Ratios in Planning-Budgeting and Bounds on Resource Requirements

Ratios in Planning-Budgeting and Bounds on Resource RequirementsAuthor(s): Jonathan HalpernSource: Operations Research, Vol. 20, No. 5 (Sep. - Oct., 1972), pp. 974-983Published by: INFORMSStable URL: http://www.jstor.org/stable/169160 .

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Ratios in Planning-Budgeting and Bounds on

Resource Requirements

Jonathan Halpern

Technion-Israel Institute of Technology, Haifa, Israel

(Received April 14, 1971)

This paper formulates and solves a budgeting problem that originates from the use of ratios of two variables as performance criteria for an organization. There are two main features to the problem: the presence of two conflicting groups or subsystems that disagree on a desirable trend for the ratio over a planning horizon, and a lag factor, which means that if one group is in a relatively unfavorable position in a given period, then an extra weight is given to its demands, with respect to changes in the ratio, in the next period. The lag factor takes the form of constraints in the model. The two conflicting objectives in the system are translated into minimization and maximization of a real- valued function. The solution produces lower and upper bonds for the value of a decision variable; an actual budget allocation has to be within these bounds. An application of the model to the use of student-to-library-seat ratios is presented.

IN MANY nonprofit organizations, it is impossible to measure the development of the organization with the criteria of the commercial world, such as total costs,

revenues, or profit. In these institutes administrators, public authorities, and others try to measure their achievements by various other performance criteria; one frequently employed is the ratio of two variables used in the organization. In institutions of higher education we find ratios such as students to teachers, and costs per hour of students' instruction or per graduated student; in hospitals, for example, we hear about averages like patients or number of beds per nurse or per doctor-there is a large number of situations in which such ratios are used. When time is also considered, we find that in many cases the accomplishments of the institution are evaluated by trends or rates of change in these ratios. The use of such measures in long- or medium-range planning brings about interesting budgeting problems. One of these is formulated and solved in this paper.

The existence of this special type of problem is mainly due to two common features that are found in many organizations. The first is the frequent presence of two forces or groups that disagree in their interpretation of what is the desirable trend or rate of change of a given ratio. For example, faculty members in a university would like to see a decreasing student-to-faculty ratio over time, while state officials, in the case of a university financed by the state, would prefer to increase the ratio. The teachers argue that a lower ratio increases the quality of education and leaves more time for research. State officials, who are responsible for the state's budget, look for a higher ratio as an indication of more 'productive' teachers and as a way to save scarce resources.

The second feature that gives our problem its special structure is the lag factor. By this we mean that, if one of the two conflicting groups finds itself during a given

974



Ratios in Planning-Budgeting 975

period, in an unfavorable situation, it provides an extra weight for the demands of this group in the next time period. To use again the example of the student-to- faculty ratio, if, in period t, this ratio is higher than some acceptable or average standard, then the pressure from the faculty may be strong enough to prevent further increase in the ratio, and possibly even to reduce it during the next period, t+ 1. The situation is reversed if initially the ratio is far below the one that is considered acceptable to the state authorities.

In this paper we consider a finite planning horizon for which the initial ratio of the two variables and the forecasts of the values of one of the variables are assumed to be given. We express mathematically, in the form of constraints, the conflicting forces that try to change the ratio in different directions over time, and the depend- ency of the set of feasible ratios in one period on the actual ratio in the previous period. This collection of constraints defines an infinite set of feasible sequences of ratios over the planning horizon. Since the values of one of the variables that form the ratio are assumed to be known (or at least estimated) for the entire planning horizon, the planning effort is directed to the values of the second variable, or equivalently, to the values of the ratio.

We present the problem with reference to a specific example where the model was used: the number of students per library seat, in a given university, where the forecasts of student enrollment are given. (See HALPERN11] for another application of the model.) We assume that, over the planning horizon, the university's budget office prefers to cut down as far as possible its allocation for new seats in the libraries; this influence tends to increase the ratio over time. Librarians and others in the university try to secure budgets for as many new seats as possible to decrease the student-to-seat ratio. We translate this over-simplified assumption about the conflicting preference orderings of the budget office and the librarians to an objective function that is to be minimized from the budget-office point of view and maxi- mized when the library's preference is followed. The solution to these two problems of minimization and maximization will provide us with lower and upper bounds, respectively, for the necessary number of new library seats over the planning horizon. It should be emphasized that any of these bounds cannot be viewed as an optimal policy for the library, even if only one point of view, that of the budget office or of the library, is adopted. These bounds may have features that, for other reasons not considered here, are not acceptable. The actual allocation of resources for new seats will probably be chosen within the range determined by these bounds. How this choice is made is beyond the scope of this paper.

The problem is solved in two steps. First, we construct a finite subset of the infinite set of feasible solutions and prove that the optimal solution is a member of this finite subset. Second, we show that this finite set of feasible solutions may be presented as a network on which the length of a path from source to sink equals the value of our objective function. The minimization (maximization) problem is then solved by using a shortest (longest) path-labeling algorithm.

FORMULATION OF THE PROBLEM

THE PLANNING HORIZON is divided into periods by a set of points t =0, 1, * , T. Period t is the interval (t-1, t]. We assume that the numbers of students and



976 Jonathan Halpern

seats are changed at the beginning of the period and fixed for the rest of it. For period t, let n (t) = number of students, x(t) = number of seats, and r (t) =n (t)/x (t) = student-to-seat ratio. We assume that the number of students is nondecreasing over time, i.e., n (t) _n (t- 1). A similar restriction is imposed on the number of library seats, namely, y (t) = x (t) - x (t-1 ) _ 0 for all t = 1, *, T. We also assume that there is a sequence s (0), *, s(T) of 'standard' ratios that is acceptable, as a general guideline or trend, for both the budget office and the library. This sequence is assumed to be either nondecreasing or nonincreasing.

Given the ratio r (t) for period t, the ratio in the next period, r (t+ 1 ), is restricted to be within bounds that are a function of r (t) and a few parameters such as s (t+ 1 ) and others. Thus, we have the constraint L[r(t)] <r(t+1) ? U[r (t)], where L[r (t)] and U[r (t)] are the lower and upper bounds, respectively, on r (t+ 1), for a given r (t). The specific form of these bounds, for which the problem is solved, is presented later. We assume that, if for some period the actual ratio equals the standard ratio s (t), then s (t+ 1) is within the bounds imposed on r (t+ 1), namely, L[s(t)]?s(t+l) ? U[s(t)].

We are interested in multiperiod planning, and therefore have to compare the value of a new seat in various time periods. This is done by introducing a discount factor a, where 0 < ?f 1. From the budget-office perspective, a new seat provided this year is costlier than a seat approved for next year. From the library's point of view, a new seat available this year is better than one promised for the future. We therefore consider the present value of the total number of new library seats available in the entire planning horizon. Analytically, we write this Et a t- (t). The lower and upper bounds for the discounted total increase in seats are found by minimizing and maximizing, respectively, this sum. The formulation of the problem is, therefore,

minimize or maximize E at y (t) subject to

L[r(t)]<r(t+1)? ~U[r(t)] and y(t)>O

for all t = O.*, T with x (O) and n (t) tO *, T known.

The solution of the problem depends on the type of bounds used; the specific form we use is presented below. It should be emphasized, however, that the class of bounds for which our method of solution is valid includes various types of bounds, linear and nonlinear, all of which have the following property. Given r (t), then the highest and lowest feasible ratios in period t+ 1, from which a desirable ratio r(t+2) in period t+2 may be reached, are members of a finite set of numbers. The values of the members of this set are determined by r(t), s(t), s(t+l), and other parameters, but are independent of r (t+2).

We describe now the specific nonlinear form of the bounds that we use in this paper. If r (t) is below s (t), then r (t+ 1) may be increased, but with some limit that depends directly on the difference s(t+l)-r(t). If r(t) is above s(t) but not too far, say r (t) < s (t) +a, where a> 0 then the ratio is not allowed to be increased unless s (t+ 1) itself is higher than r (t). If r (t) is relatively high, namely, r (t) > s (t) +a, then some reduction of the ratio is required unless the guiding ratio s(t+1) for period t+1 is higher than r(t).

To summarize, we assume:




fr(t)+a+b[s(t+1)-r(t)I, if r(t)<s(t), U[r(t)]=lmax[r(t), s(t+1)], if s(t)<r(t)?s(t)+a,

(max{r(t)-H[r(t)], s(t+1)}, if s(t)+a<r(t),

where H[r(t)J=c[r(t)-a-s(t+1)] and O<c<b<1. In short, the better the library's situation, the fewer resources it will receive to improve or keep unchanged its level of services, as measured by the student-to-seat ratio.

Similar considerations lead to the construction of the lower bounds. If the student-to-seat ratio is relatively low in a given period, the pressure of the budget office will prevent further reduction, or even force an increase, of the ratio. If the ratio is high, then some reduction may be permitted, provided it is a modest one. Thus:

fr(t)-a-b[r(t)-s(t+1)], if s(t)_r(t), L[r(t)]= min[r(t), s(t+1)], if s(t)-a< r(t)<s(t),

(mintr(t)+K[r(t)], s(t+1)}, if r(t)<s(t)-a.

Note, however, that the lower bound r(t)+K[r(t)], imposed in period t+1 when r(t)<s(t)-a, cannot be higher than q(t+ 1)=n(t+1)/x(t), namely, the ratio that will prevail in period t+ 1 if we do not add new seats. The budget office will require the increase of the ratio K[r(t)] to be at least c[s(t+1)-r(t)-a] in order to bring it closer to the standard ratios. Combining these two requirements results in K[r(t)] =mintc[s(t+l)-r(t)-a], q(t+1)-r(t) }.

Note that the increase in the number of seats in period t is

y (t) = n(t)/r (t) -n (t- 1)/r (t- 1).

Let R = [r (0), **, r.(T)] be a sequence of ratios and let (R be the set of all feasible sequences R. Denote by f(R) the value of the objective function associated with the feasible solution R:

t R) = t1 0/ '[n (t)/r (t) -73 (t- 1)/r (t- 1)] = . a- t-y(t

The problem can be stated as

minimize or maximize f(R) subject to ReGR,

where R = [r(O), *,(T)],ef if and only if [n (t)/r (t)-n (t-1 )/r (t-1 )]0 and Lr (t)] _ r (t+1 ) _ U[r (t)] for all t.

SOLUTION OF THE PROBLEM

FIRST, WE PROVE this result: LEMMA. Let R1 = [r1 (0), * * , ri (T)] and R2 = [r2 (O), * r2(T)] be two members of (R such that ri(t)=r2(t) for all tsk and r1(k)>r2(k) for some O<k<T. Then f(Ii) <f(R2), with strict inequality if 0 <a <1.

To prove the Lemma, it is sufficient to note that

f (R2) -f (R1) = (1-a)a k1ln (k)[1/r2 (k) - 1/r, (k)].

Note that r(0)=n(O)/x(0) is given and is therefore the same for all REGI. By a repeated use of the lemma, it is simple to prove this corollary:




COROLLARY 1. Let Rl and R2 be members of G& such that ri (t) _ r2 (t) for all t; then f (RI) ?f (R2)-

Note that the validity of the Lemma and Corollary 1 is independent of the existence or properties of the bounds L[r (t)] and U[r (t)].

Corollary 1 and the assumption that L[s (t)] < s (t+ 1) < U[s (t)] for all t imply this result: COROLLARY 2. If for the maximization [minimization] problem there is some k for which r(k)<s(k)[r(k)_s(k)], then the optimal ratio r*(t) in any period t=k+1,

, T satisfies r* (t)?s(t)[r*(t)n mintq(t), s(t) }]. Consider the maximization problem and assume that, contrary to the corollary

r* (t) > s (t), it follows that it is feasible to replace any set of consecutive ratios that are above s(t) by the standard ratios s(t) themselves. By Corollary 1 this new sequence of ratios will correspond to a higher value of the objective function. The proof for the minimization problem is similar.

The Lemma is used to show that, when r (t) is given, there are at most two identi- fiable feasible values for r (t+ 1), one of which is optimal. Starting from the given r (0), we can therefore construct a finite set of feasible sequences that contains an optimal sequence. We denote by U[r(t)] the highest feasible ratio at period t+1, given the ratio r(t) at period t. Thus U[r(t)]=minmq(t+1), U[r(t)]I. Note that the lower bound L[r (t)] for the ratio in period t+ 1 is the lowest feasible ratio in period t+ 1, given the ratio r (t) in period t. We state now the main result: THEOREM. Given the ratio r (t), then the optimal r (t-+ 1) for the minimization [maximization] problem is either U[r (t)] {L[r (t)] I or, if feasible, s (t+ 1).

A complete proof of the theorem requires a detailed examination of every possible form that U[r(t)] and L[r(t)] may assume, which depends on the difference r(t)-s(t). Furthermore, the cases of nonincreasing and nondecreasing s(t), and the minimization and maximization problems, all have to be studied. The proof is almost the same in all possible cases, except for some variations in the way the Lemma is used. We therefore present, as examples, the detailed proofs for only two cases.

First example. Consider the minimization problem where s (t) are nonincreasing over the entire planning horizon. Assume that r (t) = r * (t) is given for some period O 5 t < T, and that s (t) <r* (t) ? s (t) +a.

If t= T-1, then it is clear that the ratio in period T that will minimize the increase y (T) is the highest feasible ratio, namely, U[r* (t)] =r* (T- 1).

Consider the case O<t< T-1. Let r*(t+2) be the unknown optimal ratio-for period t+2, given r*(t).

We are interested in the ratios of period t+1 that are attainable from r*(t) and from which r*(t+2) is reachable. Consider the following division of these ratios: those that satisfy r(t+1) ?s(t+1), and those that satisfy r(t+1) ?s(t+1). The ratios of the first group, together with r*(t) and r*(t+2), form the set of feasible sequences represented by the shaded area K1 in Fig. 1. Similarly, the ratios of the second group form the shaded area K2. Thus, in Fig. 1 we may reach our target in period t+2 by choosing a ratio, in period t+ 1, that satisfies r* (t+2) ?

r(t+ 1) ? r*(t). This set of choices corresponds to K1. We may also choose to decrease r(t+l) to s(t+1) or lower in order to be allowed to increase it later to r* (t+2). This set of possibilities is covered by K2. Let K =K1UK2 be the set of all such feasible sequences from r*(t) to r*(t+2). Note that K is a nonempty set,




since r*(t+2) is attainable from r*(t), but it is not necessary that both K1 and K2 be nonempty. If K1 is nonempty, then, by the Lemma, the optimal ratio r*(t+1) for period t+ 1 equals r* (t). If, however, K1 is empty [for example, when r* (t+2)> r*(t)], then K2 is nonempty and r*(t+1) equals, by the Lemma, s(t+1). It follows that optimal r(t+1) equals either r*(t) = U[r*(t)] or s(t+1).

Second example. Consider the maximization problem, assuming that s(t) t=, , T is a nondecreasing sequence and r (t) = s (t) is given for some period 0< t < T. If t = T- 1, then the ratio in period T that will maximize the increase y (T) in the number of seats is the lowest feasible ratio, namely L[s (T- 1)1.

r* (t+2)

s (t+2)

t t+l t+2 Fig. 1. Feasible sequences from a given s(t) <r*(t) < s(t) +a to an arbitrary but

given r*(t+2).

Assume now t < T- 1. Let r* (t+2) be the optimal ratio for period t+2, given r (t) = s (t). Figure 2 shows an example for this case, where the shaded areas repre- sent the sequences of feasible ratios s(t), r(t+1), r*(t+1). The set K=KUK2 is the set of all feasible sequences and is nonempty, because we assume that r * (t + 2) is attainable from s (t). By the Lemma we know that if K2 is nonempty then optimal r(t+1) equals L[s(t)]. If K2 is empty, then K1 is not and optimal r(t+1) equals s (t+ 1). Thus the theorem is proved for this case.

The theorem enables us to construct a finite set of feasible sequences of ratios Gt, such that 61, is a subset of GR and contains an optimal sequence; that is, there exists an R *E(RC such that f (R *) = minRe, f (R) = minRER, f (R).

The construction of (R is simple. For example, assume that we solve the mini-




mization problem. Given r (0) =n (O)/x (0), then, by the Theorem, either r (1)= U[r (O)] or, if feasible, r2(1) =s(1) is optimal ratio for period t = 1. If r (1) = ri (1), then, by the theorem, either ri(2) =U[ri(I)] or r2(2) =s(2) is optimal for t=2, and, if r(1) =r2(1), then either r3(2) = U[r2(1) or r (2) =s(2) is optimal for t=2. We continue until we reach the last period T. An example of a four-peliod problem is shown in Fig. 3. An arrow leading from ri(t) to rj(t+1) means that the ratio rj(t+1) is possibly the optimal one for period t+1, given the ratio in period t is ri (t). Any path or a sequence of ratios that follows the arrows from the unique

s (t+2) U [s(I

r*(t+2)

s (0)K

t tI t+2 Fig. 2. Feasible sequences from a given r(t) =s(t) to a given r*(t+2).

r(0) at t=0 to a ratio r, (T) at period T is a member of the set 61, Without a rigorous presentation of its topology, we refer to the set of sequences of ratios (Ri as a network, and to an arrowed line leading from r (t) to any of the r (t+ 1)'s gen- erated by r (t) as an arc. The length of such an arc in the network is equal to aty(t+l)= at[n(t+l)/r(t+l)-n(t)/r(t)]; i.e., the contribution of this period to the length of the path from the source r (0) to one of the sinks rj (T). It follows that the length of a path in the network from source to sink, which corresponds to a sequence of ratios REcR,, equals the value of the objective function f (R) for this solution. The sequence of ratios R* that minimizes the value of the objective function is therefore the one that corresponds to the shortest path from the source




r4(3)~~~

~~~~~~~~~r 3

t=O I 2 3 4 Fig. 3. An example of the set LR, with T =4.

to the set of sinks on the network. The length of this path equals the optimal value of the objective function. A shortest-path labeling algorithm is used to identify the optimal solution.

The maximization problem is solved similarly. Using the Theorem, one con- structs a network on which the longest path from the source to the set of sinks corresponds to the optimal sequence of ratios, and its length equals the maximal value of the objective function. Note that, if both the minimization and maximization problems are to be solved, we construct a joint network on which the shortest and longest paths are optimal solutions, respectively.

A NUMERICAL EXAMPLE

A SCHOOL HAS, in a given year, t = 0, x (0) = 544 library seats for its n (0) = 7560 students. The standards of student-per-seat ratios for the initial year and five

TABLE I STANDARD STUDENT-TO-SEAT RATIOS AND ENROLLMENT FORECASTS FOR YEARS

t =-0, -. }5

1 0 1 2 3 4 5

n t 7560 8100 8800 9700 10700 11700 s(t) 14.3 13.3 12.6 12.2 12.0 12.0




13

t=O 1 2 3 4 5 Fig. 4. Network for the maximization problem of the numerical example.

years of the planning horizon, with the forecasts of student enrollment are presented in Table I. The discount rate is assumed to be a =0.9. The parameters that determine the tightness of the restrictions on the variations of the ratios are a= 0.8, b = 0.3, c = 0.2. These values are acceptable compromises to both librarians and the budget office.

Consider the maximization problem. Given r (0) = 13.9, we know that either L[r(0)] or, if feasible, s(l) is the optimal ratio for period t=1. From the defini- tion of L[r(0)] and s(0) -a<r(0) <s(0) it follows that L[r(0)] =min[r (0), s(1)] = 13.3 = s(1). Hence, there is a unique candidate for optimality in period t=1, which is s(1) = 13.3. The increase in seats associated with moving from r (0) to s (1) equals y (1) = 65.0 (see Fig. 4). Assuming now r (1) = s (1), then the optimal ratio for t=2 is either L[s(1)]=s(1)-a-b[s(1)-s(2)]=1229 or s(2)= 12.60, which is feasible. The corresponding increments in the number of seats are 107.0 and 89.4, respectively, and their discounted values, i.e., multiplied by a =0.9, are the lengths of the arcs leading from s(l) to L[s(l)] and to s(2). We continue to build the network by a repeated use of the Theorem. The network and the longest path from the source to the set of sinks are shown in Fig. 4. The sequence of ratios that is associated with the longest path is the solution to the maximization problem.

Using the theorem for the minimization problem, we construct another network on which the shortest path from the source to the set of sinks corresponds to the sequence of ratios that gives the solution to the minimization problem.

TABLE II LOWER AND UPPER BOUNDS FOR THE INCREASE IN THE NUMBER OF LIBRARY SEATS

Lower bound Upper bound

y(t) r(t) y(t) r(t)

1 13.8 14.52 65.0 13.30 2 57.6 14.30 89.4 12.60 3 75.5 14.04 132.1 11.68 4 85.0 13.79 61.2 12.00 5 85.0 13.59 152.9 11.20

Discounted sum 244.6 397.4




The solutions to the minimization and maximization problems are the lower and upper bounds, respectively, on the discounted number of new library seats over the entire planning horizon. The annual increments in library seats that give the lower and upper bounds, their discounted sum, and the ratios associated with them are presented in Table II.

ACKNOWLEDGMENT

THE INITIAL version of the problem was formulated and solved while the author was with the Ford Foundation Research Program in the office of the Vice President at the University of California, Berkeley. The author wishes to thank ROBERT

M. OLIVER for valuable comments.

REFERENCE

1. JONATHAN HALPERN, "Bounds for New Faculty Positions in a Budget Plan," Paper P-10, Research Projects in University Administration, Office of the Vice President, Univer- sity of California, Berkeley, California.



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Ratios in Planning-Budgeting and Bounds on Resource Requirements