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Simulation of Manufacture Inventory Strategy Based on Through Train Supplies
JOURNAL OF TRANSPORTATION SYSTEMS ENGINEERING AND INFORMATION TECHNOLOGY Volume 12, Issue 2, April 2012 Online English edition of the Chinese language journal RESEARCH PAPER
Cite this article as: J Transpn Sys Eng & IT, 2012, 12(2), 98104.
Simulation of Manufacture Inventory Strategy Based on Through Train Supplies HE Guoxian, Fu Zhongning* School of Traffic and Transportation, Lanzhou Jiaotong University, Lanzhou 730070, China
Abstract: This paper focus on inventory strategy of manufactures on the basis of bulk raw material supply provided by railway through trains. The outbound quantity of materials is rely on the stochastic output, such as the probability distribution of daily sales volume of certain products, production breakeven point and the designed capacity of progressive assembly line. The inbound quantity is determined by the railway train diagram and landed weight in accordance with orders. The time step method is used to obtain optimum solution by computer simulation. The model is composed of inventory holding costs, ordering costs based on order discount, procurement constant expense and material shortage costs. One example with given demand probability distribution and certain railway cargo transportation timetable is simulated by MATLAB program. The study reveals that computer simulation is more adequate for mass-production scale manufactures to make inventory management decisions. The result illustrates that order quantity and departure time-interval of through train play more important role in inventory simulation system. Key Words: logistics engineering; inventory strategy; computer simulation; railway timetable; railway through freight train
1
Introduction
The large scale manufacture always requires bulk material for production, there exists such chrematistics as long distance, large quantity, multiple times, and regular procurement in its inbounds logistics. Railway freight operation has finished transformation to modern logistics by business process re-engineering in management. And it emphasizes private railroad service relying on through freight train on the basis of strategic loading station for improving core competence. Such logistics mode can meet material requirement of large scale manufactures, railway transportation economics of scale can also be reached. For typical example, Gansu west logistics CO. LTD provides Dalian Shide branch company (located in Ningxia province) with material supply by making full use of powder tank cars. For avoiding material shortage, manufacturing enterprise and railway logistics company may sign a cooperative agreement about material supply provided by through freight train, the purpose is both for minimizing inventory level of manufacturing enterprise and for loading railway cars according to train working diagram, such operation model can ensure supplying material by through
freight train conveniently and efficiently. Here it can be seen that avoiding material shortage and minimizing inventory level is of great value for further study, in this paper we concern with simulation method for optimal relationship between order quantity and departure interval of through freight trains, both stochastic requirement quantity of end product, break even point of manufacturing, material ratio in BOM, and loading time table of through freight train originating from railway loading terminal are all taken into consideration.
2
Problem analysis
Main production plan made by large scale manufacture is always dependent on whether accumulation of daily requirement quantity t of end product˄denoted as Op˅exceeds scale production quantity Qe, the upper limit of the value Qe is designed capacity of assembly line Qm and its inferior limit is production breakeven point Qw, thus we have Qe [Qw , Qm ] . In actual production process, manufacturing enterprise make a decisive coefficient D (0, 1] according to its own market share, thus Qe can be calculated by Eq. (1) as following.
Received date: Jan 5, 2012; Revised date: Feb 13, 2012; Accepted date: Feb 21, 2012 *Corresponding author. E-mail: [email protected] Copyright © 2012, China Association for Science and Technology. Electronic version published by Elsevier Limited. All rights reserved. DOI: 10.1016/S1570-6672(11)60198-0
HE Guoxian et al. / J Transpn Sys Eng & IT, 2012, 12(2), 98104
n 1
¦U t 1
n
t
Qe d ¦ Ut
(2)
t 1
Then ex-warehousing quantity qM of Om can be expressed as Eq. (3): n
qM
k ( ¦ Ut )
(3)
i 1
Fluctuating regulation of t accompanying rate of production progress is illustrated by Fig. 1(a), and relationship between qM and ex-warehousing date is shown as Fig. 1(b). Form Fig. 1 we can see that material ex-warehousing process exits such prosperity as variable process cycle and bulk quantity, hence when enterprise organizes procurement, reasonable order quantity and actual purchasing cycle must be decided effectively. For the sake of particularity in this kind of material requirement mode, through freight train originated form loading terminal is almost the fist choice for material supplying service provided by railway 3PL. Railway freight transportation department usually construct or set one strategic loading base (private line or special railway), and provide manufacturing enterprise with through freight trains service applying uniform arrival for material supply according to its own transportation conditions and enterprise requirement. Departure interval of through train is fixed (ranges form one day to one week), whether directly through train or reverse multi-steps through train (illustrated by Figure 2) is organized depending on actual freight volume. It should be pointed out that reverse multi-steps through train differ from directly through train in organization method. When a supplier depends on reverse multi-steps through train to provide several service objects (manufacturing enterprise) with material supply, all the enterprises are not far from one another[3] and requires one special type material, besides, all enterprises are always far from loading terminal (station).In such a process the supplier makes full use of cooperative relationship with railway logistics company, facilities and equipments of strategic loading base can also be used by the supplier The problem oriented by large scale manufacture is how to make optimal ordering strategy according to arrival time interval of thought freight train decided by train working diagram and once material order quantity so as to ensure both maintaining minimal inventory and avoiding material shortage simultaneously.
3
inventory decision system as follows: Ch: unit Om holding cost (yuan/t.year); Cb: unit Om shortage cost (yuan/t.day); Tm: total production time length (day); Nq: Number of total terminal distance intervals; Vj: vector of through freight train in j distance interval (km/h); Dj: length of j distance interval (km); Nmax: maximum cars in through freight train (car); Wv: Average loading capacity per car (t); Cos : constant order cost (yuan). In this model for calculating ordering cost, both const order cost and quantity discount coat are taken into consideration, and we define essential decision variables as follows: Co: Once order cost (yuan); no: procurement (purchase) times; q: actual order quantity (t); t: the tth day accumulation of end product demand (t); f: departure interval of through freight train (day); ltm : the tth day inventory level of Om(t); j: time length of maintenance time window of j distance interval; W opt : optimal departure interval of through freight train (day); qopt: optimal order quantity (t); Cd: discount cost according to order quantity (yuan); Nd: total number of discount span; Cid : quantity discount cost of i discount span (yuan). Daily accumulation of 1(t)
(1) Once the accumulation of t exceeds Qe slightly, enterprise starts production correspondingly, and deliveries material (denoted as Qm) form warehouse to workshop, let denoted k be consumption ratio of Om according to BOM. When Ut accumulating continuously after n days, if we have Eq. (2).
1000
500
0 5 10 15 20
25 30 35 40 45 50 55 60 Date (d)
600 qM (t)
Qw D (Qm Qw )
Qe
400 200 0 14
28 41 Ex-warehouse date of Om (d)
154
(Qe=800; Qm=1200; =0.8; k=0.65) Fig. 1 Fluctuating regulation of t and qM
Model analysis First we define relevant constants about researched
Fig. 2 Timetable of reverse multi-steps through freight train generated in central cargo terminal
HE Guoxian et al. / J Transpn Sys Eng & IT, 2012, 12(2), 98104
Transportation cost (yuan/car)
Transportation cost (yuan/car)
Cd
Order quality (car)
In such a model Cd is the step function of order quantity ˄number of cars˅ˈon the assumption that transportation cost and order quantity exist such relation illustrated by Fig. 3(a), let 60 car(or slightly less) be 0 discount cost benchmark, then Cd and order quantity have such relationship indicated by Fig. 3(b). The model about material procurement supplied by thought freight train can be expressed as Eq. (4): T (4) m
Co
t 1
q
Tm
Ch w Tm lt ¦ P Cb ltw t 1 1 365
¦ t
s.t. ª º N « Tm q W j » ³ j 1 « » 1 no d « Nq D j » « ³j 1 » V j ¼» ¬«
Wv d q d N max Wv Co
Cd
q Cos Wv Nq
Wf
Tm ³ W j
m t 1
l
j 1
no
l Y q T qM m t
where P
0 ltw t 0 ® ¯1 2WKHUZLVH
Y
1 WKHt 1th day material arrive ® RWKHUZLVH ¯0
T
N 0 (baseline value). where K0 Wv ; K N N max Wv ; Cd m on the Basis of the definition, lt means the tth day real inventory level of Om, it is difficult to master its changing rule through analysis method on account of randomness. Computer simulation can be adopted to gain the solution of modelˈ starting from its initial value and keeping pace of simulation clock, the daily inventory level ltm can be analyzed according to material input/output during simulating period. Above all, for the mathematic model (4), computer simulation is more adequate than normal analysis method due to stochastic character of t, and optimal solution [opt, qopt] can be solved more effectively. d
Order quality (car)
Min Z
(5)
d
Fig. 3 Relationship between ordering cost and transportation cost plus discount
k ¦ Ut
Ko d q d K1 Cd1 ° 1 K1 q d K2 ° Cd ® ... ° ... °CdNd K N 1 q d K N ¯ d d
1 WKHt 1th day material ex-warehoue ® RWKHUZLVH ¯0
According to the cooperation framework shared by railway 3PL and manufacturing enterprise ˈunder the expression of Fig. 4, let define i as separation point between adjacent discount span, then Cd can be illustrated by Eq. (5).
4
Simulation method
Manufacturing enterprise usually ex-warehouses material depending on demand for Op and its production schedule in accordance with inventory strategy, and relies on the material supply provided by railway through freight train. In addition, we denote simulation clock step-size as t [5]. For getting the optimal solution by concrete simulation method, firstly it is necessary to deicide parameters set and variables set as well as the initial value and limit of variation for all variables involved. Then we make full use of the system image composed of parameters and variables, the optimal solution can be solved by scanning step pace of the system image as mentioned early, we denote additional parameters as follows, W a : Departure time of 1st through freight train in [1, Tm]; : mode of through freight train, =1 means there exist one trainˈotherwise let =0; M: penalty value when shortage (+ ); ª¬W sw ,W ew º¼ : Maintenance time window; Q0: Initial inventory level of Qm; 't : Simulation clock step-size; And define other variables as below; t : Simulation clock, wherer t >1, Tm @ ; t f : sub-clock for departure moment at loading terminal˗ tr : sub-clock for traveling mode of through freight train˗ Pt: accumulation of t at t, thus 5t
t
¦U
i
i 1
Then we can complete the simulation process according to the following steps, Step1: key variables initializing, let t=1; ltm Q0 ; Z 0 ; tr 0; Pt=0; t f W a . Step2: if t mod 241 then go toStep6; otherwise go to Step 3.
HE Guoxian et al. / J Transpn Sys Eng & IT, 2012, 12(2), 98104
ltm k 5 t
ltm
tr t 1; ltm
Q0 ; Z
0;
tr
0; t f
Wa;
0; Pt
Pt
0
tr 't;
?l 0 m t
? Z =1 ? 5 t t Qe
Pt Ut
Z
0;
tr
0;
Cd
q Cos Wv
tf
t f 2W f ;
? t ! Tm
Z ltm u Ch Nq
? tr ¦ j 1
? t mod 24 z 1
?t f W f ª¬W sw ,W ew º¼
ltm q;
ltm Co
Pt
Z
?t
tf
Dj Vj
tf
t f W f ;
let Z 1 t
t 't
Fig. 4 Flowchart of computer simulation process
Step 3: Pt=Pt+t. Step 4: if 5t Qe go to Step 5; otherwise let ltm ltm k 5t , 5 t 0 , and if ltm 0 , then let Z=M ( penalty value), thereafter go to Step 10. Step 5: calculating material daily holding cost by Z Z ltm u Ch . Step6: if t=tf then let =1. Step7: if =0 the go to step 9, otherwise let tr tr 't ; and if Nq
tr ¦ j 1
Dj Vj
go to Step 9; otherwise dealing with material arrival affairs: let ltm ltm q ; and calculating material ordering cost Co
Cd
q Cos Wv
let =0 for waiting next through freight train; tr=0; if t f W f ¬ªW sw ,W ew ¼º the let t f t f 2W f ; otherwise let t f t f W f . Step9: let t t 't to promote simulation process˗if t>Tm then go to Step10, otherwise go to Step 2. Step10: ª¬W f , q º¼ can be viewed as once simulation solution. The flowchart for the above steps can be illustrated by Fig. 4. We must simulate the inventory process many times according to limit of variation of ª¬W f , q º¼ , that is to say that we rely on all solution sets about f and q, the inventory strategy [W opt , qopt ] with minimized inventory costs can be obtained through designed double circulation.
5
probability density f and probability distribution F shown as Figs. 5(b) and 5(c). Table 1 Historical data of end product requirement Date (d)
t (t)
Date (d)
t (t)
Date (d)
t (t)
1
896
13
909
25
876
2
908
14
891
26
915
3
911
15
898
27
909
4
902
16
903
28
885
5
896
17
895
29
879
6
892
18
904
30
895
7
884
19
875
31
910
8
896
20
906
32
912
9
902
21
906
33
907
10
897
22
900
34
911
11
903
23
895
35
925
12
901
24
894
36
876
Case study
This study takes a steal producing enterprise with 300000 ton annual production. And the raw material is coal for steelmaking. The historical data for daily steal volumes of sale is shown as Table 1. Bar chart of historical data is illustrated as Fig. 5(a), according to Matlab function ‘hist’ we can fit curve of
Fig. 5 Bar chart of end product requirement quantity and related probability distribution (density)
HE Guoxian et al. / J Transpn Sys Eng & IT, 2012, 12(2), 98104
Table 2 Parameters of through freight train for example Category
Value
Unit
Length of 1st interval (D1)
800
km
Length of 2nd interval (D2))
600
km
Vector of train in D1 (V1)
80
km/h
Vector of train in D2 (V2)
75
km/h
Maintenance time window*
[240,252]
h h
Distance between terminals (km)
Departure time** 6 * synchronous time of simulation clock ** time of departure is 6:00am(simulation clock time)
Departure time (h)
(Tm=31 days, f==72h) Fig. 6 Timetable of through freight train generated in central cargo terminal for example
During simulation time, one random number can be generated by Matlab sentence ‘r=rand ();’then we can obtain daily product demand according to Fig. 5(c) when F=r. In addition to above necessary information, we suppose that breakeven point Qw=4,000 (t), designed capacity of assembly line Qm=6,000 (t˅, the decisive coefficient =0.65, then we get scale production quantity Qe=5,300 (t) according to formula 1.That is to say that materials for manufacturing will be ex-warehoused when the accumulation of t exceed 5,300 t. There are 2 distance intervals form strategic loading base to the manufacture enterprise when through freight train is chosen for material supply. Lengths of distance intervals, vectors of train correspondingly, maintenance time window are all illustrated by Table 2. Then one certain train working diagram with 3-day departure interval and 31-day length of total simulation time can be shown by Fig. 6. Departure interval ranges form 1 day to 10 daysˈthe maximum number of cars in one through train is 60, average loading capacity per car Wv is equal to 55 t, the basic transpiration cost per car is 3000 Yuan(included car basic price, distance basic price, tax for construction fund, payment for stamp tax) ,the relationship between quantity discount cost and order quantity is illustrated by Table 3. The scope of order quantity belong to discrete set [1, 60] (car), namely, [55, 3,300] (t),the other parameters about material inventory strategy decided by the manufacturing
enterprise are all indicated in Table 4. The time step method is presented to obtain optimum solution for the example during computer simulating process, one step-size of simulation clock is 10 min (t=10 min), the total length of simulation times is one quarter (Tm=2,160 h). then we scan 600 combinations ¬ªW f , q ¼º thoroughly, all projects of material inventory strategy can be worked out for the example. and the 3D solution space is illustrated by Fig. 7. It can be found that the longer departure interval results in the higher likelihood of material shortage(adopting penalty value M); the more order quantity lead to the lower likelihood of material shortage and objective function increasing due to extra material holding cost. The 3D coordination of optimal solution in Fig. 7 is (7, 23, 226359.75), thus by comparison and selection through simulating process the optimal project [opt, qopt] is expressed as follows: [opt, qopt]=[7, 23] That is to say that when departure interval equal to 7 days, and order quantity equal to 23 cars (1,265 t),the enterprise has lowest inventory cost, and the objective function values is 226359.75 (yuan). Table 3 Relationship between discount and quantity Quantity discount cost Cd (yuan/car)
Order quantity (car)
300
1–5
150
6–20
100
21–40
0
41–60
Table 4 Parameters of enterprise inventory strategy Category
Value
Unit
Material holding cost Ch
1.5
Yuan/t.day
Cs Const ordering cost o Material initial inventory level Q0
3000
Yuan/once
550
t
Penalty value for shortage M
15000000
Yuan
Coal injection ratio
195
Kg/t
Material consumption coefficient k
0.195
t/t
Fig. 7 3D diagrammatic sketch of solution space for inventor strategy
HE Guoxian et al. / J Transpn Sys Eng & IT, 2012, 12(2), 98104
Inventory level (t)
station should be addedDŽ
Inventory level (t)
Simulation time (d)
Simulation time (d)
Fig. 8 Curve of relationship between inventory level and simulation time for optimal order quantity
The mean value of coal inventory level is 1,366 t when departure interval is 7 days and order quantity is 23 cars, and in this case variance of coal inventory level is 0 approximately. The relationship between inventory level and simulation time is illustrated by Fig. 8, where Fig. 8(b) is the part form 30th day to 60th day of Fig. 8(a).as shown in Fig. 7 enterprise not only get rid of coal inventory accumulating phenomenon in 90 consecutive days, but make inventory level display ideal changing trend under uncertainty of product demand.
Above all, the simulating process can be divided into four steps in general. First, when simulation clock develops to another day and accumulation of t reaches material outbound breakthrough point, material ex-warehousing should be operated; secondly, Material holding cost should be added to total cost when a day coming , moreover, departure time and traveling mode can be traced by relationship between simulation clock and departure hour; finally, if the through freight train arrives ,material inbound work should be completed and ordering cost should also be added to total cost . In this paper “drawing out method” is adopted when coping with train traveling time belong to maintenance time window, and we regard material shortage cost as infinitely greatˈ therefore, departure hour being delayed due to maintenance time window and actually shortage cost need researching further.
References [1]
Porteus E L. Foundations of Stochastic Inventory Theory, US
[2]
Pen Q Y, Yang M M. The integer programming model and its
California: Stanford University Press, 2002. solution for making double-track train working graph with
6
Conclusions
We focus on the single manufacturing enterprise about flowchart of simulation and case study ˈ as to reverse multi-steps through freight train generated in loading station for several enterprises, two transformations should be adopted for optimizing inventory strategy by system simulation. firstly, total material unloaded in passed terminals should be excluded to calculating actually arriving material quantity, secondly, for caculating arriving time correctly,all times for unloading material during cargo operation works in stopping railway
computer. China Railway Science, 1994, 15(4): 60–66. [3]
Cao X M, Lin B L, Yan H X. Optimization of direct freight train service in the loading place. Journal of the China Railway Society, 2006, 28(4): 6–11.
[4]
Wang W, Zhen H, Gao J. Simulation and optimization on inventory of distribution system using theory of constraints. Journal of Transportation System Engineering and Information technology, 2009, 9(2): 115–121.
[5]
Yang Z X. Computer simulation and application. Beijing: Chinese Railway Publishing House, 2002.
Cite this article as: J Transpn Sys Eng & IT, 2012, 12(2), 98104.
Simulation of Manufacture Inventory Strategy Based on Through Train Supplies HE Guoxian, Fu Zhongning* School of Traffic and Transportation, Lanzhou Jiaotong University, Lanzhou 730070, China
Abstract: This paper focus on inventory strategy of manufactures on the basis of bulk raw material supply provided by railway through trains. The outbound quantity of materials is rely on the stochastic output, such as the probability distribution of daily sales volume of certain products, production breakeven point and the designed capacity of progressive assembly line. The inbound quantity is determined by the railway train diagram and landed weight in accordance with orders. The time step method is used to obtain optimum solution by computer simulation. The model is composed of inventory holding costs, ordering costs based on order discount, procurement constant expense and material shortage costs. One example with given demand probability distribution and certain railway cargo transportation timetable is simulated by MATLAB program. The study reveals that computer simulation is more adequate for mass-production scale manufactures to make inventory management decisions. The result illustrates that order quantity and departure time-interval of through train play more important role in inventory simulation system. Key Words: logistics engineering; inventory strategy; computer simulation; railway timetable; railway through freight train
1
Introduction
The large scale manufacture always requires bulk material for production, there exists such chrematistics as long distance, large quantity, multiple times, and regular procurement in its inbounds logistics. Railway freight operation has finished transformation to modern logistics by business process re-engineering in management. And it emphasizes private railroad service relying on through freight train on the basis of strategic loading station for improving core competence. Such logistics mode can meet material requirement of large scale manufactures, railway transportation economics of scale can also be reached. For typical example, Gansu west logistics CO. LTD provides Dalian Shide branch company (located in Ningxia province) with material supply by making full use of powder tank cars. For avoiding material shortage, manufacturing enterprise and railway logistics company may sign a cooperative agreement about material supply provided by through freight train, the purpose is both for minimizing inventory level of manufacturing enterprise and for loading railway cars according to train working diagram, such operation model can ensure supplying material by through
freight train conveniently and efficiently. Here it can be seen that avoiding material shortage and minimizing inventory level is of great value for further study, in this paper we concern with simulation method for optimal relationship between order quantity and departure interval of through freight trains, both stochastic requirement quantity of end product, break even point of manufacturing, material ratio in BOM, and loading time table of through freight train originating from railway loading terminal are all taken into consideration.
2
Problem analysis
Main production plan made by large scale manufacture is always dependent on whether accumulation of daily requirement quantity t of end product˄denoted as Op˅exceeds scale production quantity Qe, the upper limit of the value Qe is designed capacity of assembly line Qm and its inferior limit is production breakeven point Qw, thus we have Qe [Qw , Qm ] . In actual production process, manufacturing enterprise make a decisive coefficient D (0, 1] according to its own market share, thus Qe can be calculated by Eq. (1) as following.
Received date: Jan 5, 2012; Revised date: Feb 13, 2012; Accepted date: Feb 21, 2012 *Corresponding author. E-mail: [email protected] Copyright © 2012, China Association for Science and Technology. Electronic version published by Elsevier Limited. All rights reserved. DOI: 10.1016/S1570-6672(11)60198-0
HE Guoxian et al. / J Transpn Sys Eng & IT, 2012, 12(2), 98104
n 1
¦U t 1
n
t
Qe d ¦ Ut
(2)
t 1
Then ex-warehousing quantity qM of Om can be expressed as Eq. (3): n
qM
k ( ¦ Ut )
(3)
i 1
Fluctuating regulation of t accompanying rate of production progress is illustrated by Fig. 1(a), and relationship between qM and ex-warehousing date is shown as Fig. 1(b). Form Fig. 1 we can see that material ex-warehousing process exits such prosperity as variable process cycle and bulk quantity, hence when enterprise organizes procurement, reasonable order quantity and actual purchasing cycle must be decided effectively. For the sake of particularity in this kind of material requirement mode, through freight train originated form loading terminal is almost the fist choice for material supplying service provided by railway 3PL. Railway freight transportation department usually construct or set one strategic loading base (private line or special railway), and provide manufacturing enterprise with through freight trains service applying uniform arrival for material supply according to its own transportation conditions and enterprise requirement. Departure interval of through train is fixed (ranges form one day to one week), whether directly through train or reverse multi-steps through train (illustrated by Figure 2) is organized depending on actual freight volume. It should be pointed out that reverse multi-steps through train differ from directly through train in organization method. When a supplier depends on reverse multi-steps through train to provide several service objects (manufacturing enterprise) with material supply, all the enterprises are not far from one another[3] and requires one special type material, besides, all enterprises are always far from loading terminal (station).In such a process the supplier makes full use of cooperative relationship with railway logistics company, facilities and equipments of strategic loading base can also be used by the supplier The problem oriented by large scale manufacture is how to make optimal ordering strategy according to arrival time interval of thought freight train decided by train working diagram and once material order quantity so as to ensure both maintaining minimal inventory and avoiding material shortage simultaneously.
3
inventory decision system as follows: Ch: unit Om holding cost (yuan/t.year); Cb: unit Om shortage cost (yuan/t.day); Tm: total production time length (day); Nq: Number of total terminal distance intervals; Vj: vector of through freight train in j distance interval (km/h); Dj: length of j distance interval (km); Nmax: maximum cars in through freight train (car); Wv: Average loading capacity per car (t); Cos : constant order cost (yuan). In this model for calculating ordering cost, both const order cost and quantity discount coat are taken into consideration, and we define essential decision variables as follows: Co: Once order cost (yuan); no: procurement (purchase) times; q: actual order quantity (t); t: the tth day accumulation of end product demand (t); f: departure interval of through freight train (day); ltm : the tth day inventory level of Om(t); j: time length of maintenance time window of j distance interval; W opt : optimal departure interval of through freight train (day); qopt: optimal order quantity (t); Cd: discount cost according to order quantity (yuan); Nd: total number of discount span; Cid : quantity discount cost of i discount span (yuan). Daily accumulation of 1(t)
(1) Once the accumulation of t exceeds Qe slightly, enterprise starts production correspondingly, and deliveries material (denoted as Qm) form warehouse to workshop, let denoted k be consumption ratio of Om according to BOM. When Ut accumulating continuously after n days, if we have Eq. (2).
1000
500
0 5 10 15 20
25 30 35 40 45 50 55 60 Date (d)
600 qM (t)
Qw D (Qm Qw )
Qe
400 200 0 14
28 41 Ex-warehouse date of Om (d)
154
(Qe=800; Qm=1200; =0.8; k=0.65) Fig. 1 Fluctuating regulation of t and qM
Model analysis First we define relevant constants about researched
Fig. 2 Timetable of reverse multi-steps through freight train generated in central cargo terminal
HE Guoxian et al. / J Transpn Sys Eng & IT, 2012, 12(2), 98104
Transportation cost (yuan/car)
Transportation cost (yuan/car)
Cd
Order quality (car)
In such a model Cd is the step function of order quantity ˄number of cars˅ˈon the assumption that transportation cost and order quantity exist such relation illustrated by Fig. 3(a), let 60 car(or slightly less) be 0 discount cost benchmark, then Cd and order quantity have such relationship indicated by Fig. 3(b). The model about material procurement supplied by thought freight train can be expressed as Eq. (4): T (4) m
Co
t 1
q
Tm
Ch w Tm lt ¦ P Cb ltw t 1 1 365
¦ t
s.t. ª º N « Tm q W j » ³ j 1 « » 1 no d « Nq D j » « ³j 1 » V j ¼» ¬«
Wv d q d N max Wv Co
Cd
q Cos Wv Nq
Wf
Tm ³ W j
m t 1
l
j 1
no
l Y q T qM m t
where P
0 ltw t 0 ® ¯1 2WKHUZLVH
Y
1 WKHt 1th day material arrive ® RWKHUZLVH ¯0
T
N 0 (baseline value). where K0 Wv ; K N N max Wv ; Cd m on the Basis of the definition, lt means the tth day real inventory level of Om, it is difficult to master its changing rule through analysis method on account of randomness. Computer simulation can be adopted to gain the solution of modelˈ starting from its initial value and keeping pace of simulation clock, the daily inventory level ltm can be analyzed according to material input/output during simulating period. Above all, for the mathematic model (4), computer simulation is more adequate than normal analysis method due to stochastic character of t, and optimal solution [opt, qopt] can be solved more effectively. d
Order quality (car)
Min Z
(5)
d
Fig. 3 Relationship between ordering cost and transportation cost plus discount
k ¦ Ut
Ko d q d K1 Cd1 ° 1 K1 q d K2 ° Cd ® ... ° ... °CdNd K N 1 q d K N ¯ d d
1 WKHt 1th day material ex-warehoue ® RWKHUZLVH ¯0
According to the cooperation framework shared by railway 3PL and manufacturing enterprise ˈunder the expression of Fig. 4, let define i as separation point between adjacent discount span, then Cd can be illustrated by Eq. (5).
4
Simulation method
Manufacturing enterprise usually ex-warehouses material depending on demand for Op and its production schedule in accordance with inventory strategy, and relies on the material supply provided by railway through freight train. In addition, we denote simulation clock step-size as t [5]. For getting the optimal solution by concrete simulation method, firstly it is necessary to deicide parameters set and variables set as well as the initial value and limit of variation for all variables involved. Then we make full use of the system image composed of parameters and variables, the optimal solution can be solved by scanning step pace of the system image as mentioned early, we denote additional parameters as follows, W a : Departure time of 1st through freight train in [1, Tm]; : mode of through freight train, =1 means there exist one trainˈotherwise let =0; M: penalty value when shortage (+ ); ª¬W sw ,W ew º¼ : Maintenance time window; Q0: Initial inventory level of Qm; 't : Simulation clock step-size; And define other variables as below; t : Simulation clock, wherer t >1, Tm @ ; t f : sub-clock for departure moment at loading terminal˗ tr : sub-clock for traveling mode of through freight train˗ Pt: accumulation of t at t, thus 5t
t
¦U
i
i 1
Then we can complete the simulation process according to the following steps, Step1: key variables initializing, let t=1; ltm Q0 ; Z 0 ; tr 0; Pt=0; t f W a . Step2: if t mod 241 then go toStep6; otherwise go to Step 3.
HE Guoxian et al. / J Transpn Sys Eng & IT, 2012, 12(2), 98104
ltm k 5 t
ltm
tr t 1; ltm
Q0 ; Z
0;
tr
0; t f
Wa;
0; Pt
Pt
0
tr 't;
?l 0 m t
? Z =1 ? 5 t t Qe
Pt Ut
Z
0;
tr
0;
Cd
q Cos Wv
tf
t f 2W f ;
? t ! Tm
Z ltm u Ch Nq
? tr ¦ j 1
? t mod 24 z 1
?t f W f ª¬W sw ,W ew º¼
ltm q;
ltm Co
Pt
Z
?t
tf
Dj Vj
tf
t f W f ;
let Z 1 t
t 't
Fig. 4 Flowchart of computer simulation process
Step 3: Pt=Pt+t. Step 4: if 5t Qe go to Step 5; otherwise let ltm ltm k 5t , 5 t 0 , and if ltm 0 , then let Z=M ( penalty value), thereafter go to Step 10. Step 5: calculating material daily holding cost by Z Z ltm u Ch . Step6: if t=tf then let =1. Step7: if =0 the go to step 9, otherwise let tr tr 't ; and if Nq
tr ¦ j 1
Dj Vj
go to Step 9; otherwise dealing with material arrival affairs: let ltm ltm q ; and calculating material ordering cost Co
Cd
q Cos Wv
let =0 for waiting next through freight train; tr=0; if t f W f ¬ªW sw ,W ew ¼º the let t f t f 2W f ; otherwise let t f t f W f . Step9: let t t 't to promote simulation process˗if t>Tm then go to Step10, otherwise go to Step 2. Step10: ª¬W f , q º¼ can be viewed as once simulation solution. The flowchart for the above steps can be illustrated by Fig. 4. We must simulate the inventory process many times according to limit of variation of ª¬W f , q º¼ , that is to say that we rely on all solution sets about f and q, the inventory strategy [W opt , qopt ] with minimized inventory costs can be obtained through designed double circulation.
5
probability density f and probability distribution F shown as Figs. 5(b) and 5(c). Table 1 Historical data of end product requirement Date (d)
t (t)
Date (d)
t (t)
Date (d)
t (t)
1
896
13
909
25
876
2
908
14
891
26
915
3
911
15
898
27
909
4
902
16
903
28
885
5
896
17
895
29
879
6
892
18
904
30
895
7
884
19
875
31
910
8
896
20
906
32
912
9
902
21
906
33
907
10
897
22
900
34
911
11
903
23
895
35
925
12
901
24
894
36
876
Case study
This study takes a steal producing enterprise with 300000 ton annual production. And the raw material is coal for steelmaking. The historical data for daily steal volumes of sale is shown as Table 1. Bar chart of historical data is illustrated as Fig. 5(a), according to Matlab function ‘hist’ we can fit curve of
Fig. 5 Bar chart of end product requirement quantity and related probability distribution (density)
HE Guoxian et al. / J Transpn Sys Eng & IT, 2012, 12(2), 98104
Table 2 Parameters of through freight train for example Category
Value
Unit
Length of 1st interval (D1)
800
km
Length of 2nd interval (D2))
600
km
Vector of train in D1 (V1)
80
km/h
Vector of train in D2 (V2)
75
km/h
Maintenance time window*
[240,252]
h h
Distance between terminals (km)
Departure time** 6 * synchronous time of simulation clock ** time of departure is 6:00am(simulation clock time)
Departure time (h)
(Tm=31 days, f==72h) Fig. 6 Timetable of through freight train generated in central cargo terminal for example
During simulation time, one random number can be generated by Matlab sentence ‘r=rand ();’then we can obtain daily product demand according to Fig. 5(c) when F=r. In addition to above necessary information, we suppose that breakeven point Qw=4,000 (t), designed capacity of assembly line Qm=6,000 (t˅, the decisive coefficient =0.65, then we get scale production quantity Qe=5,300 (t) according to formula 1.That is to say that materials for manufacturing will be ex-warehoused when the accumulation of t exceed 5,300 t. There are 2 distance intervals form strategic loading base to the manufacture enterprise when through freight train is chosen for material supply. Lengths of distance intervals, vectors of train correspondingly, maintenance time window are all illustrated by Table 2. Then one certain train working diagram with 3-day departure interval and 31-day length of total simulation time can be shown by Fig. 6. Departure interval ranges form 1 day to 10 daysˈthe maximum number of cars in one through train is 60, average loading capacity per car Wv is equal to 55 t, the basic transpiration cost per car is 3000 Yuan(included car basic price, distance basic price, tax for construction fund, payment for stamp tax) ,the relationship between quantity discount cost and order quantity is illustrated by Table 3. The scope of order quantity belong to discrete set [1, 60] (car), namely, [55, 3,300] (t),the other parameters about material inventory strategy decided by the manufacturing
enterprise are all indicated in Table 4. The time step method is presented to obtain optimum solution for the example during computer simulating process, one step-size of simulation clock is 10 min (t=10 min), the total length of simulation times is one quarter (Tm=2,160 h). then we scan 600 combinations ¬ªW f , q ¼º thoroughly, all projects of material inventory strategy can be worked out for the example. and the 3D solution space is illustrated by Fig. 7. It can be found that the longer departure interval results in the higher likelihood of material shortage(adopting penalty value M); the more order quantity lead to the lower likelihood of material shortage and objective function increasing due to extra material holding cost. The 3D coordination of optimal solution in Fig. 7 is (7, 23, 226359.75), thus by comparison and selection through simulating process the optimal project [opt, qopt] is expressed as follows: [opt, qopt]=[7, 23] That is to say that when departure interval equal to 7 days, and order quantity equal to 23 cars (1,265 t),the enterprise has lowest inventory cost, and the objective function values is 226359.75 (yuan). Table 3 Relationship between discount and quantity Quantity discount cost Cd (yuan/car)
Order quantity (car)
300
1–5
150
6–20
100
21–40
0
41–60
Table 4 Parameters of enterprise inventory strategy Category
Value
Unit
Material holding cost Ch
1.5
Yuan/t.day
Cs Const ordering cost o Material initial inventory level Q0
3000
Yuan/once
550
t
Penalty value for shortage M
15000000
Yuan
Coal injection ratio
195
Kg/t
Material consumption coefficient k
0.195
t/t
Fig. 7 3D diagrammatic sketch of solution space for inventor strategy
HE Guoxian et al. / J Transpn Sys Eng & IT, 2012, 12(2), 98104
Inventory level (t)
station should be addedDŽ
Inventory level (t)
Simulation time (d)
Simulation time (d)
Fig. 8 Curve of relationship between inventory level and simulation time for optimal order quantity
The mean value of coal inventory level is 1,366 t when departure interval is 7 days and order quantity is 23 cars, and in this case variance of coal inventory level is 0 approximately. The relationship between inventory level and simulation time is illustrated by Fig. 8, where Fig. 8(b) is the part form 30th day to 60th day of Fig. 8(a).as shown in Fig. 7 enterprise not only get rid of coal inventory accumulating phenomenon in 90 consecutive days, but make inventory level display ideal changing trend under uncertainty of product demand.
Above all, the simulating process can be divided into four steps in general. First, when simulation clock develops to another day and accumulation of t reaches material outbound breakthrough point, material ex-warehousing should be operated; secondly, Material holding cost should be added to total cost when a day coming , moreover, departure time and traveling mode can be traced by relationship between simulation clock and departure hour; finally, if the through freight train arrives ,material inbound work should be completed and ordering cost should also be added to total cost . In this paper “drawing out method” is adopted when coping with train traveling time belong to maintenance time window, and we regard material shortage cost as infinitely greatˈ therefore, departure hour being delayed due to maintenance time window and actually shortage cost need researching further.
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6
Conclusions
We focus on the single manufacturing enterprise about flowchart of simulation and case study ˈ as to reverse multi-steps through freight train generated in loading station for several enterprises, two transformations should be adopted for optimizing inventory strategy by system simulation. firstly, total material unloaded in passed terminals should be excluded to calculating actually arriving material quantity, secondly, for caculating arriving time correctly,all times for unloading material during cargo operation works in stopping railway
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Cao X M, Lin B L, Yan H X. Optimization of direct freight train service in the loading place. Journal of the China Railway Society, 2006, 28(4): 6–11.
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