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A hierarchical production planning modelfor kanban systemsPeruvemba Sundaram Ravi aa Operations and Decision Sciences Division , School of Business& Economics, Wilfrid Laurier University , Waterloo , ON N2L3C5 ,Canada E-mail:Published online: 14 Jun 2013.

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*E-mail: pravi@wlu.ca

A hierarchical production planning model for kanban systems

Peruvemba Sundaram Ravi *

Operations and Decision Sciences DivisionSchool of Business & EconomicsWilfrid Laurier UniversityWaterloo, ON N2L3C5Canada

AbstractDuring the past few decades, a substantial amount of research has been undertaken

in the area of hierarchical production planning. Hax and Meal [19] fi rst proposed a hierar-

chical approach to the production planning process, with higher-level decisions imposing

constraints on the lower-level decision process. Besides computational advantages, a hier-

archical procedure refl ected the actual hierarchy of decisions prevalent in organizations.

Subsequently, Bitran et al., developed a hierarchical planning procedure for single-stage

production systems [10] and for two-stage production systems [11]. A hierarchical planning

model for kanban systems is developed in this paper. Solution techniques for the aggregate

problem and a disaggregation procedure are described. Application of the model to CON-

WIP and base-stock systems is also discussed.

Keywords : Hierarchical production planning, kanban systems, pull systems The author gratefully acknowledges that this research was partly supported by a Short Term Research Grant from the Off ice of Research Services at Wilfrid Laurier University.

1. Introduction

The development of production plans for a facility engaged in the

manufacture of a large number of products having diff erent seasonal

demand patterns is a formidable problem. It is a large mixed-integer pro-

gramming problem. With a few notable exceptions, researchers have not

attempted to solve the overall problem. Besides the intractability of the

problem, the infeasibility of obtaining accurate long term demand fore-

casts for several thousand products has resulted in the use of aggregation

in production planning. Aggregate demand forecasts are easier to obtain

Journal of Statistics & Management SystemsVol. 15 (2012), No. 2 & 3, pp. 369–388

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and are more accurate than detailed demand forecasts. The aggregate

problem is also smaller and more tractable. Consequently, there now ex-

ists a considerable body of literature that seeks to address the aggregate

planning problem. Independently, several researchers have focussed at-

tention on the detailed scheduling of products. Until the pioneering

work of Hax and Meal [19], few researchers (Holt et al., [21], Winters [30],

Jaikumar [22] are exceptions) attempted to tie together the development of

aggregate plans and the development of detailed schedules.

During the past few decades, however, there has been a fl urry of

research activity in this area. Hax and Meal [19] fi rst proposed a hierar-

chical approach to the production planning process, with higher-level

decisions imposing constraints on the lower-level decision process. Besides

computational advantages, a hierarchical procedure refl ected the actual

hierarchy of decisions prevalent in organizations. Subsequently, Bitran et al., developed a hierarchical planning procedure for single-stage pro-

duction systems [10] and for two-stage production systems [11]. Bitran

and Hax formulated the disaggregation problem as a convex knapsack

problem with bounded variables and developed a solution technique

for this general class of problems [9]. Graves [18] discussed the use of

Lagrangean techniques to solve hierarchical production planning problems.

Bialas and Karwan [6], Johnson et al., [23], Bard [5], Ari and Axsater [2],

and Aardal and Larsson [1] also discussed solution techniques for hier-

archical planning problems. Billington et al., [7] developed a Lagrangean

relaxation-based heuristic for multilevel lot-sizing in the presence of a

bottleneck. Issues relating to the consistency of decisions in a two-level

structure and the feasibility of aggregate plans were examined by Erschler,

Fontan and Merce [16] and by Axsater [4]. The problem of determining appro-

priate planning horizons in a hierarchical process was examined by Chung et al., [14].

Several researchers have described actual case studies in hierarchi-

cal planning. Liberatore and Miller [25] discussed the development of a

hierarchical planning system for a manufacturer of ceramic tile products.

Burch et al., [13] described the development of a hierarchical system at

Owens-Corning Fiberglass. Zijm [31] described the development of a hi-

erarchical system at Philips Electronics. Other researchers have developed

hierarchical planning procedures applicable only to certain kinds of fi rms

or goods. De Kok [15] developed a hierarchical planning procedure for the

production of consumer goods. Hendry and Kingsman [20] developed a

hierarchical planning methodology for small- to medium-sized make-to-

order companies.

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PRODUCTION PLANNING MODEL 371

Other papers that deal with issues relating to the hierarchical plan-

ning process include those by Gabbay [17], Axsater [3], Bitran and Hax [8],

Mohanty and Krishnaswamy [27], Sumichrast and Burch [29], Leong et al., [26], and Bowers and Jarvis [12].

Most hierarchical planning models assume some traditional form of

order release system at the detailed level. An exception is the paper by

Spearman et al., [28] which develops a hierarchical planning architecture

for CONWIP systems. The paper does not, however, present a general for-

mulation or solution techniques for the problem. An attempt is made in

the following sections to develop a hierarchical planning model for a fore-

cast-driven kanban system. The three-level product structure proposed

by Hax and Meal [19] is employed. The following section describes the

production system being modeled. Subsequently, the aggregate problem

is formulated. The disaggregation problem and a procedure to determine

item card counts are discussed in subsequent sections. The aggregate

problem is a large nonlinear mathematical program. Two approximate so-

lution techniques for the aggregate problem are outlined. Application of

the model to CONWIP and base-stock systems is then discussed.

2. System description

The production system being modeled is a ‘forecast-driven’ kanban

system. This is a system obtained by adding a push component to a pure

kanban system. Like other pull systems, a kanban system in its pure form

is an entirely reactive system. The system responds to demand as it occurs

and makes no use of information about future demand. As discussed by

Karmarkar (1986), the cost of maintaining an ability to respond to varying

conditions through such means as excess capacity may exceed the cost of

utilizing information about future events. In a forecast-driven kanban sys-

tem, the values of system parameters are set and are altered on the basis of

demand forecasts. Production takes place both in response to demand as

it occurs (the pull component) and in anticipation of future demand (the

push component). Demand during peak periods is met partly by drawing

down inventories built up during lean periods and partly by production

that takes place in response to the actual occurrence of demand.

It is assumed that the products being manufactured by this produc-

tion system exhibit signifi cant seasonalities of demand. The three-tier ag-

gregation scheme suggested by Hax and Meal [19] is employed. At the

bottom of the hierarchy are items, the end products delivered to custom-

ers. Items are grouped into product types. Items belonging to a product

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372 P. S. RAVI

type have similar unit production costs, inventory holding costs, work-in-

process holding costs, backordering costs, productivities (number of units

that can be produced per unit of time) and seasonalities. Each product

type consists of several families. Items within a family share a setup and

hence have the same setup cost.

An aggregate plan is formulated for the entire length of the planning

horizon. The plan determines production quantities, regular time and

overtime requirements, backorders and seasonal accumulation of inven-

tory for each product type and for each period of the planning horizon.

At the start of each period, aggregate quantities for each product type are

disaggregated into quantities for each family. Subsequently, aggregate

quantities for each family are disaggregated into quantities for each item.

The fi nal output of the system in each period is a card count for each item

and the level of seasonal accumulation of inventory or planned backorders

for each item.

The kanban system is modeled as an M/M/1 queueing system with-

out capacity restrictions. The M/M/1 assumption makes the model rela-

tively tractable. To account for the fact that there exists an upper bound on

queue length, a correction procedure is applied to the solution obtained

from this model. This procedure is based on the fact that, in an M/M/1/N

system, a certain proportion of arrivals are unable to enter the queue. The

aggregate problem is a nonlinear mathematical program. Two solution

procedures that solve the problem as a sequence of linear programs are

provided. These procedures provide a ‘good’ though not necessarily opti-

mal solution to the aggregate problem.

The aggregate planning model computes optimal production, work-

in-process, inventory and backorder levels for each product type and

for each period of the planning horizon. The inventory includes both

‘short-term’ inventory that constitutes part of the constant pool of work-

in-process and fi nished goods inventory held against kanban cards and

‘long-term’ inventory accumulated during lean periods to satisfy demand

during peak periods. Thus production may be assumed to take place ei-

ther against ‘regular kanbans’ in response to the actual occurrence of de-

mand or against ‘seasonal kanbans’

The disaggregation process consists of two stages. In the fi rst stage,

the production, work-in-process, ‘seasonal inventory’, and planned back-

order levels of each product type are disaggregated into quantities for

each family within the product type. The objective at this stage is the mini-

mization of aggregate setup time. In the next stage, work-in-process and

seasonal inventory levels for each family are disaggregated into quantities

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PRODUCTION PLANNING MODEL 373

for each item within the family. Card counts are then computed for each

item. The determination of the card count is driven by two factors. The

fi rst factor is the need to ensure that the work-in-process levels correspond

to desired item production levels. The second factor is the need to ensure

that the quantity of an item in work-in-process and fi nished goods inven-

tory is adequate to cover demand during production lead time. Card sizes

are assumed to be given. Items within a family (which share a setup) are

assumed to have the same card size.

The model incorporates safety stocks in the form of a safety factor α.

A larger value of α indicates larger values of safety stocks. This parameter

is exogenous to the model. As several researchers have noted, a reduc-

tion in the value of α over time is used as a measure of improvement.

Safety stocks would be required to account for variability in demand over

production lead time and to account for machine breakdowns and other

sources of uncertainty.

3. Aggregate planning for product types

The following notation will be used to formulate the aggregate plan-

ning problem for product types.

Decision variables:

Wit = Work-in-process of product type i at the end of period t.Iit = Finished goods inventory of product type i at the end of period t.Ipit = Finished goods inventory forming part of the kanban pool of

product type i in period t.Is

it = seasonal accumulation of inventory of product type i in period Bit

Bit = amount of product type i backordered in period t.Rit = quantity of product type i released in period t.Xit = quantity of product type i produced in period t.X1

it = Quantity of product type i produced to meet demand in period t and to build up inventory to meet peak period demand

X2it = Quantity of product type i produced in period t to account for

change in card count

Yit = Quantity of product type i produced per unit of time (.i.e. per

hour) in period t.Ot = hours of overtime production of product type i to be scheduled

during period t.

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Tqt = Average amount of time spent in period t by a batch in the queue

prior to processing

m t = Total arrival rate of batches in period t

tt = level of utilization in period t

Parameters :a = safety factor

Dit = Demand forecast for product type i in period t.

dit = Demand per unit time for product type i in period t.

Qit = Size of a batch of product type i in period t.

CAPit = Nominal production capacity (in units per hour) of product

type i in period t.

( )RT t = Total amount of regular time available during period t.

( )OT t = Total amount of overtime available during period t.

( )CW it = Work-in-process holding cost for product type i during pe-

riod t.

( )CI it = Finished goods inventory holding cost for product type i during

period t.

( )CB it = backordering cost for product type i in period t.

( )CO t = hourly cost of overtime production during period t.

The objective of the aggregate planning problem is the minimization

of the sum of overtime production costs and work-in-process and fi nished

goods inventory holding costs and backordering costs over the length of

the planning horizon. To ensure simplicity of representation, other factors

such as the cost of making releases, hiring and fi ring costs and the choice

of shift policy are not considered. The problem thus becomes:

Minimize Σi,t [(CW)itWit + (CI)itIit + (CB)itBit] + Σt (CO)tOt subject

to

0 for all ,W R W X i t+ - - =, 1i t it- it it (1)

0 for all ,I B X it I B D i t1s s- + - + - =, 1i t - it it, 1i t it- (2)

it itX X X1 2it = + (3)

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PRODUCTION PLANNING MODEL 375

X W Ip W Ip2 = + - -, 1it i t -it it , 1i t - (4)

I Ip Isit = +it it (5)

( )( / )(1 )W Ip d T Q CAPqt$ a+ + +itit it itit (6)

(( ) )X Y RT Oit it t= + t (7)

( ) for allO OT tt # t (8)

( / ) for all ,W Y T Q CAP i tit it qt it it= + (9)

(1 ) for allT t tt qt2m t- = t (10)

( / ) for allY Q tit itm R=t i (11)

( / )Y CAP for all tit itt R=t i (12)

0.99#tt (13)

, , , , , , , , , , , , , , 0 for all ,W R X X I I I B Y P P T O i tit it it it it it it it t qtS P r o1

$m tit tit it t (14)

Equations (1) and (2) are quantity balance equations. Equation (1)

states that the sum of the work-in-process of a product type at the start

of a period and the quantity of that product type released during that

period must equal the sum of the work-in-process at the end of the period

and the quantity produced during the period. Equation (2) states that the

sum of the level of seasonal accumulation of inventory of a product type

(less planned backorders) at the start of a period and production of that

product type during the period must equal the sum of the seasonal accu-

mulation of inventory (less planned backorders) at the end of the period

and demand occurring during the period. If there is a nonzero level of

seasonal accumulation of inventory of a product type at the end of a pe-

riod, the level of planned backorders of the product type at the end of that

period will equal zero (and vice versa).

Equation (3) indicates that production has two components. The com-

ponent X1it serves to satisfy demand as it occurs and to build up invento-

ries to meet demand during peak periods. The component X2it represents

the change in the size of the pool of kanban cards from one period to the

next. In other words, this is the quantity that must be produced to make

up the diff erence in card count from one period to the next. Thus if the ag-

gregate card count for a product type is increased in size from 50 to 60, a

quantity equivalent to 10 cards must be released and produced merely to

increase the size of the pool of work-in-process and fi nished goods inven-

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tory. On the other hand, if the card count for a product type is reduced,

the excess work-in-process and fi nished goods inventory is used to meet

immediate demand or to build up inventories for periods of peak demand.

Thus the variable X2it is not restricted to be positive. Equation (4) is the

defi ning equation for the variable X2it .

Equation (5) indicates that inventory has two components. The com-

ponent Ipit forms part of the kanban pool and has kanban cards associated

with it. The component Isit represents the seasonal buildup of inventory

during lean periods to satisfy demand during peak periods. Equation (6)

places a lower bound on the size of the pool of kanban cards of product

type i in period t. The size of this pool must be adequate to cover demand

for the product type during the length of time required to produce a batch.

dit represents the demand per unit time for product type i in period t. This fi gure is obtained by dividing the demand forecast fi gure for prod-

uct type i in period t by the total number of production hours employed

in period t. However, the total number of production hours (regular

time + overtime) scheduled for period t is a variable. Therefore the de-

mand per unit time is computed by dividing the demand forecast fi gure

by the number of regular production hours available in period t. This is

reasonable because the number of regular production hours available in a

period provides a lower bound on the number of production hours sched-

uled for the period. The assumption here is that the number of regular

production hours scheduled for a period is not a variable and that the total

number of production hours scheduled for a period can only be varied by

varying overtime levels. Equation (7) merely states that the production of a

product type during a period must be equal to the product of the produc-

tion per unit time during the period and the total number of production

hours used up during the period. Equation (8) states that the number of

overtime production hours scheduled during a period cannot exceed the

total number of overtime production hours available during that period.

Equation (9) represents an application of Little’s Law to this production

system. It states that the work-in-process of a product type during a period

is equal to the product of the rate of production during that period and

the production lead time for that product type during that period. The

production lead time is the sum of the time spent by a batch in the queue

waiting to be processed and the time spent in actual processing of a batch.

The time required to process a batch is the ratio of the number of units in

the batch (the batch size) and the nominal processing rate (the number of

units processed per unit time). This model does not consider setup times.

This is similar to the approach used by Hax and Meal (1975) and by Bitran

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PRODUCTION PLANNING MODEL 377

and Hax(1981), who ignore setup costs at the aggregate planning stage

but use them as the basis for disaggregation. Given that setup times are

relatively low in a kanban system, this approach seems reasonable.

Equation (10) establishes the relationship between the time spent in

the queue, the arrival rate and the level of utilization in an M/M/1 queue-

ing system. Equation (11) states that the total arrival rate of batches in

period t is the sum of the arrival rate of batches of each product type in

period t. The arrival rate of batches of a product type is the ratio of the rate

of production of that product type (expressed as units per hour) and its

batch size. Equation (12) relates the level of utilization in a period to the

rate of production of each product type and the batch size of each product

type in that period. Equation (13) represents the requirement that the level

of utilization in each period be strictly less than 1. Equation (14) enforces

the nonnegativity requirement for the decision variables.

The variable Qit is a weighted average of the batch sizes of families

within product type i. The batch size of each family is weighted by the

contribution of the family to the total scheduled production for that prod-

uct type over the planning horizon. Thus j( ( 0)/( )),Q Q D I D IjR= - -it j j i i0

where the variable Di denotes the demand forecast for product type i over

the entire planning horizon, the variable D j denotes the demand forecast

for family j (a member of product type i) over the planning horizon, the

variable Ii0 denotes the inventory level of product type i at the start of the

planning horizon and the variable I j0 denotes the inventory level of fam-

ily j (a member of product type i) at the start of the planning horizon.

The model computes optimal work-in-process, inventory levels and

backorders for each product type and for each period of the planning ho-

rizon. The inventory includes both ‘short-term’ inventory that constitutes

part of the constant pool of work-in-process and fi nished goods inventory

held against kanban cards and ‘long-term’ inventory accumulated during

lean periods to satisfy demand during peak periods. The model computes

separate values for ‘short-term’ and ‘long-term’ inventory. Production is

assumed to take place either against ‘regular kanbans’ in response to the

actual occurrence of demand or against ‘seasonal kanbans’. It is assumed

that all the production that takes place against ‘seasonal kanbans’ is com-

pleted before the end of the period. Thus none of the ‘seasonal kanbans’

are still in-process at the end of a period. All the work-in-process at the

end of a period consists of ‘regular kanbans’. The aggregate model passes

on production, work-in-process, backorder and seasonal inventory values

for each product type to the disaggregation model.

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4. The family disaggregation model

The aggregate planning model assumes that batch size is equal to card

size. In most production situations, it would be reasonable to assume that

if several batches belong to the same family and require only one setup,

only one setup would be undertaken for that family. In other words, it is

assumed that the entire quantity of a family produced in one period cor-

responds to one production run. This assumption does not hold for a kan-

ban production system. In a kanban system, production takes place in re-

sponse to withdrawal of goods from fi nished goods inventory. Production

cannot be delayed to ensure that all the batches belonging to a family are

produced in a single run. Some of the batches belonging to a family might

share a setup. The number of setups required by a family might therefore

be less than the number of batches of that family produced in that period.

However, the number of batches of a family produced in a period might be

expected to be proportional, in a sense, to the number of setups required

by that family during the period. This is true particularly when families

do not exhibit ‘seasonalities’ within a period .i.e. when the batchwise mix

of families in the demand stream remains unchanged during the period.

In other words, this is true when demand for diff erent families (measured

in number of batches per unit time) is not concentrated in diff erent parts

of the period. The disaggregation model therefore has as its objective the

minimization of the total setup time during the period, with the number

of setups for a family being assumed to be proportional to the number of

batches of that family produced during the period.

The following notation will be used to formulate the family disag-

gregation problem.

Decision variables

Yjt = Quantity of family j produced in period t.W jt = work-in-process of family j at the end of period t.

jtIs = seasonal accumulation of inventory of family j by the end of period t.

jtB = amount of family j backordered in period t.

Parameters

x jt = setup time of family j in period t.Tqt = average amount of time spent by a batch in the queue in period t.

jtQ = batch size of family j in period t.

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PRODUCTION PLANNING MODEL 379

CAPit = nominal production capacity (in units per hour) of family i in

period t.

The disaggregation problem is:

Minimize jt jt( / )Y Qj xR jt

subject to Y YjR =jt it

jt ( ) 0PY P I B D Itr o S

jt jtS

+ - + - + =t jt ,j t 1-

This is a simple knapsack problem. It is solved by initially setting

the production quantity per hour of each family equal to its lower bound

and then allocating all the remaining production per hour to the family

with the lowest value of τjt/Qjt. The solution to this problem furnishes

the values of the production per hour and the seasonal accumulation of

inventory and planned backorders of each family within the product type

during period t. The value of Isjt at 0t = is set equal to the inventory level

of family j at the start of the planning horizon. The work-in-process levels

of each family at the end of period t are then computed using the follow-

ing equation:

( / )W Y T Q CAPjt jt qt jt jtx= + + it

This procedure thus provides the values of the work-in-process and

the seasonal accumulation of inventory and planned backorders of each

family in period t. These values are then allocated to the items within the

family in proportion to the demand for that item in that period.

5. Determining item card counts

The card count for each item k in period t may be computed as fol-

lows. The number of units of an item in the kanban pool of work-in-pro-

cess and fi nished goods inventory must be adequate to cover expected

demand during production lead time. The card count Nkt for an item k is

the smallest integer that satisfi es the following inequalities:

( / )N Q d T Q CAPkt qt jt$ x+ +kt jtj it

N Q W$kt j kt

Here dkt is the expected demand per unit time for item k in period t. This is obtained by dividing the demand forecast for item k in period t by

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380 P. S. RAVI

the total number of production hours scheduled for period t. Wkt is the

work-in-process level of item k in period t.

6. Approximate solution methods for the aggregate planning problem

The aggregate planning problem for product types is a moderate-

sized nonlinear mathematical program. Two approximate solution tech-

niques for the problem will be outlined. Both techniques convert the prob-

lem into a sequence of linear programs. The fi rst method is guaranteed to

converge. The solutions obtained are not optimal. It is, however, claimed

that the methods provide ‘good’ solutions to the problem. The techniques

off er the advantage of being robust and easy to implement.

Procedure SET (Sequential examination of tradeoff s):

The essential choice in the aggregate planning problem is between

producing in excess in lean periods and holding inventories or planning

backorders on the one hand and using overtime or increased loading of

the facility in peak periods on the other. Inventories and backorders tend

to level production across periods; overtime and increased loading reduce

the extent of smoothing. The method suggested here considers these trad-

eoff s in three stages. In the fi rst stage, the choice between planned inven-

tories or backorders and increased loading of the facility is examined. The

tradeoff between backorders, inventories and overtime is considered in the

second stage. The tradeoff between facility loading and the use of overtime

is examined in the third stage of the procedure.

Part A: In this part of the procedure, the level of overtime in each period is

set equal to zero. This part of the procedure examines the tradeoff between

work-in-process holding costs and inventory holding and backordering

costs. Demand in peak periods can be met either by increasing the load on

the facility (production per unit time) during these periods or by produc-

ing during lean periods and backordering peak period demand or carry-

ing inventory to meet demand during peak periods.

Step 1. Set 0Ot = for all periods t. Set T T max1 =qt q in constraint (9) for

all periods t. Here Tqmax is an approximate upper bound on

Tq . T maxq is the expected queue length in period t if as much as

possible of the entire annual production of all product types

occurs in the period. The method used to compute T maxq is

discussed in the appendix.

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PRODUCTION PLANNING MODEL 381

Step 2. Drop constraints (10) and (11) from the set of constraints and

solve the problem as a linear program.

Step 3. Compute Tqt for all periods t using the set of constraints (10).

Step 4. Let tl be the value of t corresponding to the member of the list

of periods with the smallest value of T Tmax - qtq l .

Step 5. Repeat the following steps until :T T # f-qt qtk k1+l l

Step 5a. Initialize k to 1. Set T Tqt1

= qmax

l and set T Tqt qt=2l l.

Step 5b: Set T T=qt qtk 1+l l l in constraint (9). Set T T=qt qt

k 1+l l l in constraint (9)

for all other members t of the list of periods.

Step 5c: Drop constraints (10) and (11) from the set of constraints and

solve the problem as a linear program.

Step 5d: Compute Tqt2k + for all periods t using the set of constraints (10):

/( (1 ))T tqt2t m t= -tt

2k +

Step 5e : If T T-qt qt1k k+

l l drop period t’ from the list of periods considered

in the next iteration of step (5). If the list is now empty, go to

step 6. If the list is not empty, go to step (4). If ,T T $ f-qt qtk k1+l l

increment k by 1 and go to step 5b.

Step 6: Use the values of Yit at the stopping point of the iterative proce-

dure in steps 1 to 5 to recompute the value of Tqt for all periods t by using the set of constraints (11). Go to Part B of the procedure.

Part B:

This part of the procedure examines the tradeoff between backorder-

ing costs and overtime costs. A reduction in planned backorders can be

achieved by the use of overtime. The periods are examined in reverse

chronological order, working backwards from the last period to the fi rst.

The last period (say period t) in which backorders are observed is exam-

ined. The level of overtime in that period is increased until the cost of

any further increase in overtime outweighs the benefi ts derived from any

further reduction in backorders. Working backwards, period (t-1) is ex-

amined next. The level of overtime in period (t-1) is increased until the

cost of any further increase in overtime and inventory holding costs out-

weighs the benefi ts of any further reduction in backorders and a reduction

in overtime levels in periods (t-1) and t. Period (t-2) is examined next. The

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382 P. S. RAVI

level of overtime in period (t-2) is increased until the cost of any further

increase in overtime and inventory holding costs outweighs the benefi ts of

any further reduction in backorders and overtime levels in periods (t-2),

(t-1) and t. This process is continued until the start of the planning horizon

is reached.

Part C:

This part of the procedure examines the tradeoff between work-in-

process holding costs and overtime costs. The required level of production

in a period can be attained by reducing the load on the facility (reducing

the quantity produced per unit time) with a corresponding increase (by

the use of overtime) in the number of production hours scheduled for the

period. The level of overtime in a period Ot is increased (from its value

at the end of part B) until a point is reached when the cost of any further

increase in overtime outweighs the benefi ts resulting from any further

reduction in work-in-process holding costs. The relevant equations that

capture this tradeoff are equations (7) and (9).

(( ) )X Y RT Oit it t= +t (7)

( / ) for all ,W Y T Q CAP i tqt it it= +it it (9)

An increase of Ot from a level of k1 to a level of k2 units reduces Yit

to a value equal to Ynewit , where Ynew

it satisfi es the equation:

(( ) ) (( ) )Y RT k Y RT knewt + = +2it it t 1

or (( ) )/(( ) )Y Y RT k RT knew= + +tit it t 1 2

Work-in-process holding costs get reduced both because of a reduction in

production per unit time Yit and because of a reduction in average queue

time Tqt .

Let (( ) )/(( ) )c RT k RT k= + +t t1 2

Yit gets reduced to cYit (for all product types i). From the defi ning

equations for tm and tt (constraints (11) and (12)), it follows that these

variables get reduced by the same proportion. From constraint (10), it fol-

lows that Tqt gets reduced to Tqtnew , where Tqt

new is computed as follows:

(1 )/(1 ),T cT t cqtnew

qt t t= - - t where the value of tt used is the value exist-

ing prior to the reduction in Yit .

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PRODUCTION PLANNING MODEL 383

The appropriate value of k2 is the largest integer that satisfi es the relation:

( ) ( ) ( )CW W CO k kitnew

$R -tit 2 1i

where ( / )W Y T Q CAPitnew

itnew

qtnew= + it it

This relation can be readily written as a relation that is quadratic in a

single variable ( )k2 . Thus the appropriate value of k2 can be obtained by

solving a quadratic equation.

The Infeasible Starting Point procedure (ISP):

Variables with superscript k refer to the kth iteration of the algorithm.

Step 1: Initialize k to a value of zero. Set 0T1 =qt for all periods t. Set

Y CAP1 =it it in constraint (7) for all product types i and all peri-

ods t.

Step 2: Set 1.k k= +

Step 3: Set Y Y itk=it in constraint (7).

Step 4: Set T Tk=qt qt in constraint (9).

Step 5: Drop constraints (10) and (11) from the set of constraints and

solve the problem as a linear program.

Step 6: Compute T 1qtk + for all periods t using the set of constraints (10):

/( (1 ))T t1qtk 2t m t= -+

tt

Step 7: Set Y Y1itk

it=+ for all product types i and all periods t, where Yit

is the value obtained by solving the problem in Step 4.

Step 8: If ≤Y Yitk

it f- k for all product types i and all periods t and

≤T Tq tq f-tk k1+ for all periods t, exit. Otherwise go to step 2.

7. The correction procedure

A simple correction procedure may be used to account for the upper

bound on queue length. The product of card count and card size for a

product type (summed over all product types) provides the value of the

upper bound on queue length. The probability pN of an arriving batch

being unable to join the queue = (1/ 1)N + at levels of utilization close to

1. Two alternatives are considered. In the fi rst alternative, the production

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384 P. S. RAVI

rate for each product type is scaled up by a factor ( 1/ )N N+ . The second

alternative is to scale up the total of regular time and overtime hours by

the same factor. The lower-cost alternative is chosen.

8. Applying the procedure to other kinds of pull systems

8.1. A CONWIP system

The above procedure may be readily adapted to a manufacturing

system with a constant work-in-process (CONWIP) order release policy.

In a CONWIP system, the size of the work-in-process pool remains un-

changed. Thus the work-in-process pool may be considered to consist of

a constant number of cards. The variables Ipit in the aggregate planning

model are set equal to zero. (They are omitted from the model.) The other

part of the procedure which requires modifi cation is the item card count

determination stage. In a CONWIP system, the number of cards in the

work-in-process pool must be adequate to cover demand during produc-

tion lead time. Thus the card count Nk for an item k is the smallest integer

that satisfi es the following inequalities:

j ( / )N Q d T Q CAPkt kt qt$ r+ + itjtjt

N Q Wkt$kt j

where the variables denote the same quantities as in the kanban systems

case.

The approximate solution techniques suggested in one of the preced-

ing sections may be used to obtain a solution to the aggregate planning

problem.

8.2. A base-stock system

In a production system employing a base stock policy to regulate or-

der releases, no upper bound is placed on the levels of work-in-process.

An attempt is made to maintain a constant-sized pool of fi nished goods

inventory. Thus the fi nished goods inventory may be regarded as being as-

sociated with a constant number of kanban cards. If possible, the demand

that occurs is met entirely from the pool of fi nished goods inventory and

cards corresponding to the depletion in inventory (cards corresponding to

the occurrence of demand) are released into the production stream. If the

demand occurrence exceeds the base stock level, all the fi nished goods in-

ventory is used to meet the demand. Cards corresponding to the unfi lled

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PRODUCTION PLANNING MODEL 385

demand and to the depletion in inventory level are released into the work-

in-process stream. In general, cards equal in amount to the size of the de-

mand instance are released into the work-in-process stream. Constraint

(7) in the aggregate planning formulation for a kanban system must be

replaced by a constraint which ensures that the level of base stock is ad-

equate to cover demand during production lead time. The modifi ed con-

straint is: ( )( / ),Ip d T Q CAPqt$ +itit it it where Ipit is the base stock level for

product type i in period t, dit is the demand per unit time for product i in

period ,t Tqt is the average amount of time spent by a batch in the queue in

period t waiting to be processed and /Q CAPit it is the amount of time taken

to process a batch of product type i in period t. In the item card count de-

termination stage, the card count for an item k is set equal to the smallest

integer that satisfi es the following inequalities

( / )N Q d T Q CAPqt$ x+ +jtkt kt jt itj

N Q Ip$kt j kt

where the variables have the same denotation as in the previous cases.

The rest of the hierarchical planning procedure requires no modifi cation.

Appendix

Computation of an upper bound on Tq :

Tmaxq is the expected queue length in period t if as much as possible

of the entire annual production of all product types occurs in the period.

This is obtained by setting Xit , the quantity of product type i produced in

period t, equal to the total demand Di for that product type (less the initial

inventory Ii0 of that product type) over the length of the planning horizon.

Thus X D Iit i= - 0i

Yit , the quantity of product type i produced per unit time in period

t, is computed by dividing the quantity produced by the total number of

regular time hours available in period t. Yit is bounded above by the nomi-

nal production capacity for product type i in period t. Hence ( , / ( ) )Y Min CAP X RT=it it it t

Next, determine the value of ρt using constraint (12).

If 0.99,t 2t reduce the value of Yit corresponding to the largest

value of CAPit/Qit until the constraint 0.99t #t is satisfi ed. If this is not

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386 P. S. RAVI

achieved even by reducing this value of Yit to zero, the value of Yit corre-

sponding to the next largest value of /CAP Qitit is reduced. The process is

continued until the constraint 0.99#tt is satisfi ed. The rationale behind

this procedure is that, for a given value of ρt, minimizing tm is equiva-

lent to maximizing Tqt . For a given reduction in ,Yit a smaller value of

Qit results in a larger reduction in tm . Also, for a given reduction in ,Yit a

larger value of CAPit results in a smaller reduction in tt . This is desirable

because the objective here is to reduce the value of ρt to 0.99 as slowly as

possible, in order to enable as large a reduction of λt as possible before tt

attains the target value of 0.99.

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