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A review of optimisation models of Kanban-based production systems
E u r o p e a n Journal of Operational Research 75 (1994) 1-12 North-Holland
1
Invited Review
A review of optimisation models of Kanban-based production systems Wilson Price a,* Marc Gravel b Aaron Luntala Nsakanda a a Universitd Laval, Qudbec, Canada b Universitd de Qudbec ~ Chicoutimi, Canada Received June 1993
Abstract: This paper deals with the Kanban method of production control used initially by Toyota
Motors to replace traditional reorder point and economic lot size techniques. The method has had considerable practical success and various researchers have developed models to help production planners to choose various parameters associated with the method, principally the number of Kanban cards at a workstation and the size of the Kanban-lots. This paper reviews optimisation models of Kanban-based systems covering serial production lines, bottleneck workstations, assembly job-shop production.
I. Introduction
Within the last decade, North American manufacturers have shown much interest in the Japanese 'Just-in-Time' (JIT) production planning and control methods. JIT, according to Finch and Cox (1986), is composed of eight linked areas requiring management and continuous improvement: a focused factory, reduced set-up times, group technology, total preventive maintenance, cross-trained employees, uniform work loads, just-in-time delivery of purchased parts and materials, and the Kanban method of production control. This interest was originally sparked by the achievements of Japanese manufacturers in the late seventies and early eighties. Various authors initially attributed these successes to cultural factors and to economic conditions in Japan, but the opening in North America of successful automobile assembly plants by Honda, Toyota, Nissan, Mazda and Hyundai as well as other plants for parts manufacturing has shown this to be too simple a view. Golhar and Stamm (1991) provide an interesting and comprehensive literature review on JIT. This paper deals with the Kanban method, which was developed to regulate repetitive production and to control in-progress inventory in the plant. Some researchers and practitioners, for example Louis (1991) and Karmarkar and Pitbladdo (1990), have treated Kanban as a complement to Material Requirements Planning and to more traditional reorder-point methods. Krajewski, King, Ritzman, and Wong (1987) examined the conditions in which it is advantageous to use one system (Kanban, MRP, reorder points) rather than another. Griinwald et al. (1989) present a framework for choosing among MRP, Kanban, OPT and reorder point methods in different production situations and more recently, * Corresponding author 0377-2217/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI 0 3 7 7 - 2 2 1 7 ( 9 3 ) E 0 2 2 3 - K
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w.. Price et al. / Optimisation of Kanban-based systems
Buzacott and Shanthikumar (1992) have analysed the links between MRP and Kanban in a generalised model referred to as PAC. The practical successes of the Kanban method, as reported by Monden (1981, 1983), for example, have preceded many of the theoretical studies published in the literature. One might be tempted to say that this is merely a case of research-myopia but it seems that this is not so. The application of the technique to different or more complex situations and the generally increasing computerisation of production planning methods have engendered a desire to know more about how the method will behave and how it may be optimised in the plant. In their paper on the just-in-time philosophy, Golhar and Stamm (1991) enumerate simulation models of Kanban-based production systems. Various authors such as Badinelli (1992), Berkley (1990), Buzacott (1989), Deleersnyder et al. (1989, 1992), Jordan (1988), Mitra and Mitrani (1990, 1991), Seidmann (1988) and Wang and Wang (1990, 1991) present stochastic models of such systems. Berkley (1993) presents a classification of models of Kanban systems based on operational design criteria. This paper treats optimisation models of Kanban-based systems.
2. Operating the Kanban method Kanban, which means card, is a way of labeling production lots to obtain better control of raw materials, in-progress inventory, and finished goods. Each step of the production process must deliver its output to the following step so as to neither delay the start of production at this step nor to create excess in-progress inventory. A worker cannot start production at his work station unless a Kanban indicates that this work is required by the next downstream station. Two types of Kanban cards are used: a withdrawal Kanban is attached to waiting lots of work-in-progress and authorizes moves between work stations; a production Kanban is attached to lots being processed at a work station and serves as a workorder. Consider a plant having two work stations (Fig. 1). The first work station is a transfer line consisting of several machining operations. At the second work station, a number of assembly operations are carried out. A production run or order is divided into small, equally sized Kanban lots which are then processed individually. The work flows through the shop as follows: a Kanban lot of unprocessed parts is withdrawn from the store, machined, and then stored temporarily as work-in-progress; when called for, it is transferred to the
Ordering post
Receiving post ~
~
~- ......
~ 1 ~ ~ - - - -- ~ / / °.. l ,.......................... i ~ ~ From store
!!iiii!iiii~ili!i~/Sto re "A" ~.iiiiiiiiiiiiiiiiii::;~iii ii!iiiii!iii!iii!i!!!iiiili!i!iiiii!i!i!i!~il UPSTREAM (machining)
m
~ [ .:." " :::::: .......... "'-..
"-
~)
~
Production Kanban
iiiiiiiiiiii:~iiiili i:~:i:i~~:~T
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::#:::
Withdrawl post
B Withdrawl Kanban
'
~- I1~
~
Fig. 1. A two-card Kanban systemin a flow-shop.
DOWNSTREAM (assembly) Production unit
W.. Price et al. / Optimisation of Kanban-based systems
3
start of the assembly work station where it is again stored as work-in-progress; it is subsequently withdrawn, the assembly operations are carried out, and the lot of finished goods is then placed in finished goods storage. The Kanban cards control the work flow. In order to withdraw a waiting lot from the storage point at the head of the work station, the operator must take a free production Kanban from a 'Kanban post' at the exit of his work station. If there is no free Kanban, he must wait. If there is a free Kanban, he withdraws a waiting lot from the storage point, detaches the withdrawal Kanban, attaches the free production Kanban, and sends the newly freed withdrawal Kanban back to the storage point at the exit of the previous workstation where it authorizes the forward movement of another lot. When the operator finishes processing the lot, it is placed into the work-in-progress storage at the exit of the work station. The production Kanban remains attached to this lot until it is called downstream to the next work station by a free withdrawal Kanban. At this time the production Kanban is freed and can be used to re-start the processing cycle at the work station. The Kanban cards control material flow to minimize work-in-progress. The flow of the withdrawal Kanban is shown by the dotted lines. The operator at the assembly work station takes a lot from storage area B, removes the withdrawal Kanban attached to it, and places it in the withdrawal Kanban post. This unattached card is returned to storage area A, authorizing the transfer of another lot from A to B. Each lot waiting in storage area A is accompanied by a production Kanban which must be replaced by an unattached withdrawal Kanban before the lot moves to storage are B. The dashed lines show the movements of production Kanban cards. When detached, they go to a Kanban receiving post and are then sent to the production ordering Kanban post at the head of the machining work station to authorize processing of another lot. The Kanban cards authorize production at the machining work station only when it is required to supply the assembly work station, thus limiting the in-progress inventory that must be stored. Since the number of production Kanbans is limited, the machining work station stops production when no cards are present and resumes production when a card is sent back from the assembly work station. The number of Kanban cards in circulation fixes the absolute maximum of work-in-progress lots. Golhar and Stamm (1991) list a total of 43 refereed papers dealing with various aspects of the Kanban method, and detailed presentations of the method itself are provided by Monden (1981, 1983), Schonberger (1982a,b) and Shingo (1983). A recent treatment may be found in Mejabi and Wasserman (1992).
3. Optimisation models It would appear that Kimura and Terada (1981) were among the first to model a Kanban system and their work served as a reference for subsequent authors. They proposed a set of balance equations for a Kanban-based serial production system in which they define Ot~ as the production order quantity for the n-th stage in the t-th period, p n as the actual production quantity for the n-th stage in the t-th period, and I t as the finished goods inventory at the n-th stage at the end of the t-th period. They describe the system by means of three relations which are presented here in a simplified form that assumes production and order quantities to be multiples of the quantity represented by a Kanban card, that there is no time lag between the removal of a card and the start of production, and that production in one period is available to the following workstation in the next period. The relations are: • order quantity: O~ =Pt n-I q-ont-1 -Pin-l, which signifies that a production order at stage n will be determined by the quantities required at the next stage, n - 1, plus any 'backorders' accumulated from previous periods; • production quantity: Ptn = min{Ot~, C n, It_+11+ P~+ll}, which signifies that production is limited by the order quantity, the production capacity of the stage, and inventory available at the previous stage (n + 1);
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W. Price et aL / Optimisation of Kanban-based systems
Inventory Point
I
Fig. 2. Production and inventorypoints: Bitran and Chang (1987). • inventory: I t = 17_ 1 + P?-I - P ? - 1 , which is the material balance equation for inventory at stage n. Kimura and Terada did not directly create an optimisation model of the behaviour of the system described but they used simulation to study the amplification at workstation 'n' of fluctuations of demand at the final workstation. Their simulations permitted them to conclude that it is desirable to reduce the number of units represented by a Kanban card as much as possible. Bitran and Chang (1987) extended the work of Kimura and Terada and offer a mathematical programming model for the Kanban system in a multi-stage production setting. Their deterministic model is designed to assist in the choice of the number of Kanbans to use at each stage and thus to control the level of inventory. Fig. 2 helps to understand the notation of the model and the constraints. The Pn of Fig. 2 represent production processes and the I n represent the points at which work-in-progress is stored. The model presented here is a simplified and linearised version of the original, in which we assume that partly filled containers are not used and that only one unit from a previous stage is required at a following stage. As in the original paper, we assume that Kanbans detached in I n in period t are available to serve as production orders in Pn in period t + 1. The following notation is defined: n = The stage at which operation n is performed (n = 1, 2, 3 , . . . , N). t = Time period (t -- 1, 2, 3 . . . . . T). Xnt = Number of detached Kanbans that trigger production at Pn during period t. Ynt = Number of Kanbans detached from their containers at I n during period t. Unt = Number of detached and as yet unused Kanbans at P, at the end of period t. Vnt = Number of containers available at I n at the end of period t. Q~ = Required quantity, determined by quantity ordered and the bill of materials. tint = Production capacity in terms of full containers at Pn in period t. a = The number of parts in a full container. s(n) = Stage immediately succeeding stage n. P(n) = Set of stages immediately preceding stage n. X1 = Total production requirements expressed as a number of full containers at the last stage. The constraints of the model are as follows:
Un,t- 1 q- Yn,t- 1 --Xn, - Unt =
+x.,-
Xnt = M i n
0,
Vn, = 0,
(conservation of the flow of Kanbans at Pn) (conservation of the flow of Kanbans at In)
(1) (2)
'Un,,-I "1- Yn,t-1,
(production is limited by available Kanbans)
(3)
J~nl , (Vk,t-1 +Xkt)~,e~n),
(production is limited by capacity)
(4)
(production is limited by available parts)
(5)
(production is limited to total required by order book)
(6)
i-1
an--
E Xni, i=1
W. Price et al. / Optimisation of Kanban-based systems
Ynt = Xs(.),t.
5
(detached Kanbans equal production quantities at next stage)
(7)
The objective function is of the form Min ~'n= N lfn[Uno q- I/no]' where C n is the monetary value of a full container at stage n. This function may be interpreted as minimising the cost-weighted number of Kanbans in circulation at a given time, which is a bound on work-in-progress. By choosing to minimise the weighted total of lint and l/nt a t time t = 0, the authors have fixed the maximum number of Kanbans that can circulate in any period since this total will be carried forward through time by constraints (1) and (2). The authors reformulate constraints (3)-(6) so that the model may be solved as an ILP, and they present several special cases that simplify the formulation. Solutions are intended to guide the production managers in the choice of number of Kanbans required to operate the system and to suggest a production schedule. This paper provides a well-grounded and satisfying approach to optimisation models of Kanban systems and it has served as the starting point for a number of other researchers. Guimaraes (1991) carried out extensive numerical tests of the Bitran and Chang model. The problems tested were of simple structure, but the author's experiments show that the choice of an appropriate branching strategy can yield significant gains in solution time. Philipoom, Rees, Taylor and Huang (1990) use an integer programming approach in solving a different problem. They consider the case where it is not possible to reduce all machine setup times to the level required for smooth operation of the Kanban method. In the case that they consider, a bottleneck workstation contains more than one machine and produces more than one type of part during a production cycle. Setup times are long with respect to production times. This situation may arise in a smaller shop that uses general-purpose machine tools or in a larger plant where cost does not always allow the purchase of dedicated machines. Note that in the model of Bitran and Chang (1987), this consideration does not arise because machines are dedicated to a task and so no setups are considered. Philipoom et al. present evidence showing that in a shop where a limited number of workstations retain long setup times, the use of the unmodified Kanban method will, in general, lead to bottlenecks resulting from too frequent and too long setups. The suggested solution is to use so-called signal Kanbans at the interfaces with the workstations in question. These signal Kanbans work as illustrated in Fig. 3. As the first full containers are taken from the output store at the bottleneck workstation, no production Kanban is sent back to the ordering post of the workstation as in the usual case. However, when the container bearing the •-symbol, called the 'material Kanban' is taken, this card is sent back to the previous stage to authorise the forward movement of a container. When this container reaches the bottleneck workstation, production will not start because no production Kanban has yet arrived. As the
<--
- _
Output store Container
Container Production at bottleneck workstation
1
m x/ Container
I ]
/
-- ) to previous stage -J -- ) to this stage's -ordering post
Container Container
I L_
k I I J
• V
"Material" Kanban "Signal" Kanban
Fig. 3. The use of signal Kanbans at a bottleneck workstation.
6
W. Priceet al. / Optimisationof Kanban-based systems
container bearing the V-symbol, called the 'signal Kanban' is taken, this signal card is sent back to the start of bottleneck workstation to authorise production. One container of parts is immediately available because it was moved forward as authorised by the • - K a n b a n . The V-Kanban authorises the production of a specific number of containers and re-starts a normal exchange of Kanbans with the previous workstation. While this procedure will generate work-in-progress, which is against JIT principles, it will allow a Kanban method to be used in shops that otherwise could not implement it. The • - K a n b a n and the V-Kanban are repositioned in the stack as they move forward with containers, but one must determine the number of containers that should be produced in a run. The authors present two models, one designed to minimise inventory generated at the output of the bottleneck workstation and the second to minimise the sum of holding and setup costs. The first of these models is presented here. Using the notation of the original paper, we define: ti = The production cycle time for the i-th machine at the affected workstation (a variable). PTj = The processing time for a container of item j. n = The number of parts that can be produced at the workstation during a cycle. m = The number of machines at the bottleneck workstation. S = The setup time for the machine at the affected workstation assumed identical for all parts. qij = The lotsize (in containers) for item j processed on machine i. Qj = The total amount of item j produced during one cycle. dj = The demand per unit time for item j. Y/i = A binary variable equal to 1 if and only if item j is produced on machine i. M = A large constant required in the integer programming formulation. The model is as follows: Minimise Z : ~ Qj j=l
subject to
t i >_ ~ (qijPTi + Y / i S ) ,
(to ensure production and setup time is less than or equal to cycle time)
j=l
Y/j= 1, (to ensure that item j is assigned to only one machine)
i=
qij <-M~ij,
Qj < dj + (1 - Y//j)M,'
ej _ dj - (1 - Y/j)M,
(total production is exactly equal to the demand)
Qj = ~ qij. i=1
Note that Y/j, QJ, and qij are integer variables. The solution, in terms of the qij and the ti, will determine the production quantities and the cycle times for each product and each machine at the bottleneck workstation. The authors report using the optimisation model to determine lotsizes that are then further tested in simulation models so as to fully understand the impact on the shop. Bard and Golany (1991) examine the problem of determining the number of Kanbans in a multiproduct, multistage manufacturing system. They describe the example of a printed circuit board assembly plant where five workstations, as shown in Fig. 4, produce three end products. A workstation is, in principle, able to undertake more than one operation and so to produce more than one in-process item. The objective of the model is to determine the number of Kanbans to use at each workstation so as to
W. Price et a L / Optimisation of Kanban-based systems
7
(Workstation 1 ~ . . . . . . . . . . ~
I
A
~ ' ' - ~
Workstation5 ~
A&B
Store Workstation •
J
Workstation 31
) C
Arcs showflow of parts and work for end-itemsA, B, and C. Fig. 4. A circuit board plant example (Bard and Golany, 1991).
minimise the total cost, including setup costs, holding costs, and shortage costs, over a known time horizon. T h e initial model described contains logic operators, non-linear terms and integer variables, but the authors show how it may be converted either to a non-linear or an integer program. T h e notation defined by the authors is as follows: Sets a n d indices
I E W T
= = = = I(k) = S(i) = P(i) =
T h e set of items or parts; i = 1 . . . . . n t. Set of end items ( I ~ E). Set of workstations; k = 1 , . . . , n w. Set of time periods; t = 1 . . . . , tf. Set of items p r o d u c e d at workstation k. Set of immediate successors of i. Set of immediate predecessors of i.
Data parameters
dit
=
Ckt
hi = Si
=
f,= bi = o-i= gi
=
aii = M=
E x p e c t e d d e m a n d for item i at t. Capacity at workstation k at time t. Holding c o s t / c o n t a i n e r / p e r i o d for i. Shortage c o s t / c o n t a i n e r / p e r i o d . Setup cost for item i. P r o d u c t i o n time per lot for item i. Setup time per lot for item i. L e a d time (in time periods) for i. Containers o f / / c o n t a i n e r of j. A n arbitrarily large number.
Decision variables Xit = containers of i p r o d u c e d in t. Zit = I i f f item i is p r o d u c e d in t.
~t = Containers of i at the end of t. ~i = Containers of i in the system.
T h e optimisation m o d e l has an objective function with two main terms. T h e first term sums the holding and setup costs for in-process items. T h e second sums holding costs, shortage costs, and setup costs for end-items. It is quite reasonably assumed that a shortage cost will apply only to finished
8
W. Price et aL / Optimisation o f Kanban-based systems
products. These terms are summed over the complete time horizon. In the first constraint, we merely find the material balance equation. Note that except for end items, the dit will be zero. The second constraint has three parts and sets limits to production of a particular item. The first part states that production of an item at a workstation is limited by the availability of the parts required as produced at preceding workstations. The second part states that production is limited by the number of Kanbans in circulation at a workstation. The third part states that production is limited by the workstation capacity. The third constraint ensures that production will occur only if a setup has been done. The mOdel is as follows:
Min ~( ~ t~T
E [himax(1Tit,O)+simax(-lit, O)+fiZit])
[hiITit+fiZit]+
i~l]E
i~E
Ii,t-l+Xi,t-g,-Ijt-
E aijXjt=dit V i i i ,
t~T,
j~S(i)
min
[max(I., 1, OVai], ', ), / J1
j~P(i) t
llri - - I i , t _ 1 ,
Xit=min [Ckt-L
xi,
Xit>_O,
E
(buXat+oiZit)]//bi,
a~l(k)ti
Vi
J
I,
t
T,
~Fi>_O,
Zi~{0,1},
Iit
unrestricted
Viii,
t~T.
The authors solve this model using a standard non-linear programming code after having expressed the complex constraint on X a in terms of simple inequalities and after rendering continuous the binary variables Zir These transformations require the inclusion of penalty terms in the objective function. The objective function is nonconvex and the authors propose a three-step procedure to search for a global optimum. In the example that is presented the authors obtain numbers of Kanbans ranging from 15 to 30 and averaging 22.3 for the ten items whereas they report that use of the Toyota formula (Monden, 1981) yields an average number of Kanbans equal to 13.2. This model seems to recommend higher numbers of Kanbans because of the costs that it explicitly takes into account. Price, Gravel, Nsakanda and Cantin (1992) describe an adaptation of the model of Bitran and Chang (1987) to the assembly job-shop situation previously modeled as a simulation (Gravel and Price, 1988, 1991). The model is of similar structure to that of Bitran and Chang (1987), although further constraints and variables are required to take into account the fact that machines are used for multiple tasks. To understand the model, consider the illustration in Fig. 5. The operations diagram shows which operations are carried out on which machines as well as the sequence in which operations must be carried out. In this problem, all machines perform multiple operations and so whenever a machine becomes free, the question is to determine which of a list of waiting operations should be carried out in order to minimise makespan for an order of given size.
OperationsDiagram Machines
Free
Occupied
Fig. 5. An assemblyjob shop with Kanban.
W. Priceet al. / Optimisation of Kanban-based systems
9
In-process inventory is controlled by a parameter fixing the number of Kanbans. The notation used follows: n Index of an operation (n = 1, 2 , 3 , . . . , N). m Index of a machine-type (m = 1, 2, 3 . . . . . M). t = Index of a time period (t = 1, 2, 3 . . . . , T). The set of operations that may be executed by machine m. nrn = Available machine time per period for machines of type m. Cm Time required to execute operation n. a n P ( n ) = The set of operations preceding operation n. s(n) = The operation succeeding n. X7 = Kanbans that authorise production for operation n at time t. Vt n = Kanban-lots of the item produced by operation n available at the end of period t. Utn = Kanbans not used to authorise production at operation n at the end of period t. yt n = Kanbans removed from containers at inventory point I n during period t. Q = The total order size. The model is as follows: T
(1)
Min •
Pt XN
t=l
(2)
v, nl + v , n - x f
(3)
utN_I + X N -
(4)
Vt~, - Ytn + St" - Vtn = 0,
(5)
vtNI - ~ - X F - v t U = o ,
(6))(7
(7)
xp -
(8)
EXt
-v,"=o,
U,N = O,
0,
< o,
T
t=l T
(9)
EXN=Q, t=l
(10)
E
anX~'
n ~n m
The objective function (1) seeks to minimise makespan by encouraging the scheduling of operations so that deliveries are as early as possible. Pt is any appropriate increasing function of t. Constraints (2)-(5) ensure the conservation of Kanbans between time periods, constraint (6) allows production only if Kanbans are available, constraint (7) takes into account availability of parts for an operation, constraints (8) and (9) ensure that production is limited by demand and constraint (10) ensures that available machine time will not be exceeded. Note that constraint (7) will only allow the use of parts in the period following their production. Determining an appropriate value of T is a minor but important task because a value that is too low will render the problem infeasible. The ILP thus generated can be quite large but solvable for some problems of practical interest. Sensitivity analysis of the model can be used to seek the optimal number of Kanban cards and one can test the efficiency of alternate operation routings and shop configurations, but the main use proposed by the authors is to determine the makespan for a production run. The model has also been used to benchmark results obtained via simulation.
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W. Price et al. / Optimisation of Kanban-based systems
We end with an examination of two papers that treat the problem of determining the number of Kanbans to use at different workstations where particular conditions may apply. Moeemi and Y.-L. Chang (1990) note that a Bitran and L. Chang model of an industrial situation can be large and they therefore propose a heuristic method that gives optimal solutions for a special case: that where production capacity at the various workstations is unlimited. They also assume that demand variability is low, although if this is not the case, the authors report that their heuristic still seems to perform well. The method operates by finding an appropriate number of Kanbans for the last stage of production and then working successively back through the line towards the first operation. Problems are decomposed within the planning horizon into subproblems in which the demand in successive periods is non-decreasing. Solutions are found for each subproblem and then reconciled. The heuristic is simple and should yield rapid calculations even for large problems, but the assumption of infinite capacity diminishes its interest in practical application. In conjunction with the constraint that demand must be completely satisfied in each period, this assumption leads to numbers of Kanbans that can be quite large. In a numerical example presented by the authors, numbers of Kanbans are as high as 289. Li and Co (1991) use dynamic programming to determine the number of Kanbans to use at each stage of a production process in order to minimise inventory holding cost. Their formulation allows the treatment of both serial and tree-structured production processes. As in the previous paper the authors assume that capacity will always allow requirements to be satisfied without backorders. Given that no smoothing of demand is assumed, one may well ask if a straightforward application of MRP would not offer a better solution. Neither of the latter two papers describe an industrial setting in which the methods proposed would be appropriate.
4. Conclusions and opportunities for further research
The models reviewed in this paper seek to determine the best values for various system parameters such as the number of Kanban cards to use or the number of machines to devote to a particular operation. The mathematical complexity of many of the formulations means that the solution of large problems is not practical and could certainly not be implemented in anything approaching a real-time environment. The models, however, are important because they allow calibration of the simulation models that some authors use in conjunction with the optimisation. The formal specification of the optimisation models may also encourage the development of simple heuristics, and there appears to be room for much work in this area. The use of cost-based objective functions raises an interesting question. There is no doubt that those making manufacturing decisions seek, more than ever, to reduce costs to the lowest possible levels and a number of the models presented here respond to that concern. However, if we return to the results of the previous generation of models and their underlying principles, we may justifiably question this approach. Economic order quantity models gave us larger batch sizes and longer production runs than the JIT-Kanban implementations that sought, without seeking a direct cost justification, to lower in-process inventory to the lowest levels possible. The proponents of the JIT-Kanban methods were, in effect, acknowledging that many costs confounded the accountants' measurement systems, which didn't mean that they were any less real. Such costs include unanticipated obsolescence due to engineering changes or product cancellation, damage to work in process, earlier detection of quality problems and so on. There is a danger that cost-based models may tend to repeat the cycle and to recommend parameter choices that will neither drive down inventory or reduce total costs. One of the interesting applications of models of the type reviewed here is in real-time planning and control of a production shop. The ability to model a production run and to anticipate such matters as machine loadings, work in progress, and completion time could prove valuable to the production manager, particularly in situations where orders change frequently, where the order book is not known far in advance, or where production capacity available to the product or products being considered varies
W. Price et al. / Optimisation of Kanban-based systems
11
frequently. One of the interesting directions for further development will be in the incorporation of models such as those presented in this review in decision support systems for production control.
References Badinelli, R.D. (1992), "A model for continuous review pull policies in serial inventory systems", Operations Research 40/1, 142-156. Bard, J. and Golany, B. (1991), "Determining the number of Kanbans in a multiproduct multistage production system", International Journal of Production Research 29/5, 881-895. Berkley, B.J. (1992), "A decomposition approximation for periodic Kanban controlled flow shops", Decision Sciences 23/2, 291-311. Berkley, B.J. (1993), "A review of the Kanban production control research literature", Working Paper, School of Business, University of Wisconsin, Madison, WI. Bitran, G., and Chang, L. (1987), " A mathematical programming approach to a deterministic Kanban system", Management Science 33/4, 427-441. Buzacott, J.A. (1989), "Queueing models of Kanban and MRP controlled production systems", Engineering Costs and Production Economics 17/1, 3-20. Buzacott, J.A., and Shanthikumar, J.G. (1992), Stochastic Models of Manufacturing Systems, Prentice-Hall, Englewood Cliffs, NJ. Deleersnyder, J., Hodgson, T.J., Muller-Malek, H., and O'Grady, P.J. (1989), "Kanban-controlled pull systems: An analytic approach", Management Science 35/9, 1079-1091. Deleersnyder, J., Hodgson, T.J., King, R.E., O'Grady, P.J., and Sawa, A. (1989); "Integrating Kanban-type pull systems and MRP-type push systems: Insights from a Markovian model", liE Transactions 24/3, 43-56. Finch, B.J., and Cox, J.F., (1986), "An examination of Just-In-Time management for the small manufacturer, with an illustration", International Journal of Production Research 24/2, 329-342. Golhar, D.Y., and Stamm, C.L. (1991), "The JIT philosophy: A review", International Journal of Production Research 29/4, 657-676. Gravel, M., and Price, W.L. (1988), "Using Kanban in a job-shop environment", International Journal of Production Research 26/6, 1105-1118.
Gravel, M., and Price, W.L. (1991), "Visual Interactive Modeling shows how to use the Kanban in small business", Interfaces (ORSA/TIMS) 21/5, 22-33. Griinwald, H., Striekwold, P.E.T., and Weeda, P.J. (1989), "A framework for quantitative comparison of production control stategies", International Journal of Production Research 27/2, 281-292. Guimares, L. (1991), "Urn algoritmo 'branch and bound' para um modelo de otimizqao de mu sistema 'Kanban' '", M.Sc. Thesis of the Faculdade El6ctrica da Universidade Estadual de Campinas, Brazil, March, 1991. Jordan, S. (1988), "Analysis and approximation of a JIT production line", Decision Sciences 19/3, 672-681. Karmarkar, U.S., and Pitbladdo, R. (1990), "Integrating MRP with Pull/Kanban shop control", ORSA/TIMS Philadelphia, October, 1990. Kimura, O., and Terada, H. (1981), "Design and analysis of Pull System, a method of multi-stage production control", International Journal of Production Research 19/3, 241-253. Krajewski, L.J., King, B.E., Ritzman, L.P., and Wong, D.S. (1987), "Kanban, MRP and shaping the manufacturing environment", Management Science 33/1, 39-57. Li, A., and Co, H.C. (1991), "A dynamic programming model for the Kanban assignment problem in a multistage multiperiod production system", International Journal of Production Research 29/1, 1-16. Louis, R.S. (1991), "MRP Ill material acquisition system: A combination of MRP, Kanban, bar coding and EDI", Production and Int'entory Management Rer,iew 11/7, 26-27. Mejabi, O., and Wasserman, G.S., (1992), "Basic concepts of JIT modeling", International Journal of Production Research 30/1, 141 149. Mitra, D., and Mitrani, I. (1990), "Analysis of a Kanban discipline for cell coordination in production lines I", Management Science 36/12, 1548-1566. Mitra, D., and Mitrani I. (1991), "Analysis of a Kanban discipline for cell coordination in production lines II: Stochastic demands", Operations Research 39/5, 807-823. Moeemi, F., and Chang, Y.-L. (1990), "An approximate solution to deterministic Kanban systems", Decision Sciences 21/3, 596-607. Monden, Y. (1981), "Adaptable Kanban system helps Toyota maintain Just-ln-Time production", bzdustrial Engineering 13/5, 28-46. Monden, Y. (1983), Toyota Production System, Industrial Engineering and Management Press, Atlanta, GA. Philipoom, P.R., Taylor, B.W., Huang, P.Y., and Rees, L.P. (1990), " A mathematical programming approach for determining workcentre lotsizes in a just-in-time system with signal Kanbans", International Journal of Production Research 28/1, 1-15.
12
W. Price et al. / Optimisation of Kanban-based systems
Price, W.L., Gravel, M., Nsakanda, A.L., and Cantin, F. (1992), "A mathematical programming model of a Kanban job-shop", Working Paper, Facult6 des sciences de l'administration, Unversit6 Laval, November, 1992. Schonberger, R.J. (1982), "The transfer of Japanese manufacturing management approaches to US industry", Academy of Management Review 7/3, 479-487. Schonberger, R.J. (1982), Japanese Manufacturing Techniques, Free Press, New York. Seidmann, A. (1988), "Regenerative pull (Kanban) production control policies", European Journal of Operational Research 35/3, 401-413. Shingo, S. (1983), Ma~trisede la Production et M~thode KANBAN, Editions d'Organisation, Paris. Wang, H.M., and Wang, H.-P.B. (1990), "Determining the number of Kanbans: A step towards non-stock production", International Journal of Production Research 28/11, 2101-2115. Wang, H.M., and Wang, H.-P.B. (1991), "Optimum number of Kanbans between two adjacent workstations", InternationalJournal of Production Economics 22/3, 179-198.
1
Invited Review
A review of optimisation models of Kanban-based production systems Wilson Price a,* Marc Gravel b Aaron Luntala Nsakanda a a Universitd Laval, Qudbec, Canada b Universitd de Qudbec ~ Chicoutimi, Canada Received June 1993
Abstract: This paper deals with the Kanban method of production control used initially by Toyota
Motors to replace traditional reorder point and economic lot size techniques. The method has had considerable practical success and various researchers have developed models to help production planners to choose various parameters associated with the method, principally the number of Kanban cards at a workstation and the size of the Kanban-lots. This paper reviews optimisation models of Kanban-based systems covering serial production lines, bottleneck workstations, assembly job-shop production.
I. Introduction
Within the last decade, North American manufacturers have shown much interest in the Japanese 'Just-in-Time' (JIT) production planning and control methods. JIT, according to Finch and Cox (1986), is composed of eight linked areas requiring management and continuous improvement: a focused factory, reduced set-up times, group technology, total preventive maintenance, cross-trained employees, uniform work loads, just-in-time delivery of purchased parts and materials, and the Kanban method of production control. This interest was originally sparked by the achievements of Japanese manufacturers in the late seventies and early eighties. Various authors initially attributed these successes to cultural factors and to economic conditions in Japan, but the opening in North America of successful automobile assembly plants by Honda, Toyota, Nissan, Mazda and Hyundai as well as other plants for parts manufacturing has shown this to be too simple a view. Golhar and Stamm (1991) provide an interesting and comprehensive literature review on JIT. This paper deals with the Kanban method, which was developed to regulate repetitive production and to control in-progress inventory in the plant. Some researchers and practitioners, for example Louis (1991) and Karmarkar and Pitbladdo (1990), have treated Kanban as a complement to Material Requirements Planning and to more traditional reorder-point methods. Krajewski, King, Ritzman, and Wong (1987) examined the conditions in which it is advantageous to use one system (Kanban, MRP, reorder points) rather than another. Griinwald et al. (1989) present a framework for choosing among MRP, Kanban, OPT and reorder point methods in different production situations and more recently, * Corresponding author 0377-2217/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI 0 3 7 7 - 2 2 1 7 ( 9 3 ) E 0 2 2 3 - K
2
w.. Price et al. / Optimisation of Kanban-based systems
Buzacott and Shanthikumar (1992) have analysed the links between MRP and Kanban in a generalised model referred to as PAC. The practical successes of the Kanban method, as reported by Monden (1981, 1983), for example, have preceded many of the theoretical studies published in the literature. One might be tempted to say that this is merely a case of research-myopia but it seems that this is not so. The application of the technique to different or more complex situations and the generally increasing computerisation of production planning methods have engendered a desire to know more about how the method will behave and how it may be optimised in the plant. In their paper on the just-in-time philosophy, Golhar and Stamm (1991) enumerate simulation models of Kanban-based production systems. Various authors such as Badinelli (1992), Berkley (1990), Buzacott (1989), Deleersnyder et al. (1989, 1992), Jordan (1988), Mitra and Mitrani (1990, 1991), Seidmann (1988) and Wang and Wang (1990, 1991) present stochastic models of such systems. Berkley (1993) presents a classification of models of Kanban systems based on operational design criteria. This paper treats optimisation models of Kanban-based systems.
2. Operating the Kanban method Kanban, which means card, is a way of labeling production lots to obtain better control of raw materials, in-progress inventory, and finished goods. Each step of the production process must deliver its output to the following step so as to neither delay the start of production at this step nor to create excess in-progress inventory. A worker cannot start production at his work station unless a Kanban indicates that this work is required by the next downstream station. Two types of Kanban cards are used: a withdrawal Kanban is attached to waiting lots of work-in-progress and authorizes moves between work stations; a production Kanban is attached to lots being processed at a work station and serves as a workorder. Consider a plant having two work stations (Fig. 1). The first work station is a transfer line consisting of several machining operations. At the second work station, a number of assembly operations are carried out. A production run or order is divided into small, equally sized Kanban lots which are then processed individually. The work flows through the shop as follows: a Kanban lot of unprocessed parts is withdrawn from the store, machined, and then stored temporarily as work-in-progress; when called for, it is transferred to the
Ordering post
Receiving post ~
~
~- ......
~ 1 ~ ~ - - - -- ~ / / °.. l ,.......................... i ~ ~ From store
!!iiii!iiii~ili!i~/Sto re "A" ~.iiiiiiiiiiiiiiiiii::;~iii ii!iiiii!iii!iii!i!!!iiiili!i!iiiii!i!i!i!~il UPSTREAM (machining)
m
~ [ .:." " :::::: .......... "'-..
"-
~)
~
Production Kanban
iiiiiiiiiiii:~iiiili i:~:i:i~~:~T
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::#:::
Withdrawl post
B Withdrawl Kanban
'
~- I1~
~
Fig. 1. A two-card Kanban systemin a flow-shop.
DOWNSTREAM (assembly) Production unit
W.. Price et al. / Optimisation of Kanban-based systems
3
start of the assembly work station where it is again stored as work-in-progress; it is subsequently withdrawn, the assembly operations are carried out, and the lot of finished goods is then placed in finished goods storage. The Kanban cards control the work flow. In order to withdraw a waiting lot from the storage point at the head of the work station, the operator must take a free production Kanban from a 'Kanban post' at the exit of his work station. If there is no free Kanban, he must wait. If there is a free Kanban, he withdraws a waiting lot from the storage point, detaches the withdrawal Kanban, attaches the free production Kanban, and sends the newly freed withdrawal Kanban back to the storage point at the exit of the previous workstation where it authorizes the forward movement of another lot. When the operator finishes processing the lot, it is placed into the work-in-progress storage at the exit of the work station. The production Kanban remains attached to this lot until it is called downstream to the next work station by a free withdrawal Kanban. At this time the production Kanban is freed and can be used to re-start the processing cycle at the work station. The Kanban cards control material flow to minimize work-in-progress. The flow of the withdrawal Kanban is shown by the dotted lines. The operator at the assembly work station takes a lot from storage area B, removes the withdrawal Kanban attached to it, and places it in the withdrawal Kanban post. This unattached card is returned to storage area A, authorizing the transfer of another lot from A to B. Each lot waiting in storage area A is accompanied by a production Kanban which must be replaced by an unattached withdrawal Kanban before the lot moves to storage are B. The dashed lines show the movements of production Kanban cards. When detached, they go to a Kanban receiving post and are then sent to the production ordering Kanban post at the head of the machining work station to authorize processing of another lot. The Kanban cards authorize production at the machining work station only when it is required to supply the assembly work station, thus limiting the in-progress inventory that must be stored. Since the number of production Kanbans is limited, the machining work station stops production when no cards are present and resumes production when a card is sent back from the assembly work station. The number of Kanban cards in circulation fixes the absolute maximum of work-in-progress lots. Golhar and Stamm (1991) list a total of 43 refereed papers dealing with various aspects of the Kanban method, and detailed presentations of the method itself are provided by Monden (1981, 1983), Schonberger (1982a,b) and Shingo (1983). A recent treatment may be found in Mejabi and Wasserman (1992).
3. Optimisation models It would appear that Kimura and Terada (1981) were among the first to model a Kanban system and their work served as a reference for subsequent authors. They proposed a set of balance equations for a Kanban-based serial production system in which they define Ot~ as the production order quantity for the n-th stage in the t-th period, p n as the actual production quantity for the n-th stage in the t-th period, and I t as the finished goods inventory at the n-th stage at the end of the t-th period. They describe the system by means of three relations which are presented here in a simplified form that assumes production and order quantities to be multiples of the quantity represented by a Kanban card, that there is no time lag between the removal of a card and the start of production, and that production in one period is available to the following workstation in the next period. The relations are: • order quantity: O~ =Pt n-I q-ont-1 -Pin-l, which signifies that a production order at stage n will be determined by the quantities required at the next stage, n - 1, plus any 'backorders' accumulated from previous periods; • production quantity: Ptn = min{Ot~, C n, It_+11+ P~+ll}, which signifies that production is limited by the order quantity, the production capacity of the stage, and inventory available at the previous stage (n + 1);
4
W. Price et aL / Optimisation of Kanban-based systems
Inventory Point
I
Fig. 2. Production and inventorypoints: Bitran and Chang (1987). • inventory: I t = 17_ 1 + P?-I - P ? - 1 , which is the material balance equation for inventory at stage n. Kimura and Terada did not directly create an optimisation model of the behaviour of the system described but they used simulation to study the amplification at workstation 'n' of fluctuations of demand at the final workstation. Their simulations permitted them to conclude that it is desirable to reduce the number of units represented by a Kanban card as much as possible. Bitran and Chang (1987) extended the work of Kimura and Terada and offer a mathematical programming model for the Kanban system in a multi-stage production setting. Their deterministic model is designed to assist in the choice of the number of Kanbans to use at each stage and thus to control the level of inventory. Fig. 2 helps to understand the notation of the model and the constraints. The Pn of Fig. 2 represent production processes and the I n represent the points at which work-in-progress is stored. The model presented here is a simplified and linearised version of the original, in which we assume that partly filled containers are not used and that only one unit from a previous stage is required at a following stage. As in the original paper, we assume that Kanbans detached in I n in period t are available to serve as production orders in Pn in period t + 1. The following notation is defined: n = The stage at which operation n is performed (n = 1, 2, 3 , . . . , N). t = Time period (t -- 1, 2, 3 . . . . . T). Xnt = Number of detached Kanbans that trigger production at Pn during period t. Ynt = Number of Kanbans detached from their containers at I n during period t. Unt = Number of detached and as yet unused Kanbans at P, at the end of period t. Vnt = Number of containers available at I n at the end of period t. Q~ = Required quantity, determined by quantity ordered and the bill of materials. tint = Production capacity in terms of full containers at Pn in period t. a = The number of parts in a full container. s(n) = Stage immediately succeeding stage n. P(n) = Set of stages immediately preceding stage n. X1 = Total production requirements expressed as a number of full containers at the last stage. The constraints of the model are as follows:
Un,t- 1 q- Yn,t- 1 --Xn, - Unt =
+x.,-
Xnt = M i n
0,
Vn, = 0,
(conservation of the flow of Kanbans at Pn) (conservation of the flow of Kanbans at In)
(1) (2)
'Un,,-I "1- Yn,t-1,
(production is limited by available Kanbans)
(3)
J~nl , (Vk,t-1 +Xkt)~,e~n),
(production is limited by capacity)
(4)
(production is limited by available parts)
(5)
(production is limited to total required by order book)
(6)
i-1
an--
E Xni, i=1
W. Price et al. / Optimisation of Kanban-based systems
Ynt = Xs(.),t.
5
(detached Kanbans equal production quantities at next stage)
(7)
The objective function is of the form Min ~'n= N lfn[Uno q- I/no]' where C n is the monetary value of a full container at stage n. This function may be interpreted as minimising the cost-weighted number of Kanbans in circulation at a given time, which is a bound on work-in-progress. By choosing to minimise the weighted total of lint and l/nt a t time t = 0, the authors have fixed the maximum number of Kanbans that can circulate in any period since this total will be carried forward through time by constraints (1) and (2). The authors reformulate constraints (3)-(6) so that the model may be solved as an ILP, and they present several special cases that simplify the formulation. Solutions are intended to guide the production managers in the choice of number of Kanbans required to operate the system and to suggest a production schedule. This paper provides a well-grounded and satisfying approach to optimisation models of Kanban systems and it has served as the starting point for a number of other researchers. Guimaraes (1991) carried out extensive numerical tests of the Bitran and Chang model. The problems tested were of simple structure, but the author's experiments show that the choice of an appropriate branching strategy can yield significant gains in solution time. Philipoom, Rees, Taylor and Huang (1990) use an integer programming approach in solving a different problem. They consider the case where it is not possible to reduce all machine setup times to the level required for smooth operation of the Kanban method. In the case that they consider, a bottleneck workstation contains more than one machine and produces more than one type of part during a production cycle. Setup times are long with respect to production times. This situation may arise in a smaller shop that uses general-purpose machine tools or in a larger plant where cost does not always allow the purchase of dedicated machines. Note that in the model of Bitran and Chang (1987), this consideration does not arise because machines are dedicated to a task and so no setups are considered. Philipoom et al. present evidence showing that in a shop where a limited number of workstations retain long setup times, the use of the unmodified Kanban method will, in general, lead to bottlenecks resulting from too frequent and too long setups. The suggested solution is to use so-called signal Kanbans at the interfaces with the workstations in question. These signal Kanbans work as illustrated in Fig. 3. As the first full containers are taken from the output store at the bottleneck workstation, no production Kanban is sent back to the ordering post of the workstation as in the usual case. However, when the container bearing the •-symbol, called the 'material Kanban' is taken, this card is sent back to the previous stage to authorise the forward movement of a container. When this container reaches the bottleneck workstation, production will not start because no production Kanban has yet arrived. As the
<--
- _
Output store Container
Container Production at bottleneck workstation
1
m x/ Container
I ]
/
-- ) to previous stage -J -- ) to this stage's -ordering post
Container Container
I L_
k I I J
• V
"Material" Kanban "Signal" Kanban
Fig. 3. The use of signal Kanbans at a bottleneck workstation.
6
W. Priceet al. / Optimisationof Kanban-based systems
container bearing the V-symbol, called the 'signal Kanban' is taken, this signal card is sent back to the start of bottleneck workstation to authorise production. One container of parts is immediately available because it was moved forward as authorised by the • - K a n b a n . The V-Kanban authorises the production of a specific number of containers and re-starts a normal exchange of Kanbans with the previous workstation. While this procedure will generate work-in-progress, which is against JIT principles, it will allow a Kanban method to be used in shops that otherwise could not implement it. The • - K a n b a n and the V-Kanban are repositioned in the stack as they move forward with containers, but one must determine the number of containers that should be produced in a run. The authors present two models, one designed to minimise inventory generated at the output of the bottleneck workstation and the second to minimise the sum of holding and setup costs. The first of these models is presented here. Using the notation of the original paper, we define: ti = The production cycle time for the i-th machine at the affected workstation (a variable). PTj = The processing time for a container of item j. n = The number of parts that can be produced at the workstation during a cycle. m = The number of machines at the bottleneck workstation. S = The setup time for the machine at the affected workstation assumed identical for all parts. qij = The lotsize (in containers) for item j processed on machine i. Qj = The total amount of item j produced during one cycle. dj = The demand per unit time for item j. Y/i = A binary variable equal to 1 if and only if item j is produced on machine i. M = A large constant required in the integer programming formulation. The model is as follows: Minimise Z : ~ Qj j=l
subject to
t i >_ ~ (qijPTi + Y / i S ) ,
(to ensure production and setup time is less than or equal to cycle time)
j=l
Y/j= 1, (to ensure that item j is assigned to only one machine)
i=
qij <-M~ij,
Qj < dj + (1 - Y//j)M,'
ej _ dj - (1 - Y/j)M,
(total production is exactly equal to the demand)
Qj = ~ qij. i=1
Note that Y/j, QJ, and qij are integer variables. The solution, in terms of the qij and the ti, will determine the production quantities and the cycle times for each product and each machine at the bottleneck workstation. The authors report using the optimisation model to determine lotsizes that are then further tested in simulation models so as to fully understand the impact on the shop. Bard and Golany (1991) examine the problem of determining the number of Kanbans in a multiproduct, multistage manufacturing system. They describe the example of a printed circuit board assembly plant where five workstations, as shown in Fig. 4, produce three end products. A workstation is, in principle, able to undertake more than one operation and so to produce more than one in-process item. The objective of the model is to determine the number of Kanbans to use at each workstation so as to
W. Price et a L / Optimisation of Kanban-based systems
7
(Workstation 1 ~ . . . . . . . . . . ~
I
A
~ ' ' - ~
Workstation5 ~
A&B
Store Workstation •
J
Workstation 31
) C
Arcs showflow of parts and work for end-itemsA, B, and C. Fig. 4. A circuit board plant example (Bard and Golany, 1991).
minimise the total cost, including setup costs, holding costs, and shortage costs, over a known time horizon. T h e initial model described contains logic operators, non-linear terms and integer variables, but the authors show how it may be converted either to a non-linear or an integer program. T h e notation defined by the authors is as follows: Sets a n d indices
I E W T
= = = = I(k) = S(i) = P(i) =
T h e set of items or parts; i = 1 . . . . . n t. Set of end items ( I ~ E). Set of workstations; k = 1 , . . . , n w. Set of time periods; t = 1 . . . . , tf. Set of items p r o d u c e d at workstation k. Set of immediate successors of i. Set of immediate predecessors of i.
Data parameters
dit
=
Ckt
hi = Si
=
f,= bi = o-i= gi
=
aii = M=
E x p e c t e d d e m a n d for item i at t. Capacity at workstation k at time t. Holding c o s t / c o n t a i n e r / p e r i o d for i. Shortage c o s t / c o n t a i n e r / p e r i o d . Setup cost for item i. P r o d u c t i o n time per lot for item i. Setup time per lot for item i. L e a d time (in time periods) for i. Containers o f / / c o n t a i n e r of j. A n arbitrarily large number.
Decision variables Xit = containers of i p r o d u c e d in t. Zit = I i f f item i is p r o d u c e d in t.
~t = Containers of i at the end of t. ~i = Containers of i in the system.
T h e optimisation m o d e l has an objective function with two main terms. T h e first term sums the holding and setup costs for in-process items. T h e second sums holding costs, shortage costs, and setup costs for end-items. It is quite reasonably assumed that a shortage cost will apply only to finished
8
W. Price et aL / Optimisation o f Kanban-based systems
products. These terms are summed over the complete time horizon. In the first constraint, we merely find the material balance equation. Note that except for end items, the dit will be zero. The second constraint has three parts and sets limits to production of a particular item. The first part states that production of an item at a workstation is limited by the availability of the parts required as produced at preceding workstations. The second part states that production is limited by the number of Kanbans in circulation at a workstation. The third part states that production is limited by the workstation capacity. The third constraint ensures that production will occur only if a setup has been done. The mOdel is as follows:
Min ~( ~ t~T
E [himax(1Tit,O)+simax(-lit, O)+fiZit])
[hiITit+fiZit]+
i~l]E
i~E
Ii,t-l+Xi,t-g,-Ijt-
E aijXjt=dit V i i i ,
t~T,
j~S(i)
min
[max(I., 1, OVai], ', ), / J1
j~P(i) t
llri - - I i , t _ 1 ,
Xit=min [Ckt-L
xi,
Xit>_O,
E
(buXat+oiZit)]//bi,
a~l(k)ti
Vi
J
I,
t
T,
~Fi>_O,
Zi~{0,1},
Iit
unrestricted
Viii,
t~T.
The authors solve this model using a standard non-linear programming code after having expressed the complex constraint on X a in terms of simple inequalities and after rendering continuous the binary variables Zir These transformations require the inclusion of penalty terms in the objective function. The objective function is nonconvex and the authors propose a three-step procedure to search for a global optimum. In the example that is presented the authors obtain numbers of Kanbans ranging from 15 to 30 and averaging 22.3 for the ten items whereas they report that use of the Toyota formula (Monden, 1981) yields an average number of Kanbans equal to 13.2. This model seems to recommend higher numbers of Kanbans because of the costs that it explicitly takes into account. Price, Gravel, Nsakanda and Cantin (1992) describe an adaptation of the model of Bitran and Chang (1987) to the assembly job-shop situation previously modeled as a simulation (Gravel and Price, 1988, 1991). The model is of similar structure to that of Bitran and Chang (1987), although further constraints and variables are required to take into account the fact that machines are used for multiple tasks. To understand the model, consider the illustration in Fig. 5. The operations diagram shows which operations are carried out on which machines as well as the sequence in which operations must be carried out. In this problem, all machines perform multiple operations and so whenever a machine becomes free, the question is to determine which of a list of waiting operations should be carried out in order to minimise makespan for an order of given size.
OperationsDiagram Machines
Free
Occupied
Fig. 5. An assemblyjob shop with Kanban.
W. Priceet al. / Optimisation of Kanban-based systems
9
In-process inventory is controlled by a parameter fixing the number of Kanbans. The notation used follows: n Index of an operation (n = 1, 2 , 3 , . . . , N). m Index of a machine-type (m = 1, 2, 3 . . . . . M). t = Index of a time period (t = 1, 2, 3 . . . . , T). The set of operations that may be executed by machine m. nrn = Available machine time per period for machines of type m. Cm Time required to execute operation n. a n P ( n ) = The set of operations preceding operation n. s(n) = The operation succeeding n. X7 = Kanbans that authorise production for operation n at time t. Vt n = Kanban-lots of the item produced by operation n available at the end of period t. Utn = Kanbans not used to authorise production at operation n at the end of period t. yt n = Kanbans removed from containers at inventory point I n during period t. Q = The total order size. The model is as follows: T
(1)
Min •
Pt XN
t=l
(2)
v, nl + v , n - x f
(3)
utN_I + X N -
(4)
Vt~, - Ytn + St" - Vtn = 0,
(5)
vtNI - ~ - X F - v t U = o ,
(6))(7
(7)
xp -
(8)
EXt
-v,"=o,
U,N = O,
0,
< o,
T
t=l T
(9)
EXN=Q, t=l
(10)
E
anX~'
n ~n m
The objective function (1) seeks to minimise makespan by encouraging the scheduling of operations so that deliveries are as early as possible. Pt is any appropriate increasing function of t. Constraints (2)-(5) ensure the conservation of Kanbans between time periods, constraint (6) allows production only if Kanbans are available, constraint (7) takes into account availability of parts for an operation, constraints (8) and (9) ensure that production is limited by demand and constraint (10) ensures that available machine time will not be exceeded. Note that constraint (7) will only allow the use of parts in the period following their production. Determining an appropriate value of T is a minor but important task because a value that is too low will render the problem infeasible. The ILP thus generated can be quite large but solvable for some problems of practical interest. Sensitivity analysis of the model can be used to seek the optimal number of Kanban cards and one can test the efficiency of alternate operation routings and shop configurations, but the main use proposed by the authors is to determine the makespan for a production run. The model has also been used to benchmark results obtained via simulation.
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W. Price et al. / Optimisation of Kanban-based systems
We end with an examination of two papers that treat the problem of determining the number of Kanbans to use at different workstations where particular conditions may apply. Moeemi and Y.-L. Chang (1990) note that a Bitran and L. Chang model of an industrial situation can be large and they therefore propose a heuristic method that gives optimal solutions for a special case: that where production capacity at the various workstations is unlimited. They also assume that demand variability is low, although if this is not the case, the authors report that their heuristic still seems to perform well. The method operates by finding an appropriate number of Kanbans for the last stage of production and then working successively back through the line towards the first operation. Problems are decomposed within the planning horizon into subproblems in which the demand in successive periods is non-decreasing. Solutions are found for each subproblem and then reconciled. The heuristic is simple and should yield rapid calculations even for large problems, but the assumption of infinite capacity diminishes its interest in practical application. In conjunction with the constraint that demand must be completely satisfied in each period, this assumption leads to numbers of Kanbans that can be quite large. In a numerical example presented by the authors, numbers of Kanbans are as high as 289. Li and Co (1991) use dynamic programming to determine the number of Kanbans to use at each stage of a production process in order to minimise inventory holding cost. Their formulation allows the treatment of both serial and tree-structured production processes. As in the previous paper the authors assume that capacity will always allow requirements to be satisfied without backorders. Given that no smoothing of demand is assumed, one may well ask if a straightforward application of MRP would not offer a better solution. Neither of the latter two papers describe an industrial setting in which the methods proposed would be appropriate.
4. Conclusions and opportunities for further research
The models reviewed in this paper seek to determine the best values for various system parameters such as the number of Kanban cards to use or the number of machines to devote to a particular operation. The mathematical complexity of many of the formulations means that the solution of large problems is not practical and could certainly not be implemented in anything approaching a real-time environment. The models, however, are important because they allow calibration of the simulation models that some authors use in conjunction with the optimisation. The formal specification of the optimisation models may also encourage the development of simple heuristics, and there appears to be room for much work in this area. The use of cost-based objective functions raises an interesting question. There is no doubt that those making manufacturing decisions seek, more than ever, to reduce costs to the lowest possible levels and a number of the models presented here respond to that concern. However, if we return to the results of the previous generation of models and their underlying principles, we may justifiably question this approach. Economic order quantity models gave us larger batch sizes and longer production runs than the JIT-Kanban implementations that sought, without seeking a direct cost justification, to lower in-process inventory to the lowest levels possible. The proponents of the JIT-Kanban methods were, in effect, acknowledging that many costs confounded the accountants' measurement systems, which didn't mean that they were any less real. Such costs include unanticipated obsolescence due to engineering changes or product cancellation, damage to work in process, earlier detection of quality problems and so on. There is a danger that cost-based models may tend to repeat the cycle and to recommend parameter choices that will neither drive down inventory or reduce total costs. One of the interesting applications of models of the type reviewed here is in real-time planning and control of a production shop. The ability to model a production run and to anticipate such matters as machine loadings, work in progress, and completion time could prove valuable to the production manager, particularly in situations where orders change frequently, where the order book is not known far in advance, or where production capacity available to the product or products being considered varies
W. Price et al. / Optimisation of Kanban-based systems
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frequently. One of the interesting directions for further development will be in the incorporation of models such as those presented in this review in decision support systems for production control.
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