OPERATIONS RESEARCH SYMPOSIUM
paper machine is used to convert wet pulp into paper
A and thus enjoys a position of central importance in the paper mill. I t is important that maintenance and
*
breakdown time be kept low since the output of the mill is directly determined by the production rate on the machines. The machines should also be maintained in an efficient condition so that a high proportion of the production is good paper. The repulping of “broke,” the term used for off-specification paper, is undesirable since this recycle increases the probability of producing more “broke” with the consequent introduction of an undesirable instability into the production process. Many types of paper machines exist and a particularly simple type, used mostly for the production of tissue paper, has been chosen for this exploratory study. Of the great many parts which comprise such a paper machine, we single out for special attention the wire and the first drying felt. The other parts are grouped together and are all inspected and maintained during maintenance periods. Figure 1 is a diagram of the machine. The wire is an endless mesh passing over a horizontal table of rollers onto which the pulp is fed as a thin layer. The water drains from the pulp to form a web which moves away from the feed point. After a short time on the wire, the web is sufficiently strong to support its own weight and is fed continuously onto the first drying felt which, like the wire, is endless and supports the web through a series of rollers which squeeze the web and felt to express water. A continuous length of paper is thus made and is passed to subsequent drying and rolling stages.
Figure 7.
Optimal Replaceme nt and Maintenance Policies P. R.
KING
Strategies for the maintenance
of a paper machine demonstrate
how to schedule downtime so as to minimize over-all production cost
Diagram of the papermachine VOL. 6 0
NO. 2
FEBRUARY 1968
29
The importance of these two components to the paper machine is at once apparent, and some replacement policy for each must be in operation because the wire is subject to random failure and the felt to aging. The felt ages with time because of a reduced ability to take up water, with a consequent loss in production rate. Because of the extreme delicacy of the papermaking operation a t high production speeds, the entire paper machine must be kept in a good state of maintenance. Replacement and maintenance must inevitably be costly, and operation under an optimal replacement and maintenance policy can be expected to produce a significant gain over nonoptimal policies. The problem of deciding in what sense a policy should be optimal is of course very difficult, and not every requirement can be met simultaneously. For example, in a widely used text (5), the papermaker is urged to “use his felts to produce the most paper in the shortest time at the highest quality and the lowest over-all cost”-a tall order indeed ! I n this paper we attempt to specify policies which minimize cost for wire and felt replacements and for machine maintenance. Maintenance is here taken to refer to maintenance of the other machine parts and in no way renews or modifies the wire or felt. Operating Model
An operating model for the paper machine must be postulated before any rational basis for the specification of an optimal policy can exist. This model should be capable of describing each operation which can influence the cost of production, and it should be possible to assess the cost of these operations fairly precisely. This operating model, like all quantitative descriptions of a technical process, is at best an approximation, albeit a good one if sufficient pertinent data can be obtained. If decisions are to be made concerning the operation of the machine we must necessarily make assumptions, but we should be conscious that incorrect assumptions must inevitably lead to incorrect decisions. I t is also important that the model should be continually reviewed and improved with the accumulation and assessment of current operating data. Indeed, the availability of an operating model often serves to indicate which plant data are worthy of record and which might safely be consigned to oblivion. The operation of our paper machine is here assumed to be according to the following model. T h e wire is subject to random failure with an adequate estimate of the time-to-failure distribution. The machine must be stopped and the wire replaced on failure since the machine cannot make paper if the wire fails. The felt ages with time in such a way that the equivalent loss in production rate increases monotonically with time. The felt is assumed never to fail. Finally, the other 30
INDUSTRIAL AND ENGINEERING C H E M I S T R Y
parts of the machine must be maintained regularly and this must take place at least once per week, an estimate which has been obtained by long operating experience on the machine. Clearly this is subject to a more rational decision only if the effect of maintenance can be realistically assessed in terms of cost, but the information is at present lacking. The state of the machine is specified completely by the age of the felt, t l ; the time, t z , for which the wire has been in operation; the time since last maintenance, t 3 ; and the state of the wire, whether good or failed. An operating policy requires the specification of replacement times for the felt and the wire, and maintenance times for the machine. Because the items to be replaced are costly and because the machine must be stopped, with loss in production, all paper mills have some type of replacement policy in force. An optimal operating policy can be expected since we hope to take advantage of economies of scale by performing two or more of the above operations simultaneously if the shutdown is planned, an option not available during an unplanned shutdown, unless a shutdown is scheduled during the repair time. In addition, the cost of a planned shutdown is on the average less than an unplanned shutdown because repair crews are not immediately available and labor charges are higher outside the day shift. The cost of felt replacement must also be balanced against the benefits of increased productivity. Associated with each of these actions are labor and replacement charges, designated C1 for felt replacement, C2 for wire replacement, and Cs for maintenance. All costs are specified in units of equivalent production hours lost. I n addition, the time down, C, means a loss of production which is the same for each of the operations. Time down for simultaneous operations is equal to that for single operations. Unscheduled replacements increase cost by a factor of 0 and result in a downtime loss, C. The loss in production rate due to felt aging is a strictly monotonically increasing function W(tl) of the felt age. The felt is assumed to continue to age while the machine is down for repair or maintenance. Wire Failures and Associated Renewal Process
The failure of wires is a random process, and the sequence of wire lives and replacement times after each failure constitutes a renewal process. It is clear that it is impossible to predict precisely the pattern of failures that will occur. The best we can hope for is an expectation of what will happen when operating under any given policy. This is no great disadvantage as the average behavior of such a process over a fairly extended period of time is quite close to the expected behavior predicted from a probabilistic description of the process.
T o implement the methods of renewal theory, it is necessary to have the time-to-failure distribution for the wire type in use on the particular machine. If the random variable, X , represents the age at which the wire fails, it has a distribution density f i ( x ) which for many wires has been found to be of the type given by Equation 1. fi(x)
=
b2x exp (- bx)
(1)
I t is here assumed that the time to repair a broken wire is constant, and if the random variable, Y , denotes the time between one failure and the next, it will have the distribution density shown in Equation 2.
fdY)
= b2(y
- c) exp
-b(y
- c)
=o
for y
2
for y
t2 + c )
where c is the repair time. The corresponding cumulative distribution is given by F,(y) = 1 - [bO,
- c)
+ 11 exp - b ( y
=o
- c) for y
2c
for y
Y tt + c)Irf,(tt'
1
t8'
1, ts' 3
The terms ,-I
ufil
+
+ 1, + 1))
(14)
ta'
+ C represent the direct cost in
capital outlay and downtime associated with any a, # 0. The term L:&dt
represents the loss due to wire aging
+
1. The complicated integral represents over stage r the integrated probability of a wire renewal at any time t multiplied by the d m c t cost of this renewal 8Cs C, plus the m i n i u m cost of the remaining I stages, starting with the machine in an appropriate state. The last term is the probability of passing thmugh stage r 1 without wire failure multiplied by the minimum cost of operation over the I remaining stages, commencing in the state (I]' 1, t:' 1,k' 1) which is the state resulting from inmmenting each age at the beginning of stage r 1 by 1 day. Notice that the costs are divided by r 1 and so repment cost per stage and that the length of a stage is reduced by c as far as the wire renewal profess is concerned, if any shutdown is scheduled, in which case a = 1. The iterative solution of Equation 14 is quite straightforward and a three-way tabulation o f f , ( t ~ ,tb t J against
+
+
+
+
+
+ +
VOL 60
NO. 2
FEBRUARY 1968
33
its arguments is constructed for successive values of r using the condition f&l,
ts,
tr) = 0
to start. Only integral values of tl and f a need be considered. T o evaluate the integral, f, must be known as a continuous function of the argument I t including negative values. In practice, j,is tabulated for integral values of ts and one negative value, -c, and others obtained by interpolation. The minimization is achieved by comparing each ofthe eight possible combinations of a1, u2, and U S for each combmation of the arguments. Arbitrary upper limits must be fixed for tl and tt while tr need not exceed 7. T o obtain the minimum cost and optimal policy for an unbounded process it is assumed that the sequence j&i, tz, ta) converges to a unique functionf(t1, h, ta), and the calculation is continued until the functionf, and the optimal policy no longer change from day to day.
cF 'i
2 a function of
4. Aclionr lo bc taken = 5 days, r = 5
the cumcnt stak of the
I M C ~ ~ M t. i
Example
T o illustrate the method of solution, the following hypothetical u t data were used
W(fi)= 1.5 ti 0 = 1.3, Ci = 10, Cr = 20, Ca = 6
C = 8,b
= 0.3 day-:
c =
'/a day
<
