Leadtime Reduction in a (Q, r) Inventory System: An Agency Perspective

Transcription

1 eadtime Reduction in a (Q, r) Inventory System: An Agency Perspective Jun-Yeon ee 1 School of Business Administration University of Houston-Victoria Victoria, TX eroy B. Schwarz Krannert Graduate School of Management Purdue University West afayette, IN ABSTRACT We examine the problem of designing and implementing a continuous-review (Q, r) inventory system from an agency perspective, in which the agent s effort influences the item s replenishment leadtime. Our results are as follows: If the agent is risk-neutral, a linear or quadratic contract achieves first-best. For a risk-averse agent with an exponential utility function, assuming a normal leadtime distribution, we determine the optimal linear contract. Extensive numerical experiments suggest that ignoring the possible influence of the agent on the replenishment leadtime can be costly, but that the cost penalty of ignoring agency can be significantly reduced by a simple contract. Key Words: (Q, r) inventory model, leadtime reduction, agency model 1 Corresponding Author: Jun-Yeon ee. 603 ornmead Dr., Houston, TX Phone) ) leej@uhv.edu.

2 1. INTRODUCTION Traditionally, inventory-replenishment models have taken the perspective of the owner-manager; that is, decisions are modeled as if the owner of the business is directly involved in designing the inventory-management system to be used and/or directly involved in implementing that system. However, in all but the smallest businesses, owners neither design management systems nor implement them. In this paper we examine a stochastic-demand inventory-replenishment model from an agency perspective, in which the owner (principal) delegates its design and implementation to an agent (manager, supplier, or consultant). We focus on the case where the agent s hidden effort influences the item s replenishment leadtime. We chose leadtime to be the performance measure of the agent s effort, because it is well known that leadtime has a significant impact on inventory-related cost; and because it is easy to measure. In our model the agent exerts a one-time effort in designing and implementing a continuous-review, single-item, (Q, r) inventory system. The result is a fixed leadtime that we model as a single drawing from a known probability distribution whose mean and/or variance are influenced by the agent s effort. Corresponding to his one-time effort, the agent receives a onetime payment based on this realized leadtime. In choosing his effort, the agent maximizes the expected value of his utility for his one-time payoff. The agent can be either risk-neutral or riskaverse. The principal s objective is to choose the values of Q and r and a payment scheme that minimize her expected long-run average total cost/time (cost per unit time), where the one-time payment to the agent is amortized over an infinite horizon as in Porteus (1985). If the agent is risk neutral, then, as expected, a simple linear or quadratic contract achieves first best. For a risk-averse agent, if the agent has an exponential utility function and the leadtime is drawn from a normal distribution, then we provide the parameters of the optimal linear 1

3 contract. Then, based on an extensive numerical experiment we observe that: Ignoring the possible influence of the agent on the replenishment leadtime can be costly; i.e., the principal s total cost/time given no agent effort can be significantly higher than the principal s total cost/time under first-best. The average increase in total cost/time was 3.6%; and the maximum was 7.%. The cost penalty of ignoring agency can be significantly reduced by a simple contract. The optimal linear contract recaptured 91.95% of the cost penalty on average (minimum 77.78%). Three categories of literature are relevant to our work: (i) the vast literature on (Q, r) inventory models, for which we will cite the most relevant references, (ii) the leadtime-reduction literature, and (ii) agency models in managing supply chains. (Q, r) Inventory Models: Federgruen and Zheng (199) provide the cost/time function for a (Q, r) policy and develop a simple and efficient algorithm for computing an optimal (Q, r) policy, which is used in our study. Zheng (199) examines the corresponding continuous demand model and proves that if one uses Q = Qd = λk( h + p) ( hp) and r = r( Q), where ( Q) r is the optimal value of r given Q, then the cost/time penalty is no more than 1.5%. Axaster (1996) provides an improved bound of approximately 11.8% on the penalty for using the EOQ heuristic. eadtime-reduction iterature: Fisher (1997) and Chopra and Meindl (001) argue that for functional make-to-stock products, management should focus on reducing operating costs. Safety stock, which is often a major component of such operating costs, depends on the variance of demand during leadtime. Although in some cases, management is able to influence the demand process, generally speaking, management has more influence over the leadtime. Karmarkar (1993) discusses factors that determine leadtimes in manufacturing systems and the impact of leadtimes on system performance. Hill and Khosla (199) provide a framework for

4 comparing these costs and benefits. Several researchers have examined leadtime reduction in inventory models (e.g., van Beek and van Putten (1987), iao and Shyu (1991), Ben-Daya and Raouf (1994), Gerchak and Parlar (1991), Paknejad, Nasri, and Affisco (199), and Choi (1994)). However, to our knowledge, this paper is the first attempt to examine leadtime reduction in inventory models from a principal-agent perspective. Agency Models in Managing Supply Chains: One of the earliest models of incentives in an operations-management environment is Porteus and Whang (1991), who examine incentive structures designed to maximize the residual return to the owner of a firm, where a manufacturing manager and several marketing managers act in their self-interest; and their hidden efforts influence production capacity and customer demand, respectively. Grout and Christy (1993), Grout (1996), Grout (1998), Van Miegham (1999), Grout and Christy (1999) among others examine incentive issues in various buyer-supplier settings. More recently, Corbett (001) and Corbett, Zhou and Tang (004) examine contracting problems under hidden information between a buyer and a supplier. Plambeck and Zenios (000) provide a dynamic principal-agent model for a general Markov decision process. Plambeck and Zenios (003) examine a make-to-stock single-server queueing system from an agency perspective. The rest of this paper is organized as follows: In section we present and analyze the (Q, r) agency model. In section 3 we provide computational results. Finally, section 4 summarizes the main results.. THE MODE AND ANAYSIS Consider a continuous-review, single item, (Q, r) inventory-replenishment system over an infinite horizon. The demand process is stochastic with fixed mean rate λ units/time. A fixed order cost, K, is associated with every replenishment order. Inventory-holding cost is incurred at 3

5 the rate of $ h per unit stock per unit time; backorder-penalty cost is incurred at the rate of $ π per backorder per unit time. Federgruen and Zheng (199) provide the following formula for the expected long-run average cost/time of this policy: (, ) C Q r = λk + Q+ r y= r+ 1 Q H y (1) where H( y ) is the rate at which the expected inventory-holding and backorder-penalty costs accumulate at time t + leadtime when the inventory position at time t equals y ; that is, y ξ D ξ π ξ D ξ () ξ= 0 ξ= y = ( ) + ( ) H y h y f y f where f () is the probability function of demand during leadtime. This expression is exact, D provided leadtime is fixed, inventory position is uniformly-distributed, and inventory position is independent of leadtime demand (Zheng (199)). In our model the agent exerts a one-time effort, e e, for system design and implementation, where e represents the smallest possible effort level that he can choose. The agent s effort results in a fixed replenishment leadtime,, that we model as a single drawing from a known probability distribution whose mean ( e) e the agent s effort. We assume that μ ( e), σ 0 and ( e), σ 0 μ and variance σ () e are influenced by μ e for e e ; that is, the agent s effort reduces the mean and standard deviation of the leadtime distribution at a marginally-decreasing rate. The agent incurs a cost of effort, G ( e), and receives a one-time payment, W( ), based on the realized leadtime. We assume G ( e), () e e G, and G () e > 0 for > e ; i.e., the agent s cost of effort is positive and increasing at an increasing rate with his effort level. We also assume G( e) G ( e) 0 = = for convenience. 4

6 In choosing his effort level, e, the agent maximizes the expected utility for his one-time payoff, i.e., ( () ) U ew E u W G e e (3) where u ( ) is the agent s utility function and the expectation is taken with respect to given e. et u( w ) be his reservation utility, which means the agent will accept the contract as long as he can get u( w ) from it. After is realized and observed, the principal chooses the values of Q and r to minimize her long-run average inventory-related cost/time: (, ) C Q r = λk + Q+ r y= r+ 1 Q H y (4) where H y is the same as () with f ( ) replaced by f D D, which is the probability function for demand during the fixed leadtime. et Q( ) and r( ) be the values of Q and r that minimize C( Qr, ) for a given. Define, C C Q r (5) as the corresponding minimum long-run average inventory-related cost/time given. The principal s objective is to design a contract that specifies a strategy for choosing an { } inventory policy, Q, r( ), and a payment, W convert this one-time payment to its amortized value i P W of capital. Then, the principal s expected total cost/time is: ( ( ), ) = + P, to the agent. As in Porteus (1985), we, where i P is the principal s cost TC W e E C i W e (6) Hence, the principal s optimization problem is to choose W ( ) and e to 5

7 (, e) Minimize TC W ( ) subject to U ew u( w) U( ew ) (7) e maximizes () over all e e (8) The participation constraint, (7), assures that the agent will accept the contract, and the incentivecompatibility constraint, (8), induces the agent to voluntarily choose the effort level desired by the principal. Note that in the first-best case where the agent s effort is observable (i.e., contractible), the incentive-compatibility constraint, (8) is unnecessary, while in the second-best case where his effort is unobservable, both of the constraints, (7) and (8), are required. Proposition 1 If the agent s effort is observable, then for any given e e, a fixed payment, w+ G e, minimizes the principal s total cost/time. This result is expected. Substituting the above fixed payment into (6) yields = + + FB TC e E C e ip w G e (9) The following proposition states that if the agent is risk-neutral, i.e., u ( ζ ) = ζ, and if the agent s effort is unobservable, then a simple linear or quadratic payment scheme is optimal. Further, if the risk-neutral agent influences the mean of the leadtime distribution, then a linear payment scheme is optimal. However, if the agent s effort only influences the variance of leadtime, then a quadratic payment is optimal. Since the expected amount of the payments given by (10) and (11) is w G( e) +, which is equal to the first-best optimal payment, Proposition implies that a linear or quadratic contract achieves first-best if the agent is risk-neutral. Proposition Suppose the agent is risk-neutral and that his effort is unobservable. μ ~ e < for all e e, then for any given e e, a linear payment of the form (a) If 0 6

8 W α β = is optimal. The corresponding parameters are: (b) If μ ( e ) = 0 G e α = w+ G e + μ ( e) μ ( e) () G e and β = (10) e and σ ~ e ) < 0 for all e e, then for any given e e, a quadratic payment of ( the form W α β ( μ ) = is optimal. The corresponding parameters are: σ σ ( e) G e e α = w+ G( e) + Proof. All the proofs are in Appendix. μ G e and β = (11) σ () e σ () e If the agent is risk averse with an exponential utility function, u ( ζ ) exp( γζ ) =, where γ represents his risk aversion. (The exponential utility function has been widely used in Economics. See, e.g., Holmstrom and Milgrom (1987).) We also assume that the fixed leadtime,, is a realization from a normal distribution with mean μ ( e ) and variance σ ( e ) form of optimal payment scheme is unknown, we focus on linear payment schemes. where Assuming a linear payment scheme, W, then it can be shown that ( ) = E u W G e e u M e. Since the, (1) γ M ( e) = E W e Var( W e) G( e) (13) See al and Srinivasan (1993). Therefore, the agent s goal is to maximize M ( e ) in (13). Correspondingly, the principal s optimization problem is: (, e) Minimize TC W ( ) Subject to M ( e) w (14), M e M e e e (15) 7

9 Proposition 3 Suppose the agent has an exponential utility function and that his effort is unobservable. Assume the fixed leadtime is drawn from a normal distribution. If the agent s payment is of the form W α β =, then, for any given e e, the parameters of the optimal linear payment are: γ α = w+ G( e) + β σ( e) + βμ( e) and (16) μ ( e) ( μ ( e) ) 4γσ( e) σ ( e) G ( e), if σ ( e) 0 γσ ( e) σ ( e β = ) G ( e), if σ ( e) = 0 and μ ( e) 0 μ ( e) (17) Substituting the above payment into (6), SB γ TC ( e) = E C e + ip w + G ( e) + βσ ( e) (18) Proposition 4 An upper bound e on the optimal effort e * in the first- and second-best case can be obtained by solving E C e e i w; otherwise, if = > P 1 G( e) = E C e= e w i (19) * e P = e. Although the parameters for the optimal linear payment do not have a closed form, they can be computed by using Proposition 3 and 4. In particular, Proposition 4 limits the range of possible values for * e. 3. NUMERICA EXPERIMENT We conduct an extensive numerical experiment for a risk-averse agent. The primary goal of the numerical experiment is to compare the principal s minimum total cost/time in the no-effort case (i.e., e = e ), the first-best case, and the second-best case in order to assess the possible impact of agency and the effectiveness of the optimal linear contract in recapturing the cost 8

10 penalty of ignoring agency. We also examine the sensitivity of the optimal solutions to model parameters. A secondary goal is to assess the goodness of the EOQ heuristic in choosing Q (see, e.g., Zheng (199) for the EOQ heuristic). We assume that the demand process is Poisson with rate λ and that a fixed leadtime is realized from a normal distribution. The functions representing the impact of the agent s effort on the parameters of the leadtime distribution are μ b = 0 + and σ d e a ae e ce =, where e e = 1; i.e., the mean and variance of the leadtime distribution are reduced at a decreasing rate by the agent s effort. The leadtime parameters are chosen so that the probability of negative leadtime is quite small, even in the most-likely case (i.e., a 0 = 0.5 and c = 0.5 ). The agent has an exponential utility function with risk-aversion factor γ, and his cost-of-effort function is κ ( e ) G e =. The value of w is set to zero. 1 v In the numerical experiment, we begin with a base case; that is, a set of values for all the parameters, intended to represent a realistic business scenario. We then examine alternative values for each of the most important parameters around the base case. In the base case, the holding-cost rate, h = $1/unit-month, and the backorder-penalty cost, π = $99/unit-month, in order to yield a target service level over the replenishment leadtime (i.e., a newsvendor ratio) of Given these, the Poisson monthly demand rate, λ, and the fixed order cost, K, are chosen so that there are approximately 1 replenishment orders per year. Table 1 summarizes the functions and their parameter values. We use Federgruen and Zheng (199) s algorithm to compute the optimal values of Q and r and the corresponding minimum inventory-related cost/month, C( ), for a given fixed leadtime. Note that in this scenario we have to calculate the minimum inventory-related 9

11 cost/month for each possible fixed leadtime, and then evaluate the expected total cost/month over all the values of given e. Since this is not possible for a continuous distribution of, we made a discrete approximation as follows: Divide the interval from 0 to μ ( e) 4σ ( e) + into 100 small intervals, and assign to the midpoint of each small interval the probability that falls within that small interval. We define IMPACT%, EFFECTIVENESS%, and EOQ Performance% as follows: where TC ( FB) TC ( FB) TC NE IMPACT % = 100 TC ( SB) TC ( FB) TC NE EFFECTIVENESS% = 100 TC NE * TC ( Q ) * TC ( Q ) TC EOQ EOQ Performance% = 100 TC J, J = NE, FB, SB, are the principal s minimum total cost/month in the no-effort, the first-best, and the second-best cases; and * TC EOQ and TC Q are her total cost/month when using the EOQ heuristic and the optimal Q, respectively. From Table, we observe: Ignoring the possible influence of the agent on the replenishment leadtime can be very costly. The average value of IMPACT% was 3.6%; and the maximum was 7.%. The cost penalty of ignoring agency can be significantly reduced by a simple contract. Under the optimal linear contract, the average value of EFFECTIVENESS% was 91.95% and the minimum was 77.78%. We also observe that the EOQ heuristic performs very well. The average cost penalty for setting Q Qd = and r = r( Qd ), where r Q is the value of r that minimizes the 10

12 inventory-related cost given Q and, was only 0.16%; and the maximum was 0.7%. This compares with the Axsater (1996) s lower bound of approximately 11.8%. As expected, the optimal effort level in the second-best case, e * SB, is smaller than in the first-best case, e * FB model parameters:. The following paragraphs discuss the sensitivity of our results to important As the backorder-penalty cost, π, increases, the optimal e increases (in order to reduce expected backorders), and IMPACT% increases (because of larger opportunity for inventorycost reduction, i.e., larger portion of the inventory costs that can be reduced by the agent s effort). As the fixed order cost, K, increases, the optimal e decreases. This is because a larger K leads to a larger Q, and, hence, to less frequent ordering. Hence, increasing K decreases the impact of long or highly variable leadtimes on the expected inventory cost/time (since there are fewer replenishment cycles/time). Therefore, increasing K reduces the cost impact of the agent s effort. So, all other things equal, the principal desires a lower level of effort from the agent as K increases. We also observe that as K increases, IMPACT% decreases (because of less frequent ordering). As the agent s risk aversion, γ, increases, the optimal e and EFFECTIVENESS% decreases, because a higher risk-premium has to be paid. 4. CONCUSION The owner-manager perspective of traditional inventory-replenishment models ignores the fact that, in all but the smallest businesses, the owner delegates the design and implementation of an inventory-management system to an agent. Failure to take the principal-agent relationship into account in particular, the hidden effort of the agent therefore, poses the risk of unnecessary extra cost to the principal. In this paper, we have examined a continuous-review, single-item, (Q, 11

13 r) inventory system from an agency perspective, in which the hidden effort of an agent influences the item s replenishment leadtime. We have demonstrated that the possible influence of the agent on the replenishment leadtime can be large, but that a simple linear contract is capable of recapturing most of the cost penalty of ignoring agency. 1

14 APPENDIX. PROOFS OF RESUTS IN SECTION Proof of Proposition. For any given e, the principal has to choose W ( ) so as to (a) et Minimize E W e subject to E W G e e w (A1) e maximizes E W μ μ ( e) G e e over all e e (A) ( e) G e e G e * W w+ G e + μ μ μ ( e) ( e) G e e G e = + + μ ( ) ( ). * U e E W G e e w G e e G e μ U ~ e at e Then, it can be easily shown that = 0 the agent s expected utility is maximized at * E W e w G e e~ = and < 0 U e ~ for all e. This means that e~ = e and hence (A) is satisfied. Further, note that = + and hence (A1) is satisfied. Since the expected payment is the same as the optimal (fixed) payment in the first-best case, W * (b) et Then, = 0 W * σ is an optimal payment scheme. G e e G e + + * W w G e σ e σ e σ e * ( μ ) U e E W G e e G ( e) σ ( e) G ( e) = w+ G( e) + σ ( e ) G( e ). σ ( e) σ( e) σ ( e) U ~ e at e e~ = and U ( ) < 0 * for all e. Further, E W e = w+ G( e) e ~ is an optimal payment scheme.. Hence, 13

15 Proof of Proposition 3. For any given e, the principal has to choose α and β so as to Minimize α ( e) βμ γ e e G e w (A3) Subject to α βμ β σ α βμ γ () e β σ () e G() e α βμ γ ( e~ ) β σ ( e~ ) G( ~ e ), ~ e (A4) et U ( e~ ) α βμ ( e~ ) β σ ( e~ ) G( ~ e ) γ. Then, ( e~ ) = βμ ( e~ ) γβ σ ( e~ ) ( e~ ) G ( e~ ) U σ and G < 0 ( e~ ) = μ ( e~ ) γβ ( σ ( e~ )) + σ ( e~ ) σ ( e~ ) U β ~ e for all e. Hence, in order for W to satisfy (A4), β should be chosen so that U ( e) = 0 e that β given by (17) satisfies U () = 0 given by (16). J Proof of Proposition 4. Minimizing. It is easy to see. Now, it is easy to see from (A3) that the optimal α is TC e, J = FB, SB is equivalent to maximizing J J J =, where TC NE TC e CR e i AC e P TC NE E C e = e, SB =, AC ( e) w + G ( e) + ( e), and FB AC ( e ) w G ( e ) J CR e TC NE E C e J FB Note that CR ( e) TC ( NE) for any e and that γ βσ = +. AC e is strictly increasing and convex in e. This means that the optimal effort in the first-best case is no larger than e, where FB = P, which is equivalent to (19), if TC ( NE) P TC NE i AC e Further, since SB FB AC ( e) AC ( e) no larger than e. > i w. Otherwise, clearly * e = e. for any e, the optimal effort in the second-best case is also 14

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19 Table 1. Functions and Parameters in Numerical Experiment Functions Parameters Base Case Value Alternative Values ( h π ) λk + Q d = hπ μ σ a ( e ) = a + 0 b ( e ) = d e c e () e = κ ( e ) v G 1 exp( γζ ) K $0/order 5, 10, 30, 50, 100 λ 50 units/month - h $1/unit-month - π $99/unit-month 19, 4, 33, 49, 199 a month - a 0.5 month - b 1 0.3, 0.5,, 3, 5 c 0.5 month - d 1 0.3, 0.5,, 3, 5 κ 10 0, 30, 50, 70, 100 v 3, 4, 5, 7, 10 u ζ = γ , 0.03, 0.05, 0.1, 0., 1 Others w 0 - i P e 1-18

20 Table. IMPACT% and EFFECTIVENESS% Base Case and Alternative Parameter Values IMPACT% EFFECTIVENESS% Base Case p = 19, 4,33, 49, ~ ~ 95.0 K = 5,10,30,50, ~ ~ b = 0.3, 0.5,,3, ~ ~ d = 0.3, 0.5,,3, ~ ~ κ = 0,30,50, 70, ~ ~ 9.09 v = 3,4,5,7, ~ ~ γ = 0.0, 0.03, 0.05, 0.1, 0., ~ Average IMPACT% = the % increase in cost of ignoring agency (no-effort vs. first-best) EFFECTIVENESS% = the effectiveness (%) of the optimal linear contract 19