤ j¯ ≤ J¯

(22)

where vt¯ ¯ ∈ {−1, 0, 1} for 1 ≤ j¯ ≤ J¯ − 1 and the last element of V¯t ij

i

follows vt¯ ¯ ∈ [Vmin , Vmax ], and V¯t is initialized randomly when t = 0. iJ

i

To clear express the initialization process, we would take a fouritem case for example without considering discounts, and the procedure for the case is given as Step 1. The domains of different elements in each particle’s position are given. For example, the domains of ki s are [1, 5] and the domain of T is set as [0.0 0 01, 1]. The total number of locusts (or the population size). Step 2. The positions together with the corresponding velocities of the particles are generated randomly. For example, X t=0 = [3, 5, 1, 2, 0.1368] and V t=0 = [−1, 1, −1, 0, 0.007]. Step 3. The fitness of each locust is computed. The total cost function (Eq. (8)) is taken as the fitness function. Meanwhile, the minimum fitness and its corresponding locust are output as the pbest, and pbest = gbest. 3.2.2. The particle searching campaign of sub-swarms 1 Different from the original PSO in updating discrete integervalued positions from binary-valued velocities [50], the conventional velocity updating formulation is applied to increase the searching step size (is not just limited to ± 1) for ki s, see Eq. (23). Then, the updated position of the particle is given in Eq. (24).

vt+1 = round (ω · vti¯ j¯ + c¯1 r1 ( pbesti¯t − xti¯ j¯ ) + c¯2 r2 (gbesti¯t − xti¯ j¯ )) (23) i¯ j¯ xt+1 = xti¯ j¯ + vt+1 ¯¯ ¯¯ ij

ij

(24)

where round() is only applied to round the value if and only if 1 ≤ j¯ ≤ J¯ − 1, otherwise, round() takes no effect, ω is the inertia weight, c¯1 and c¯2 are two constants between 0 and 1 such that c¯1 + c¯2 ≤ 1 [52] for fine search of T(T ≤ 1), and c¯ − 1 = c¯2 = 1.494 [42] for coarse search of ki s, to represent the cognitive and social behavioral factors of the particles, and r1 and r2 are two random numbers in (0, 1). In order to illustrate the processes of updating the present best and global best particles and eliminating illegal particles, also the former case is applied here to address the details. Step 4. The velocity is updated using Eq. (23) and its former 4 elements are rounded. In above case, the velocity is updated as V t=1 = [−1, 1, −1, 0, 0.007] with ω = 0.8. Step 5. Two vectors are predefined as the upper bound ‘upb’ and lower bound ‘lob’ of each decision variable. Thus, in the case, we have upb = [5, 5, 5, 5, 1] and lob = [1, 1, 1, 1, 0.001]. Step 6. The old particle is updated according to Eq. (24), and the illegal element(s) in the new updated locust is(are) eliminated with the corresponding bounds. For example, if the new updated particle is X t=1 = X t=0 + V t=1 = [2, 6 , 0 , 2, 0.1375], and the illegal elements are the numbers with box, then, taking the illegal number 6 for example, 6 is replaced with its upper bound 5 if it is larger than its upper bound ( 6 > 5), vice versa. Step 7. The new fitness is computed with the new updated positions of each locust and the pbest and the corresponding particle information is output. Specifically, if the new minimum fitness (nmf) is smaller than the current global minimum fitness (gmf), fmf is replaced by nmf, and gbest is updated with the new position of

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the locust. If gmf < nmf < the present fitness pmf of the former pbest, pbest is updated with the new position of the particle. Step 4 to 7 is repeated till the preset maximum iteration number is reached, then the procedure of Sub-Swarms 1 is finished and the results are output for Sub-Swarms 2. 3.2.3. The locust searching campaign of sub-swarms 2 In the stage of Sub-Swarms 2, the ‘scout’ process is realized based on the results (the pbest and gbest of the last optimum) from Sub-Swarms 1. However, considering the nature of decision variables, it is necessary to make adjustments to original parameters. Correspondingly, the procedure of Sub-Swarms 2 is divided into two phases, in the first phase, new scouting ranges and positions of ‘smart points’ are generated randomly, and the other phase is to perform the searching iterations (fine search) to find a lower fitness. In specific, the two processes are illustrated using the above case within below steps. Step 8. New (‘smart’) locusts are generated to explore local spaces of last optimal. In this step R candidates are generated randomly and S¯ of the best ones are chosen for further searching. Substeps are presented below: (a) The minimum variation range of each decision variable and the gap, and spacing are predefined. In the case, range and gap are two vectors and defined as range = [4, 4, 4, 4, 0.001]/4 and gap = [1, 1, 1, 1, 0.001], respectively. (b) Compute delta and velocity0 by referring to Eqs. (18) and (19), respectively. Illegal numbers in delta and velocity0 are eliminated based on Steps 4–6. (c) Compute the new fitness based on the data such as pbest and gbest of the last optimum, delta and velocity0 , output S¯ best fitness and the corresponding particles, which then is taken as the candidates for further exploration. Step 9. Run the iterations as those in Steps 4–7 based on the new ‘smart’ locusts and velocity0 till the maximum iteration is researched. Step 10. Output gbest and best-found fitness. Above 10 steps are the main procedure of LS for solving JRP without considering quantity discount. Comparing to conventional JRP and JRP with discounts, the differences are mainly concentrated on obtaining the fitness. One of the most important superiorities of beta-heuristic algorithms is reflected in its direct representation of solutions, especially in comparing with the heuristic algorithms in solving JRP with quantity discount [10]. Hence, LS is very easy to implemented to compute the fitness of JRP considering quantity discounts. In order to speed up the trivial process of positioning the order quantity in the discounted price interval, given the ordering quantity Qi and Ui , the following procedure is adopted. (a) T emp1 = Ui − Qi . For example, if Ui = [0, 100, 200, 500] and Qi = 300, then T emp1 = [−30 0, −20 0, −10 0, 20 0]. (b) T emp2 = f ind (T emp1 < 0 ), where find() is to output the element(s) meet the condition in its brackets. Thus, T emp2 = [−30 0, −20 0, −10 0] in the example. (c) index = length(T emp2 ), where index is the right position of Qi , and also the index for the corresponding discounted price. The flow chart of LS for solving JRP with two quantity discounts is given in Fig. 2 below. 4. Numerical experiments

Fig. 2. Flow chart of LS for the proposed JRP .

In this section, a numerical example is introduced to address the efficiency and effectiveness of LS in contrast to genetic algorithm (GA), differential evolutionary (DE) algorithm and PSO. All Please cite this article as: L. Cui et al., A novel locust swarm algorithm for the joint replenishment problem considering multiple discounts simultaneously, Knowledge-Based Systems (2016), http://dx.doi.org/10.1016/j.knosys.2016.08.007

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Table 1 Input data for the JRP case.

GA DE PSO LS

9.8 1

2

3

4

5

6

Di hi si S ci

10,0 0 0 1 45 200 3.25

50 0 0 1 46 200 3.25

30 0 0 1 47 200 3.25

10 0 0 1 44 200 3.25

600 1 45 200 3.25

200 1 47 200 3.25

9.7

log(Fitness)

Item i

Table 2 Discount distribution. Item i

Discount type

Intervals

Price ($)

1

All-unit

2

Incremental

0 ≤ Q1 < 500 500 ≤ Q1 < 1, 000 10 0 0 ≤ Q1 < 2, 0 0 0 Q1 ≥ 2, 0 0 0 0 ≤ Q 2 < 500 500 ≤ Q 2 < 1, 000 10 0 0 ≤ Q 2 < 2, 0 0 0 Q 2 ≥ 2, 0 0 0 0 ≤ Q 3 < 500 500 ≤ Q 3 < 1, 000 Q 3 ≥ 1, 0 0 0 — 0 ≤ Q 5 < 300 Q 5 ≥ 300 0 ≤ Q 6 < 150 Q 6 ≥ 150

3.25 3.20 3.15 3.10 3.25 3.20 3.15 3.10 3.25 3.20 3.15 3.25 3.25 3.20 3.25 3.20

3

All-unit

4 5

No discount Incremental

6

All-unit

9.6

9.5

9.4

9.3 0

100

200

300

400

500

Iteration Fig. 3. Searching processes of GA, DE, PSO and LS .

crossover mutation is adopted. Correspondingly, the parameter settings for DE follow those in Wang et al. [31] such that mutation factor F = 0.5 and the cross factor Cr = 0.1. The initial parameter settings for PSO and LS are mainly referred from Chen [41] such that w = 0.8 and V = 0.4, other parameter settings are similar with those in Section 3.2. Each algorithm is run 20 times, the population size (number of particles of PSO and locusts of LS) of each algorithm is 30 and the maximum iteration of GA, DE and PSO is 500. For LS, the maximum iteration for Sub-Swarms 1 is 200, and then Sub-Swarms 2 is run 300 iterations. The computation results are given in Table 3, their evolutionary processes are illustrated in Fig. 3. The computational results in Table 3 reveal us that in the searching stableness, LS outperforms the other three algorithms, the following-up is PSO, the third one is DE, the worse one is GA. Of all the algorithms, all four algorithms can converge to the best-found result, such as the fitness, ki s and T. By comparing the searching time, PSO is the fast algorithm converging to its bestfound, LS is the following-up, which reflects that the improvements of LS is at the cost of certain computation time. Fig. 3 confirms some features of four algorithms in the searching processes. Fig. 3 indicates us that the convergence process of PSO and LS are close to each other in their early stage, DE runs the third, while GA is the last that converges to its stable stage. From above computational results, LS can be taken as a promising algorithm for further experiments.

algorithms (GA, DE, PSO and LS) are encoded in Matlab 2014a and run on a personal computer with 2.7 GHz CPU and 2 GB RAM. Then, the numerical analyses are conducted to investigate the sensitivities of different parameters to LS and the best-found results of JRP. Finally, randomness of problem-related parameters and large scale cases are raised to test the performance of LS in solving large scale cases. 4.1. Comparison of GA, DE, PSO and LS A typical JRP case with 6 items is presented, the basic data of which is listed in Table 1 and the data of quantity discounts is presented in Table 2. The comparing algorithms are GA, DE and PSO, the reasons for selecting those comparing algorithms are that, GA and DE have already been testified effective and efficient in solving JRP cases, the experience of which is very convenient for testing other algorithms; while PSO is the father algorithm of LS, it is reasonable to apply to test the performance of LS. What’s more, in light of the experiment for the 6 item case is an initial test to investigate the basic performance of LS, the parameter settings of above mentioned algorithms are shared with their original settings. The performance of GA and DE has been extensively investigated by Moon et al. [10] and Wang et al. [31] in solving JRP, respectively. Specifically, the basic parameter settings for GA are inherited from Moon et al. [10] such that the uniform crossover factor Pc and mutation factor Pm are set to 0.6 and 0.2, respectively, also the single point

4.2. Parameter sensitivity analyses Parameters here are of two kinds: the ones that related to LS, and the others that related to the JRP problem. In the following

Table 3 Comparison of performance of GA, DE and LS. Fitness

GA DE PSO LS

Best-found results

AVE. CPU Time (s)

Mean

Std.

Total cost

Schedule of ki s

T

10710.86 10647.86 10649.86 10642.88

71.6489 12.3927 16.5999 3.7009

10641.86 10641.96 10641.84 10641.84

1, 1, 1, 1,

0.0738 0.0740 0.0738 0.0737

1, 1, 1, 1,

1, 1, 1, 1,

2, 2, 2, 2,

2, 2, 2, 2,

4, 4, 4, 4,

0.5449 0.5335 0.4409 0.4862

Please cite this article as: L. Cui et al., A novel locust swarm algorithm for the joint replenishment problem considering multiple discounts simultaneously, Knowledge-Based Systems (2016), http://dx.doi.org/10.1016/j.knosys.2016.08.007

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1.35

4

PSO

x 10

1.5 pbestfit gbestfit

LS

x 10

pbestfit gbestfit

1.45

1.3 1.4 1.35

Fitness

Fitness

1.25

1.2

1.15

1.3 1.25 1.2 1.15

1.1 1.1 1.05

0

20

40

60

80

100

1.05

0

Iteration

20

40

60

80

100

Iteration Fig. 4. Evolutionary processes of PSO and LS .

contents, sensitivities of these two classes of parameters are extensively investigated in two sequential separated parts below. 4.2.1. Sensitivity analyses of LS-related parameters LS-related parameters may have significant impacts on the final results of the proposed JRP problem, especially the parameters in generating ‘smart’ start points that are designed to ‘launch’ away from their previous optimum acquired by a typical PSO. Thus, investigations on the performance of LS are divided as an overall investigation of evolutionary processes of LS and PSO, and effects of LS under different parameter combinations. (1) Evolutionary processes of LS Considering the close relationship of LS and PSO, PSO is chosen as the comparing algorithm. The evolutionary processes of two algorithms, especially the present best fitness (pbestfit) and the global best fitness (gbestfit), can reveal us more details on the searching processes of LS and PSO. To observe the evolutionary processes clearly in the initial stages of LS and PSO, their maximum iterations are all set as 100, respectively. It is noted that Sub-Swarms 1 and 2 of LS are run 50 iterations, respectively. Here below Fig. 4 is illustrated the evolutionary processes two algorithms of for our analysis. Fig. 4 shows that thanks to the local search ability improvements in Sub-Swarms 2, LS converges to its best-found optimal earlier than that of PSO, see the evolutionary processes of gbestfit. Taking the in-depth search for discussion, LS has lower pbestfit than that of PSO before they come to their convergent results. From the evolutionary processes in Fig. 4, we can find that LS has been explained the superior searching abilities comparing to those of PSO. (2) Effects of LS under different parameter combinations Several parameters are comprised in LS, such as the inertia factor w, gap factor between the ‘smart’ points and the last optimal, each of them may cause significant impacts to the whole searching efficiency of LS. Thus, the performance of LS is extensively investigated under different parameter settings. The maximum iterations

of LS under different parameter settings are all set as 100, each setting is run 15 times. The statistical results are listed in Table 4 below. From the results in Table 4, we can safely conclude that: a. The setting in No 0∗ is the most expectable setting for the experiments in stableness and searching effects. b. The fluctuations of parameters of LS show quite different influences on the results. For example, the increasing number of locusts for searching has slightly impacts on the final results, while the disturbances of c¯1 , c¯2 , gap and v cause significant impacts to standard errors of final results. c. We also find that the searching time of LS under different parameter settings, except when the number of locusts is increased, are quite close to each other. Hence, we believe that if LS keeps the number of searching locusts unchanged, the searching time of LS is considered being equal. 4.2.2. Sensitivity analyses of problem-related parameters (1) Problem-related parameter sensitivity analyses The fluctuations of problem-related parameter impact the final results of the total cost of the proposed JRP model. Furthermore, the disturbances of yearly demand, hold cost per unit of item, and ordering cost might cause different level of impacts to the total cost. Thus, the flowing experiments are performed on these parameters. To acquire more generalized results, LS is run 20 times in each parameter settings. The comparison results are listed in Table 5 below. The results in Table 5 reveal us in three aspects: a. The disturbances of all parameters show positive relations with the changes final results. As the magnitude of parameter increases, the final results are increases, but the change rates of final results are smaller than that of parameters. b. The change rates of demand is closely related to that of the holding cost per unit and the cost of items per unit, so are the change rates of major ordering cost and minor ordering cost, which dues to their direct linear relationships in the proposed JRP model. c. The disturbances of demand (and holding cost per unit) have more substantial impacts than those of ordering cost to the

Please cite this article as: L. Cui et al., A novel locust swarm algorithm for the joint replenishment problem considering multiple discounts simultaneously, Knowledge-Based Systems (2016), http://dx.doi.org/10.1016/j.knosys.2016.08.007

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Table 4 Computational results under different parameter combinations of LS. No. ∗

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

Locusts

w

c¯1

c¯2

R

gap

spacing

v

Mean

Std.

30 10 50 100 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30

0.8 0.8 0.8 0.8 0.4 1.2 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8

1.494 1.494 1.494 1.494 1.494 1.494 0.5 1 2 1.494 1.494 1.494 1.494 1.494 1.494 1.494 1.494 1.494 1.494 1.494 1.494 1.494 1.494 1.494

1.494 1.494 1.494 1.494 1.494 1.494 1.494 1.494 1.494 0.5 1 2 1.494 1.494 1.494 1.494 1.494 1.494 1.494 1.494 1.494 1.494 1.494 1.494

50 0 0 50 0 0 50 0 0 50 0 0 50 0 0 50 0 0 50 0 0 50 0 0 50 0 0 50 0 0 50 0 0 50 0 0 200 500 10 0 0 50 0 0 50 0 0 50 0 0 50 0 0 50 0 0 50 0 0 50 0 0 50 0 0 50 0 0

[1,1,1,1,1,1,0.001] [1,1,1,1,1,1,0.001] [1,1,1,1,1,1,0.001] [1,1,1,1,1,1,0.001] [1,1,1,1,1,1,0.001] [1,1,1,1,1,1,0.001] [1,1,1,1,1,1,0.001] [1,1,1,1,1,1,0.001] [1,1,1,1,1,1,0.001] [1,1,1,1,1,1,0.001] [1,1,1,1,1,1,0.001] [1,1,1,1,1,1,0.001] [1,1,1,1,1,1,0.001] [1,1,1,1,1,1,0.001] [1,1,1,1,1,1,0.001] gap × 2 gap/2 gap/4 [1,1,1,1,1,1,0.001] [1,1,1,1,1,1,0.001] [1,1,1,1,1,1,0.001] [1,1,1,1,1,1,0.001] [1,1,1,1,1,1,0.001] [1,1,1,1,1,1,0.001]

0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.4 0.6 0.8 1.0 0.2 0.2

0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.8 1.0

10645.21 10651.14 10645.09 10648.42 10649.62 10648.46 10668.34 10648.58 10648.60 10688.18 10648.36 10651.60 10651.84 10645.33 10649.16 10650.23 10645.48 10658.58 10654.33 10654.74 10670.33 10710.39 10662.72 10673.69

7.1943 8.3096 7.2604 8.8155 8.9276 8.7923 41.1170 8.6837 7.7193 62.9452 8.8715 7.9097 8.5880 8.8372 7.3275 18.7633 7.6961 22.4357 17.7645 26.1665 32.2957 97.9712 20.1420 66.5191

∗ means the settings have already applied in above experiments. Table 5 Total cost changes under parameter fluctuations. %

D h S s c

TC r TC r TC r TC r TC r

−30%

−20%

−10%

0%∗

10%

20%

30%

8913.33 -16.24 8913.33 -16.11 9794.36 -7.96 9803.39 -7.88 8938.68 -16.00

9524.65 -10.50 9524.66 -10.46 10084.62 -5.24 10099.42 -5.10 9578.40 -9.99

10098.80 -5.10 10098.80 -5.13 10366.90 -2.58 10374.91 -2.51 10105.42 -5.04

10641.84 0.00 10641.84 0.00 10641.84 0.00 10641.84 0.00 10641.84 0.00

11158.34 4.85 11158.34 4.88 10909.96 2.52 10902.34 2.45 11152.90 4.80

11651.99 9.49 11651.844 9.41 11183.878 5.09 11156.862 4.84 11686.8 9.82

12125.18 13.94 12125.27 13.85 11425.87 7.37 11405.84 7.18 12111.34 13.81

r = (T C − T C ∗ )/T C ∗ , TC∗ is the total cost when there is no parameter fluctuation.

Table 6 Computational results under different discount strategies. Fitness

JRP JRP JRP JRP

with with with with

N A I A&I

Best-found results

Mean

Std.

Total cost

Schedule of ki s

T

11418.73 10642.26 10684.00 10647.88

6.3107 7.7611 6.9514 7.7009

11407.34 10641.81 10666.86 10641.89

1, 1, 1, 1,

0.0693 0.0737 0.0737 0.0737

1, 1, 1, 1,

1, 1, 1, 1,

2, 2, 2, 2,

2, 2, 2, 2,

4, 4, 4, 4,

total cost. Thus, to decrease the total cost, a reasonable way is to control the amount of items, decrease the holding cost per unit or offer a lower purchasing price. (2) Different quantity discount effects One main aim of this research is to investigate different quantity discount strategies to the total cost decisions. Below experiments are conducted to investigate the JRP under different discount considerations. Considering the former experiments on the same case with 6 items, we try to find the global best answers to investigate different quantity discount effects to the total cost, not to investigate the effectiveness and efficiency of LS,. Henceforth, we decrease the running times as 10 but prolong the iterations as 10 0 0. The computational results are given in Table 6 below. The computational results in Table 6 indicates us that

a. The standard errors of fitness under different cases are very small, which means that the searching process of LS are stable, and the results are effective for the analyses. b. The total cost for multi-item without considering any discount is larger than that of the other situations, which tells us that the discount strategy (no matter what type of discount strategy is considered) can help decrease total cost. c. The magnitudes of reduced total cost are different considering different quantity discount strategies. For example, of all average fitness with quantity discount(s), JRP with A has the smallest computational results, JRP with I owns the highest total cost, and the results of JRP with A & I lies in the medium, which means that JRP with A & I coordinates the roles of JRP with A and JRP with I. Management insights reveal us that, JRP with A can create benefits for the buyer, but JRP with I creates benefits for the supplier, JRP with A & I contributes an ‘equilibrium’ of these two strategies and output a satisfied total cost. 4.3. Further test of LS In this part, two groups of experiments are conducted to investigate robustness of LS in solving JRP with A & I with randomly generated parameters and the searching ability of LS in solving JRPs under 5 larger scale JRP cases.

Please cite this article as: L. Cui et al., A novel locust swarm algorithm for the joint replenishment problem considering multiple discounts simultaneously, Knowledge-Based Systems (2016), http://dx.doi.org/10.1016/j.knosys.2016.08.007

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L. Cui et al. / Knowledge-Based Systems 000 (2016) 1–12 Table 7 Computational results of LS with random parameters of JRP. No.

1 2 3 4 5 6 7 8 9 10

S

s

[10 0, [50 0, [50 0, [10 0, [10 0, [10 0, [10 0, [10 0, [50 0, [50 0,

50 0] 10 0 0] 10 0 0] 50 0] 50 0] 50 0] 50 0] 50 0] 10 0 0] 10 0 0]

[1, 50] [1, 50] [50, 100] [1, 50] [1, 50] [50, 100] [50, 100] [1, 50] [50, 100] [50, 100]

D

[10 0 0, [10 0 0, [10 0 0, [10 0 0, [10 0 0, [10 0 0, [10 0 0, [10 0 0, [10 0 0, [10 0 0,

c

50 0 0] 50 0 0] 50 0 0] 10 0 0 0] 50 0 0] 50 0 0] 50 0 0] 10 0 0 0] 10 0 0 0] 10 0 0 0]

Fitness (PSO)

[1, 10] [1, 10] [1, 10] [1, 10] [11, 20] [1, 10] [11, 20] [11, 20] [1, 10] [11, 20]

Fitness (LS)

IPR

Means

Std.

Means

Std.

21207.92 18728.01 22556.91 23039.32 21182.12 18996.09 27748.00 31981.11 41790.00 49577.52

109.59 98.77 113.71 121.80 109.34 51.48 126.30 134.25 179.16 78.18

21081.32 18635.30 22373.64 22846.00 21043.25 18862.68 27591.01 31759.86 41487.21 49207.13

39.39 56.50 64.93 94.50 77.10 68.42 81.19 92.69 74.23 71.79

0.60% 0.50% 0.82% 0.85% 0.66% 0.71% 0.57% 0.70% 0.73% 0.75%

IPR = (MeansPSO − MeansLS )/MeansLS , IPR denotes to the improvement rate of total cost. Table 8 Input data for 5 larger cases. Items

Discounts

Di in [200, 20000]

S

si in [40, 60]

ci in [1, 20]

10

A:5, I:4, N:1

200

[58,55,60,43,41,50,55,47,56,57]

[17,7,3,7,7,2,18,16,2,12]

15

A:2, I:8, N:5

200

20

A:6, I:5, N:9

[52,47,60,44,42,53,46,60,46,47, 48,57,53,48,47] [46,48,49,59,47,58,43,47,50,44, 44,56,56,48,45,47,45,59,58,47]

[20,14,14,13,14,18,9,19, 17,14,10,1,8,15,6] [11,9,8,8,8,8,6,15,1,6,5, 19,9,19,15,1,20,15,16,20]

30

A:10, I:10, N:10

200

[41,52,58,57,59,51,46,60,51,43, 52,55,42,49,40,43,46,47,47,55, 40,47,52,58,55,59,44,40,60,43]

[5,3,4,12,3,2,9,8,4,1, 3,3,9,7,8,10,3,6,5,16,18, 5,7,1,16,17,20,10,20,2]

50

A:20, I:20, N:10

[1922,1016,10965,18946,14776,15676,4049, 16698,17713,7499] [8562,15214,16958,11232,8446,18057,15944, 7781,11580,15579,14359,3329,7877,6329,14352] [327,19705,18840,5462,9237,4656,1458,4058, 2983,16836,1879,6051,994,19602,16158,7568, 12243,9151,5598,6483] [19067,7826,8713,8178,11371,15067,17082, 15529,17769,5410,8783,16101,5089,19005, 7804,13862,878,11447,12278,5055,15077,9169, 8343,8060,2401,1642,8492,5502,19446,16248] [7243,2181,7822,13985,14059,16117,11477, 14822,513,13155,3656,19097,6795,1948,9229, 17284,2965,15068,8042,5994,13309,10527,8668, 11413, 15334,874,16991,13601,14925,8159,11781, 18809,15984,731,6558,6829,16599,13906,4827, 8498,9360,7782,4537,18237,5975,7757,1713, 3467,4164,14620]

200

[52,49,51,58,54,45,58,41,41,43, 47,52,58,46,44,52,48,49,42,56, 59,53,47,50,46,59,54,42,40,54, 48,52,46,46,51,59,54,51,45,42, 40,40,43,53,51,50,51,52,48,47]

[52,49,51,58,54,45,58,41, 41,43,47,52,58,46,44,52, 48,49,42,56,59,53,47,50, 46,59,54,42,40,54,48,52, 46,46,51,59,54,51,45,42, 40,40,43,53,51,50,51,52, 48,47]

4.3.1. Tuning test of LS under random problem-related parameters To verify the effectiveness of LS in solving the proposed JRP, LS is tested in a more randomness environment on proposed JRP problem. PSO is also chosen as the comparative algorithm. The parameters are generated uniformly in their corresponding domains, each algorithm under a specific setting is run 20 times and their average value and standard errors of fitness are collected. Moreover, we assume the discount intervals are equally divided in length considering the number of intervals in Table 2, and the discounted prices are randomly given with stepped-down sequences, correspondingly. The maximum iteration of PSO and LS is set as 10 0 0. The random parameters and computational results are given in Table 7. The results in Table 7 tell us that LS outperforms PSO both in the searching best-found results (mean value of fitness) and stableness (standard errors of fitness) in random parameter situations. In addition, as the magnitude of parameters increases, IPR are increased correspondingly, which indicates that even in the more general situations, the searching efficiency and effectiveness of LS is superior to that of PSO, especially when the magnitude of parameters is larger. 4.3.2. Performance of LS under some larger cases To further investigate the performance of LS, we provide 5 larger cases, for below experiments, the numbers of items is 10, 15, 20, 30 and 50, respectively. To ease our analysis, we assume that there have four intervals for items offered with all-unit discount, three intervals for items offered with incremental discount. In addition, the lengths of each interval are assumed equally divided over the domain of corresponding item. The parameters for the

200

5 larger cases are generated randomly and listed in Table 8. The comparing algorithm is PSO, the remaining parameter settings for JRP and two algorithms are inherited from those in above experiments. Each algorithm is run 30 times and the maximum iteration is set as 10 0 0. The computational results are listed in Table 9 below. From Table 9, we can draw that: (1) Both PSO and LS can fast converge to their best-found results. However, LS outweighs PSO at the searching ability and stableness, but inferior to PSO at the searching time. (2) As the scale of the proposed JRP problem increases, the stableness of both PSO and LS is getting weaker, but the searching time increases tremendously. (3) The basic cycle time T becomes smaller as the scale of the proposed JRP problem increases. That is to say, the more items is considered in one batch, the more grouping coordination cost will it take, and more items is to be replenished in per order. (4) Obviously, when the amount of items comes to very large scale, e.g. 50, whether the performance for grouping all items randomly (DGS) is efficient and effective than grouping all items using IGS is questionable, which is not reflected in our research, but left spaces for our future study. 5. Conclusions In this paper, a new JRP model is constructed considering two types of discounts, all-unit quantity discount and incremental quantity discount, simultaneously. Considering the NP-hard nature of JRP, a novel meta-heuristic algorithm, namely, LS, is introduced

Please cite this article as: L. Cui et al., A novel locust swarm algorithm for the joint replenishment problem considering multiple discounts simultaneously, Knowledge-Based Systems (2016), http://dx.doi.org/10.1016/j.knosys.2016.08.007

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18.18 571500.69 528363.96

32564.81

The multiple discount impacts to the proposed JRP and the performance of JRP is vigorously investigated though, there still leaves some spaces for improvements, for example, the main drawback of LS is that it has multiple algorithm-related parameters which cost lots of energy to find a more effective and efficient parameter combination. Therefore, in our future research, the comparisons of two grouping methods, DGS and IGS, in large scale JRP cases considering order quantity discounts by developing new LS, which integrates the superiorities of ABC in ‘scout’ generation. Acknowledgment This research is partially supported by Social Science Planning Project of Chongqing (2015YBGL117; 2015BS027), National Natural Science Foundation of China (71602015; 71371080; 71471024), Scientific and Technological Research Program of Chongqing Municipal Education Commission (No. KJ1500523), and Fundamental and Frontier Research Project of Chongqing (cstc2016jcyjA0530).

16.24

0.0262 0.0160 0.0205 0.0162

2,3,1,1,1,1,1,1,1,1 1,1,1,1,1,1,1,1,1,1, 1,5,5,1,1 4,1,1,2,1,2,2,1,5,1, 5,1,4,1,1,5,1,1,1,1 1,3,3,1,2,2,1,1,2,3, 3,1,2,1,1,1,4,1,2,2, 1,5,2,4,3,2,1,1,1,4 1,1,1,1,1,1,1,1,1,1, 1,1,1,1,1,1,1,1,1,1, 1,1,1,1,1,1,1,1,1,1, 1,1,1,1,1,1,1,1,1,1, 1,1,1,1,1,1,1,1,1,1 532546.48

568128.16

33509.55

References

5

CPU T (s) is the average computational time.

0.0105

48259.91 94328.84 93228.01 130308.13

48669.58 96693.80 94272.18 135058.36

539.56 2647.50 2719.92 2938.10

3.68 5.34 7.32 9.26 T

0.0259 0.0180 0.0201 0.0203

1,4,2,1,1,2,1,1,2,1 1,1,1,1,1,1,1,1,1,1, 1,5,2,1,1 5,1,1,2,1,2,1,1,3,1, 3,1,3,1,1,5,1,1,1,1 1,2,2,1,2,2,1,1,1,2, 3,1,1,1,1,1,5,1,1,1, 1,2,2,5,2,3,1,2,1,2 1,1,1,1,1,1,1,1,1,1, 1,1,1,1,1,1,1,1,1,1, 1,1,1,1,1,1,1,1,1,1, 1,1,1,1,1,1,1,1,1,1, 1,1,1,1,1,1,1,5,1,1 1 2 3 4

Std. Means ki

Total cost

Fitness

(1) LS is introduced and redesigned to solve the proposed JRP for the first time, and has been testified as an efficient and effective algorithm for solving the proposed JRP model through extensive numerical experiments. (2) Parameter analysis indicates that most of LS-related parameters except the number of locusts can cause significant impacts to the searching ability and searching stableness of LS, while the JRP problem-related parameters have positive influences on total cost of the proposed JRP problem, which give us some hints for decreasing the total cost. (3) The quantity discount strategy in JRP operations can help decrease the total cost. While a more flexible quantity discount offer that combines different discount strategies according to specific items, might benefit both the supplier and the buyer.

0.0097

48455.16 95271.58 93526.09 130969.23 48224.57 94059.54 92012.22 129949.32

195.70 1119.10 1796.62 750.77

4.28 5.76 9.16 11.30

CPU T (s) Std. Means

Fitness

Total cost T ki

CPU T (s)

LS Best-found result

11

and redesigned to solve the proposed JRP model. From the results of numerical experiments, we can conclude that,

PSO Best-found result

Table 9 Computational results of 5 larger cases.

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Please cite this article as: L. Cui et al., A novel locust swarm algorithm for the joint replenishment problem considering multiple discounts simultaneously, Knowledge-Based Systems (2016), http://dx.doi.org/10.1016/j.knosys.2016.08.007