Workload-dependent capacity control

Workload-dependent capacity control Gergely Mincsovics • Nico Dellaert March 20, 2006 Department of Technology Management, Technische Universiteit Eindhoven, P.O. Box 513, NL-5600 MB Eindhoven, The Netherlands [email protected] • [email protected]

Abstract The development of job intermediation and the increasing use of Internet allow companies to carry out quicker and quicker capacity changes. In many cases, capacity adaptation can be driven according to the present workload to minimize idle capacity. We introduce a set of Markov chain models to represent workload-dependent capacity control policies. In our analysis we assume exponential interarrival and service times, although the model permits extension to phase-type distributions. We present two analytical approaches to stationary analysis in order to evaluate the policies due-date performance. One provides an explicit expression of throughput time distribution, the other is a fixed-point iteration method that calculates the moments of the throughput time. Eventually, we compare due-date performance, capacity, capacity switching, and lost sales costs to select optimal policies. We also give insight into which situations a workload-dependent policy is beneficial to introduce. Our results and tools can be used by manufacturing and service industries when declaring a static policy for dynamic capacity planning. Keywords: capacity planning, workload-dependency, due-date performance, Markov chain, state-dependent service, switching cost, lost sales.

1

1

Introduction

Most make-to-order companies do not have a constant flow of orders, which leads to a queue of customers. Despite the uncertainty in both arrival and service of orders, customers would like to have a due date both short and strict. To compensate for the variations in the orders’ frequency, a good practical solution is to try to match the production rate to the actual workload or to the length of the queue. Such a policy may bring benefits in labor cost reduction or in a better due date performance. This paper investigates the value of using an optimal workload-dependent policy, by where “workload” means the number of customers in the system. Our motivation for studying workload-dependent capacity planning originates from the area of engineer-to-order (ETO). As a definition for ETO, we quote that of Gelders [18]. “In an engineer-to-order environment a company designs and produces products to customer order.” In the same paper, Gelders conclude that ETO companies need “fastresponse capacity”, which is to be considered as a general characteristic of competitive ETO production. One motivating example for ETO manufacturing is shoe production in China. After a new shoe has been designed, a new order arrives to the production line. The production line receives orders of more designing companies, often unpredictably. Depending on the workload, the production line works in zero to three shifts. To each shift a quality controller is assigned. For this reason, shifts are always kept full. One can observe similar situations by building houses or by manufacturing situations with an expensive bottleneck machine. Companies in the service industry also often face unforeseen variations in the number of orders. One example is a special translator service, where topic leaders share translation tasks and assign them to specialized free-lance translators. A second example is a data recording company, where audio tapes are transcribed to digital text format in order to make later searches possible. While workers listen to the tapes, they type the text into a computer. In all four examples, “fast-response capacity” is not just desirable, but also admissible. ETO manufacturing does not usually require labor of high educational background; labor acquisition lead times are often very short. Although, in most service industry situations labor has specific knowledge, acquisition lead time substantially decreased in the last decade due to the increasing use of the Internet. In the case of the translator company, contact information of free-lance translators is carefully maintained, so distributing new tasks takes just hours. In our paper, we assume that we can afford instantaneous capacity changes as an abstraction of opportunity for “fast-response capacity”. We consider stationary Poisson jobs arrival process, exponentially distributed service time, and FCFS-type service. The number of employees (servers) is assumed to be a decision variable that depends on the number of jobs in the system. Our objective is to minimize the total average cost per time unit, where the costs consist of labor costs per employee, costs of hiring and firing employees, costs related to order acceptance and costs related to due-date deviations. For this problem, we present our model taking two evaluation approaches. One approach aims at exact calculation of the distribution of the throughput time (time spent in the system). Another approach, which can deal with problems of a larger scale, is based on a moment-approximation of the throughput time. With respect to the effectiveness of the flexible capacity it is essential to increase and decrease capacity at the right moment. We determine the proper switching states for small scale problem instances by searching exhaustively. The optimal strategy is compared in terms of costs with the simple fixed 2

capacity rules. Previously, a number of researchers pointed out relations among service rate, workin-process and due date performance. However, there is no model in the literature that incorporates all these three closely related notions. We believe that our paper makes valuable contribution by being the first to study a queuing system with adaptive capacity to satisfy the objective of having short and accurate due-dates. Moreover, our examples and their analysis provide insight into when workload-dependent strategy is effective, what characteristics this strategy has. The paper is organized as follows. In the next section, we describe the related literature. Then, in section 3, we give a mathematical formulation of our problem. Results on the effectiveness and characteristics of the strategy are shown in section 4. Finally, conclusions and plans for future extensions are presented.

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Related literature

Decisions on capacity changes were studied first in capacity expansion problems. In the case of deterministic demand with positive trend, Chenery [6] found that gas pipelines have extra capacity, permanently. This was the basis of his “excess capacity hypothesis” that says capacity is always larger than demand; optimal overcapacity is to be investigated by looking at economies of scale. Manne [7] revised Chenery’s hypothesis when extending his model. The extension of the model with a backlog option created an environment in which the “excess capacity hypothesis” no longer held. Manne also studied stochastic, stationary demand without the backlog option. This model resulted in a smaller deviation from what Chenery’s model suggested. Luss [8] gave a comprehensive review of the literature on capacity expansion. Models with capacity expansion/reduction decisions and hiring/firing costs are usually studied by means of dynamic programming. In the deterministic demand case, a discrete time DP model was established by Lippmann et al. [5], who considered a production to stock system with an overtime option. Pinder [1] deduced approximative randomized hiring/firing policy for stationary demand. Here, capacity is treated as discrete, which is a particular common point with our paper. However, Pinder did not consider due date performance in any form. Hiring/firing costs also appeared in the continuous time DP model of Bentolila et al. [4], where first of all sensitivity analysis on firing costs was presented. A production-to-order environment with non-stationary demand, including both capacity expansion/reduction decisions and hiring/firing costs, was studied by Rocklin et al. [2]. Rocklin et al. showed the optimality of the (S 0 , S 00 )-policy known from inventory theory by the means of discrete time DP. In their model, demand must always be met; if demand exceeds available capacity, the capacity must be increased to overcome the deficit. The work of Rocklin et al. was extended to production-to-stock environments by Angelus et al. [3], where another (S 0 , S 00 )-policy was optimal. The difference from the model of Rocklin et al. is the addition of a backlog option and the removal of hiring/firing costs. Contrary to the work of Rocklin et al., in this new model production lead time has been already represented. Too long lead times were penalized by backlog cost, though indirectly. When lead time is stochastic in nature, it is more appropriate to use a due date performance measure to model the related costs. This is the notion we used in this paper. One application of due date performance measures is the evaluation of batch scheduling rules. In the scheduling rules, capacity level and work-in-process are both consistently 3

present as parameters showing the strong relation to the due date performance (see e.g. Philipoom et al. [9]). Due date setting is a topic where work-in-process is also recognized as an adequate parameter that improves the performance of the rules (see e.g. Bertrand [10]). These two examples reveal the connection among capacity level, work-in-process and due date performance. Queuing models made use of servers with load-dependent service times for approximate performance analysis since the fundamental work of Avi-Itzhak et al.[11]. These servers are apt to represent a sub-network having nearly the same characteristics. Marie [12] introduced a similar approximation technique. A comparison is given in Baynat et al. [13]. In simple workload-dependent policy classes, the optimal (capacity) control policy can be analytically determined. Faddy [14] introduced the class of PλM policies for the control of water reservoirs. Namely, the policy PλM sets the output rate to M if the water level in the reservoir reaches (or exceeds) λ, and the output rate is set to zero if the reservoir is empty. This policy is still a subject of research (see Kim et al. [15]). Another simple policy class, the two-step service rule, was defined in Bekker et al. [16]. A policy of this class has a lower service rate, r1 , when the workload is not more than K, and a higher service rate, r2 , if the limit K is exceeded. This policy class has the drawback that the undesired frequent service rate changes are not excluded. Tijms et al. [17] proposed a different class of policies for inventory control, called two switch-over level rule. A rule is characterized by two inventory levels, y2 ≤ y1 . Inventory decrease to y2 /increase to y1 triggers compensating raise/reduction. This design restrains frequent service rate changes. In our paper, we specify a class of control policies for managing capacity with a higher degree of freedom, which allows to describe more characteristics than the three simple classes discussed, however we do not present an analytical form of the optimal solution.

3

Model formulation

In the context of this paper, a workload-dependent capacity planning policy is defined by two “switching points” for each pair of neighboring capacity levels. Switching points are specific workload values, for which a job arrival/departure can trigger a switch in capacity. If the system is at an up/down switching point, and a job arrives/departs, we switch from the lower/upper capacity level to the upper/lower. The switches, however, result in additional hiring/firing cost each time (cswitch ). Besides, the firm has an admissible domain of capacity levels; that is between Cmin and Cmax . Naturally, the firm also needs to pay for used capacity. The cost of having capacity level c is given by c · ccapacity per time unit. Although this paper do not investigate the impact of efficiency aspects, one may also take into account the psychological effect of lower or higher workload [19], and the efficiency of using different levels of capacity [20]. The service rate can be measured and given for each workload and capacity level (µw,c ), and include in our model. The firm works with a fixed due date (d). It has to pay charges for each time unit when a job is late (ctardiness ). A smaller cost is due for holding if a job is ready before the due date (cearliness ). The number of jobs with which the firm can deal is constrained by a constant number, Wmax . This is the maximum number of jobs the firm can afford. All the new jobs are refused if the firm has already Wmax number of jobs. Therefore, lost sales may occur, which incurs a virtual expense of clostsales for each occasion. To sum up, we have a five element vector of cost coefficients γ = (ccapacity , cswitch , clostsales , cearliness , ctardiness ). 4

In this section, we derive mathematical formulas for the main characteristics of workload-dependent capacity planning policies, and formalize the firm’s capacity control problem.

3.1

Definitions, assumptions and problem formulation

We assume a stationary environment in which both arrivals and departures are Poisson processes. Arrivals have a constant rate of λ; departures have, in general, capacity level and workload-dependent rates. We denote by µw,c the service rates for workload w, and capacity level c. For the sake of better comprehension, we assume µw,c = cµ for all w and c. We define the feasible capacity planning policies. A policy can be characterized by a triple π =(Γmin , Γmax , Θ). The terms Γmin , and Γmax stand for the minimum, and maximum capacity levels which are used by the policy. Necessarily, these terms have to satisfy the inequalities, Cmin ≤ Γmin ≤ Γmax ≤ Cmax . The term Θ denotes a (Γmax − Γmin )-by-2 matrix, which describes all the switching points. For the cases when Γmin = Γmax , we have an empty matrix. We use indices c/c + 1 or c + 1/c with c ∈ [Γmin . . . Γmax − 1] to show that the switch occurs between the capacity levels c and c + 1 either up or down. Particularly, Θc/c+1 , and Θc+1/c are workloads, where the capacity is changed from c to c + 1, upwards, or from c + 1 to c, downwards, if a job arrival/departure occurs. The elements of Θ are constrained by Wmax , and by each other. On the one hand, 1 ≤ ΘΓmin +1/Γmin , and ΘΓmax −1/Γmax ≤ Wmax − 1. On the other hand, Θc+1/c ≤ Θc/c+1 + 1 needs to hold for all c ∈ [Γmin . . . Γmax − 1], and Θc+1/c ≤ Θc+2/c+1 , and Θc/c+1 ≤ Θc+1/c+2 for all c ∈ [Γmin . . . Γmax − 2]. We assume that capacity adjustments can be done instantaneously, in parallel with the change in workload. For the five cost coefficients mentioned in the model description, we have the related costs. These costs can be separated into two groups. The costs of capacity, switching, and the lost sales penalty are functions of the policy only. The costs of earliness and tardiness are functions of the due date, additionally. Now, we are able to formulate the capacity control problem as min

costgroup1 (π) + costgroup2 (π, d)

π∈Ω(Cmin ,Cmax ,Wmax )

(1)

where Ω (Cmin , Cmax , Wmax ) is the set of constraints, d is the fixed due date, costgroup1 (π) is the sum of costcapacity (π), costswitch (π), costlostsales (π), and costgroup2 (π, d) is the sum of costearliness (π, d), and costtardiness (π, d).

3.2

Evaluation of cost functions

In this section, we aim to express the costs one by one. First, we start with the ones that are independent from the due date. After that, we derive the due date dependent ones. We create a Markov chain, M C π,λ,µ , according to the policy, the given arrival rate, and the given service rate unit. In this Markov chain, we have the states labeled by (w, c), the workload (number of jobs in the system), and the capacity usage. A policy π can be given by the arrival matrix (Aπ ), and the departure matrix (Dπ ). These matrices are squared; rows and columns are indexed by the states of the Markov chain. Elements are ones at the related arrival or departure arcs of the Markov chain and zeros elsewhere. The state space of M C π,λ,µ depends on the policy. As an example, the transition rate diagram of M C π,λ,µ can be seen in Figure 1 when the policy is π = (1, 3, [3, 1; 4, 2]).

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Figure 1: M C π,λ,µ for π = (1, 3, [3, 1; 4, 2]) Via steady state analysis, the first cost group can be easily evaluated. If we add λA to µDπ weighted in rows (w, c) by capacity c, we can obtain Qπ , the transition rate matrix of the Markov chain. π

π Qπ(w1 ,c1 ),(w2 ,c2 ) = λAπ(w1 ,c1 ),(w2 ,c2 ) + c1 µD(w 1 ,c1 ),(w2 ,c2 )

(2)

π π π We define the diagonal elements ½ πof πQ as the negative state leaving rates: vi := diag (Q )i = P Q T = 0 − Qπi,j . Solving the LP , where e is the all one vector, we have the eT π = 1 j6=i distribution of time being in a state (T π ). Eventually, we can specify all the cost of the first group. P P π costcapacity (π) = c ccapacity Tw,c c

costswitch (π)

=

w

2cswitch λ

P

(w,c)∈U

π Tw,c

(3)

costlostsales (π) = clostsales λP (lost sale) π where P (lost sale) = TW is the probability that a new job is rejected, and U is max ,Γmax the set of states from which the up-switches in capacity are made. Note, that it is enough to account costs for up-switches (or down-switches) twice because of the stationarity. In what follows, we deduce the earliness and tardiness costs in two different ways. Both approaches aim at identifying the throughput time (or sojourn time) as a random variable.

3.2.1

Derivation of the throughput time distribution

In order to derive an explicit formula for the throughput time distribution, we extend our Markov chain. We call the extended Markov chain EM C π,λ,µ . We suppose that the Markov chain M C π,λ,µ is in an arbitrary state, corresponding to the stationary distribution, T π . Then, a new, pointed, job arrives. We are going to follow the pointed job while it is in the system. We observe its position in the queue in addition to the system’s workload and the used capacity. Consequently, EM C π,λ,µ has states labeled by (w, c, q), the workload, capacity, and queue position of the pointed job, respectively. First note that Γmin ≤ c ≤ Γmax , and 0 ≤ q ≤ w ≤ Wmax holds for every state. Now suppose that the pointed job arrives. Its queue position remains the same when new jobs arrive, but it 6

decreases when jobs depart. Hence, in EM C π,λ,µ we can define both A¯π extended arrival ¯ π extended departure matrix based on Aπ , and Dπ , as follows. matrix, and D ½ 1 , if Aπ(w1 ,c1 ),(w2 ,c2 ) = 1, and q1 = q2 A¯π(w1 ,c1 ,q1 ),(w2 ,c2 ,q2 ) = 0 , otherwise ½ π 1 , if D(w = 1, and q1 − 1 = q2 1 ,c1 ),(w2 ,c2 ) ¯π D = (4) (w1 ,c1 ,q1 ),(w2 ,c2 ,q2 ) 0 , otherwise for all (w1 , c1 , q1 ), and (w2 , c2 , q2 ). The transition rate diagram of EM C π,λ,µ for the policy π = (1, 3, [3, 1; 4, 2]) can be seen in Figure 2.

Figure 2: EM C π,λ,µ , for π = (1, 3, [3, 1; 4, 2]) ¯ π , can be given in two steps like before. The extended transition rate matrix, Q ¯π ¯π ¯π Q (w1 ,c1 ,q1 ),(w2 ,c2 ,q2 ) = λA(w1 ,c1 ,q1 ),(w2 ,c2 ,q2 ) + µc1 D(w1 ,c1 ,q1 ),(w2 ,c2 ,q2 ) if (w1 , c1 , q1 ) 6= (w2 , c2 , q2 ). Next, we define the vector of state leaving rates, v¯π . P ¯π v¯sπ = Q s,r

(5)

(6)

s6=r

¯ π can be defined as Q ¯ π = −¯ Now, the remaining diagonal elements of Q vsπ . s,s ¯ π we can derive the extended transition probability matrix, P¯ π . From Q ( ¯π Qr,s , if r 6= s π v¯rπ P¯r,s = 0 , if r = s

(7)

To make the Markov chain EM C π,λ,µ more manageable, we use uniformization. Let v¯U π = max v¯sπ be the uniformized state leaving rates. The uniformized extended transis ½ v¯rπ π , if r 6= s P¯ Uπ v¯U π r,s = tion probability matrix, elementwise, is P¯r,s . v¯rπ 1 − v¯U π , if r = s Using the matrix, P¯ U π , we can determine the probability of getting from state r to s in time t. ∞ ¡¡ ¢ ¢ −¯vU π t (v¯U π t)n P Uπ n π ¯ ¯ P e P (t) = (8) r,s

n=0

r,s

n!

Since the states (w, c, 0), where the pointed job can exit the system, are absorbing states we can express the throughput time CDF as the sum of probabilities of getting to certain exit states as follows

7

F r,π (t) =

P

π P¯r,s (t) is the throughput time CDF if the pointed job arrives when

s=(w,c,0)

the system is in state r = (wr , cr , qr ) with qr = wr . On the one hand, when the pointed job arrives, we have somewhat deviated from the stationary distribution (T π for M C π,λ,µ ). Using the arrival matrix (Aπ ), we can get the distribution of being in a state right after the arrival (TAπ ) in a straightforward manner; that is TAπ = T π Aπ . On the other hand, when the pointed job arrives, its queue position is equal to the system’s workload. Based on this fact, we can express the after arrival distribution of the extended Markov chain, EM C π,λ,µ (we call it T¯Aπ ) from the after arrival distribution of ½ M C π,λ,µ . π , if q = w TA,(w,c) π ¯ (9) TA,(w,c,q) = 0 , if q 6= w Once we have the starting distribution T¯Aπ , the throughput time CDF (F π ) can be expressed. P π F π (t) = T¯A,r F r,π (t) (10) r=(wr ,cr ,qr ) qr =wr

Finally we can give formulas for the due date dependent cost functions. Rd costearliness (π, d) = cearliness λ (1 − P (lost sale)) (d − t) dF π (t) costtardiness (π, d) = ctardiness λ (1 − P (lost sale))

0 ∞ R

(t − d) dF π (t)

(11)

d

3.2.2

Moment approximation of the throughput time

Apart from the distribution function, we also derive a moment approximation for the throughput time as an alternative for the steps from (8) to (10). This approach allows evaluation of large scale problems, as it both speeds up the calculations and needs less memory. Besides, throughput time moments can be determined with higher accuracy than extracting them from the throughput time distribution. We evaluate the moments based on the equation that describes the relation of the conditional expected throughput times. We denote the kth moment of the throughput time if starting from state r by (Xr )k and the duration of the visit to state r by Zr . Then we have h i P h i k k π ¯ E (Xr ) = Pr,s E (Zr + Xs ) (12) s

where P¯ π is the extended transition probability matrix, containing the probabilities of going from state r to some neighboring state s. Similar to the approach in the previous subsection, a state is described by the components (w, c, q) and therefore the definition of P¯ π is identical to the one in the previous subsection. For small state spaces, we can solve this linear system by matrix inversion, but for bigger state spaces, we need a vector iteration to determine the moments for the relevant states. This vector iteration can be established by indexing the equation by the iterator n, and declaring the starting state. h i h i P  π  En+1 (Xr )k = P¯r,s En (Zr + Xs )k s h i (13)  E (X )k = 0 0 r Using independency of conditional throughput time and visit duration, we can write

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i P i h i h i P k ¡ ¢ h k ¡ ¢ h k−j j k−j j k k E (Zr + Xs )k = E (Z ) (X ) = E (Z ) E (X ) r s r s j j j=0

(14)

j=0

From this, we notice that for the evaluation of higher moments all the previous ones are needed. In the general form, we evaluate the first K moments. We can use the following algorithm. We increase the moment iterator, k from 1 to K. Forh each ik we take limit in ∞ for n with the vector iteration to evaluate the moments E (Xr )k one after the other, inductively. Similarly to (10), we weight the conditional moments by the after arrival distribution of the extended Markov chain, EM C π,λ,µ (that £ ¤ is T¯Aπ ), which gives the kth moment of the throughput time, E X k . h i £ k¤ P k π ¯ (15) E X = TA,r E (Xr ) r=(wr ,cr ,qr ) qr =wr

We show that the vector iteration (13) is a fixed-point iteration for each k. Firstly, the solution of the conditional expected throughput time equation (12) is a fixed-point of the iteration (13). Secondly, from (14) we can rewrite the iteration (13), h i P i h i h i P k−1 P ¡k ¢ h E (Zr )k−j E (Xs )j (16) En+1 (Xr )k = P¯ π En (Xs )k + P¯ π s

r,s

r,s

j=0

j

h i´ ³ h i´ k k By substituting xn+1 = En+1 (Xr ) , xn = En (Xs ) and h i h i P π k−1 P ¡k¢ c = P¯r,s E (Zr )k−j E (Xs )j , we have j s

³

s

j=0

xn+1 = P¯ π xn + c (17) ° ° ¡ ¢ with P¯ π having a spectral radius ρ P¯ π ≤ °P¯ π °∞ = 1. In practice, we can evaluate the equation for the first two moments, E [X] and E [X 2 ], and fit a suitable distribution, e.g. in the class of gamma distributions to gain the throughput time CDF, F π (t).

3.3

Model extensions

Extensions to our model are numerous. We summarize the most relevant ones only. • More general interarrival and service time distributions can be handled by using phase-type distributions instead of Poisson. • In some situations only one server can be assigned to a job. • In many cases service rate efficiency has a decreasing trend when more and more capacity is used. • Capacity costs are usually increased for higher capacity levels. • Switching costs for starting/suspending production are usually higher than increasing/decreasing production rate when production already runs.

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Results

We have two goals while giving insight. On the one hand, we would like to show in which situations workload-dependent capacity planning is worth to use. We study the effect of setting high/low switching cost coefficient as well as the constellations of due date, and 9

arrival rates for a fixed service rate. On the other hand, we would like to characterize the workload-dependent policies as compared to the fixed capacity policies, the policies that can use one capacity level only. Our model has its limitations; it cannot be evaluated in acceptable time to optimality for systems that can either use many capacity levels, or handle high number of orders economically. In these cases the number of workload-dependent policies, and their state space increases to large values. We included the number of policies in Table 1. The state space of the policy having the most states is (Wmax − Cmax + 1) (Cmax + 1) if Cmin = 0. Cmax \ Wmax 0 1 2 ¤ ¡ £3 ¢ 4 5 6 Table

1 2 3 1 1 1 3 5 8 5 10 21 7 15 35 9 20 49 11 25 63 13 30 77 1: Number of

4 5 1 1 12 17 40 69 78 157 117 260 156 364 195 468 policies for

¤ ¡ £6 ¢

7 8 9 10 11 1 1 1 1 1 1 23 30 38 47 57 68 110 ¥ 165 236 325 434 565 ¨ 785 1198 1757 2493 §288 ¦ 490 531 1005 1783 2996 4809 7425 795 1626 3132 5719 9962 16648 1060 2275 4642 9037 16844 30162 Cmin = 0; initial setting is marked

As a starting setting we take Cmin = 0 and Cmax = 3, having in mind shifts from 0 to 3 in number. Besides, we assume that more than 6 order is too risky to take on, meaning Wmax = 6, and also assume a service rate of µ = 0.04. In our cost coefficient test-bed the capacity and switching cost coefficients have a pointed role. First, we fix the capacity cost coefficient to 100, to normalize the test-bed. We observe the models behavior for different switching cost coefficients, interarrival rates, and due date settings. For each of the remaining cost parameters we set up three values. To the coefficient, clostsales 3000, 4000, and 5000 are assigned. For cearliness we take 1, 2, and 5. Finally, ctardiness has values 10, 25, and 100, respectively.

4.1

Value of workload-dependent capacity control

We investigate the value of workload-dependency in capacity control in different environments. We compare workload-dependent capacity policies with fixed capacity policies. The value of workload-dependency is defined as the expected total cost ratio of the optimal fixed capacity and the optimal workload-dependent policy. As fixed capacity policies form a subset in the set of workload-dependent capacity policies, this ratio is never less than one. Therefore, we consider only the cost excess percentage (CE%) due to not using the optimal workload-dependent policy. E.g. a CE% value of 5 means that taking the costs of the optimal workload-dependent policy 100%, one has to pay 105% if using the optimal fixed policy. We conduct numerical experiments to circumscribe the cases where high CE% values are to be expected. In Figure 3 we depicted contour lines of CE% value functions for cswitching values 1000 and 3000. We take the average of the 33 = 27 outcomes for the different settings of the rest of the test-bed variables. Contour lines are drawn at values 0.25, 1, 4, 8, and 12, respectively. In what follows, we discuss seven major properties of the CE% value that can be observed in this figure. 1. When cswitch decreases, CE% value increases As fixed capacity policies use a constant capacity level, they do not incur switching cost. All the rest of the policies do have switching cost. Thus, the switching cost 10

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Figure 3: Average CE% values for Cmin = 0, Cmax = 3, Wmax = 6, and µ = 0.04 coefficient affects the CE% value directly. The lower the switching cost coefficient, the higher the CE% value. 2. When d decreases, an increased capacity usage is optimal Tight due dates (d) imply higher capacity usage. However, capacity ≡ Cmax is not always optimal among the workload-dependent policies. 3. When d → ∞, CE% value decreases Long due dates penalize less the long waiting times. Long waiting times have a utilization smoothing effect similar to workload-dependent capacity flexibility. This means that for high enough due date (d) values CE% value is low. Due to the utilization smoothing effect, a high due date can be a substitute of a smaller interarrival rate (λ) in average effective workload. This λ/d substitution effect is limited by the lost sale costs and not by ergodicity. 4. When λ & 0, a decreased capacity usage is optimal; CE% value increases As interarrival rate (λ) decreases, we use less and less capacity. Γmin also decreases for the optimal policy. This way, there is more and more room for workloaddependent policies to use capacity levels. This results in an increasing trend in the CE% value. 5. When λ → ∞, an increased capacity usage is optimal; capacity ≡ Cmax is also optimal; CE% value decreases When λ increases, we use more and more capacity, and Γmin also increases for the optimal policy. In a similar way as before, we can see that this gives a decreasing trend in the CE% value. 6. Cuts at fitting regions It would make a difference, if we could set an optimal fixed capacity level on a continuous basis. Discreteness creates regions where the optimal fixed policies get close to the continuous fixed optimum, and regions where they are rather far. We call these regions fitting, and not fitting regions. To identify the fitting and not fitting regions, we plot the optimal continuous fixed policies in Figure 7. The contour lines of the continuous optimum also appoints the lines of λ/d substitution mentioned in property 4. Fitting regions create cuts in the CE% value, as there 11

fixed policies perform much better, while the workload-dependent ones are not particularly affected. 7. Cut at λ = µ, non-closable systems Constant zero capacity creates only lost sales which is extremely not economical. Consequently, the optimal fixed policy is the constant one capacity, even for small λ, and high d values (in the test-bed, for all the cost and due date values when λ = µ4 ), which is not competitive with the optimal flexible policy. An environment where the λ = µ cut vanishes is the non-closable systems. We define closable systems which have Cmin = 0, and non-closable systems which have Cmin > 0. CE% values of the non-closable system with Cmin = 1, and Cmax = 3 are depicted in Figure 4. 180 cswitching=1000

0. 25 1

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Figure 4: Average CE% values for Cmin = 1, Cmax = 3, Wmax = 6 , and µ = 0.04 As we can see, there is a high similarity with Figure 3 when µ < λ. The major difference with the closable systems is that the constant one substitutes the constant zero capacity policy, so we substitute an extremely not economical policy with a normal one. Therefore, for low λ values CE% value start to decrease again in the non-closable case. To sum up, property 4 is different for the non-closable systems. 4.n When λ & 0, capacity ≡ Cmin is optimal for non-closable systems; CE% value decreases Moreover, capacity ≡ Cmin is the only optimal policy for small interarrival rate (λ) values, or even for slightly larger interarrival rates if d is higher. We also performed sensitivity analysis in Wmax and Cmax . We observed an increase of the CE% value in both cases. The reason is that the larger the Wmax or Cmax value, the more workload-dependent capacity policies there are, while the number of fixed capacity policies remain the same or increase only by one. The increase in Wmax induces an overall new old = 4 affects = 3 to Cmax increase in CE% values, while the increase of Cmax from Cmax old the graph of CE% values only at λ values above around µCmax .

4.2

Characterization of workload-dependent policies for uncapacitated, large scale settings

We take the uncapacitated setting, Cmin = 0, Cmax = ∞ with λ = 0.10, µ = 0.04, d = 30, and cost coefficients, γ = (100, 1000, 3000, 1, 10), and observe changes of the optimal workload-dependent policy for an increasing Wmax value.

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Figure 5: Characteristics of optimal workload-dependent policies for changing Wmax We have found that the capacity/lost sales cost component is monotone increasing/decreasing in Wmax for the optimal policy. Moreover, the simultaneous stagnation of these two costs entails the stagnation of the total costs. Figure 5 shows that • the total cost is decreasing in Wmax to a certain value; thereafter it gets stagnating. As a consequence, it is safer to allow high workload values in the uncapacitated case. • the number of used capacity levels shows a monotone increasing, concave shape. Γmin is unaffected; here it is always one. • optimal switching points of the first levels tend to change less and less. (We measure a Cauchy-type convergence of the switching points in the first four levels. We count the not matching switching points of optimal policies of consecutive Wmax values.) 7 6

capacity

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6 8 workload

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Figure 6: First six pair switching points of optimal policies for 14 ≤ Wmax In Figure 6, the first six pair of switching points can be seen, in which optimal policies for 14 ≤ Wmax had no difference. We can see that up- and down-switching points limit the attainable states in a closely linear manner.

4.3

Characteristics of workload-dependent policies vs. policies

fixed

For a given policy, π, we define permanent capacity as Γmin . Note, that for a fixed capacity policy all the capacity is permanent. We can observe in Figure 7 that the permanent capacity of an optimal flexible policy is bounded from above by that of an optimal continuous fixed policy (in this policy class, the fixed capacity can be set to any real values between Cmin and Cmax ). In addition, optimal flexible permanent capacity

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fixed continuous cswitching=1000

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Figure 7: Permanent capacity for optimal continuous fixed and workload-dependent policies (Cmin = 1, Cmax = 3, Wmax = 6, µ = 0.04) follows the contour lines of the continuous optimum even for low switching cost coefficients (cswitching ), when the bound is not strict. We try to answer the question, why workload-dependent policies outperform the fixed policies. Workload-dependent policies often compensate switching costs using less capacity, avoiding lost sales, and providing a throughput that gives less earliness-tardiness costs. There is an example shown in Table 2 where the optimal workload-dependent policy beats the fixed optimum in all the costs but costswitching . λ µ 0.07 0.04 opt. π (1,3,[3,1;4,2]) fixed π (2,2,[ ]) cont. fixed 1.88

d 30 total cost 232.0 CE% 8.9 CE% 8.5

ccapacity 100 costcapacity 182.0 costcapacity 200 costcapacity 187.9

cswitching 1000 costswitching 18.7 costswitching 0 costswitching 0

clostsales 4000 costlostsales 12.6 costlostsales 25.9 costlostsales 32.0

cearliness 2 costearliness 0.7 costearliness 0.9 costearliness 0.8

ctardiness 25 costtardiness 18.1 costtardiness 25.9 costtardiness 31.2

Table 2: An example, where all cost components are decreased by workload-dependency, except for the switching cost. We emphasize that costearliness and costtardiness both decreased. Thus, the throughput time distribution adopts better to the due date (d). The reason is twofold. With workload-dependency the expected value of the throughput time gets closer to the actual due date. In addition, the variance of throughput time decreases. Table 3 shows the expected value and standard deviation of throughput times of the previous example. Policy π = (1, 3, [3, 1; 4, 2]) π = (2, 2, [ ]) continuous fixed 1.88 Table 3: Expected value and standard deviation

E Std 35.5 20.4 39.0 30.3 43.8 32.9 of throughput time in the same setting.

The transition rate diagram of the corresponding Markov-chain, M C (1,3,[3,1;4,2]),λ,µ can be seen in Figure 1.

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5

Conclusions

Under the restriction that we can afford instantaneous capacity changes, we have defined the set of workload-dependent capacity planning policies. These policies have the ability to react on dynamic workload changes. We have introduced a model that gives insight into the value of workload-dependent capacity control, conducting stationary analysis. We have shown that non-closable systems differ from the closable ones, but the structural difference is limited to demand arrival rates below the unit service rate. We have measured the value of using workload-dependency. Large switching cost coefficients, high demand rates, and long due dates are detrimental, while being unable the set appropriate fixed capacity, high workload limits (Wmax ), and low demand rates above the unit service rate are beneficial for the value. If the demand arrival rate is below the unit service rate, for closable systems, the value of workload-dependency further increases towards zero, whereas for non-closable systems it decreases. We have observed λ/d substitution effect, which means that a long due date (d) can be a substitute of a smaller interarrival rate (λ) in average effective workload. We have shown continuous fixed optimum contours that determine lines for λ/d substitution. Furthermore, we have observed that these lines approximately coincide with the contour lines of permanent capacity usage of the optimal workload-dependent policies. In the uncapacitated case, we have observed that using a higher order-acceptance rate, or equivalently a high workload limit (Wmax ), is safe for the workload-dependent strategy. We have found that for high workload limits, the optimal up- and down-switching points tend to change less and less, and appear to form two lines. Finally, we have revealed that compared to the optimal fixed capacity policies, the optimal workload-dependent capacity planning policies can achieve a better due date performance thanks to their throughput time being closer to the due date. In particular cases they can also spare capacity, and decrease lost sales probability at the same time. Further research may reveal to what extent the assumption of instantaneous capacity changes is restrictive in workload-dependent capacity planning.

References [1] J.P. Pinder; An approximation of a Markov decision process for resource planning; Journal of the Operational Research Society, Jul. 1995, 46(7), p819-830. [2] S.M. Rocklin, A. Kashper, G.C. Varvaloucas; Capacity expansion/contraction of a facility with demand augmentation dynamics; Operations Research, Jan.-Feb. 1984, 32(1), p133-147. [3] A. Angelus, E.L. Porteus; Simultaneous capacity and production management of short-life-cycle produce-to-stock goods under stochastic demand; Management Science, Mar. 2002, 48(3), p399-413. [4] S. Bentolila, G. Bertola; Firing costs and labour demand: How bad is eurosclerosis?; The Review of Economic Studies, Jul. 1990, 57(3), p381-402. [5] S.A. Lippman, A.J. Rolfe, H.M. Wagner, J.S.C. Yuan; Algorithms for optimal production scheduling and employment smoothing; Operations Research, Nov.-Dec. 1967, 15(6), p1011-1029.

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[6] H.B. Chenery; Overcapacity and the acceleration principle; Econometrica, Jan. 1952, 20(1), p1-28. [7] A.S. Manne; Capacity expansion and probabilistic growth; Econometrica, Oct. 1961, 29(4), p632-649. [8] H. Luss; Operations research and capacity expansion problems: a survey; Operations Research, Sep.-Oct. 1982, 30(5), p907-947. [9] P.R. Philipoom, M.K. Malhotra, J.B. Jensen; An evaluation of capacity sensitive order review and release procedures in job shops; Decision Sciences, Nov.-Dec. 1993, 24(6), p1109-1133. [10] J.W.M. Bertrand; The effect of workload dependent due-dates on job shop performance; Management Science, Jul. 1983, 29(7), p799-816. [11] B. Avi-Itzhak, D.P. Heyman; Approximate queuing models for multiprogramming computer systems; Operations Research, Nov.-Dec. 1973, 21(6), p1212-1230. [12] R.A. Marie; An approximate analytical method for general queueing networks; IEEE Transactions on Software Engineering, Sept. 1979, SE-5(5), p530-538. [13] B. Baynat, Y. Dallery; A unified view of product-form approximation techniques for general closed queueing networks; Performance Evaluation, Nov. 1993, 18(3), p205-224. [14] M.J. Faddy; Optimal control of finite dams: discrete (2-stage) output procedure; Journal of Applied Probability, 1974, 11, p111-121. [15] J. Kim, J. Bae, E.Y. Lee; An optimal PλM -service policy for an M/G/1 queueing system; Applied Mathematical Modelling, Jan. 2006, 30(1), p38—48. [16] R. Bekker, O.J. Boxma; An M/G/1 queue with adaptable service speed; SPORReport 2005-09, Eindhoven University of Technology, The Netherlands [17] H.C. Tijms, F.A. van der Duyn Schouten; Inventory control with two switch-over levels for a class off M/G/1 queueing systems with variable arrival and service rate; Stochastic Processes and Their Applications, Jan. 1978, 6(2), p213-222. [18] L.F. Gelders; Production control in an ‘engineer-to-order’ environment; Production Planning and Control, Jul.-Sept. 1991, 2(3), p280-285. [19] J.W.M. Bertrand, H.P.G. van Ooijen; Workload based order release and productivity: a missing link; Production Planning & Control, 2002, 13(7), p665-678. [20] M. Schlichter; Stork Fokker - The production planning and control system; unpublished postgradutate dissertation, Logistics Management Systems program, Technische Universiteit Eindhoven, The Netherlands, 2005, p55.

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