Resource allocation in organizations: an optimality result

Transcription

1 Resource allocation in organizations: an optimality result Arun Sundararajan New York University, Stern School of Business 44 West 4th Street, K-MEC 9-79 New York, NY I thank Abraham Seidmann for his guidance and suggestions, to Marshall Freimer for comments that led to more succinctproofs,andtoharrygroeneveltforvaluablediscussiononbottlenecksandtheconvexityofsojourntimes in queuing systems. 1

2 Abstract This paper studies the problem of constrained resource allocation across multiple tasks, stages or divisions in an organization. The allocations suggested by balancing marginal resource productivity, and by balancing processing speed (or eliminating bottlemecks), are compared to the optimal organizational allocation. It is shown, under fairly general assumptions, that the former two approaches never result in an allocation that is optimal, and that the only exception to this result is when the different stages are identical. The implications and sample applications of the problem formulation and result are discussed. 2

3 1 Introduction The allocation of a single resource is analyzed in a general, multi-stage system, for which the efficiency of the different stages may not be identical,or equivalently,the amount of work at each stage may be different. The objective is to optimize the additive non-linear objective function, subject to a single linear constraint corresponding to the constrained resource. This problem is similar to those studied by Luss and Gupta (1975) and Zipkin (1980). Apart from its obvious relevance to capacity planning in manufacturing and computer networks, the formulation of the problem has a wide range of applicability, in search theory (Charnes and Cooper, 1958, Koopman, 1958), process design (Chand et al., 1996, Seidmann and Sundararajan, 1997) and finance (Elton, Gruber and Padberg, 1976). A characterization of the optimal solution is developed, and the resulting optimal allocation is compared to those obtained from two well-known rules of thumb for resource allocation. The first rule is to choose an allocation that balances the marginal product of the resource at each stage. This greedy approach, which is intuitively consistent with price-driven resource allocation, is exactly equivalent to maximizing the total (or average) processing speed of the entire system, or equivalently, the total resource productivity. The second rule of thumb is to allocate the resource in a manner that balances the productivity of each stage, and eliminates bottlenecks. This intuitively fair approach chooses an allocation such that the net processing speed at each stage is identical, thereby equalizing the expected queue lengths and sojourn times at each stage. It seems likely that one or both of these heuristics should yield an optimal allocation, depending on the objective function of the planner. Before describing the analysis in detail, a brief overview of the model and salient results is provided. A system with n stages is modeled. Each stage has an identical and general arrival process. Each stage has a general processing time distribution. The processing rate at each stage depends on the technology of the system (which is identical for all stages), the efficiency of the stage, and the amount of a single resource allocated to each stage. Hence, depending on the productivity 3

4 and resources at each stage, different stages may have different processing rates (and consequently, different expected sojourn times). There is a limited amount m of the resource available, and the problem is to allocate the resource in a manner that minimizes the total (or average) cycle time of the system, which is the sum of the expected sojourn times at each stage. This kind of problem is commonly faced by business process managers, who have the task of frequently reallocating scarce labor or information technology resources between the different agents or teams that comprise their processes, and whose performance is evaluated based on cycletime-related measures such as time to market, market to completion time, average customer service time and so on. In addition, the model is immediately generalizable to any situation in which the objective function of each stage is decreasing an convex in it s inputs, and the value of these inputs in turn are increasing and concave in the amount of resource allocated to them. The key result of the paper is that the optimal allocation of resources almost never coincides with either the allocation which eliminates bottlenecks, or the allocation which balances the marginal product of the resource at each stage. In fact, there is only one situation in which either of them coincides with the optimal allocation when the productivity at every stage is identical and in this single case, both rules of thumb yield identical allocations which coincide with this optimal allocation. The result suggest that eliminating bottlenecks or maximizing total resource or factor productivity methods that are widely advocated are pretty much never optimal solutions to the resource allocation problem, and that careful attention must be paid to the exact effects of queuing and individual efficiency. The rest of the paper is organized as follows. Section II describes the model and derives the main results. Section III discusses the results, illustrates them graphically, and describes future research. 4

5 2 Model Consider an n-stage sequential system, with an identical, general arrival process at each stage. The system has available a total amount m of a single resource, which must be allocated between the n stages. In general, x =(x 1,x 2,..., x n ) is called an allocation. In the remainder of the paper, an upper case subscript implies a specific vector, and a lower case subscript implies the component of a vector. For instance, y A is a vector, y i is the i th component of vector y, and y Ai is the i th component of vector y A. Theinter-arrivaltime ateachstagehasageneraldistributionwithmean1/λ. The processing rate at each stage is α i µ(x i ), where α i is the efficiency of the agent/processor at stage i, andx i is the amount of the resource allocated to stage i. µ(.) is the technology of the system, and has the following properties: µ(0) = 0, and for all w>0, µ(w) > 0,µ 0 (w) > 0andµ 00 (w) < 0. In other words, the technology of the system is such that the processing rate of any single agent or processor is strictly increasing and strictly concave in the quantity of the resource allocated. This is a fairly standard assumption, and is motivated by observed diminishing returns to any single input factor. The actual processing time is distributed according to some (general) distribution g(. µ), which is of the same form at each stage. Each stage of the system is therefore a FCFS G/G/1 queue. The expected sojourn time corresponding to these arrival and processing time distributions, as a function of the arrival and processing rates is denoted τ(µ, λ). Since λ is identical at every stage, we drop the second variable, and denote the sojourn time as τ(µ). It is known that the sojourn time of an FCFS G/G/1 queue is strictly positive, strictly decreasing, and is convex in its processing rate (Harel, 1990). Therefore, τ(µ) > 0, τ 0 (µ) < 0andτ 00 (µ) > 0. Also, τ(µ) isfinite for all µ>λ, and lim τ(µ) =. µ λ + Consequently, the expected sojourn time at stage i is τ(α i µ(x i )). One may or may not know the exact functional form of τ. However, both µ(.) andα are known. The set of feasible allocations F is defined as the set of allocations such that the processing rate at each stage is at least as large as the arrival rate, and the resource is completely used: 5

6 F = {x : x < n P +, n x i = m, α i µ(x i ) λ i}. The following lemma is immediately true from the fact that F is constructed from a set of linear constraints: Lemma 1 F is compact and convex. The objective is to findthefeasibleallocationthatminimizesthetotalcycletimeofthesystem. This is termed Problem D, and has the following formulation: min D(x) = X n τ(α i µ(x i )). (Problem D) x F Two other problems Problem P and Problem B are also defined: max P (x) = X n α i µ(x i ). (Problem P) x F Find x F such that α i µ(x i )=α j µ(x j ) i, j. (Problem B) Problem P maximizes total resource (factor) productivity, and Problem B balances the processing rate at each stage, and thereby eliminates bottlenecks. Before proceeding further, a couple of assumptions are required: Assumption 1: The relative interior of F (termed F )isnon-empty. Assumption 2: There exists x F which is feasible for Problem B. Since µ(.) is continuous and strictly monotone, it is invertible. Define a i = µ 1 (λ/α i ). Since µ(.) is strictly increasing, a i > 0 i. The definition is simply for notational convenience. It is easily verified that α i µ(x i ) λ x i a i (and similar equivalence holds for the strict and reverse inequalities). F can now be defined as: F = {x : np x i = m, x i >a i i}. If one were to similarly redefine F, it is clear that there is no need to independently constrain the values of x to be in < n +. Therefore, these constraints are dropped in the subsequent analysis. Lemma 2 (Interior solution) If x solves Problem D, then x F. 6

7 Proof: lim τ(µ) =, which implies that lim τ(α + i µ(x i )) =. This implies that for any µ λ x i >a i x F and x/ F P, n τ(α i µ(x i )) is infinite. Also, if µ>λ, τ(µ) <. Sinceα i µ(x i ) > λ x i >a i, this implies that for any point x F np, τ(α i µ(x i )) <. Since by Assumption 1, there exists at least one point in F, the result follows. Given Lemma 2, and the focus of the paper, it makes sense to only consider cases where Problems P and D could have the same solution. This necessitates a third assumption: so far. Assumption 3: Problem P has an interior solution, i.e. a solution in F. The next lemma establishes uniqueness for each of the problems, under the assumptions made Lemma 3 (Uniqueness) Problems B, D and P all have unique solutions. Proof: The proof is established independently for each problem. Problem B: Assume the converse, i.e., there exists x, y F such that x 6= y and α i µ(x i )= α j µ(x j ), α i µ(y i )=α j µ(y j ). Assume, without loss of generality, that x p <y p for some p. This requires that x q >y q for some q 6= p. Since µ(.) is strictly increasing, this implies that α p µ(x p ) < α p µ(y p ). Since y solves Problem B, α p µ(y p )=α q µ(y q ). Since µ(.) isstrictlyincreasing,α q µ(y q ) < α q µ(x q ). Combining these three expressions yields: α p µ(x p ) < α p µ(y p )=α q µ(y q ) < α q µ(x q ), which contradicts α p µ(x p )=α q µ(x q ). The result follows. Problem P: Since µ(.) isstrictlyconcave,itiseasytoverifythatp (x) is strictly concave in x all the diagonal terms in the Hessian are negative, and all non-diagonal terms are zero. Also, the set F is compact and convex. The result follows. Problem D: Again, the non-diagonal terms of the Hessian of D(.) are zero. The i th diagonal term is 2 τ(α x 2 i µ(x i )) = (α i µ 0 (x i )) 2.τ 00 (α i µ(x i )) + α i µ 00 (x i ).τ 0 (α i µ(x i )), which by inspection is i strictly positive (τ 00 is positive, and both µ 00 and τ 0 are negative). Therefore, D(.) isstrictlyconvex in x. Also,the set F is compact and convex. The result follows. 7

8 Lemma 4 Problem D has a unique solution which satisfies: α i τ 0 (α i µ(x i )).µ 0 (x i )=γ D i, and x F Proof: The complete formulation of Problem D is: nx min D(x) = τ(α i µ(x i )), subject to nx x i m = 0 x i a i 0 i. Lemmas 1, 2 and 3 ensure that the Karash-Kuhn-Tucker (KKT) conditions are sufficient (strictly convex objective, compact and convex constraint set, interior solution). The Lagrangian for Problem D is: P L D = n µ np τ(α i µ(x i )) γ D P x i m n δ i (x i a i ), where γ D and δ 1, δ 2,..., δ n are the n + 1 multipliers. The KKT conditions, in their most explicit form, for this problem are therefore: L D x i = α i τ 0 (α i µ(x i )).µ 0 (x i ) γ D δ i =0 i; np x i m = 0; x i a i 0 i; δ i 0 i; P γ D ( n x i m) = 0; δ i (x i a i ) = 0 i. P Lemma 2 implies that x i a i > 0 for all i, and hence δ i =0foralli. Since n x i m =0,γ D can take any constant value. The definition of F completes the proof. Lemma 5 Under assumption 3, Problem P has a unique solution which satisfies: α i µ 0 (x i )=γ P i, and x F 8

9 Proof: The complete formulation of Problem P is: nx max P (x) = α i µ(x i ), subject to nx x i m = 0 (1) x i a i 0 i. Lemma 2 and Assumption 3 ensure that the KKT conditions are sufficient. The Lagrangian for Problem P is: P L P = n µ np α i µ(x i ) γ P P x i m n δ i (x i a i ), where γ P and δ 1, δ 2,..., δ n are the n + 1 multipliers. The KKT conditions, in their most explicit form, for this problem are therefore: L P x i = α i µ 0 (x i ) γ P δ i =0 i; np x i m = 0; x i a i 0 i; δ i 0 i; P γ P ( n x i m) = 0; δ i (x i a i ) = 0 i. Sincewehaveaninteriorsolution,δ i =0foralli. Since constant value. The definition of F completes the proof. n P x i m =0,γ P can take any This lemma should clarify that maximizing total resource or factor productivity is exactly equivalent to balancing the marginal product of the resource across the n stages. The preceding lemmas have established uniqueness for each of the three problems, and have characterized the solutions for the two optimization problems. The main results of the paper can now be presented namely, that if the stages are all equally productive, that the solutions to all three problems B, D and P are identical, and that if there is any imbalance in productivity between stages, then all three solutions are necessarily different. 9

10 Let x B,x D and x P be the solutions to Problem B, D and P respectively. Proposition 1 If α i = α j for all i and j, thenx B = x D = x P. Proof: Trivially, if α i = α j for all i, the solution to problem B is x Bi = m n for all i. By Lemma 5, x P must satisfy, component-wise, α i µ 0 (x Pi )=α j µ 0 (x Pj ). Since α i = α j,thisimplies that µ 0 (x Pi )=µ 0 (x Pj ). Since µ(.) is strictly concave, implying that µ 0 (.) is strictly monotone, this implies that x Pi = x Pj for all i, and consequently, x P = x B. Similarly, based on Lemma 4, to show that x D = x B, all one needs to show is that a i τ 0 (α i µ(x i )).µ 0 (x i ) is strictly monotone in x i. However, this follows immediately, since its first derivative with respect to x is (α i µ 0 (x i )) 2.τ 00 (α i µ(x i )) + α i µ 00 (x i ).τ 0 (α i µ(x i )), the i th term of the Hessian, which Lemma 2 showed is strictly positive. The result follows. Proposition 2 If a i 6= α j for any i, j, thenx P 6= x B. Proof: Without loss of generality, let us say that α i < α j for some i, j. Now assume the converse of the proposition, i.e., that x P = x B = x. This implies that α i µ(x i )= α j µ(x j ), since x solves Problem B. Since α i < α j,thisimpliesthatµ(x i ) >µ(x j ). Now, since µ(.) is strictly increasing, this implies that x i >x j. This in turn implies that µ 0 (x i ) <µ 0 (x j ), since µ 0 (.) is strictly decreasing. Now, since µ 0 (x i ) <µ 0 (x j ), and α i < α j, and all terms are positive, clearly α i µ 0 (x i ) < α j µ 0 (x j ), which contradicts Lemma 5. The result follows. Proposition 3 If a i 6= α j for any i, j, thenx P 6= x D. Proof: Without loss of generality, let us say that α i < α j for some i, j. Assume the converse of the proposition, i.e., that x P = x D = x. Therefore, by Lemmas 5 and 6, x i and x j simultaneously satisfy: α i τ 0 (α i µ(x i )).µ 0 (x i )=α j τ 0 (α j µ(x j )).µ 0 (x j ), 10

11 and α i µ 0 (x i )=α j µ 0 (x j ). Comparing these two expressions yields τ 0 (α i µ(x i )) = τ 0 (α j µ(x j )), which implies that α i µ(x i )= α j µ(x j ), since τ 0 (.) is strictly monotonic (τ(.) is strictly concave). However, since α i < α j, this contradicts Proposition 2. The result follows. Proposition 4 If a i 6= α j for any i, j, thenx B 6= x D. Proof: Without loss of generality, again let us say that α i < α j for some i, j. Assume the converse of the proposition, i.e., that x B = x D = x. Therefore, by Lemma 5 and the definition of Problem B, x i and x j simultaneously satisfy: α i τ 0 (α i µ(x i )).µ 0 (x i )=α j τ 0 (α j µ(x j )).µ 0 (x j ), and α i µ(x i )=α j µ(x j ). Since τ 0 (.) is strictly monotonic, the latter equality implies that τ 0 (α i µ(x i )) = τ 0 (α j µ(x j )). Canceling these from the former equality yields α i µ 0 (x i )=α j µ 0 (x j ), which contradicts both Proposition 2 and 3. The result follows. 3 Discussion and Conclusion Figure 1 illustrates the result graphically for a two-stage system. The x-axis represents the amount ofresourceallocatedtoeachstage,withthedifferent allocation to stage 1 denoted x P, x D and x B. The corresponding allocations to stage 2 are evidently m x P, m x D and m x B. When α 1 = α 2 = 1, the three solutions coincide at m/2; however, when α 1 = α < 1, then the three solutions differ. x P is the point where the slope of αµ(x) equals the slope of µ(m x), while x B is thepointwherethevalueofαµ(x) equals the value of µ(m x). It is fairly easy to demonstrate that, in the two-stage case, x P <x D <x B in the interest of brevity, this proof is omitted. 11

12 (m-x) (x) (x) x P 0 0.5m m x B Figure 1: Illustration for a two-stage system

13 Similar results have been observed in queuing systems with finite buffers that balancing flows in a manufacturing system may not be optimal (the shallow bowl effect). However, these results are a consequence of the finiteness of the queues in tandem. The results of this paper suggest that this effect will carry over to the case where the queues have infinite buffers. While the formulation is in terms of a multi-stage system, the result is applicable to a system of parallel processors, where each processor has the same independent arrival process, and the organizational objective function is to minimize the average sojourn time across processors. In fact, it is applicable to any situation for which the objective function of each part is strictly decreasing and convex in its inputs, the value of these inputs in turn are strictly increasing and concave in the amount of resource allocated to them, and the organizational objective is the sum of the objectives of the parts. Such problems are widely observed, in a variety of contexts. For instance, most sequential process redesign problems involve resource allocation across asymmetric sequential stages, and aim to minimize total cycle time. Capacity allocation in mixed web server clusters, where the objective is to minimize the average response time to the user, is also a problem whose formulation is similar to Problem D. The resource could be either processor speed, or dollars spent on each new system. Individual effort allocation in problems of Internet search also fall under this formulation, if the search is either sequential, or done simultaneously in multiple windows. Other applications include information acquisition by portfolio managers where the marginal benefit of additional information is diminishing, and media planning in marketing. Resource allocation under asymmetric information has been studied fairly actively (see, for instance, Groves, 1973, Hurwicz, 1973, Marschak and Radner, 1972, Radner, 1972, among others). The author s current research is extending the results of this paper to include the incentive problems caused by asymmetric information about the true productivity of each stage, and to include multiple customer classes (Mendelson and Whang, 1990, Wein, 1992). Since transfer pricing systems are likely to favor allocations based on individual marginal productivity, and centrally planned allocations could favor allocations that balance productivity, this framework of this paper is well 12

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