OPERATI ONS RESEARCH
Application in the Chemical Industries This operations research study represents an initial effort b y a chemical manufacturer to use formalized optimization techniques and apply them to production scheduling and inventory control. A step-by-step account of the process conceptualization for a fermentation operation i s described from statement of the problem to construction of a general cost equation in terms of the pertinent variable factors. Evaluation of the mathematical mode’ necessitated an alternate approach in the form of a simpler scheduling procedure through use of a controlled scheduling factor. Retrospective scheduling for the three years preceding the study shows that a substantial savings i s afforded b y this method. PAUL STILLSON
Rumo- Wooldridge Corp., 8820 Belluncu Ave., los Angeles 4 5 , Calif.
I
S RECEST years t1iei.r h:is beeii :L g r o x i n g :i\vart.~ie>s i t l i i i i the chemical industry of the nerd for developing improl-ocl methods of production scheduling aiid inventor)- control. T h e inrreased competition in virtuallj- all branches of thv in. has emphasized the necessity of inipro\-ing oper:itionnl veness and decreasing operating costs. -4ttainiiig tlirar objectives, however, is not merely a technical problem within a chemical process but rathcr a blending of techriologicnl adv:iric+ ment and managerial skill in utilizing the process t o realize :in optimum gain from its output. T h e modern chrniist, then, i b charged with an additional responsibility which goes lxyoiid his conventional fnnctions in the laboratory, pilot plant, or on the, production line and extends into the planning, control, and t i c s cision-making phases of the company’s operation. The 111: of these comhined fields of the scientist and the ticv%ion-m:iliei, has become knoxl-n as “Operations Resrarch.” T h e operations research study discusst>d here reprwmts surli an effort by a chemical manufacturing Concern to use these t w h niques and apply them t o production and inventory contrd in a chemical process. Although the cost equations, mathematic:il model, and scheduling procedures are specific for the prwe9s under discussion, the conceptualization of that niodrl :inti t h e approach t o the scheduling and control prohlems nxiy rc:itlily be adapted t o most chemical processes. The company t h a t sponsorrd this research project is a. w l l knon-n chemical producer with an average annual gross sales of about S55,OOO:OOO. Its production facilities are contained i i i seven plants distrihuted throughout the rountr approximately 2500 people. . i n-id? vtirirty of c.licmic:il products is made. utilizing lmth chemical und hnctc~i~iolopic:il pro(’iiatrire of thcx esses. EIovever, hecmiir of the highly comp(~titivc~ process discussed. this company has r t ~ c l i i w t c ~t1i:it ~ l i1c.ithc.r it,. name nor the name of th:, protluc4 lw m r ~ i i t i o i i ~ t l .
Statement of the Problem The specific area of the company’s prociuctioii wtivities selrctecl for this study concerns a fermentation prorws for the production of a basic ran- materid which is t1ic.n furthcr processed t o three dc>rivative products. For the purpose of this presrntatioii. lve refcr t o these as crude &, finished Q, and R. Each of thcse three products has its ox-n individual process cycle, production capncitir., and market potentialities. S o n c of these products is producwl to order or for the fulfillmerit of a backlog h u t is prodneed ior
402
inwiltor>. in ;iiiti~ip:itioii(if
:I
f u t u r e tleni:iiitl.
.-iniiiitinl
stnte-
1, \\-h:it qiittntitj- of r:i~v feriiieiitation p r o d u ~ ~should t be riled per day i n order to enslire over-all niininiriiii opcrating
the opliniuni :illocation of this h ? i c i’tiw iii:itcrial 1c.s rrqiiirenientP for ciic~hof the t h r c ~deriv:itive T o appreciate the ~irolilemmore fiil!y; it might be \\ c.11 to ni?rition some of the cornpiexities of the owr-all process \\-hich \ v ~ r e inimetliately appnrcnt in the initial stages of the study imd which d l lie described in greater detail. First, although the three derivativc products are inventoried separately in t h i r salable lornis, there is vithiri the proccw the possibility oi’ conversion from one find product t o :inother-for exampl(3, c.riitle Q to finished Q to X or finished Q t o X. However, the reverse conversions are not ec,ononiically feasihlc, :tnd the commitment of either raw niateri:il or Q produrts t o the R process is ii,rr:vocahle. Then thcre is a n iritcrnietliatrs stage at ivhich t h r . p:irti:ill~material may he inventoried as a salnhlc~ product, or from r\-hich it ma?- he furthcr processed t o finished ivcraelj-. the finished Q product m a y I)(, oiily p x t i d l y and roturnrd to the intermediate stag(, : i t i d -o!d as cycled for t h r sccoiid time, to the finished proc1iic.t de1,ivative. I n ndtiition! tliere is always :I certain :mount oi p r o d u c t I? i\-liich ma>- ijr cconoinicall>- recovered as a procesP;rl)le tiyproduct of the production of the other tn-o derivatives. Thew :ire oth(it, recycling opcrations within the over-all process xhicli tied latrr, hut these nrc sufficient to illustrate soni(1 ltiei involved in ronceptualizing the prolilem :ind tievr.lopirig a n optimum scheduling procc~lrire. Adtiitional romp1ic:itiona are iritroduccd l)y ~qiiii~nic~rit limitations in t h a t portioiis of tn-o processes, criidc : r n d finishotl (2, contain common equipment. Thereforc, a decision to all any portion of t h r raw material t o product crudc Q will ne tatc :i holdup in the production of finished Cj even though material schrdulcd for the latter derivative. ire a1terii:itcl methods of procwsing the fci,mentetl matcrial nhivh mny increase the captcity of the systcm to produce both forms oi‘ 0, !)lit thcse also nwess:irily affect the unit c o s t s of thc products. 13ric.fly. then, there are three f h l products:
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
Vol. 48,No. 3
Operations Research -1means of converting, a t least to a limited degree, from one product to another -4lternate processes for producing two of these products vihich affect both capacity and cost T h e inadvertent production of one product as the by-product of the production of the other tn-o IVe can now reformulate the problem on the basis of the decisions which must be made during the over-all process operation. T h e first decision is concerned n-ith the total number of fermentors to be scheduled per day and fixes the quantity of raw material to be processed. The second dccision involves the distribution of this fermented rax7 material to either product Q or product R which, as shown later; is a natural division point in the process. A third decision is necessary to further subdivide the material allocated to product Q in order to produce the crude form of that product. Then, a fourth decision must be made as to the amount of crude Q which should be removed from the crude Q inventory and further reprocessed to the finished Q derivative. These are, a t the minimum, the major decisions which must be made, either intuitively or quantitatively, t o schedule the process completely. HOT t o make each decision contribute to a more effective operation, in terms of minimizing costs and optimizing inventory levels, is the problem of the operations research.
Description of the Process Since the complete fermentation process involves approximately 100 separate operations, it was necessary a t the outset to group these operations in such a way as t o identify each stage of the process (numbers), shon- all possible channels of material flow (solid lines), indicate product separation points (solid circles),
and define the control decision points discussed in the preceding section (letters). The resulting process diagram is shown in Figure 1. States 1, 2, and 3 represent the production of the basic raw material from the seed preparation phase through the fcrmcntation. Unfortunately, bacteriological processes are subject to contamination and a certain percentage of the fermented material is not suitable for further processing. I n addition, theit, is a portion of the raw fermented product n-hich is uneconomical to process through the Q products because of Ion- potericy, but it may be profitably recovered for the X product. These two product separation points are denoted by the small black circlw directly beneath the third stage (Figure 1). At this point a control decision is required as to the portion 01 materials suitable for both Q arid X production t h a t is to t)o sent through the Q process. This control decision is shown as A in Figure 1. The material allocated t o the Q process is sent to the crude conversion stage, 4C, and the rcmainder is sent, to the R process, 4 F . If we continue t,o follow the Q process stream, \ye firid III allel,nate process, 4C, through which the material may be processed during the Q process crude conversion stage. This altwnate process increases the capacity of the Q system as well as the unit process cost. Obviously, a second control decision, C, is required to determine the quantity of material to be directed through t h i k special process. The output of both processes, /iC’ arid 4C,, are directed t o 4Cz, and the crude conversion is completed. LIaterial loss is encountered throughout thcse operations; it i c represented by the product separation point and the solid linc. labeled “sex-er.” I n addition, a portion of the stream is inadvertently diverted from the Q process and may either he recycled
GENERAL FLOW MODEL
FERMENTATION
--------_--
---------
CRUDE CONVERSION
---------
PRODUCT
March 1956
CRUDE INVENTORY
- - - - - - - -- -
FINISHED
CONVERSION
PR?,DUCT R”
.P”
Figure 1
INDUSTRIAL AND ENGINEERING CHEMISTRY
403
Table I.
A 100 100 100 100 73.S5 73.Sb
B 100 100 100 100 100 100
Percentage of Fermentation Yields for Various Control Decisions
C 0 100 100 100 0 100
D
E
0 0 100 100 0 100
0 0 0 100 0 100
Finished Q % of Product 25.7 38 35.2 56 37.5 60 39.7 65 20.2 29 31.2 49
% of X
Crude Q
70 of X
R
% of Product 4 5 6 6 3
2.5 3.4 3.6 3.8 2.0 3.1
% of X 39.3 24.8 21.3 17.8 48.1 29.0
5
% of Product 58 39 34 29 68 46
Wastea 70 of
X
32.5 36.6 37.6 38.7 29.7 36.7
a This quantity is total of expected losses in system consisting of process-phase losses plus unusable quantities which are sewered. In addition to this waste, there i s also an expected waste of 0.054 fermentor per day at fermentation stage. Since number of fermentors that can be sent to Q per day is limited by nature of process, if maximum number of daily settings i s made, A can be no greater than 73.8%.
*
Table II. Control Decisions A B C D E 1 1
1 1
0 1
0 0
0 0
1
1
0
1
0
1
1
0
0
1
1
1
1
1
1
Daily Fermentor Settings
Finished
Min. Max. Min. Max. Min. Max. Min. Max. Min. Max.
Q
Recycling Cost Analysis
Crude
Q
R
Units
Units
17.0 39.1 23.3 53.6 18.2 41.7 18.0 41.4 27.2 62.5
1.7 3.9 2.3 5.3 1.8 4.1 1.8 4.1 2.7 6.2
Units
Total Units
Process cost
22.9 92.7 13.3 70.6 20.5 87.1 20.7 87.6 5.0 51.6
41.6 135.7 38.9 129.5 40.5 132.9 40.5 133.1 34.9 120.3
$2247 4638 2294 4732 2273 4658 2305 4694 2305 4744
through 4C2 or sent t o the R product process. The control decision t o determine which of these alternatives should prevail for a specific process batch is represented by t,he letter D. The remainder of the Q process stream is now available for either the crude or finished product. T h a t portion of the stream t h a t is allocated t o the finished product (decision point B ) is sent through the sequence of operations shown on Figure 1 as 6C1 through 6C4. T h e process loss associated with this finished conversion stage as n-ell as the recoverable loss to the R process is shown immediately following step 6C1. Decision point E again represents the choice of recycling through an earlier stage of the Q process. Meanwhile, a t each successive stage in the finished conversion series the stream may inadvertently he directed t o the crude Q product because of lon-er-than-specification purity. The portion of material allocated to the crude Q product jcontrol decision B ) is sent through stage 6C4 and recovered as crude Q inventory. This material may be sold as such or removed from inventory and reprocessed through the finished conversion series. A s already mentioned. all of the process streams directed t o the R process must folloi7 the series of operations, ,$FJ 5 F , arid 6F. Material losses resulting from this recovery process :ire shown by the product separation points following both the crude and finished conversion. The crude R inventory stage merely represents an intermediate depository t o facilitate storage-not a salable product.
Developing the Mathematical Model One of the major objectives in any operations research project is t o quantify the operating variables in terms of a measure of effectiveness. This is done through the formulation of a mathematical model representing the over-all procrss. I n this case, the measure of effectiveness is the minimum production cost t o
404
Income $ 3,939
13,062 3,588 12,255 3,814 12,750 3,816 12,773 3,137 11,219
Related Net Income
Per Cent
$1692 8424 1294 7523 1514 8092 1511 8073 832 6475
100.0 100.0 76.5 89.3 89.5 96.1 89.3 95.9 49.2 76.9
satisfy t,he demand requirements of the three salable products. Having constructed the flow diagram (Figure l), it is now possible t o derive an equation for the processing cost of each product as well as t'he over-all cost of the process. This total cost of processing is, of course, dependent on the number of fermentors scheduled per day arid the values established for each of the six control decisions. The effect of these control decisions on process yields, based on fermentor output, is shown in Table I. Two sets of values were assumed for each control decision in t h a t either all the material stream is directed to the finished Q product (100) or all the raw material is diverted a a a y from this product (0). Thus, the sis cases defined in Table I represent extreme decisions a t each control point. For example, in the first case all suitable material a t decision point A is sent to the Q process (A = 100) and all suitable material a t decision point B is directed to the finished Q product ( B = 100). LIean\Thile, the material rrhich is not proccssible for the finished Q product xithout recycling a t decision points C, D,and E is sent directly to the R process (C, D, and E = 0). In a similar manner, the other five cases are s h o w ~ ifor various decision comhinations with their expected yields of each product expressed as percentages of the fermentation yield (X). The results of this initial analysis sliow that the more material recycled, the greater is the amount of process losses within the system. Specifically, a comparison oi no recycling with complete reel-cling for daily settings of one or two fermentors (case I cs. case IV) indicates that there is 6.2r0 more fermentor yield availahlc for processing from the same amount of basic raw matcrinl. For a maximum daily fermentor setting (case V us. case VI) there is 77, more fermentor yield available. orid consideration is to determine how this reduction of m-aste through the elimination of recycling afiects production costs and income. Therefore, costs were computed for various
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 48, No. 3
Operations Research
SIMPLIFIED MODEL FERMENTATION
----------------------
CRUDE CONVERSION
-----
-----I---
CRUDE INVENTORY
-----------FINISHED CONVERSION
.25r Am:a PRODUCT
'a'
PRODUCT
'R'
Figure 2
degrees of recycling and for the minimum and maximum daily fermentor settings. These are shonm in Table 11. The allocation of material to the finished Q process is again represented b y the extreme cases of all or none-e.g., 1 and 0. The income was computed by assuming t h a t all the material produced in a salable form is sold a t average market prices. The difference between income and process cost is defined as a related net income. Once again, these results point out t h a t production without recycling yields twice 8s much related net income for minimum fermentor settings and one and a third times as much for masimum fermentor settings. It appeared quite evident t h a t an over-all savings could be realized by eliminating both recycling and more costly alternate processes. The only remaining question to be ansivered is whether the process capacity is sufficient to meet the sales requirements under normal process operations. I n this connection, a comparison of annual process capacity with product shipments for the three previous years was made. The results are shorvn in Table 111, where 19-50 shipments of finished Q are equated t o 1, and a11 other figures are expressed as multiples of this base figure. Obviously, the process capacity n-as more than sufficient to meet the requirements for each product during the three years prior to this study. A simplified flow model of the process is shown in Figure 2. All recj-cling operations and alternate process procedures have been eliminated, thus reducing the number of control decisions t o a minimum. I n addition, the quantity of process material flowing between any tn-o stages of the process is identified as a function of fermentor output and the tn-o remaining control decisions. T o illustrate, let us suppose t h a t X ' units of fermentation product is schetliiled per day. The number of these fermentors
March 1956
sent t o sewer because of contamination was obtained from an analysis of past data which showed t h a t this was independent of the number set per day and could be represented by a constant in any one scheduling period. Thus, X ' - 0.054 represents the number of fermentor units expected to appear at t,he srcond separation point. For convenience, the quantity X' - 0.0-54 is denoted as X. Further process analysis has established t h a t an average of 8% of this stream can be expected t o be of low potency and unprocessable by the Q process. Thrrefore, 0.92X is available for allocation at decision point A , and 0.08X is automatically directed toward the R process. Since the decision a t point -4is expressed as a fraction of the stream channeled t o either process, the resulting flow toward the Q process nil1 be 0.92AX and ton-ard the R process, 0.02(1 - A ) X . B y inserting the individual process yields and the remaining decision control variable, it is then possible t o represent the complete flow diagram in terms of a series of algebraic expressions. A general cost equation can n o v be constructed which ex-
Table Ill.
Maximum Annual Yields without Recycling Compared with Annual Shipments
Product Finished Q Crude Q R Total
Annual Yield (for A = 0.738, B = 0.5) 3.6 5.6 16.6 25.8 ~
INDUSTRIAL AND ENGINEERING CHEMISTRY
Annual Shipments 1950
1951
1952
1.0 1.8 5.3 1.2 6.6 13.4 -~ 12.9 16.4
1.0 4.1 11.0 __ 16.1
~
405
Date
Table IV. Scheduling Work Sheet (Scheduling factor: 2) Expected Yields From From Scheduled Fermentor From finished crude Production Settings fermentors Q process Q process
Inventory
Actual Sales
Finished Q Product
..
October November December January 1 January 2 January 3
Total Units Available
.. ..
..
..
283 214 196
106 164
.. ..
.. ..
3 6
51 102 170
7
..
..
..
..
..
.. ..
.. ..
334 316 366
..
.. ..
105 330 315 120 120 120
Crude Q Product October November December January 1 January 2 January 3
.. .. ..
..
870 1081 1297
289 73
.. .. 276 276 50
11 11 2
..
..
5
1151 1367 1364
10 17
975 1080 930 70 70 70
R Product October November December January 1 January 2 January 3
.. ..
..
.. ..
2060 1650 1308
0 792
*.
..
presses the total cost of the process as a function of a set of variables completely describing the three production processes. These processes are represented in the simplified model as a series of stages from seeding (1) to finished conversion [ S ) . There is, of course, a cost corresponding t o each of these stages. I n addition t o the t,otal production cost, K,-,, there is a n in-process inventory cost, Ki, and the cost of finished inventory, Kg. T h e total cost, K T , may then be expressed in its simplest form as
KT =
KI-6
f K,
+ h8
T h e expansion of the total cost eqwtion into a computational form involves a detailed mat,hematical analysis that we need riot consider here. For example, the cost of production, k ' l - 8 , appeared as
Ki--6
=
0.788 - 0.35OA - 0.119AB (X 0.788 - 0.063A - 0.05AB
+ O.O54)h'i23 +
0'376 - o'llSB (0.92dX) K A C If 0.663 - 0.05B
+
0'376 - o'llSB (0.616AXj KAC;: 0.79 (0.3'i6.iBX) IiOscl 0.424 - 0.05OB
+
0.91 (0.282ABX) Ksc? - (0.376AX - 0.376.1BXj K m
(0.788X - 0.726AX
+
+ COnStant)h'4F~6F- constant
T h e inventory t,erms were consideyably more complex and would easily fill a printed page. Unfortunately, the length and complexity of the resulting general cost equation raised critical analytical and practical problems. First, the distrihution of orders for two of the three salable products did not lend itself t o a convenient mathematical expression. Secondly, the general cost equation was of a form n-hich did not permit an exact solution. Finally, even if a method could be evolved for approximating a solution i t would result in a scheduling procedure which rroultl be too costly and complex to he used \I>-company pcrsonnel in production scheduling.
406
..
..
..
..
49
69 137 229
22 1 22 1 40
.. ..
0 0 1
..
.. .. ..
2350 2008 1626
940 840 810 700 700 700
The Second Approach It is not iincomnion in operations research studies or any scientific invrstigatioii that the first approach t o a problem is unsuccessful. I n this case, a great deal of information had been accumulated concerning the process stages and costs, a simplified flow model of the process had been obtained which eliminated the costly recycling operation, and the material flow through any process stage had been quant,ified. It xyas certainly not a >Tasted effort. Severtheless, a second solution m-as necessitated which combined the desired features of optimum production allocation anti inventory control with an easier implementat,ion of the scheduling procedure. The complexity of the preceding equations was primarily due t o the fact t h a t inventory costs lvere expressed as a function of the distribution of sales. It occurred t o the operations research team, then, that there might be another propert,y of sales (other than the distribution of quantities) on which t o base scheduling. Such a property was found in a study of changes in sales from period to period and the characterization of the distribution of these changes. I n order t o study changes in sales from period t o period, it was necessary to define the length of a scheduling period. T h a t is, how frequeiitly should the production process be rescheduled or adjusted t o maintain a pre-established inventory level based on an anticipated demand? Although the fermentors are set every day, the level of production cannot easily be changed on a daily basis because of the costs and efficiencies associated x i t h these short-t,erm changes. Therefore, t,he shortest possible practical period between adjustments v a s investigated and established as the length of the average production cycle. .4 study of process times for production viithout recycling showed t h a t the average process cycle is approximately 10 days; thereby the scheduling period was set a t 10-day intervals. Extensive d e s data vere collected and arranged according to the three 10-day periods in each month. Changes in sales were expressed as a multiple of a standard based on past sales. I n
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
Vol. 48,No. 3
Operations Research this cade, the most convenient standard was found t o be the highest sale in the previous period since scheduling Ixwxl on this previous high n-odd ti,nti to minimize i i i v e n t ~ r yshortnger. An analysis of t h e ratios olit:iiued 11). dividing each nlonthly sale for each of' tlie three proclucts I>>. tlica Iiigliest wlc, iii t h P p 1 two, thrcle, four: :iiid five niuiitli~rrFiiIte(1 i n r c ~ ~ p w t i rtli,tribue tions vhich apl~e:ired quite siniihr. H o ~ w r e r !the tlietribution of t h e previous three months' s d e s ratios was found t o be approximately normal and v a s used as tlie standard interval. It m i s also ohserved t h a t not more than i',3yoof the ratios exceeded 2.0 in any of the distributions, and consequently thif v:ilue TWS srlccteti ;is u possible scheduling factor.
The Scheduling Procedure The first model of the scheduling procediire n a s h cia>-scheduling period, a st:indard o1it:iinrd from the highest sale in tlie preceding nine 10-(13>- periods., and s scheduling factor of 2 . In applying this procedure it is only necessary t o knon- thc, amount of inventory on hand :it the tieginning of the scheduliiig period, the enlee for the previous three months, the amount of in-process material from the last scheduling adjustment: ~111~1 t hf, process yields hnsed 011 any level of fermentor output. ;in example of this method of production scheduling in operation is s h o n n in the scheduling n-ork sheet of Table 11-. Let UP assume that it is January I and t h e inventories O I I Iinnd are iis s h o x n in the first column for each product. T h e schedulctl production for the first 10-day period, established at the lieginning of the prcvious 10-day period, culls for 3 fermentor outputs t o he allocnted t o t h e finished C j product, 11 fermentor olltputs are for crude Q. and 0 fermentors are set for R. This total of 11 fermentors is t o be sct and processed during the period of Jailnary 1 t o January 10. T h e process ?-ielda resulting from this :~110catio11c a n rendily I)? cn1cul:ited n-ith the :lid of the simplified flax diagrmi (Figure 2 1 . in which each of the stre:ims is quantitatively cxprewxl in tcrnis of scheduling decisions. =is \\-ill he recalled, the production of X is expected although no fermentor o:itputs n-ere directly allocated t o t h a t product. Similarly, a small amount of crude Q is expected from each processing of niatcrial for f i n i r h d Q . These process yields :ire shown under their appropriate hwtiingr in Tahle Is'. T h e total amount of each product iivailahlc for sales a t the end of t h e 10-day period is s h o ~ ~ in - n the co!unilis labeled "Total Units .ivailatile" .&;suniing an eqi1:~1distribution of sales during t h e month, the highest IO-day periods of sales during the preceding t h r r r months are 110, 360, and 330 units, respectively. T h e scheduling factor of 2 is then applied. The rem:iinder of the scheduling procedure is pure arithmetic. Let 11s take finished Q :E an example. If 334 units are available for sales and 220 unit? are espected to be released for shipmrnt, there n - o d d remain 114 units a t the end of the first 10-day pcriod. I n order t o prepare for the same amount of sales during the nest scheduling period. it ~ o u l dhe necensarj- t o produce I06 iinits t o hrinp the nmoiint of
March 1956
product :iv:til:ible for sales back u p t o 220. T h e production reqnirc>nientsof finished Q is then t h a t number of fermentors \\-hose p r ( ~ c c wyield would he 106 units. I n a similar manner each of the othc~rproduc~tris scheduled, taking into account the inter>-ieldsdescribed. .kdjiistnients in the scheduling procediire :IW mnde as the actual sales w e reported :ifter ear11 whetluli1ig prriod, mid the differences I~ct\vcenthe sales espcvtm c i w :iiid tlic :ictu:il mioiint of products shipped nre corrcctrcl.
Procedure Evaluation
In thc first scheduling attempt using this wiipie plan, no cHort ~ r a smade t o limit the change in production Icvel for suei r e scliediiliiig periotib. T h e results of :ippIyiiig this procedure rctros;wctivrl>-to the preceding ihrce-ycar opcrntion n-rre ns follo\vs: 1. S o s1iort:iges occurred 2. Finished (I average inventory decreased by . ? ~ 8 7 ~ 3. Crude C) average inventory decreased b\- 63% 1. K average inventory decreased b y 5670 3 . Total nuniber of fermentors set decreased by 17% hlthough thesr results v e r e very rncow:iging, it m ~ i s t I)P rcniemk)ered t h a t they werp hased on severd :i pi,iori assumptions wliicah coiild not lie justified in actual operntions. First. a study of the daily shipments shoved t h a t t h r distribution of sales for groups of 10-day periods v-ere not e q d l y spread through the month, hut rather folloiwd a 70-15-15yo distribution. Secondly continuous fluctuations in production lrvel over short periods of time n-ould present serious difficulties t o even the best run production team. Finally, n severe fluctuation hetween any two scheduling periods n-ould necessitate a change in the number of v o r k c r e w employed. Such changes were unacceptahle to the company hecause of the lahor problems which mould be precipitated by this type of operation. BJ- instituting the constraints requircd t o eliminate these objections, a revised scheduling program was developed and restrospectively applied t o the aforementioned threr-J-ear period. -4 summary of the final results showed: 1. 2. 3. 4. 5. 6.
-
I .
S o shortages occurred Finished Q average inventory decreased by Crude Q average inventory decreased bj- 57% I: average inventory decreased by 57Y0 Total number of fermentors set decreased by I8y0 Recl-cling and alternate processes eliminated with increnscd production savings T h e scheduling procedure obtained is simple to operate, can he applied by esisting company pereonnpl, and is easily implemented Acknowledgment
This research project was carried out by :in operations research team from the Case Institute of Technology, Clevel:tnd, Ohio, under t h e direction of Riissell L. -4ckoff. T h e author vas a member of the team as well :is the conipmy representative. R E C E I Y EfD o r re\-i-w M a y 2 4 ,
lQ.ii.
INDUSTRIAL AND ENGINEERING CHEMISTRY
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J a n u a r y 19, 1'156.
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