Operations research in analytical chemistry - American Chemical Society

One of the principal preoccupations of the analytical chemist is to devise optimal analytical methods. To do this, he must very often make a choice between many possible combinations, even for very simple procedures such as a colorimetric determination, in which the result depends on the values of one or (usually) more parameters. The analyst’s task is then to select the optimal combination of parameter values so that the determination is optimised with respect to some criterion such as the signal-to-noise ratio. Such a problem can be solved by many different optimisation techniques. One of the most attractive, by virtue of its simplicity, is the simplex method described in an earlier REPORT by Deming and Morgan ( I ) . This is an evolutionary operations method which is part of the vast science called operations research (OR). There are many definitions of OR. For our purpose, we define it simply as a collection of 1244A

mathematical methods permitting the selection of the “best” combination from a large set of possibilities. There are many instances in analytical chemistry in which the best combination should be selected and in which OR can be useful. Several examples will be given in this report. The main applications of OR deal with organisation problems with economical or social implications; therefore, applications to problems of this nature in the management of analytical laboratories are rather straightforward. Goulden’s ( 2 )recent article about OR and other management techniques in analytical research, development, and service covers part of these applications. Therefore, only one such application will be described here, namely, the choice of the optimal combination of apparatus and manual methods in a clinical laboratory ( 3 ) . We will try to show that OR can

ANALYTICAL CHEMISTRY, VOL. 47, NO. 14, DECEMBER 1975

also be used in the optimisation and/ or selection of analytical procedures or programs by introducing some of the more important methods used in OR with typical examples from management science and by applying these to analytical situations. To permit easy understanding, these situations are greatly simplified, but in fact, more complex problems can be tackled. In most cases, because of space limitations, we have omitted mathematical details and have only described the construction of the OR model. Although the examples used reflect the authors’ interest in chromatography, the methods are equally applicable to other branches of analytical chemistry. Optimal Configuration of a Clinical Laboratory (Integer Programming) De Vries ( 3 ) ,in investigating some of the economical aspects of clinical

Report

laboratories, has shown that the problem of minimising costs in the chemical laboratory can be reduced to an integer programming problem. I t is supposed that I different determinations must be carried out with a number of apparatus or manual methods. There are J apparatus (manual, 1-,2-, 4-, 6-, 8-, 12-channel apparatus with or without direct digital readout) available, and the problem consists of making a selection among these to carry out all the determinations at total minimal cost. All determinations of the same substance must be carried out by the same apparatus. Costs taken into account are fixed, Cj, and variable (the cost per analysis), Vj, of each apparatus. The number of determinations with each apparatus, N,, and the maximal capacity, L,, of each apparatus are also given. The problem can be formulated mathematically as follows: Minimise J j=1

(C,+ NjVj)Xj

The constraints (Equation 3) express that no apparatus can be bought if its capacity is smaller than required. There are a number of mathematical methods to solve such linear integer programs. - Essentially, these methods can be divided into two groups. In the first group, called cutting plane methods, the solution is found by solving the problem as a linear program (the variables XIare considered as real instead of integer). At the same time a new constraint is added a t each iteration. In the second group of methods, called enumeration procedures, the set of possible solutions is divided into sub-

sets which are examined successively until the best solution is found. The branch and bound method (discussed in a later section) is part of the latter group. A complete discussion of these methods can be found in Zionts ( 4 ) . Distribution Problem (Graph Theory)

Suppose that a production unit is connected with a number of clients through a pipeline. One must know how to interconnect the clients and the production unit so that the pipeline has a minimal length. The distances between the clients and the production units and between the

Figure 1. Examples of trees in a graph. Uppermost figure is minimal spanning tree of complete graph given in Figure 2

(1)

subject to the constraints J j=1

mijXj = 1 i = 1,. . . , I

(2)

where X , is a 0-1 variable equal to 1 if apparatus j is part of the selected (optimal) configuration and equal to 0 when this is not the case, and mij is a coefficient equal to 1 if it is possible to use apparatus j for determination i and 0 if this is not the case. The economic or objective function (Equation 1)expresses the total costs which are to be minimised. The constraints (Equation 2) express that exactly one apparatus must be used for each kind of determination, exclude that one kind of determination should be carried out on more than one apparatus, and require that each kind of determination should indeed be carried out. ANALYTICAL CHEMISTRY, VOL. 47, NO. 14, DECEMBER 1975

1245A

1

Table I. Distance Between Points in Figure 2 A

A

B C D E F G

B

C

D

E

F

G

0 28 0 32 23 0 35 40 60 0 100 80 103 75 0 119 104 128 90 29 0 127 105 126 105 30 35 0

Figure 2. Complete graph for distribution problem

clients themselves are known. In Figure 1, A is the production unit and B-G are the clients. Two possible configurations of the pipeline are given in the figure. It is clear that (a) is a better solution than (b). Let us now see how to find the optimal one. By drawing all possible interconnections between the points, a graph is obtained where clients and production unit are the nodes and the interconnections are the edges, the values of which are given in Table I. Both (a) and (b) of Figure 1are graphs which are actually part of the graph in Figure 2. These graphs are connected (all points are linked directly or indirectly to each other) and contain no cycle (if F were connected to G, EFG would constitute a cycle). This is called a tree, and the tree for which the sum of the values of the edges is minimal is called the minimal spanning tree. In our example, the minimal spanning tree also yields the shortest distribution pipeline. The problem of finding the latter is therefore reduced to finding the minimal spanning tree in Figure 2. Several algorithms allow this, the conceptually simplest one being Kruskal’s algorithm ( 5 ) ,although other algorithms are better adapted for computer calculation. Kruskal’s algorithm can be stated as follows: “Choose from the edges that are not yet part of the tree, the one with the smallest value which does not form a cycle.” This is applied to Figure 2 with the values of Table 1. One starts by selecting the smallest value in the table: Step 1: edge BC (23) Step 2: edge AB (28) Step 3: edge E F (29) Step 4: edge EG (30) Step 5: not edge AC (cycle), but AD (35) Step 6: not FG, BD, or CD (cycle), but DE (75). The optimal distribution network is therefore given by Figure 1 (a). How can such a distribution problem be used in analytical chemistry? If one carefully considers Figure 1 (a) (which 1246A

is drawn on scale), one observes that two clusters can be distinguished, A-D and E-G. These clusters can be obtained formally by breaking up the longest edge (DE) in the tree. Clustering techniques can aid in de-

veloping optimal analysis procedures. One example is the selection of optimal combinations of thin-layer chromatographic systems. When one elaborates a qualitative identification scheme with this technique, one has to

Figure 3. Minimal spanning tree for eight TLC systems for basic drugs. Distances between systems are (1-p) values. Values of correlation coefficient obtained from ref. 7

Figure 4. Communications network


combine two or more TLC systems so that they do not yield too much correlated information. By clustering the available systems according to their similarity and selecting one system from each cluster, this purpose can be attained. This was applied by De Clercq and Massart (6) to the selection of the best combination of three TLC or PC systems from a data set containing R f data for 100 basic drugs in eight systems ( 7 ) .By using the correlation coefficients between each pair of systems as calculated by Moffat, and considering (1-0)as the distance between systems, a graph such as in Figure 2 was obtained. The minimal spanning tree derived from this graph is given in Figure 3. If one accepts the convention that edges should be broken only when the resulting clusters consist of a t least two members, the best combination must consist of one system from each of the groups 2 314 5 611 7 8. By measuring the information content (8)of each separate system and selecting the best system from each group, one obtains in a rather simple way the presumably best combination 3, 6, 7 . This is in fact. the optimal solution as shown by Moffat and Smalldon ( 7 ) ,who calculated how many pairs of

+

+ +

+ +

drugs could be separated with each possible combination. This is, of course, the most accurate way of selecting the optimal combination. I t is also a rather more elaborate method than most practising separation chemists are prepared to carry out. In contrast, the complete OR technique, correlation coefficient calculations included, can be carried out in one afternoon with the aid of a pocket calculator and yields almost certainly the best or second best possible combination. Communication Network Problem (Graph Theory) Psychologists and sociologists have used graphs to represent the communications between individuals in certain organisations and in this way to deduce certain characteristics about the organisation in question. An interesting application for analytical chemistry in this respect is the work by Allen (9) and Frost and Whitley (10) about communication patterns in research and development laboratories. A typical problem in this area is the determination of the groups of individuals between whom communications exist. Suppose some of a population of eight people, A-H, are directly in contact with each other, whereas

others are not. The problem is to distinguish the sets of people between whom a communication (direct or indirect) exists. For example, in Figure 4,C and F communicate directly, B and D indirectly (by way of A or E), and there is no communication a t all between the sets ABDE and CFGH. In graph theoretical terms (ABDE]and {CFGHJare connected graphs, whereas {ABCDEFGH] is not, and the problem is reduced to finding the connected components of the latter graph. One of the possibilities for applying this in the development of analytical methods is: Let us suppose that we want to do research on the thin-layer chromatographic separation of a rather large group of substances and, in particular, we want to develop better methods than the existing ones. One of the possible strategies is to select those groups that are particularly hard to separate by existing TLC procedures and to concentrate on finding better methods for those substances. A group of substances that are hard to separate is defined as a group consisting of substances difficult to separate from a t least one of the other substances of the group. There are several possibilities for determining which pairs of substances answer this de-

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1247A

Check the Nicolet NIC-I180 Applications The NIC-7 780 currently offers software packages for these applications: APPLICATION: General Use CAPA8ILITIES:This includes: an assemblereditor program, both for paper tape and diskbased systems, for user programming in machine lang uage; d i ag no st i cs for mac h i ne mai nt enance; a floating point package consisting of subroutines for most arithmetic functions; and a disk monitor program for both the floppy and cartridge disk systems. APPLICA TION: N M R Spectroscopy CAPABILITIES: Four programs are currently offered in the NMR applications software package: 1) FT-NMR-1180.This package includes the capability to perform Fourier transforms on time domain data blocks from 256 data points to 64K data point?. A 4K transform takes 2V2 seconds; a 32K transform takes 24 seconds. Routines are included for real-time phase correction, zoom display, peak picking, inverse FFT, 9 point smooth (per Savitsky-Golay)’ exponential or trapezoidal windowing functions, constant speed plotting while acquiring new data, integration with baseline and slope correction, and addition and subtraction of spectra. 2) Quadrature FT-NMR. This package includes the above plus the ability to acquire and transform quadrature data (per the technique of Shaeferl Stejskal).* 3) ITRCAL. This program for the analysis of complex-coupled NMR spectra uses the method of Castellano and Bothner-By3to calculate a theoretical spectrum which is then iterated for a best fit with a set of experimentally observed lines. 4) T1/1180. This program is for the measurement of the spin-lattice relaxation time, T1, using either inversion recovery, progressive saturation or McDonald-Lei g h4 p u Ise sequences.

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A PPLICA TION: Laser Raman Spectroscopy CAPABILITIES: The Laser Raman package for the NIC-1180 uses the Multi-Channel Scaling acquisition mode to collect data from the spectrometer discriminator. The spectrometer is driven by pulses from the data processor which also monitors the progress of the spectrometer’s stepper motor to provide reproducible signal averaging. Processing software includes a routine to divide spectra (for depolarization ratios), a full scale expansion routine which allows spectra of different intensities to be compared to each other, and calibration commands. 1248A

ANALYTICAL CHEMISTRY,

VOL.

47, NO. 14, DECEMBER 1975

F E A T U R E : A hardware Read Only Memory (ROM) program loader is included.

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1249 A

Table II. hRf Values of Antibiotics (from Ref. 7 7) System

6

7

8

9

10

11

1 2 3

Actinomycin Aminocidin sulphate Chloramphenicol

92 0 86

25 87 85

100 4 91

84 77 90

43 6 93

90 23 88

89 3 89

88 0 45

20 27 86

92 0 85

15 0 9

4 5 6

Chlortetracycline hydrochloride Colistin sulphate Cycloheximide

28 0 72

51 94 87

81 65 92

82 96 91

31 49 91

64 54 80

56 24 80

0 14 59

43 14 93

0 0 70

0 0 13

7 8 9

Demethyltetracycl ine Dihydrostreptomycin Etamyci n

25 0 93

31 95 33

60 8 90

73 95 94

30 0 92

45 57 95

43 0 94

0 0 85

28 0 39

27 0 93

0 0 15

10 11 12

Filipin Griseofulvin Leukomycin tartrate

95 90 94

0 0 82

97 96 96

87 91 95

94 80 94

93 90 93

92 92 94

0 94 93

0 0 28

75 95 95

0 89 15

13 14 15

Lincomycin hydrochloride Mikamycin B Misi on i n

38 91 72

100 13 0

90 95 93

95 81 86

89 94 67

71 96 79

66 95 72

65 55 50

100 25 0

0 97 0

0 27 0

16 17 18

Neomycin Nystatin Oxacil I i n

0 6 46

95 0 88

0 84 93

77 68 96

0 0 95

45 55 .66

0 53 68

0 0 0

0 0 94

0 0 58

0 0 0

19 20 21

Pa ro mom yci n Penicillin G-Na Penicillin V-K

0 40 37

92 94 88

0 88 88

81 94 94

0 91 94

22 62 57

0 64 62

0 0 0

0 100 100

0 0 0

0 0 0

22 23 24

Polymixin B sulphate Puromycin Pyr rol id in e methyltetracycline

0 49

97 65

67 92

96 95

0 86

67 73

52 67

0 43

0 18

0 0

0 0

7

65

83

82

43

55

36

25

37

0

0

25 26 27

Rifamycin 0 Rifamycin S Rifamycin SV-Na

96 95 91

0 28 43

94 93 94

100 94 93

93 92 81

100 93 92

100 94 91

100 83 91

65 77 79

100 92 100

54 69 50

28 29 30

Spiramycin base Staphylomycin Streptomycin sulphate

86 93 0

94 45 100

100 100 0

100 0 85

93 93 0

94 94 36

88 92 0

0 18 0

7 34 4

99 100 0

0 0 0

31 32 33

Terra myci n base Ty I osi n base Viomycin sulphate

25 92 0

74 92 100

92 92 0

100 100 100

54 92 0

67 87 49

55 85 0

10 35 0

52 43 0

0 100 0

0 0 0

No.

Antibiotic

1

2

3

4

5

scription. One of the simplest, though not necessarily the best, seems to be to determine the euclidean distance. For substances A and B this is equal to I

DAB=

d 5[ ( h R f )-~( h R f ) ~ ] ' 1

(4)

where n is the number of systems for which hRf values are given. One can then consider difficult those separations for which this value is lower than a predetermined value, draw a graph in which those pairs of solutes catalogued as difficult to separate are connected, and look for the connected components of the graph, exactly as in the communication network described. When the number of components is too large, the graph is too 1250A

complex and a matrix notation is used in which a 1means that there is a direct link between two components and a 0 means that this is not the case. As an example, this was carried out for 33 antibioti'cs. Their Rf values taken from ref. 11 in 11 TLC or PC systems are given in Table 11. Substances are difficult to separate when DAB< 50. The resulting matrix is given in Table 111. Using a very simple algorithm, one concludes that the following group of chromatographically similar substances are present: Aminocidin sulphate, dihydrostreptomycin, neomycin, paromomycin, streptomycin sulphate, viomycin sulphate Chlorotetracyclin, cycloheximide, tylosin base


Chlorotetracyclin hydrochloride, colistin sulphate, demethyltetracycline, pyrollidinemethyltetracycline, terramycin base Etamycin, mikamycin B Penicillin G-Na, Penicillin V-K Rifamycin 0, rifamycin S,rifamycin SV-Na. Such a classification also has some significance where the structure of these antibiotics is concerned: the oligosaccharides dihydrostreptomycin, neomycin, paromomycin, and streptomycin are found in one group and so are the tetracyclines (chloro-, demethyl-, pyrollidinemethyltetracycline, and terramycin), the penicillins, and the rifamycins. When the sets are large, the probability increases that the complete graph is connected so

that no groups can be separated. In this case, one can proceed by decreasing the value used as a criterion for qualifying the separation as difficult. The algorithm proposed here is a classification algorithm which constitutes a pattern cognition or clustering technique. Therefore, the classification problem presented earlier as a distribution network problem can also be solved in this way and vice versa. In fact, classification can be carried out by a variety of techniques, some of which are not generally considered as OR methods. Some of these have been used in analytical chemistry: a statistical technique called pattern cognition was introduced for the classification of GLC stationary phases by Wold (12), and numerical taxonomy was proposed (13)for the same purpose and for the combination of TLC systems problems (14).In terms of the latter technique, the algorithms described here are single-linkage algorithms. They are related to pattern recognition techniques which in the last few years have become accepted techniques in analytical chemistry (1517). There is a difference between pat-

tern cognition and recognition techniques: in pattern recognition one must first distinguish between two or more given classes (clusters) according to patterns (representing, for example, mass spectra) with the aid of a training set of known compounds and then use the pattern of an unknown compound to identify it as belonging to one of the classes. In pattern cognition, one determines which classes can be distinguished without the use of a training set. Here, we discuss OR classification techniques with a TLC example because this is a rather simple application which can be solved by simple algorithms. However, applications are also possible in GLC (13, 16) and infrared and mass spectrometry. More sophisticated algorithms are then necessary. Shortest Path Problem (Graph Theory and Dynamic Programming)

The theory of graphs can be applied to the optimisation of chromatographic separation schemes for multicomponent samples by using a shortest path

algorithm (18). The same result can be obtained by using dynamic programming (19) which is a method of sequential optimisation based upon Bellman’s (20) principle of optimality: “A policy is optimal, when a t a given stage and whatever the preceding decisions, the decisions which remain to be taken constitute an optimal policy taking into account the preceding decisions” or, more succinctly, an optimal policy is composed of optimal subpolicies. This principle can be applied to problems in which many decisions are required to obtain an optimal result in a system composed of sequential steps, on the condition that the later stages do not influence the results obtained after the earlier stages. It is used quite often in chemical reactor designs such as the design of multistage crosscurrent liquid-liquid extraction or distillation (21). Analogous applications can be found in analytical chemistry. One can, for example, apply this to the optimisation of gradient elution chromatography or to certain countercurrent distribution applications. Since the mathematics tend to be complex, we will use two simple applications to il-


1251A

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Curtin Matheson Scientific Company, Inc. 12101 Centron Place Cincinnati, OH 45246 Area (513) 671-1200 Curtin Matheson Scientific Company, Inc. 4540 Willow Parkway Cleveland, OH 44125 Area (216) 883-2424 Fisher Scientific Co. ~.~~ _ _. ~ . . . 548 1 Creek Road Cincinnati, OH 45242 Area (513) 793-5100 Fisher Scientific Co. 26401 Miles Avenue Warrensville Heights Cleveland, OH 44128 Area (216) 292-7900 Sargent-Welch Scientific Co. 10400 Taconic Terrace Cincinnati, OH 45215 Area (513) 771-3850 Sargent-Welch Scientific Co. 9520 Midwest Avenue Garfield Heights Cleveland, OH 44 125 Area (216) 587-3300 VWR Scientific P.O. Box 855 2042 Camero Avenue Columbus, OH 432 15 Area (614) 445-8281

NEW JERSEY

Beckman Instruments, Inc. Science Essentials Operation U.S. Highway 22 at Summit Road Mountainside, NJ 07091 Area (201) 232-7600 Curtin Matheson Scientific Company, Inc. Mid-Atlantic Industrial Park 1571 Imperial Way Thorofare, NJ 08086 Area (609) 848-1500& (215) 462-4700 Curtin Matheson Scientific Company, Inc. 357 Hamburg Turnpike Wayne, NJ 07470 Area (201) 278-3300 Fisher Scientific Co. 1 Reagent Lane Fair Lawn, NJ 07410 Area (201) 796-7100 Fisher Scientific Co. 52 Fadem Road Springfield, NJ 0708 1 Area (201) 379-1400 Sargent-Welch Scientific Co. 35 Stern Avenue Springfield, NJ 07081 Area (201) 376-7050 ~~

NEW YORK

Bioclinical Laboratories 375 Central Avenue Bohemia, NY 11716 Area (516) 567-6677 Fisher Scientific Co. 15 Jet View Drive Rochester, NY 14624 Area (716) 464-8900 VWR Scientific P.O. Box 23 Bronx, NY 10452 Area (212) 294-3000 VWR Scientific P.O. Box 1050 39 Russell Street Rochester, NY 14603 Area (716) 288-5881 VWR Scientific P.O. Box 182 1130 Military Road Buffalo,NY 14240 Area (716) 874-3072

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OKLAHOMA

Curtin Matheson Scientific Company, Inc. Box 747 Tulsa, OK 74101 Area (918) 622-1700 PEN NSY LVAN IA

Fisher Scientific Co. 191 South Gulph Road King of Prussia, PA 19406 Area (215) 265-0300 Fisher Scientific Co. 7 11 Forbes Avenue Pittsburgh, PA 15219 Area (412) 562-8300 TENNESSEE

Preiser Scientific, Inc. CSI Building 2162 Courtland Place Memphis, TN 38104 Area (901) 276-6112

Curtin Matheson Scientific Company, Inc. P.O. Box 1546 4220 Jefferson Avenue Houston, TX 77001 Area (713) 923-1661

Area (713) 523 GAC Laboratories P.O. Box 29641 San Antonio, TX 78229 Area (512) 684-2797 Sargent-Welch Scientific Co. 59 15 Peeler Street Dallas, TX 75325 Area (214) 357-9381 WEST VIRGINIA

Preiser Scientific, Inc. 900 MacCorkle Avenue S.W. Charleston, WV 25322 Area (304) 343-5515 Preiser Scientific, Inc. Jones & Oliver Streets St. Alban, WV 25177 Area (304) 727-2902 WISCONSIN

Curtin Matheson Scientific Company, Inc. Area (414) 793-22 CANADA

Fisher Scientific Co., Limited Box 3840-Station D Edmonton, Alberta T5L 4K1 Area (403) 455-3151 Fisher Scientific Co., Limited 194 West Third Avenue Vancouver 10, B.C. Area (604) 872-7641 Fisher Scientific Co., Limited 184 Railside Road Don Mills, Ontario M3A 1A9 Area (416) 445-2121 Fisher Scientific Co., Limited 1830 Walkley-Station E Ottawa, Ontario K1S 5A9 Area (613) 731-0470 Fisher Scientific Co., Limited 8555 Devonshire Road Montreal 9, Quebec Area (514) 735-2621 Fisher Scientific Co., Limited 21 Gurholt Drive Dartmouth Industrial Park Dartmouth, Nova Scotia Area (902) 469-9891 Sargent-Welch Scientific of Canada, Ltd. 285 Garyray Drive Weston, Ontario Area (416) 741-5210

TEXAS

Beckman Industries, Inc. Science Essentials Operation 5810 Hillcroft Ave. Houston, TX 77036 Area (713) 781-0810 Curtin Matheson Scientific Company, Inc. P.O. Box 5304 1103-07 Slocum Street Dallas, TX 75222 Area (214) 747-2503 ANALYTICAL CHEMISTRY, VOL. 47, NO. 14, DECEMBER 1975

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Figure 5. Map showing possibilities for construction of highway between points A and K

column. These ions are considered to be initially present together on a column (notation ABC//). One can then elute one of the elements. If one first elutes C(AB//C), the next step in the separation scheme must be the separation of A and B, for example, by elution of A. This leads to the situation B//C/A. The last step in this case is the elution of B, leading to //A/B/C. If one knows [for example, from a data library (18)]the time necessary to carry out each step, a graph such as the one in Figure 7 results (19). The shortest path from ABCII to //A/B/C yields the desired optimal separation scheme. Another example of problem solving by dynamic programming is the following: a company disposes of a certain amount of money which can be invested in four projects. The expected profit as a function of the amount invested is known. The question is: what is the combination of investments which yields the highest expected profit? This is a well-known problem to the OR specialist. The same problem arises in analytical chemistry when a program for chemical analysis must be composed so that the amount of material consumed is minimal (and a t most equal to a given quantity) and the total information obtained about a number of compounds or elements present is maximal. If there is a way in which the results obtained by the methods in an analytical program can be evaluated numerically, then the optimal program can be obtained. The difficulty lies in the numerical evaluation (see Conclusions). Location Problem (Heuristic and Branch and Bound Methods)

Figure 6. Graph representing map given in Figure 5, as used in dynamic programming

lustrate the principles of the method, namely, a shortest path and an investment problem. Let us suppose that a highway must be built between towns A and K. There are several possibilities depicted in Figure 5. The construction costs for each section are known, and the least expensive route must be determined. A graph is constructed as in Figure 6. The points are represented on subsequent levels (I, 11, . . . V), and the costs are calculated level by level. The calculations for B-D are trivial. For E there are three possibilities. By way of B the cost will be 10, by way of C, 9, and by way of D, 11. Whatever the path chosen from E to K, the best route through E will be by way of C, and one can conclude that if the optimal solution passes through E, it will pass also through C. We have applied Bellman’s principle 1254 A

of optimality: ACE is an optimal subpolicy. In the same manner, the best subpolicy for all other points on level I1 is calculated. Thereafter, one passes on to the next level. The best substrategy for H is, for example, ADGH (cost: 19). Optimal subpolicies are selected in this manner until arrival at the last level where the selected subpolicy constitutes the complete optimal policy (ADGHK). This calculation procedure eliminates the consideration of other possible combinations such as, for example, the possibility ABEIK. Quite complex multielement separation problems (7-11 compounds or elements on several columns) can be resolved by shortest path calculations. The simplest possible case is the separation of three ions A, B, and C on one


Suppose that the problem is to locate a series of p new supermarkets or other service centers in a country in which there are n villages. Each supermarket must be located in one of the villages, and the problem is to minimise the total distance from the villages to the nearest supermarket. To visualise the problem, consider Figure 8 in which 10 villages are represented by points on a map. In this case, if two supermarkets are wanted, they should be built in A and B. The effect is to split up the set by what is called a two-median, in two halves, so that villages A, C, D, E, F buy in A and G, H, I, J, B in B. At this stage it is not so easy to visually select the optimal location of three supermarkets or, in OR language, to find the three-median. Furthermore, in the economic version of this problem, additional conditions can be introduced such as unequal sizes of the villages and the possibility for the customers to fulfill a fraction of their demand in different service centers. Luckily, we do not

Figure 7. Shortest path for separation of A, B, and C

Figure 8. illustration of location model

need to consider these complications in the analytical version which will be explained later. The problem of finding a p-median can be solved in two different ways: one can find the optimal solution by a branch and bound method or a “good” solution (not necessarily the ideal one) by a heuristic method. The latter is usually employed as a starting solution for the former. A heuristic method can be defined as a technique used to solve programming problems by the search for a feasible solution which is not necessarily the optimal one. The principal advantage of a heuristic method is the speed with which it is possible to find a “good” solution. This makes i t possible to obtain acceptable solutions of much larger problems than those solvable by exact techniques. In the heuristic solution, one starts with a 0-median, i.e., a solution with 0 chosen elements. One element is then

iteratively added to the solution, until p such elements or, in other words, a p-median has been obtained. This is carried out by adding the element which causes the largest decrease in distance (distance is defined here as the sum of the distances from each element to the nearest chosen element). One then investigates whether i t is possible to obtain a better solution by changing one of the elements. If this is possible, the change which causes the largest decrease is carried out and repeated until no further decrease is observed. Branch and bound methods are fairly recent. Proposed by Land and Doig (22) in 1960 for solving the linear integer programming problem, these methods have since found many uses such as in the renowned traveling salesman problem. The basic idea of the branch and bound method is the following. Suppose that the given objective function

is to be minimised and assume that a solution is available (this solution was found by a heuristic method). At first the set of all solutions is partitioned into several subsets (branch). Then for each subset a lower bound (i.e., the lowest value that the objective function could obtain) is computed for the value of the objective function for the solutions of that subset (bound). The subsets, the bounds of which exceed the value of the known solution, are then excluded from further consideration. One of the remaining subsets is then partitioned further into smaller subsets. Lower bounds are computed on the new subsets, and the process is repeated until a subset contains only one solution. If this solution is better than the best one found previously, it replaces it. When all subsets have been excluded, the method terminates. This was applied to a GLC problem. In GLC there is a large variety of stationary phases. To characterise them, one measures the retention indices of a number of standard solutes or functional “probes”. This is, for example, the basis of the Rohrschneider (23) index. A question which recurs regularly in the literature is how much and which probes should be used. Most authors answer this question on the basis of a retention index library, which is thought to be representative for the whole retention index universum, i.e., the retention indices for all compounds on all known GLC phases. Such a library can take the form of a data set containing the retention index of some 60 compounds on 25 GLC phases ( 2 4 ) .Of course, there can be some doubt concerning the validity of considering such a library representative for all possible compounds and GLC phases. However, we need not go into this here, and we can restate the question of the selection of the probes: how can one choose a number of standard solutes so that they are as representative as possible for a given set? An OR approach to this problem is to construct a graph, the nodes of which are the compounds from which the probes must be selected. If the similarities between the solutes are expressed numerically in one way or another (here as 1-p, where p is the correlation coefficient between the retention indices observed for the solutes on a selected set of stationary phases), then these values can be thought of as distances. In this way, one obtains a complete graph with edges the values of which are the distances. This is a situation which can be represented as in Figure 8 (the edges are not drawn for the sake of clarity, but their values are supposed to be known). If A . . . J are the solutes and one would want to represent them by two probes, then


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ferent from all other solutes in the set. The one which resembles it most is dioxane. This result, obtained in an entirely independent way, confirms the judiciousness of Rohrschneider's choice. This correspondence indicates that the location model proposed here gives valid results and that it can be applied with success to the choice of sets of p = 4 or less functional probes and to the selection of representative substances in other branches of analytical chemistry.

one would choose A and B. Mathematically, the problem can be described as: Minimise

C C dijxij 1

J

subject to xij

=1

i

xij

< Yi

Sequencing Problem (Heuristic and Branch and Bound Methods)

Yi40,11 Xij40,11

(i = 1,.. . , n)G = 1,. . . , n ) p = number of probes. dj; = distance between substance j and probe i. Xij = a coefficient that permits distinguishing which probe is representative for substance j . Xij = 1 if j is closest to probe i and is therefore represented by i and = 0 when this is not the case. Yi = a coefficient that permits distinguishing whether a substance was selected as a probe. Yi = 1 when this is the case and 0 when it is not. Because of the complexity of the mathematics, a detailed account of methods and results will be presented in a later article. As a sample result, the p-medians (p = 1, , . . , 6 ) that are obtained with the heuristic method for Rohrschneider's data set (23)are given in Table IV. This discussion will be confined to the p = 5 result. Rohrschneider proposed ethanol, methyl ketone, nitromethane, pyridine, and benzene as functional probes, whereas in Table IV one finds ethanol, propionaldehyde, acetonitrile, dioxane, and thiophene. Ethanol is found both among our probes and Rohrschneider's, and there is very little difference between methyl ethyl ketone and propionaldehyde ( p = 0.9995), acetonitrile and nitromethane ( p = 0.9988), and benzene and thiophene ( p = 0.9989). The difference between dioxane and pyridine is somewhat larger. However, pyridine is dif-

Table IV. Functional Probes Selected from Rohrschneider's Set Using Heuristic Method p = 1 ethylbromide p = 2 dioxane, cyclopentanol p = 3 benzene, crotonaldehyde, cy-

clopentanol

I

p = 4 ethanol, crotonaldehyde, di-

oxane, thiophene

p = 5 ethanol, propionaldehyde, acet on i t r i le, dioxane, thio-

1

pnene

p = 6 benzene, ethanol, phenylacetylene, p r o p i o n a l d e h y d e , ace-

tonitrile, n-dibutylether

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1

In many instances, analytical tests are applied according to some predetermined sequence with the aim of identifying a compound. There are several problems for consideration in the optimisation of such a scheme. First is the minimisation of either the average number of tests or the maximal number of tests in a dichotomous scheme. Katona (25) investigated this problem by using noiseless encoding or combinatorial search theory. These techniques are not generally considered as being part of OR and will therefore not be discussed here. Secondly, imagine a toxicological laboratory whose purpose is to detect and identify poisons in toxicological samples. Only one poison is contained in each sample, and the probability of occurrence pi, i = 1 . . . n is known. The poison is identified by carrying out one by one a number of selective identification methods. Each method identifies only one compound. If the mean execution time ti, i = 1 . . . n for each method is known, determine the sequence of methods that minimises the mathematical expectation of the sum of the times of execution of the methods that have to be carried out. A simple heuristic method was developed which, as was proved subsequently, gives the exact optima1,answer. The method can be described as: Range the methods in the order of decreasing pi coefficients Invert methods i and i 1 if piti+l - pi+lti < 0, and repeat this until no such inversions are possible.

+

Conclusions From the examples given, the conclusion is that OR techniques do have applications in analytical chemistry. These techniques have already been used to solve real analytical problems (the location and distribution problem, for instance). This is only to be expected for a science such as analytical chemistry in which many possibilities must often be compared or combined. At first sight it is rather surprising that more applications have not been proposed in the literature.


Apart from the fact that these techniques are either unknown to analytical chemists or else that they have not realised that such techniques can be applied in this domain, there are a few .reasons why operations research techniques have not been used more frequently. In many of the examples given, it is necessary to evaluate numerically properties of analytical procedures such as the information yield (the investment problem) or the similarity of requisites such as TLC phases (the communication network problem). In some instances such as the investment problem, this is not easy a t all. In fact, the utility of the solution obtained in such a problem can be questionable because of the lack of an accurate scale of values. If the application of OR techniques in analytical chemistry is more closely investigated, one often encounters the difficulty of according meaningful figures of merit to a technique. This conclusion stresses the need felt by many authors [Kaiser (26), Wilson (27) Dupuis and Dijkstra (28), and us (7)] to develop more scientific ways of expressing the performance characteristics of analytical methods. The other proble'm which hampers the application of such techniques is that analytical problems are often too complex to be cut down to the size which can be handled by models such as the graphs which have been used in this report. This is also the main difficulty encountered in more classical, i.e., economical, applications. In the same way that the optimal solution of a personnel allocation problem in industry is likely to be rejected by management for fear of trouble with the trade unions, the analytical chemist will reject an optimal solution which directs him to add strong perchloric acid to a hot ether extract or an optimal set of GLC phases because he wants to work at high temperatures and one of the phases is known to have insufficient thermal stability. In such cases, the solution obtained by OR techniques is used more as a touchstone for evaluating more realistic solutions, i.e., to see how close to optimality these solutions come. Notwithstanding these difficulties, OR techniques could have many more applications in analytical chemistry than a t present, particularly in instances such as the combination, selection, classification, or comparison of spectroscopic, chromatographic, or extraction systems, etc., for which it is relatively easy to obtain meaningful descriptors. OR techniques also have the advantage of working with models that are easily visualised. They should therefore offer an opportunity for formalising our way of thinking about analytical chemistry problems and for

eliminating some of the empirism with which we solve such frequently occurring and important questions as the selection of the best analytical method for a given problem.

(12) S. Wold, Dept. of Statistics, University of Wisconsin, Rept. No. 357, Madison, Wis. (13) D. L. Massart, P. Lenders, and M. Lauwereys, J. Chromatogr. Sei., 12,617 (1974). (14) D. L. Massart and H. De Clerca. Anal. C h e m , 46,1988 (1974). (15) T. L. Isenhour, B. R. Kowalski, and P. C. Jurs, Crit. Reu. Anal. Chem., 4, I (1974). (16) A. Eskes, F. Dupuis, A. Dijkstra, H. De Clercq, and D. L. Massart, Anal. Chem., 47,2168 (1975). (17) B. R. Kowalski, ibid., p 1152A. (18) D. L. Massart, C. Janssens, L. Kaufman, and R. Smits, ibid., 44,2390 (1972). (19) D. L. Massart, C. Janssens, L. Kaufman, and R. Smits, Z. Anal. Chem., 264, 273 (1973). (20) R. Bellman, “Dynamic Programming,” Princeton Univ. Press, Princeton, N.J., 1957. (21) G.S.G. Beveridge, and R. S.Schechter, “Optimisation: Theory and Practice,” McGraw-Hill, New York, N.Y., 1970. (22) A. H. Land and A. G. Doig, Econometrica, 28,497 (1960). (23) L. Rohrschneider, J. Chromatogr., 22, 6 (1966). (24) W. 0. McReynolds, ibid., 12,113 (1974). (25) G.O.H. Katona, in “A Survey of Combinatorial Theory,” J. N. Srivastava et al., Eds., p 285, North-Holland, Amsterdam, The Netherlands, 1973. (26) H. Kaiser, Anal. Chem., 42 (2), 24A (1970). (27) A. L. Wilson, Talanta, 20,725 (1973). (28) F. Dupuis and A. Dijkstra, Anal. Chem., 47,379 (1975). \

Acknowledgment

The authors thank P. Hansen, H. De Clercq, M. Detaevernier, J. Smeyers-Verbeke, E. Blockeel, and other collaborators for permission to use unpublished ideas and results. References

(1) S.N. Deming and S. L. Morgan, Anal. Chem., 45,278A (1973). (2) R. Goulden, Analyst, 99,929 (1974). (3) T. De Vries. Het Klinisch-Chemisch Laboratorium in economisch perspectief, H. E. Stenfert-Kroese, Leiden, The Netherlands, 1974. (4) S. Zionts, “Linear and Integer Programming,” Wiley, New York, N.Y., 1974. (5) J. B. Kruskal, Proc. A m . M a t h . SOC.,7, 48 (1956). (6) H. De Clercq and D. L. Massart, presented at the VIIIth International Symposium on Chromatography and Electrophoresis, Brussels, Belgium, 1975. (7) A. C. Moffat and K. W. Smalldon, J. Chromatogr., 90,9 (1974). ( 8 ) D. L. Massart, ibid., 79,157 (1973). (9) T. J. Allen, Res. Deu. Manage., 1, 14 (1971). (10) P. A. Frost and R. D. Whitley, ibid., p 71. (11) J. Souto and A. Gonzalez de Valesi, J. Chromatogr., 46,274 (1970).

-

,

Financial assistance by FKFO and FGWO

New Ampholytes from Bio-Rad

New Bio-Lytea carrier ampholytes for isoelectric focusing are now available from stock in one wide working pH range (Bio-Lyte 3/10) and in six narrow pH ranges, Bio-Lyte 315, 416, 517, 618, 719 and 8/10, (The product designations are indicative of the working pH range.) Made of polyamino-polysulfonic acid, the Bio-Lytes are ideal for use with a polyacrylamide gel as the stabilizing medium, either by substituting directly for the ampholytes you are now using, or by following the suggested formulations in Bio-Rad’s Bulletin 1030 or in the instructions that accompany each Bio-Lyte shipment. Bulletin 1030 has all the details, including pH profiles, actual separations and complete pricing. It also contains information on the new Gel Pro-pHiler described below.

Gel Pro-pHiler With the new Gel Pro-pHiler, miniature pH electrodes and a pH meter, you can take accurate pH readings every 5 mm along the entire gel length. No more gel slicing and soaking. The Bio-Rad Gel

DBsirB L. Massart (left) was born in Ghent, Belgium, where he obtained his chemistry degree and later his PhD (in 1969) at the local university as a member of the staff of Professor Hoste’s research team on activation analysis. In 1968 he was appointed at the then new Flemish University of Brussels (Vrije Universiteit Brussel). Currently, he teaches general analytical chemistry and food analysis a t the university’s Pharmaceutical Institute and is director of the laboratory of analytical chemistry. Current research topics are applications of atomic absorption spectrometry in the medical sciences and of ion selective electrodes in food, pharmaceutical, and environmental analysis, and fundamental studies on ion exchange and ion selective electrodes. He is author and coauthor of over 60 scientific papers. Dr. Massart’s personal research ambition is to develop objective (Le., mathematical) methods for the characterisation, selection, or optimisation of analytical methods. Leonard Kaufman obtained a degree in mathematics at Brussels University in 1970 and a PhD in operations research in 1975. At present he is an assistant in the Department of Statistics and Operations Research of Brussels University and teaches a course in operations research at the Polytechnicum of Lille, France. His research interests include location problems in 0-1 programming and the application of optimisation techniques to analytical chemistry. In these fields he is author or coauthor of 10 scientific papers.

holds a cylindrical gel in position so you can measure the pH profile of a gel as soon as it is removed from its tube. When you are finished, the gel emerges virtually undamaged and ready for staining. If you are using isoelectric focusing, or if you suspect you should be, then write for Bulletin 1030. You’ll find everything you need for this proven method of separating proteins.

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(7, NO. 14, DECEMBER 1975

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Operations research in analytical chemistry - American Chemical Society

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