College and Research Libraries S. K. GOYAL Allocation of Library Funds To Different Departments of a University- An Operational Research · Approach Library governing boards are faced with many administrative ques- tions affecting services offered. Rather than using explicit analysis of such questions, answers are implicitly based on a reluctance to change past practices. This paper describes the operational research approach to the problem of allocation of funds to different departments of a university for purchase of books and journals. A linear programming model is suggested for solving the allocation problem. INTRODUCfiON uNIVERSITY LIBRARY FUN~S available for purchase of books and periodicals are commonly much less than the de- mand for such funds by different de- partments. Each department vigorously asserts its claim for funds to. the li- brary. A library administrator confront- ed with this problem may not realize that the techniques of operational re- search may be useful in developing a solution. Some questions raised include: What fraction of the yearly budget should be allocated to. the purchase of books? To periodicals? How should this be allocated among various disciplines covered by the collection? Which books (or periodicals) within a discipline should be purchased? How and when does one decide to buy a duplicate? How Mr. Goyal is a lecturer in Operational Re- search at Glamorgan Polytechnic, South Wales, United Kingdom. can one evaluate alternative decision procedures? This paper discusses only an opera- tional research approach to the alloca- tion of funds to different departments for purchases of books and periodicals. We assume first that the funds available ( M) are known in advance. MATHEMATICAL TECHNIQUE USED FOR REsoURCE ALLOCATION A linear programming model will be used to find the optimum allocation of funds. AssuMPTIONs oF LINEAR PROGRAMMING RELEVANT TO THE MonEL (a) Non-negativity: Funds allocated must occur at a positive level or not at all. Negative levels are not included. (b) Linear objective function: The importance "Ci" of department ci' is a measure of the social benefits per unit of amount spent on purchase of books I 219 220 I College & Research Libraries • May 1973 and periodicals in that department. The objective function follows the rule of proportionality as regards funds allocat- ed within one department, as well as the role of ~dditivity in determining the to- tal social benefits_ .. due to the allocated funds. The importance of the department ( Ci) is expressed as follows: The general formulation is: Maximize · n Z = ~ Ci ·Xi, i = 1 where Xi > Li, Xi< Ui, n ~Xi 0 (1) (2) (3) (4) (5) C. · f(s· T. o·) (si+Ti) o· h 1 = 1, 1, 1 = - 2 - x 1, w ere GROUPED CoNSTRAINTS Si =importance which society attaches In actual resource allocation problems to . the work of the department... there may be other constraints, termed Ti = importance which the university grouped constraints, e.g., the total funds gives to the work of the depart- allocated to departments 3 and 4 should ment. · -. not exceed U 34, and tota'I of funds al- : .. Oi = importance due to the size of the · located to departments\ 5, 6, and 7 department. This depends . on 'should not be less than ·L567· The ad- , · the number of students (under- O.itional constraints will he described as: graduate I postgraduate I research) and the number of parttimelfull- · X3 + X4 < Ua4, and time students. X5 + X6 + X7 > L567· Any mathematical expression relating An original formulation, after de- Ci with Si, Ti and Oi can be considered. partment discussion, might require re- thinking and modification : of many con-.This expression was chosen because it eliminates those variables for which Oi straints. In such cases it is essential that is zero. On the other hand, it does not the problem be formulated correctly af- eliminate those cases for which either ter consultations with the departments ·cSi' and/ or cTi' is zero. and the library administrators. The op- erational research ( O.R.) approach to FORMULATION OF FUND ALLOCATION PRoBLEM: Let: · ( 1) the number of departments (or subjects), for which the funds are to be allocated for pur- chase of books and periodicals, be en'; ( 2) the total funds available for purchase of books be . eM'; ( 3) the funds allocated for de- partment ci' be CXi' where i = 1, 2, 3 ... , n; ( 4) the importance of depart- t ,., b cc·' ( · 1 2 3 men 1 . e 1 1 = , , , . . . , n); ( 5) the lower limit and the up- .per · limit of funds as furnished by department Ci' be eLi' and CUi'; this problem will help to standardize de- partmental formulation of book and periodical requirements . .. SoLUTION TEcHNIQUE When a number of grouped con- straints are present, .then the problem can become complex; depending on the number of variables and constraints, a computer may be required to solve the problem. The most commonly employed and ma.st efficient method for solving this problem is the Simplex Method. For a description of this method see any text .on O.R. techniques. In order to use the Simplex Method, the inequality con- straints must be converted into equali- ties. This is done by introducing slack ' variables for all inequalities having ( <) signs. In cases where inequalities are of the type ( > ) then we introduce slack and artificial variables. The introduc- tion of artificial variables complicates and lengthens the computation work. LOGICAL STEPS FOR CARRYING OuT THE OPERATIONAL RESEARCH PROJECT 1. D esign a form for all departments to complete when submitting their esti- mates for purchase of books and jour- nals. It should be divided into three categories. (a) Books and journals that the de- partment needs to function effec- . tively (this is the lower limit of funds for the department) . (b) Books and journals that are de- sirable but not absolutely essential. (c) Books and journals that the de- partment would like to have if funds are available. The department should specify the number of copies required for each book. There may be cases when a single copy of a particular book comes under category (a) but additional copies may come under category (b) or even under category (c). The total of (a) and (b) will determine the upper limit of funds required for a department. In some cases category (c) books will also be added for obtaining "Ui." 2. Check that the summation of eLi' for all departments does not exceed the available funds or that the summation of 'Ui' for all departments is less than the funds available. In the former case, if minimum funds required exceed funds available, then there is no other alternative (if M cannot be increased), but to ask the departments to cut down their estimates and allocate the mini- mum possible. In the latter case, the maximum funds required by all the de- partments can be allocated. 3. Determine the importance 'Ci' of Allocation of Library Funds I 221 each department. The 'following factors should influence Ci: (a) Social importance of the subjects taught within a department; (b) Departmental support of other departments; (c) Importance which the university attaches to the work · of the depart- ment; (d) Number of students ( fulltime/ parttime) in the department (under-:- graduate, postgraduate, research stu- dents); (e) Number of staff in the depart- ment, and the number of staff active- ly engaged in research and consultant work. For determination of Ci see Appen- dix . 4. If. there are grouped constraints in the problem then apply the Simplex Method for determining optimum allo- cation of funds. Otherwise use the fol- lowing method: First allocate the lower limit of re- quired funds to all the departments. Remainder of the funds ( M -1 ~1Li) will be allocated by selecting that de- partment with the highest Ci and al- locating the maximum permissible funds. (Difference between the upper limit and lower limit for this depart- ment.) Pick up the next department from the remaining departments having highest Ci and allocate the maximum permis- sible funds to this department. Repeat this procedure until the funds are ex- hausted. CoNCLUDING REMARKS Allocation problems can be answered by applying operational research tech- niques. For most libraries, a significant amount of funds are not being used op- timally; many journals and books sel- dom have been read or shortage of copies exist for those books in great de- mand. 222 I College & Research Librar·ies • May 1973 APPENDIX Determination of Ci: This aspect of the project, always subject to criticism, is dif- ficult, if not impossible, to determine. How- ever, the problem of determining Ci is greatly simplified, if we assume that the so- ciety and the university attach equal impor- tance to the work of each department. Even if they are biased in favor of certain depart- ments, then it will be reflected in the size and growth of such departments. So deter- mination of Ci is synonomous to determin- ing Oi-the importance due to the size of a department. Determination of Oi: For each depart- ment compile the information in the given tabular form; the numbers are easy to ob- tain. To determine the weights for each category, a survey can be conducted. The weights can be assumed as constant. The sum of all the total weights for a depart- ment is the value of Oi importance due to size for that department. TABLE 1 FoR DEPARTMENT 'r' Category Number Weight Total Attached Weight Fulltime Students Undergraduate Ri Ui Ri.Ui Postgraduate Pi Vi Pi. Vi Research Student Di Wi Di.Wi Parttime Students Undergraduate ri ui ri.ui Postgraduate pi vi pi. vi Research di wi d i.wi Staff Member Engaged in teaching ei ki ei.ki Registered for higher studies fi li fi.li Engaged in approved consultant work gi mi gi.mi Total= Oi REFERENCES 1. P. M. Morse, Library Effectiveness: A Sys- tems Approach (Cambridge, Mass.: M.I.T. Press, 1968). 2. D. J. White, W. A. Donaldson, and N. L. Lawrie, Operatioool Research Techniques (Business Books Limited, 1969). 3. W. W. Garvin, Introduction to Linear Pro- gramming (New York: McGraw-Hill, 1960). 4. C. W. Churchman, R. L. Ackoff, and E . L. Arnoff, Introduction to Operational Re- search (New York: Wiley, 1966). 5. S. K. Goyal, "A Systematic Method for Re- ducing Over-ordering of Books," Library Re- sources and Technical Services 16:26- 32 (Winter 1972) .