stochastic choice hazard and incentives in a common service facility

ELSEVIER European Journal of Operational Research 81 (1995) 324-335

• EUROPEAN JOURNAL

OF OPERATIONAL RESEARCH

Theory and Methodology

Stochastic choice hazard and incentives in a c o m m o n service facility

Suresh Radhakrishnan a, Kashi R. Balachandran b0,

a Graduate School of Management, Rutgers University, Newark, NJ 07102, USA b LeonardN. Stern Schools of Business, New York University, New York, NYIO003, USA

Received March 1993; revised September 1993

Abstract

This paper examines the pricing and incentives in an M / G / 1 queue. A setting wherein, two division managers who share a common production facility and decide on the demand (usage) rates is developed. Stochastic choice hazard created by the unobservability of demand rates chosen by the division managers leads to incentive issues. Specifically, the headquarters needs to design incentive schemes such that the use of the common facility is optimal for the firm. When the incentive schemes are based on divisional performance measures, a charge based on usage ensures firm-wide efficient use of the common facility as a unique equilibrium. The charge is such that it encourages use of the facility when a division's expected benefit increases. Increases in the capacity of the common facility are not shared proportionately by the division managers, and therefore, the charge is not monotone in the capacity of the common facility.

1. Introduction

In this paper, we consider a common production facility (CF) that is shared by two division managers (DMs). We model this setting as an M / G / 1 queue, where the profits of each division depends on the number of units demanded and the delays experienced in processing and fulfilling the order. The DMs choose the demand rates and the unobservability of demand rates leads to stochastic choice hazard (i.e., congestion). The realized divisional profits and the realized demands are observable and can be used for designing incentives. The question is, how should the Headquarters (HQ) design incentives for the DMs and price the usage of the facility, such that the DMs use the facility in the best interests of the firm? Note that each DM can increase her division's profits by increasing her demands, but as a consequence cause congestion for the other DM.

Balachandran and Schaefer (1979a) analyze an M / M / 1 setting with a first-come-first-serve queue discipline. The users have information on the average number of customers in the system, when making their decisions whether to join the queue. They derive a pricing scheme that equalizes the attractiveness

* Corresponding author.

0377-2217/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 7 7 - 2 2 1 7 ( 9 3 ) E 0 3 1 7 - Q

S. Radhakrishnan, KR. Balachandran / European Journal of Operational Research 81 (1995) 324-335 325

for each class to use the facility. Balachandran and Schaefer (1980) consider an M / G / 1 setting with a first-come-first-serve queue discipline. They show that when expected delay at the facility is fixed, a pricing mechanism, incorporating an entry charge and a rebate depending on the actual waiting time in the queue, can be constructed that ensures equal access to all the classes. Naor (1969), Yechiali (1971, 1972), Balachandran and Schaefer (1979b) and Alperstien (1988) are some of the other studies that examine the conflicts that arise between publicly optimal behavior and privately optimal behavior in the use of a common facility. In contrast to the above models, we consider incentive issues that arise due to the unobservability of demand rates chosen. Furthermore, in this study, we consider a non-preemptive priority production order (wherein the high-benefit-high-cost DM is provided non-preemptive priority for production).

Whang (1989) shows that charges based on average costs are superior to charges based on marginal costs, in a two part game where, (a) the capacity of the common facility is based on the users, requirements as communicated to the Headquarters, and (b) the users then determine their usage. Mendelson and Whang (1990) consider a M / M / 1 setting, when the DMs have pre-contract private information on the demand rate and the benefit derived is not observable. They show that a cost application scheme based on marginal delay costs induces the truth for homogeneous expected production times. When expected production times are heterogeneous, the truth inducing cost application scheme depends on the usage and production time. In these models, the users do not influence their demand rates and incentive effects that arise solely due to hidden information are examined. In contrast to these studies, we focus on incentive issues that arise due to hidden action (i.e., the choice of demand rates).

The rest of the paper is organized as follows: in Section 2 we develop the model; Section 3 contains the results; and Section 4 contains some concluding remarks.

2. The model

Consider a firm with two product divisions A and B that receive production orders for homogeneous units of products which are to be produced at a common facility. The production orders (number of units demanded), x k, arise from independent Poisson Processes, for each k = A, B; with demand rate (mean) X k"

The revenues realized from the i-th unit, g/k for k = A, B, is random. 1 The revenues from each order is independent of the Poisson Process generating the order. The expected revenue from a unit of product demanded is denoted G e = E[g g] where, E[-] is the expectations operator.

The units that are to be produced experience delays at the common production facility (CF), due to (a) the waiting time in queue that depends on the congestion at the CF, and (b) the production time. The production time p/k for each unit of demand has a general distribution with mean P and finite second moment M and the realized waiting time of the i-th unit be wp. A cost of H k per unit time of delay is incurred. The total realized net profit for division k is given by

x k x k

rk= E r i k= g [gik--(Wik +Pik)Hg] • i ~ l i= 1

1 Superscript k is used to denote the divisions, i.e., k = A, B.

326 S. Radhakrishnan, K.R. Balachandran / European Journal of Operational Research 81 (1995) 324-335

The profit for the H Q and the incentive for the DMs are given by

Ue(") = E (ri k - s k ( ' ) ) - C(P, M), k

(1)

where sk( . ) are the i ncen t ives / compensa t ions provided by the H Q to the DMs, and C(P, M) is the opera t ing cost of the p roduc t ion facility. The expected compensa t ion to the DMs is denoted Sk( . ) (where the expectat ion is taken over the a rguments of s~(.)). The realized net profit r k and the realized demands x g are observable, and can be used to provide incentives to the DMs. We assume that incentive schemes are l inear in the observed variables. 2

N o t e that the total revenues of each division follows a c o m p o u n d Poisson Process and the expected revenue for each division is GkX k. The sum of the units d e m a n d e d by the two divisions is Poisson distr ibuted with m e a n (E~Xk) . Let Dk(X k, X j, P, M) for k ~ j , denote the expected waiting time for an arbitrary unit - then the expected delay cost is given by [HgDk(.)X ~ +HkPX~]. 3 The expected net profit, Rg(-) , for the divisions and the firm are

Rk( ' ) = [ G k - H k D k ( x k , X i , P , M ) - H k P ] X k for k , j = A , B , kq=j, (2)

R( ' ) = E R k ( ' ) -- C(P, M). (3) k

The capaci ty of the c o m m o n facility is assumed to be fixed (i.e. P and M are exogenous and c o m m o n knowledge).

Note that each D M ' s net profits can be affected by increased demand rates and increased congest ion (ei ther by her own use or by the use of the CF by the o ther DM). In effect, the disutility that a D M derives is th rough expected delay costs that one D M can impose on the other. The demand rate is the expected value of actual demand and thus is not verifiable perfect ly at any time by the H Q or the o ther agent, since at best only one: real izat ion of the r andom variable is observable.

W e restrict analysis to situations when a non-preempt ive p r o d u c t i o n priority is provided to the high-benefit=high-cost division. Wi thout loss of generality, we assume that A is the high-benefi t-high-cost division 4 (i.e., GA(") > GB(") and H A > HB).

The sequence o f events unfold as follows: first, the H Q designs incentive schemes based on the realized demands (x k) and realized net profits (rg), the observable variables; the DMs choose the d e m a n d rates ( X k) based on the incentive schemes provided by the HQ; finally, the demands and net benefits are realized and payments are made by the H Q to the DMs based on the agreed incentive scheme.

2 Merchant's (1989) survey of incentive schemes offered to Profit Center Managers (like the DMs in this model) shows that seventy five percent of the corporations use linear short term incentive schemes. In the case with no moral hazard, it is shown that the linear incentive scheme belongs to the optimal class of incentive schemes in the sense that the first-best solution obtains.

3For an M/G/1 queue, the expected waiting time for an arbitrary unit of demand in steady state is given by the Pollaczek-Khintchine Formula as Dk( • )= [(X k + XJ)M][2(1- Xkp)]-1 and D j = [(X k + XOM][2(1- XkP)(1- X~P - XJP)]-1 for k ~ j, when division k is granted non-preemptive priority for production. See Cooper (1981 pp. 215-224) for a derivation of the Pollaczek-Khintchine formula.

4 The conditions for the non-preemptive priority production order being superior to the first-come-first-serve production order is derived in Balachandran and Radhakrishnan (1992).

S. Radhakrishnan, K.R. Balaehandran / European Journal of Operational Research 81 (1995) 324-335 327

3. The results

The H Q designs incentive schemes based on which the DMs choose their demand rates, ( x k ) , which are unobservable. The realized demand rates ( x k) and the realized net benefits ( r k) are observed and can be used for designing incentive schemes.

The problems of congestion and the problem of stochastic choice hazard (unobservability of demand rates) exist. The problem for the H Q is that it must ensure that incentives are provided such that the DMs expect to get a compensat ion which they would otherwise get in an alternative employment. In addition, the H Q should recognize that the DMs will choose the demand rates to maximize their own compensation. In effect, the DMs will act as self maximizing individuals, given an incentive scheme. The problem of the H Q is given below.

(Program 1)

max E[R~(.) - s k ( - ) ] - C ( P , M ) (4) sk, Xk k

s.t. s k ( . ) > ~ k for k = A , S , (i)

ash(.) aX k - 0 for k = A , B , (ii)

where R k ( . ) = (G k - H K D k ( X A, X B, P, M ) - H k p ) x k for k = A, B and ~k is the minimum compensation that needs to be provided.

The H Q wants to maximize the firm's net profits after making the incentive payments to the DMs. Constraints 4(i) ensure that the incentive schemes designed by the H Q provides the DMs with at least their minimum levels of compensat ion that they could obtain elsewhere. Constraints 4(ii) recognize that given any incentive scheme the DMs would act as self interest maximizers, i.e., choose the demand rates such that they maximize their payoffs. Constraints 4(ii) are the first order conditions for a maximum with respect to the demand rates, s

Note that if the choice of demand rates were observable to the HQ, then the H Q will choose the demand rates by maximizing the firm's net profits after making payments to the DMs, i.e., ~ k [ R k ( " ) --

~ k ] _ C ( P , M ) . In this case, constraint 4(ii) need not be considered by the HQ. Consequently, the H Q can pay the DMs exactly the minimum level of compensat ion ~k and maximize the firm's net profits. We define firm-wide efficient demand rates as those demand rates that achieve the same maximum as the case when the H Q chooses the demand rates (i.e., when the demand rates are observable).

Definition 1. Finn-wide efficiency: The demand rates X k are firm-wide efficient, when X k* maximizes the unconstrained expected total net profits, Ek[Rk(") -- U k] -- C(P , M ) .

First, consider incentive schemes based on divisional performance, i.e. sk ( r k, X k ) = [r k - ~ k x k ] l k where 6 k are the price charged for the use of the facility, and I k are the proport ion of the divisional net profits after the charge based on usage given to the DMs as bonus. The expected payoffs to the DMs are Sk( . ) = Ik[ R k -- ~ k x k ] for k = A, B.

5 Note that Sk( • ) is concave in X ~, when st( • ) is linear in rk ( ' ) . ~.kRk( • ) is not concave in {X k, X j} in a first come first serve queue (see Balachandran and Schaefer, 1980). However, with a non preemptive priority to one class ~,kRk(") is concave in {X k, X j} (see Balachandran and Radhakrishnan, 1992). Hence, the objective function given in (4) is concave in {X k, XJ}.


The solution to Program 1 when incentive schemes are based on divisional performance is character- ized in the following Theorem.

Theorem 1. In Program 1, when incentive schemes are based on divisional per formance:

1) The expected payo f f s to the D M s do no t exceed the reservation utility.

2) Firm-wide eff icient d e m a n d rates are chosen by the DMs .

3) A price needs to charged f o r every usage o f the c o m m o n facil i ty to mit igate congestion costs, and the price equals the marginal expected delay cost imposed by one division on the other.

Proof. The constraints 4(ii) are obtained by setting the partial derivatives equal to zero: 6

Skx k = G k - H k p - H X D k - H k X k D ~ k - 6 k = O.

The Lagrangian of Program 1 is

L = Y'~ { ( G k - H k O k - H k p ) x k ( 1 -- I k) + 6 k X k I k} -- C ( P , M )

k

+ ~ _ , { A k [ ( G k - - H k D k - - H k p - a k ) X k I ~ - - U ~ ] } k

+ E { I z ~ [ G k - - H g D k - - H I ¢ P - - H k X g D ~ k - - a ~ ] } (5) k

where A k a n d / x k are the Lagrange multipliers of 40) and 4(ii) respectively. To establish Theorem 1.1 and Theorem 1.2 we need to show that A ~ > 0 and ]z k = 0. Differentiating the Lagrangian with respect to I k and 6 k and setting them equal to zero we have

Lx k = _ ( G k _ H k O k _ H k p _ ~ k) X k + Ak( G k _ H k D k _ H k p _ (5 k) X k = O, (6)

L~k = I k X k -- 2t~IkX k --/x k = 0. (7)

I t is direct from (6) and (7) t ha t /x k --- 0 and A k = 1. This establishes Theorem 1.1 and 1.2. To establish Theorem 1.3 we have to show that 6 k = H~XJDixk for j" v~ k. Differentiating (5) with

respect to X k and using A k = 1, /x k = 0 and constraint 4(ii)

L x k = ( G k - H k D k - H k p -- H k X k D ~ x k ) -- H i X J D i x k = 0 for j * k. (8)

Substituting again f rom 4(ii), 6k _ HJXi DJx ~ = 0 for j ¢ k establishing Theorem 1.3. []

Theorem 1.3 states that a price that equals the marginal expected delay cost that each user imposes on the other is required. Theorem 1.2 states that the unobservability of demand rates chosen by the DMs is mitigated by the charge (~k), and consequently, the optimal expected demands ( X k*) maximize the unconstrained expected net profit R(-). It follows that since the H Q can induce the DMs to choose the same demand rate as it would have if the demand rates were observable (by charging 6 ~') the H Q does not incur any additional cost due to unobservability. Hence, the solution is firm-wide efficient. The assumed linear incentive scheme induces the DMs to choose the firm-wide efficient solution. This implies that the linear incentive scheme leads to the global opt imum for Program 1. In effect, the linear incentive scheme does not impose any restriction on the solution to Program 1.

6 We use the notation Skx~(.) to denote (aSk(.)/bXg).

S. Radhakrishnan, K.R. Balachandran / European Journal of Operational Research 81 (1995) 324-335 329

We next analyze incentive schemes based on other observable variables. Group incentives are based on the firm's total profits. Specifically, s k ( ' ) = (r ~ + rJ)I k for k vaj, and I k is the proportion of the total net profits that is provided as bonus to the DMs.

Corollary 1. A solution to Program 1 with incentive schemes based on group performance are equivalent to incentive schemes based on divisional performance.

Proof. The firm-wide efficient solution, i.e. when X k are observable, satisfies

R x k ( ' ) = G k - H k D k - - H k p - - H k X k D k k - H k X J D J x ~ = 0 for k v~j. (9)

The compensation to the DMs is a constant ~ k if X k = X k* and 0 otherwise. When X k are unobservable and the contracts are of the form s k = (r k + r O I k constraints 4(ii) will be

R~ckI k + RJxkI ~ = 0 ~ RkxkI k - HJXSD~cklk = 0 for k 4:j. (10)

The solution obtained by (9) is ,~ to the solution of (10). This implies that the firm-wide efficient X k* is obtained as a solution to (10). Choose I ~ such that the DMs are paid no more than their reservation wages. From Theorem 1.2 a contract of the form sk(r ~, x k) is equivalent to (9). Hence, a group incentive scheme and a divisional incentive scheme with a charge are equivalent. This establishes Corollary 1. []

Corollary 1 shows that when the demand rate is unobservable, charging a price is sufficient but not necessary to implement the firm-wide efficient solution. A group incentive scheme would do as well. 7 However, this might not be feasible for the firm. The divisions could be engaged in various activities. Only one activity could involve the use of the common production facility. The group incentive scheme would suggest that the DMs should be compensated on the total net profits that the firm derives from the common facility. That means the HQ should determine the profits on those activities of the divisions that utilize the common facility. This could be costly and confusing to administer in practice. Hence, divisional incentive schemes are easier to use in practice. In the rest of the paper we analyze the properties of the price that needs to be charged with divisional incentive schemes.

The DMs can implicitly collude to choose demand rates and obtain strict gains, s For such implicit collusion to occur at least one DM would have to reduce her demand rate. However, the DM who reduces the demand rate in the implicit collusion solution could obtain strict gains by deviating from the implicit collusion solution, since the action that induces the demand rate is unobservable. Therefore, the equilibrium in the DMs sub game induced by the divisional performance measure and the group incentive measure should be unique. This conjecture is established in the following theorem.

Theorem 2. In Program 1, when incentive schemes are based on divisional performance or group performance, the DMs will not implicitly collude and deviate f rom the f irm-wide efficient demand rates.

7 It can be shown that compensation schemes for the DMs such as sA(r A, r B, X A) for A, and sB(r A, r B, x B) for B are also equivalent to incentive schemes based on divisional or group performance. Similarly, including x B in A's compensation and x A in B's compensation is also equivalent to the divisional or group performance based incentive schemes.

8 We define implicit collusion as a situation where both DMs choose a dominating saddle point and stay with it in spite of non-verifiability of demand rate chosen.


Proof. Let the non-cooperative solution to Program 1 be {ik*, 8k*, X~*}. Given the incentive scheme {I k*, 8 ~*} if the DMs collude then they will maximize:

max • ( R k - 61"*xk)I k*. (11) Xk k

Denote the solution to (11) by J ~ . Let S denote the expected payoffs to both DMs when they collude, i.e.

= ~ (R ~ - 8~ '2k) I ~*. (12) k

Let S* denote the expected payoff to the DMs in the solution to Program 1. Clearly the DMs would collude iff S > S* = EkU k with strict inequality for at least one k and, where the second equality follows from Theorem 1.1. The first order conditions for (11) will be

(Gk--gkp--gkJOk--gkxkDkk--(sk*)Ik*--HJ2JDjrklJ=O for k :~j. (13)

Constraints 4(ii) are

(G k - H k p - H k D k* -H~Xk*D~*k)I k* - 6 k* = 0 for k 4=j. (14)

Note that )(~ > X k* for k = A, B is not possible. To see this, assume that J ( e > X k* which implies 1) k > D k* and Dx% > Dxk2 since Dxkk > 0, and D~:~x~ > 0 for k = A,B. Subtract (14) from (13) to obtain a contradiction.

Therefore the only possibilities are: Case 1: j~g > xA*; j~B < xB*; Case 2: )~g < xA*; 3 ~ > xB*; and Case 3: )~A < xA*; )~B < X B*. This implies that at least one of the DMs will have to necessarily reduce expected demands. Without loss of generality let us assume that division A reduces expected demands in the collusion solution. From (13) we have

[ I~A -- aA*2A ] IA* = H A ( 2 A )2d~AxalA* + HB2Aj~B/)xBa/B*

< H A ( X A*) 2D~ Ia* + HBxA*2BD~*I B*, since X A* > )~A and DxAxAA, DBBxA > 0. This implies A will obtain strict gains from deviating from the collusion solution. We can similarly argue that B will also deviate from j~B whenever X B* > j~B.

The collusion solution cannot be sustained and therefore, the firm-wide efficient demand rates will be chosen by the DMs when incentives are based on divisional performance.

To see that collusion will not occur with group incentives it is sufficient to note that both DMs choose their demand rates based on the total expected net benefits (as the H Q desires). Since, both DMs maximize the same objective function the collusion solution will be the same as the non-cooperative solution. Hence, with group incentive schemes the DMs will not implicitly collude. []

We next study the properties of the price that is required when incentives are based on divisional performance.

3.1. Properties of the price

The relationship between ~k and expected revenue is examined. 9 Intuitively, as the expected revenues (G k) of one division increases the price should encourage that division to increase use of the

9 We suppress the * for sake of brevity. The 6 ~ is the optimum price given as a solution to Program 1.


common facility. Therefore, the price of a division should be inversely related to its expected revenues and the price should increase as the expected revenues of the other division increases.

Theorem 3. In Program 1, when incentive schemes are based on divisional performance, and the expected revenues of one division increases, 1) the price for usage of the common facility of that division decreases and the price for usage of the

common facility of the other division increases; 2) the demand rate of that division increases and the demand rate of the other division decreases.

Proof. Constraint (4c) using ~k = HiXJDJxk = _Rix, from Theorem 1.3 will be

Rx~=RA~÷RBk=O for k = A , B . (15)

Differentiate (15) implicitly with respect to G k for k = A, B to get

1 + (oxg/OGk)Rxkx~ + (OXJ/OGg)Rxkx ~ = 0, (16)

(OX~/OGk)RxJxJ + (oxk /oGk)Rxkx ~ = O. (17)

Rearranging (17)

(oxk/OG k) = -- (OXJ/OGk )( R x , x J R x k x O . (18)

Substituting (18) into (16) we have

( ~ x j t ( 2 [ -1 Rxkxk Rx~xJ - R x k x J = - - - < 0 ,

1 OG~ J Rx~x, !

since Rx~xkRxJxJ - R2kxj > 0 by second order sufficient condition and Rx~x, < 0. Hence, (aXJ/OG k) < 0.

From (18), (RxJxJRxkxJ) > 0 = ( 0 x k / 0 G k) > 0, establishing Theorem 3.2. To show that 6 A decreases with GA: From Theorem 1.3 we know that

~A = HBD~AX B. (19)

Using the Pollaczek-Khintchine formula (see Cooper, 1981, p. 216), i.e.

D E k X k M D A = D B ( 1 - - x A p - - x B p ) and D B= where D =

1 - - x A p 2(1 - F_,kXk)p '

in R k some algebraic manipulations yield

( H A - H B ) M 2HBDx A RxAxB = --

2(1 - - x A p ) 2 (1 - x A p - X B P ) '

2HBDx A RxBx" = -- (1 - x A p - x B p ) "

This implies RXBxB/RxAx B < 1 and from (18) ](oxB/oGA)[ > [(oxA/OG A) [, i.e. for a change in G k, the effect on X B is greater than the effect in X A. Specifically when G A goes up, the decrease in X a is greater than the increase in X A.

332 S. Radhakrishnan, K.R. Balachandran /European Journal of Operational Research 81 (1995) 324-335

Differentiating (19) implicitly with respect to G A,

o~A HB[DBAxAXBOXA_[_DBAxBXBOXB| _ _ B 0 x B ] / aG N = [ OG A OG + D x A - ~ -X ]

< B B B ~()xB _D~dAxBXB B ] H ]DxAxAX oxB B oxB -Dx I- J

OX B = o---G- X HB[ DBAxAX B -- D B A x , X B -- DBx,].

DXA DP

Note that

DBA -- + 1 - X A p ( 1 - X A p ) 2 ,

B DxAxA 2 DxA P 2 DP 2 DxAxA -- q- -b

1 - x A p (1 - x A p ) 2 (1 - - x A p ) 3 '

DxAx A DxAP DBAxS -- 1 -- x A ~ + (1 -- x A p ) 2,

since DxAxA = DxBxB = DxAxB.

(20)

Substituting the above in (20) and rearranging,

[DBxAxAXB--DBxAxBXB--DBA] - DxA (1 - x A p ) 2 [ x B p - 1 + x A p ]

DP + (1 - - x A p ) 3 [ 2 x B p - 1 + x A s ] < 0,

since 1 - - x A p - x B p > 0. Using this in (20) (06A/OG A) < O. Similarly (o6A/OG B) > 0 can be established since RxAxA/RxAxB > 1, and I (oxB/OG A) I <

[ (axA/aGA) I.

(ii) To show that •B increases with GA: since X A increases with G A it is sufficient to show that 6 s increases with X A. Note that 6 B = HAXADAB from Theorem 1(4). Therefore,

d6 B d X B LrAvA r'~A d X A = HADAB + HAXADABxB~-X--X + 11 A UxAxB > 0,

since DABx B = 0, DAB > 0 and DAAx B > O. []

This property is illustrated in Table 1 using the following paramete r values G B = 9.00; H A = 1.00; H s = 0.80; P = 0.01; M = 0.50.

The price equals the marginal expected delay cost that one division imposes on the other (see Theorem 1.3) and the expected delays depend on the expected production time. If the expected production time is inversely related to committed costs 10 (i.e. Cp(P, M ) < 0), then it would seem that the price for both DMs should increase with increases in committed costs, since an equitable scheme, in

10 Note that an increase in expected production time implies reduced capacity.

S. Radhakrishnan, IcLR. Balachandran /European Journal of Operational Research 81 (1995) 324-335

Table 1 Sensitivity to G A

333

G A X A* X B* D A* D B* R* 6 A* 6 B*

9.90 3.66 13.01 4.33 5.19 83.32 4.49 0.95 9.91 4.35 12.27 4.34 5.21 83.36 4.20 1.14 9.92 5.02 11.54 4.36 5,22 83.41 3.93 1.32 9.93 5.69 10.82 4.38 5.24 83.46 3.66 1.51 9.94 6.35 10.10 4.39 5.26 83.52 3.39 1.70 9.95 7.00 9.40 4.41 5.27 83.59 3.14 1.88 9.96 7.64 8.70 4.42 5.29 83.66 2.89 2.07 9.97 8.27 8.02 4.44 5.30 83.74 2.65 2.25 9.98 8.90 7.34 4.45 5.32 83.83 2.42 2.44 9.99 9.51 6.67 4.47 5.33 83,92 2.19 2.63

10.00 10.12 6.01 4.49 5.35 84,02 1.97 2.82

Parameter values: G B = 9.00; H A = 1.00; H B = 0.80; P = 0.01; M = 0.50,

the sense of Loehman and Whinston (1974), would increase charges for both DMs. A similar argument may lead us to believe that the price should be inversely related to the second moment of the production time (M). Theorem 5 will show that these are not true. The next Theorem establishes that the demand rates of B (the low priority division) are inversely related to the expected production time and the second moment (i.e. variability) of production time.

Theorem 4. In Program 1, when incentive schemes are based on divisional performance, the demand rate of the low priority (i.e. low-benefit-low-cost) division is inversely related to: 1) the expected production time, P; and 2) the variability of production time, M.

Proof. Differentiating (15) with respect to P, we have

~X A ~X B RxA P + R x A x A OP + R x A x B OP = O,

~X A OX B RxBe + R x B x A - - ~ + RX~XB OP = O.

(21)

(22)

Note that RxAp, RxB P < 0 and RxAxA, RxBxA, RXBXB < 0. It is therefore clear that both (oXA/OP) > 0 and ( ~ x B / a P ) > 0 is not possible. Solving the equations, we have

~X B RxAxBRxAp - RxAxARx~ e

O---if - = R x A x A R x , x , _ R 2 x A x " , (23)

OX A RxBxBRxA P -- R x A x B R x ~ P

OP = R x A x A R x B x B _ R2Ax B (24)

We will show that ( ~ x A / o P ) < 0 and ( o X B / O P ) > 0 is not possible. Assume that ( o x A / o P ) < 0 and ( o x B / o P ) > 0. The denominator of (23) and (24) is positive by concavity of R(.). Hence, by assumption we must have from (23) RxAxBRxA e -- RxAxARxB P > 0, which implies

R x A p / R x B p > R x A x A / R x A x B . (25)

Similarly, from (24)

R x A p / R x B P < RXAxB/RxBxB. (26)


Table 2 Sensitivity to expected service time, P

P X A* X s* D A* D B* R* 6 A* 6 B*

0.004 2.209 17.424 4.952 5.374 92.993 4.467 0.557 0.006 8.605 9.361 4.736 5.308 89.697 2.579 2.268 0.008 9,824 7.129 4.600 5.321 86.739 2.143 2.665 0.010 10,119 6.006 4.485 5.347 84.017 1.971 2.815 0.012 10.115 5.288 4.383 5.376 81.494 1.893 2.878 0.014 9.987 4.769 4.289 5.406 79.145 1.858 2.903 0.016 9.805 4.367 4.202 5.434 76.951 1.849 2.907 0.018 9.598 4.039 4.121 5.462 74.896 1.855 2.901 0.020 9.381 3.765 4.046 5.489 72.964 1.872 2.887 0.022 9.164 3.529 3.974 5.514 71.146 1.898 2.869

The committed cost C(P, M) is monotone decreasing in P. Parameter values: G A = 10.00; G B = 9.00; H A = 1.00; H B = 0.80; M = 0.50.

S u b t r a c t (25) f r o m (26) t o g e t RZxax B > R x a x A R x B x B a c o n t r a d i c t i o n w i t h t h e c o n c a v i t y o f ( ~ x B / o p ) < O.

S i m i l a r l y w e c a n e s t a b l i s h t h a t (OXB/OM) < O. []

R . ~

W e s h o w b e l o w t h a t t h e p r i c e n e e d n o t b e m o n o t o n e in e i t h e r t h e e x p e c t e d p r o d u c t i o n t i m e o r t h e

v a r i a b i l i t y o f p r o d u c t i o n t i m e , a n d t h e r e f o r e t h e p r i c e is n o t a lways m o n o t o n e in c o m m i t t e d cos t s o f t h e

c o m m o n faci l i ty . T h e d e m a n d r a t e o f t h e h i g h - b e n e f i t - h i g h - c o s t d i v i s i o n a l so n e e d n o t n e c e s s a r i l y b e

m o n o t o n e in t h e e x p e c t e d p r o d u c t i o n t i m e .

T h e o r e m 5. In Program 1, when incentive schemes are based on divisional performance, 1) the price need not always be monotone in the committed costs o f the common facility; 2) the demand rate o f the high-benefit-high-cost division need not always be monotone in the expected

production time.

Proof. A n e x a m p l e d e m o n s t r a t i n g t h e n o n - m o n o t o n i c i t y is s u f f i c i e n t to e s t a b l i s h t h e t h e o r e m . T h e

e x a m p l e in T a b l e 2 e s t a b l i s h e s t h e n o n - m o n o t o n i c i t y o f p r i c e a n d t h e e x p e c t e d d e m a n d o f A w i t h r e s p e c t

t o t h e e x p e c t e d p r o d u c t i o n t i m e , P ( a n d c o n s e q u e n t l y t h e c o m m i t t e d c o s t ) w h e r e G A = 10.00; G B = 9.00;

H A = 1.00; H s = 0.80; M = 0.50.

Table 3 Sensitivity to variability of production time, M

M X A* X B* D A* D B* R* sA* 3B*

0.10 34.07 9.30 3.29 5.81 268.72 1.61 2.58 0.15 26.91 8.62 3.64 5.66 209.23 1.62 2.76 0.20 22.20 8.00 3.88 5.56 171.94 1.65 2.85 0.25 18.83 7.47 4.05 5.50 146.17 1.69 2.90 0.30 16.27 7.04 4.18 5.45 127.23 1.73 2.92 0.35 14.26 6.70 4.28 5.41 112.68 1.78 2.91 0.40 12.63 6.41 4.36 5.39 101.16 1.84 2.89 0.45 11.27 6.19 4.43 5.36 91.78 1.90 2.86 0.50 10.12 6.01 4.49 5.35 84,02 1.97 2.82 0.55 9.12 5.87 4.53 5.33 77,47 2.04 2.76

The committed cost C(P, M) is monotone decreasing in M. Parameter values: G A = i0.00; G B = 9.00; H A = 1.00; H B = 0.80; P = 0.01.


T h e example in T a b l e 3 es tab l i shes the n o n - m o n o t o n i c i t y of p r ice wi th r e spec t to the var iabi l i ty of p r o d u c t i o n t ime, M (and consequen t ly t he c o m m i t t e d cost) w h e r e G A = 10.00; G B = 9.00; H A = 1.00; H B = 0.80; P = 0.01. []

This p r o p e r t y of t he p r ice is cons i s ten t wi th the cause and effect c r i t e r ia for charging. I n c r e a s e d capac i t i es a re no t sha red p r o p o r t i o n a t e l y by bo th divisions. T h e H Q could r equ i r e one division to r educe d e m a n d s as p r o d u c t i o n capac i ty inc reases - as the example demons t r a t e s , the h igh-benef i t -h igh-cos t division could b e r e q u i r e d to d e m a n d less of the facility. T h e pr ice is r e d u c e d for the division tha t is r e q u i r e d to inc rease d e m a n d s and vice versa . Hence , the cause - to i n t e rp re t the cause and effect c r i t e r ia - is t he o p t i m u m d e m a n d ra t e which is sus ta ined by the pr ice .

4. Concluding remarks

This p a p e r ana lyzed the re levance of charg ing a p r ice for each usage when a p r o d u c t i o n facil i ty is sha red by two divisions and de lays a re costly. W h e n the d e m a n d ra tes chosen by the division m a n a g e r s a re unobse rvab l e de f i ned as s tochas t ic choice haza rd , and incent ive schemes are b a s e d on divis ional p e r f o r m a n c e , we e s t ab l i shed tha t a charge b a s e d on usage is r e q u i r e d to a l loca te the use of the c o m m o n facili ty. Incen t ives b a s e d on divis ional p e r f o r m a n c e achieve f i rm-wide eff iciency as a un ique equi l ibr ium. T h e p r ic ing scheme encou rages use of the c o m m o n facility, when the benef i t s of one division increases . However , any inc rease in the capac i ty of the c o m m o n facil i ty is no t sha red p r o p o r t i o n a t e l y by the divisions. Consequen t ly , the pr ices a re necessar i ly no t m o n o t o n e in the capac i ty of the c o m m o n facility.

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stochastic choice hazard and incentives in a common service facility

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