July Abstract. theoretical results have been published on approximation methods for these problems. To

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1 Fully Polynomial Approximation Schemes for SingleíItem Capacitated Economic LotíSizing Problems C.P.M. van Hoesel æ A.P.M. Wagelmans y July 1997 Abstract NPíhard cases of the singleíitem capacitated lotísizing problem have been the topic of extensive research and continue to receive considerable attention. However, surprisingly few theoretical results have been published on approximation methods for these problems. To the best of our knowledge, until now no polynomial approximation method is known which produces solutions with a relative deviation from optimality that is bounded by a constant. In this paper we show that such methods do exist, by presenting an even stronger result: the existence of fully polynomial approximation schemes. The approximation scheme is ærst developed for a quite general model, which has concave backlogging and production cost functions and arbitrary èmonotoneè holding cost functions. Subsequently we discuss important special cases of the model and extensions of the approximation scheme to even more general models. Subject classiæcation: Analysis of algorithms, suboptimal algorithms: fully polynomial approximation schemes. Dynamic programmingèoptimal control: lotísizing models. Inventoryèproduction: singleíitem capacitated lotísizing. In the singleíitem capacitated economic lotísizing problem we consider a production facility which manufactures a single product to satisfy known integer demands over a ænite planning horizon of T periods. At each period, the production and holdingí backlogging cost functions are given, and the amount of production is subject to a æ Department of Quantitative Economics, Maastricht University, P.O. Box 616, 6200 MD Maastricht, The Netherlands; eímail: s.vanhoesel@ke.unimaas.nl y Econometric Institute, Erasmus University Rotterdam, P.O. Box 1738, 3000 DR Rotterdam, The Netherlands; eímail: wagelmans@few.eur.nl 1

2 capacity limit. The problem is that of determining the amounts to be produced in each period such that all demand is satisæed and the total cost is minimized. Florian, Lenstra and Rinnooy Kan è1980è and Bitran and Yanasse è1982è have shown that the singleíitem capacitated lotísizing problem is NPíhard, even for many special cases. For notable exceptions, we refer to Florian and Klein è1971è, Bitran and Yanasse è1982è, Rosling è1993è, Chung and Lin è1988è and Van Hoesel and Wagelmans è1996è. NPíhard cases of the problem have been the topic of extensive research and continue to receive considerable attention. The proposed solution methods are typically based on dynamic programming èfor instance, Kirca 1990; Chen, Hearn and Lee 1994a, 1994b; Shaw and Wagelmans 1995è, branchíandíbound èfor instance, Baker et al. 1978; Erenguc and Aksoy 1990è, or a combination of the two èfor instance, Chung, Flynn and Lin 1994; Lofti and Yoon 1994è. It should also be mentioned that a lot of research has been devoted to ænding a èpartialè polyhedral description of the set of feasible solutions of lotísizing problems; see, for example, Pochet è1988è and Leung et al. è1989è, Pochet and Wolsey è1993, 1995è and Constantino è1995è. The main motivation for studying the polyhedral structure of capacitated singleíitem models is to use the results to develop branchíandícut methods for more complicated problems, such as multiíitem problems, that contain this model as a substructure. However, the branchíandícut approach has not èyetè resulted in competitive algorithms for the capacitated singleíitem problems themselves. Surprisingly, very few theoretical results have been published on approximation methods for capacitated singleíitem problems. The only notable exceptions are Bitran and Matsuo è1986è and Gavish and Johnson è1990è. The ærst article considers approximation formulations which are solvable in pseudoípolynomial time. The optimal solution of an approximation formulation can be used as an approximate solution of the actual problem. For special cases of the problem, it can be shown that the relative error of the approximate solution value can be bounded by an expression which depends on the input data. The authors argue that this bound will be satisfactory for practical purposes. Gavish and Johnson present a fully polynomial approximation scheme which is applicable to a large class of capacitated singleíitem scheduling problems. Their approach, however, appears to be more suitable for continuous time models, than for discrete times models, such as those considered in this paper. The reason is that in calculating an approximate solution, the discrete nature of the problem is ignored. Therefore, ëthe translation from this solution back to an equivalent discreteítime model may be diæcult" èp. 74è. Another drawback of the approach is that the error of the approximate solution is not measured as the usual relative error with respect to the optimal value, but as ratio of the value of the approximate solution and an upperbound on value of any feasible solution. èthe ratio of this upperbound 2

3 and the optimal value may be arbitrarily large.è Gavish and Johnson justify the alternative error measure by pointing out that the usual relative error is inadequate for a minimization problem if there is a possibility that the optimal value is zero. Although this is true in general, we will explain in Section 1 why this is not a relevant argument for the lotísizing problems considered in this paper. To summarize the above discussion: to the best of our knowledge, until now no polynomial approximation method is known for the singleíitem capacitated lotísizing problem which produces solutions with a relative deviation from optimality that is bounded by a constant. In this paper we will show that such methods do exist, by presenting an even stronger result: the existence of fully polynomial approximations schemes. Recall that such algorithms determine for any æé0 and any problem instance, a solution of which the relative deviation from optimality is at most æ, in a running time which is polynomial in both 1=æ and the size of the problem instance. This paper is organized as follows. In Section 1, we ærst deæne the model for which the approximation scheme will initially be developed. It assumes concave backlogging and production functions. The holding cost functions are only assumed to be noní decreasing. In Section 2 we present an exact dynamic programming procedure for this model. This algorithm diæers from and is more complicated than the standard DP approach presented by Florian, Lenstra and Rinnooy Kan è1980è. Two approximation methods, one of which is based on the DP algorithm, are described in Section 3, and in Section 4 we show how these methods can be combined to yield a fully polynomial approximation scheme. In Section 5 we discuss two important special cases, namely the model without backlogging èwhich allows the concavity assumption on the production cost functions to be droppedè and the model in which all cost functions are pseudoílinear èwhich allows an improved complexityè. Furthermore, we will show in this section that our results can be extended to models with features such as bounds on the inventory levels, piecewise concave cost functions and startíup and reservation costs. Section 6 contains concluding remarks. 1 Problem deænition In this section we deæne the model for which the approximation scheme will initially be developed. Let T denote the length of the planning horizon. For each period t 2f1;...;Tg we deæne: d t : demand in t; x t : production level in t; c t : production capacity in period t; 3

4 I t : inventory level at the end of t; p t èx t è: production costs in t, a function of x t ; h t èi t è: holdingíbacklogging costs in t, a function of I t. Furthermore, I 0 is deæned to be 0 and we make the following assumptions: All demands, capacities, production and inventory levels are integer. The production cost function p t is nonídecreasing and concave in the integers of the interval ë0;c t ë, t 2f1;2;...;Tg.Furthermore, p t è0è=0. The holdingíbacklogging cost functions are nonídecreasing on ë0; 1è and noní increasing and concave on è,1; 0ë. If backlogging is not allowed, then the costs are equal to 1 for all negative inventory levels. Furthermore, h t è0è = 0 for all t 2f1;2;...;Tg. All cost functions can be evaluated in polynomial time at any value in their domain and are scaled such that they are integer valued. The objective is to satisfy all demand at minimal cost, subject to the capacity constraints. Hence, the problem can be formulated as z æ = min P T t=1 èp tèx t è+h t èi t èè s.t. I t = I t,1 + x t, d t t =1;2;...;T x t çc t I 0 =0 t=1;2;...;t x t ç0 integer t =1;2;...;T I t integer t =1;2;...;T The assumptions h t è0è = 0 and p t è0è = 0 for all periods t, imply that we are only considering the costs which depend on the production plan, i.e., constant costs are ignored. Although adding the same positive constant to the cost of every feasible solution does not change the cost ordering of the solutions, it would decrease the relative error of every solution. Hence, the assumptions can be viewed as a normalization of the problem. With respect to the issue of zero cost solutions, as raised by Gavish and Johnson è1990è, we note the following. In Subsection 3.2, it will be shown that under very mild conditions èmonotonicity of the cost functionsè it is possible to determine in polynomial time whether or not there exists a zero cost solution of a given instance of the singleíitem capacitated lotísizing problem. Moreover, if it exists, such a solution is found. Hence, the issue of polynomial approximation is only relevant for those problem instances for which we do not ænd a zero cost solution. Of course, for these problem 4

5 instances the relative error with respect to the optimal value is a meaningful measure for the quality of approximate solutions. In the next section, we will describe an exact solution method for the above problem. 2 A dynamic programming algorithm In the standard dynamic programming approach to the capacitated dynamic lot sizing problem, one computes èin a forward or backward fashionè for every period t 2 f1; 2;...;Tg and all possible inventory levels in t the minimal cost of achieving that level. The running time of this approach is proportional to P T t=1 c t æ P T t=1 d t èsee Florian, Lenstra and Rinnooy Kan, 1980è. It is not easy to base an approximation scheme on this DP approach, since the running time can only be decreased if both cumulative capacity and cumulative demand are rescaled, which means that the set of feasible solutions is changed. As a consequence, it may not be trivial to translate an optimal solution of a rescaled problem instance into a feasible solution of the original instance. It may even be possible that one instance is feasible while the other is infeasible. Therefore, we will present a diæerent, more complicated, dynamic programming approach of which the running time mainly depends on an upperbound on the optimal value z æ. This approach can be viewed as being ëdual" to the standard dynamic programming approach. i.e., the ending inventory is maximized subject to a budget constraint. 2.1 Preliminaries To facilitate the exposition, we will assume from now on that it takes constant time to evaluate any of the cost functions which we deæned in the previous section. The reader will have no problem in verifying that every polynomial running time obtained in this paper, will remain polynomial if the function evaluations take polynomial time instead. Furthermore, we will assume from now on that all capacities are strictly positive. The adaption of our algorithms for zero capacities is straightforward. The following lemmas are wellíknown, and will be frequently used in our exposition. Lemma 1 If two functions f and g are both nonídecreasing, then f + g is also noní decreasing. Lemma 2 If two functions f and g are both concave, then f + g is also concave. 5

6 Lemma 3 If a function f deæned on the interval of integers ëa; bë, is nonídecreasing and can be evaluated inconstant time, then we can ænd for any number y the values min ff èxè ç yg and açxçb max ffèxè ç yg; açxçb and the corresponding values of x, in Oèlogèb, aèè time by applying binary search. Lemma 4 If a function f deæned on the interval of integers ëa; bë, isconcave and can be evaluated inconstant time, then we can ænd for any number y the values min açxçb ffèxè ç yg; max ff èxè ç yg; açxçb min ff èxè ç yg and açxçb max ffèxè ç yg açxçb and the corresponding values of x, in Oèlogèb, aèè time by binary search. 2.2 The recursion formulas Let B be any integer upperbound on z æ.for t 2f1;2;...;Tgand b 2f0;1;...;Bgwe deæne F t èbè as the maximum value of I t which can be achieved by production in the ærst t periods if the total cost incurred in these periods is at most b. Hence, b can be viewed as the total budget that we are allowed to spend in the ærst t periods. F t èbè is deæned to be,1 if there does not exist any value of I t with a corresponding feasible production plan costing at most b. Note that z æ is equal to the smallest value of b for which F T èbè ç 0. By deænition, the following holds for t =1: F 1 èbè= max 0çx 1 çc 1 fx 1, d 1 j p 1 èx 1 è+h 1 èx 1,d 1 èçbg for b =0;...;B For any b 2 f0;1;...;bg the value of F 1 èbè can be calculated as follows. Deæne m 1 = minfd 1 ;c 1 g. The function p 1 èx 1 è+h 1 èx 1,d 1 è is concave on the interval ë0;m 1 ë, and on the interval ëm 1 ;c 1 ë it is nonídecreasing. Therefore, we can ænd è1è maxfx 1 2f0;1;...;m 1 gjp 1 èx 1 è+h 1 èx 1,d 1 èçbg è2è and maxfx 1 2fm 1 ;m 1 +1;...;c 1 gjp 1 èx 1 è+h 1 èx 1,d 1 èçbg in time Oèlogèc 1 èè, by Lemmas 3 and 4. If both maxima exist, we take the second, i.e., the maximum of the two; if none exists we set F 1 èbè =,1. We have shown the following. Proposition 5 Determining the values of F 1 èbè for all b 2f0;1;...;Bg can be done in OèB log c 1 è time. 6

7 Now consider a period t 2f2;3;...;Tg and a æxed budget b 2f0;1;...;Bg. A correct recursion formula which links F t èbè to the values F t,1 èaè, a 2f0;1;...;Bg,is not trivial. Consider a æxed value of a, 1çaçb, and suppose we want to determine the maximum value of I t such that the total cost incurred in the ærst t, 1 periods is at most a and the cost incurred in period t is limited by b, a. We ærst discuss two situations between which we will distinguish. By deænition, with the given budget the maximum ending inventory of the ærst t,1 periods is I t,1 = F t,1 èaè. The remainder b,a of the budget is available for production and inventory costs in period t. The ærst situation is the one in which it is possible to extend the production plan corresponding to I t,1 = F t,1 èaè, to a plan also including period t, i.e., there exists an x t 2f0;1;...;c t g such that p t èx t è+h t èf t,1 èaè+x t,d t èçb,a We will show that in this case I t,1 can be assumed to be F t,1 èaè. In the second situation we can not extend a plan corresponding to I t,1 = F t,1 èaè, i.e., for all x t 2f0;1;...;c t g wehave p t èx t è+h t èf t,1 èaè+x t,d t èéb,a For this case we can show that x t may be assumed to be 0. Thus, in both situations, we can restrict the value of one of the variables x t and I t,1. This is proved in the following two propositions, which are valid even if the backlogging and production cost functions are not concave, but only monotone. Proposition 6 If there exists an x t 2f0;1;...;c t g such that p t èx t è+h t èx t +F t,1 èaè,d t èçb,a then only production plans with I t,1 = F t,1 èaè need tobeconsidered when computing the maximum value of I t, given budget a for the ærst t, 1 periods. Proof. Let ^x t be the maximum feasible production level in period t given I t,1 = F t,1 èaè, i.e., ^x t = maxfx t 2f0;1;...;c t gjp t èx t è+h t èf t,1 èaè+x t,d t èçb,ag è3è Suppose that yéf t,1 èaè and that there also exists a feasible production plan with I t,1 = y; let çx t be the corresponding maximum production level in period t, i.e., çx t = maxfx t 2f0;1;...;c t gjp t èx t è+h t èy+x t,d t èçb,ag To prove the proposition, it suæces to show that F t,1 èaè+^x t,d t çy+çx t,d t, since this means that taking I t,1 = F t,1 èaè isalways at least as good as I t,1 = yéf t,1 èaè. 7

8 Deæne ~x t =çx t,f t,1 èaè+y.thus, ~x t é çx t ç c t, since yéf t,1 èaè. We will prove that ^x t ç ~x t, which immediately implies F t,1 èaè+^x t,d t ç F t,1 èaè+~x t,d t = y+çx t,d t. The proof is by contradiction. Suppose that ~x t é ^x t, then 0 ç ^x t é ~x t é çx t ç c t. Furthermore, ~x t is a feasible production level for I t,1 = F t,1 èaè. To see this, we note that p t è~x t è ç p t èçx t èby the monotonicityofp t, and h t èf t,1 èaè+~x t,d t è=h t èy+çx t,d t è, since the arguments are equal. Hence, p t è~x t è+h t èf t,1 èaè+~x t,d t èçp t èçx t è+h t èy+çx t,d t èçb,a Thus, x t = ~x t, and I t,1 = F t,1 èaè satisfy the budget constraint. We nowhavea contradiction with the deænition of ^x t in è3è. Hence, ^x t ç ~x t, which completes the proof. 2 Proposition 7 If for all x t 2f0;1;...;c t g p t èx t è+h t èx t +F t,1 èaè,d t èéb,a è4è then only production plans with I t,1 2fd t ;d t +1;...;F t,1 èaè,1g and x t =0need to beconsidered when computing the maximum value of I t, given budget a for the ærst t, 1 periods. Proof. Let yéf t,1 èaè and çx t be such that taking I t,1 = y and production in period t equal to çx t is feasible, and y +çx t is maximal. If y +çx t çf t,1 èaè, then because of the monotonicity ofp t it is also feasible to take I t,1 = F t,1 èaè and production in period t equal to ~x t =çx t,f t,1 èaè+y. Note that 0 ç ~x t é çx t ç c t. However, è4è states that such a feasible plan does not exist. Hence, we have a contradiction, which implies y +çx t éf t,1 èaè. Now assume that F t,1 èaè ç d t. Then, p t èçx t è+h t èy+çx t,d t èçh t èy+çx t,d t èç h t èf t,1 èaè,d t èéb,a, where the second inequality follows from the fact that h t èi t è is noníincreasing on è,1; 0ë, while the last inequality is è4è for the case x t = 0. Again we have a contradiction. So, besides y +çx t éf t,1 èaè, we may assume F t,1 èaè éd t in the sequel. Any level of I t,1 in the interval ëd t ;F t,1 èaèë can be attained at total cost at most a in the ærst t, 1 periods. To see this, take a production plan for the ærst t, 1 periods with I t,1 = F t,1 èaè and total cost at most a. Change this production plan by lowering the production level in the last production period until the desired value of I t,1 is reached or the production level becomes 0. In the latter case, repeat the procedure with the new production plan. Iterate until a production plan with the desired value 8

9 of I t,1 is obtained. This production plan has cost at most a, because in the process of changing the production plan, both the production and holding costs do not increase. Hence, in particular, we have that I t,1 = d t can be attained at cost at most a. In combination with zero production in period t we get I t = 0. Clearly, this is feasible, because there are no additional costs in period t. Hence, the maximum value of I t is nonínegative, which implies y +çx t çd t. Wenowhave derived that d t ç y +çx t é F t,1 èaè. But this means that also I t,1 = y +çx t can be attained at cost at most a. In combination with zero production in period t, we get a production plan with total cost in period t equal to h t èy +çx t,d t èçp t èçx t è+h t èy+çx t,d t èçb,a. Since this means that the production plan is feasible, we now have shown that it suæces to consider only production plans with d t ç I t,1 éf t,1 èaè and zero production in period t. 2 The above two propositions lead to the following recursion formula for b = 0;...;B and t =2;...;T: F t èbè= max max 0çaçb 8é é: or, equivalently, F t èbè = max 8é é: max max ff t,1 èaè+x t,d t jp t èx t è+h t èf t,1 èaè+x t,d t èçb,ag; 0çxtçct max 0çaçb 0çxtçct max fi t j h t èi t è ç b, ag 0çItéFt,1èaè,dt ff t,1 èaè+x t,d t j p t èx t è+h t èf t,1 èaè+x t,d t èçb,ag; maxfi t ç 0 j9a2f0;1;...;bg: I t éf t,1 èaè,d t ; h t èi t èç b,ag 9é= é; 9é= é; è5è Once more, we would like to mention that we have used the monotonicity, but not the concavity of the cost functions to derive the above recursion formula. 2.3 Complexity Using è5è, F t èbè can be computed from the values F t,1 èaè; a2f0;1;...;bg, as follows. For the evaluation of the ærst expression we propose a procedure similar to the procedure for t = 1, described at the beginning of the preceding subsection. Consider a æxed value of a 2f0;1;...;bg and deæne m a t = 8 éé : 0 if d t, F t,1èaè é 0 d t, F t,1 èaè if 0 ç d t, F t,1 èaè ç c t c t if d t, F t,1 èaè éc t 9

10 Now p t èx t è+h t èf t,1 èaè+x t,d t è is concaveonë0;m a të and nonídecreasing on ëm a t ;c të. Thus, because of Lemmas 3 and 4, the largest achievable value of I t = F t,1 èaè+x t,d t can be determined by binary search inoèlog c t è time for each a 2f0;1;...;bg. Hence, the ærst expression can be evaluated in Oèb log c t è time. For the second expression, we ærst note that there is no value of I t satisfying the conditions in this expression if F t,1 èbè ç d t, since F t,1 èaè is nonídecreasing in a. Let ça b be the smallest value of a in f0; 1;...;bgsuch that F t,1 èça b è éd t and h t èf t,1 èça b è,d t è é b,ça b. The value of ça b is nonídecreasing in b, since for any a the value minfi t j h t èi t è é b, ag is nonídecreasing in b. Therefore, the total computational eæort for ænding ça b is OèBè for all b 2f0;1;...;Bg simultaneously, i.e., we can compute ça b in constant amortized time per b 2f0;1;...;Bg. Toevaluate the second expression once ça b is known, we note that for each a 2f0;1;...;bg there are two functions that bound the inventory I t, namely I t éf t,1 èaè,d t and h t èi t è ç b, a. Now F t,1 èaè, d t is nonídecreasing in a, and maxfi t j h t èi t è ç b, ag is noníincreasing in a. Therefore, if the value of I t that we are looking for exists, it belongs to the interval ëf t,1 èça b, 1è, d t ;F t,1 èça b è,d t è. To be more precise, it is the largest nonínegative value in the interval for which h t èi t è ç b, ça b. Hence, this value can be determined by binary search in OèlogèF t,1 èça b èèè = Oèlogè P t,1 ç =1 c çèè time. To summarize, the evaluation of the second expression takes Oèlogè P t,1 ç =1 c çèè amortized time for all b 2f0;1;...;Bg. Wehavenow derived the main result of this section. Theorem 8 The complexity of the dynamic programming algorithm based on formulas è1è and è5è is OèB 2P T t=1 log c t + B P T t=1 logèp t,1 ç =1 c çèè. 3 Two approximation algorithms In this section we discuss two approximation algorithms. The ærst one is based on the dynamic programming algorithm presented in the preceding section. It yields a feasible solution whose absolute deviation from optimality is bounded, but dependent ont. The second approximation algorithm is quite simple and yields a feasible solution whose relative deviation from optimality is less than 2T. Both approximation algorithms are part of our approximation scheme to be presented in the next section. The ærst algorithm forms the basis of the approximation scheme, the second algorithm merely provides an appropriate upperbound B on the optimum value z æ. 10

11 3.1 Approximation based on DP algorithm This approximation algorithm is based on scaling, an idea which is often used in approximation schemes. However, instead of scaling the cost functions, we are going to scale the budgets of the periods. Cost scaling is not a good idea, since it destroys concavity, i.e., functions such asbp t èx t è=kc, where K is a positive integer, are in general not concave. As before, let B be any integer upperbound on z æ.furthermore, let K be a positive integer such that 1 ç K ç B. For t 2f1;2;...;Tg and b 2f0;K;2K;...;èbB=Kc + TèKg we deæne G t èbè as the maximal value of I t which can be achieved by production in the ærst t periods under the restriction that the total budget for these periods is at most b and the budget allocated to each individual period is a multiple of K. From the preceding section it should be clear that we can compute G t èbè for all t 2f1;2;...;Tg and all b 2f0;K;2K; 2P...;èbB=Kc + TèKg P in a logèp total computational eæort which is OèèB=K + Tè T log c T t,1 ç=1 ç +èb=k + Tè t=1 ç c =1 çèè. The idea is to take the smallest value of b 2f0;K;2K;...;èbB=Kc + TèKg for which G T èbè ç 0 as the value of the approximate solution of the lot sizing problem. We will show the existence of such a solution and give a bound on the absolute diæerence between the value of the approximate solution and the optimal value in the following proposition. Proposition 9 There exists a b 2f0;K;2K;...;èbB=Kc + TèKg with G T èbè ç 0. Moreover, the smallest such value is less than or equal to z æ + TK. Proof. Consider an optimal solution and let r t denote the associated cost incurred in period t, t 2f1;2;...;Tg. Clearly, the solution is feasible if we would allocate a budget of èbr t =Kc +1èK to each period t 2f1;2;...;Tg. Because these budgets are multiples of K, this implies that G T è P T t=1 èbr t=kck + Kèè ç 0. The proposition now follows from P T t=1 èbr t=kck + Kè çb P T t=1 r t=kck + TKè ç èbz æ =Kc + T èk and the fact that the last expression is bounded from above by both èbb=kc + TèK and z æ + TK A simple polynomial approximation algorithm We will now show how to compute an upperbound on z æ which is at most 2Tz æ. This approximation algorithm is quite simple and it can also be used if the cost functions are not concave, but only monotone. It is based on the fact that there are 2T diæerent cost functions. The idea of the algorithm is to ænd the smallest value L for which there 11

12 exists a feasible solution if all cost functions are restricted to contribute at most L to the total cost. Hence, such a feasible solution has cost at most 2TL. Clearly, in any optimal solution of the original problem, each cost function contributes not more than z æ. Therefore, it holds that L ç z æ. This implies that ç B ç 2TL is an upperbound on z æ such that ç B ç 2Tzæ. To show that L can be found in polynomial time, we ærst show that it is possible to determine in polynomial time whether or not there exists a feasible solution if the contribution of each cost function is at most some given value l. For each period t we deæne an upperbound on the production level by çc t = maxfx ç c t j p t èxè ç lg, and alower and upperbound on the inventory level by u t = minfi ç 0 j h t èiè ç lg and v t = maxfi ç 0 j h t èiè ç lg, respectively. These bounds can be determined using binary search. A feasible solution in which each cost function contributes at most l exists if and only if there exists a feasible solution which satisæes the above upper and lowerbounds on the production and inventory levels. We can use dynamic programming to check this. Let M t denote the largest value of I t,achievable by production in the ærst t periods by a production plan satisfying all upper and lowerbounds. In particular, we have M 1 = minfçc 1, d 1 ;v 1 g.ifm 1 éu 1, then there does not exist a feasible solution. Otherwise, we proceed using the recursion formula M t = minfm t,1 +çc t,d t ;v t g for t =2;...;T and we stop as soon as we ænd a t for which M t éu t. There exists a feasible solution if and only if we reach T and M T ç 0. Clearly, L is nonínegative and a trivial upperbound on L is given by U ç max fèp tèc t è;h t è, 1çtçT t,1 X i=1 d i è;h t è TX i=t+1 èor max 1çtçT fèp t èc t è;h t è P T i=t+1 d ièg if backlogging is not allowedè. Now it should be clear how L can be determined using binary search. Note, however, that the value of any feasible solution is also an upperbound on L. Suppose such avalue, say ~ B,is known èfor instance, ~ B could be the value of any heuristic solutionè, then we can do the following. We ærst check whether there exists a feasible solution in which each cost function contributes at most d ~ B=2Te. If this is not the case, then d ~ B=2Te élçzæ. Hence, ~ Bé2Tzæ, and we are done. Otherwise, we carry out the binary search for L on ë0; ~ Bë. The running time of the above heuristic is easily seen to be OèT log 2 Uè. Note that this heuristic can also be used to check in polynomial time whether there exists an optimal solution with zero cost. 12 d i èg

13 4 The fully polynomial approximation scheme We will ærst describe a straightforward version of our approximation scheme, and then discuss possible ways to improve its complexity. 4.1 Description and correctness Our fully polynomial approximation scheme consists of two steps and combines the approximation algorithms discussed in the preceding section. Let æé0 be given. 1. Use the simple approximation algorithm to calculate in polynomial time an upperbound B which satisæes B ç 2Tz æ. 2. Apply the DP based approximation algorithm; use the calculated B as the upperbound and K = maxfbæb=2t 2 c; 1g. We now state the main result of this paper. Theorem 10 The above procedure has a complexity which is polynomial in both the size of the problem instance and 1=æ, and determines a feasible solution with a value not larger than è1 + æèz æ. Proof. For the ærst part of the proposition, we only have to analyze the complexity of 2P Step 2. As already mentioned P in logèp Subsection 3.1, its running time is OèèB=K + Tè T log c T t,1 ç=1 ç +èb=k + Tè t=1 ç c =1 çèè. Clearly, this is a polynomial bound if B=K ç T. Therefore, let us assume B=K é T. IfæB=2T 2 é 1, then K é æb=4t 2 ; otherwise, K ç æb=2t 2. In both cases, it is easily veriæed that the running time is OèT 4P T ç =1 log c ç =æ 2 + T 2P T t=1 logèp t,1 ç =1 c çè=æè, which is polynomial in the size of the problem instance and 1=æ. If K = 1, then a solution with value z æ is found in Step 2. If K = bæb=2t 2 c,we can use the fact that this step yields a solution whose value exceeds z æ by at most KT, which is less than or equal to æb=2t ç æz æ. This completes the proof Complexity In the proof of Theorem 10, we mentioned the complexity bound OèT 4P T ç =1 log c ç=æ 2 + T 2P T t=1 logèp t,1 ç =1 c çè=æè for Step 2 of the approximation scheme. There are several ways to improve this bound. An obvious approach is to apply the DP based approximation algorithm not once, but twice. First it is applied with K = maxfb^æb=2t 2 c;1g, 13

14 where ^æ is a relatively large error. This yields an upperbound, say ^B. Subsequently, the approximation algorithm is applied with K = maxfb^æ ^B=Tè1 + ^æèc; 1g, yielding a solution with the required quality guarantee. A good choice for ^æ is one for which the complexity of the ærst and second execution of the approximation algorithm is about the same. For instance, if we 2P take ^æ= p Tæ, the overall complexity, P including Step 1, is OèT log 2 U +èt+t p Tæè T log c ç=1 ç =æ 2 +èt+t p logèp T t,1 Tæè t=1 ç c =1 çè=æè. Another way to improve the complexity is due to Kovalyov è1995è, to whom we refer for details. Given the lowerbound L, the upperbound B and the fact that B=L ç 2T,it can be shown 2P that a lowerbound P ^L and logèp an upperbound ^B with ^B=^L ç 3 can be found in Oèlog T T èt log ç c T t,1 =1 ç + T t=1 ç c =1 çèèè time. The idea is to iteratively apply the DP based approximation algorithm with K = maxfbl 0 =T c; 1g, starting with L 0 = L. If the approximation algorithm does not ænd a feasible solution, the value of L 0 is doubled and the algorithm is repeated. When a feasible solution is found, the procedure terminates. ^B is equal to the value of the feasible solution and ^L is equal to the current value of L 0. Since ^B ç 3zæ,we can subsequently apply the DP based approximation algorithm with K = maxfbæ ^B=3Tc;1g to obtain a solution with the desired accuracy. The overall complexity of this approach isoètlog 2 U + log T èt 2P T ç =1 log c ç + P logèp T t,1 + T t=1 ç c =1 çè=æè. Further improvements of the complexity maybeachieved for certain special cases T P T t=1 logèp t,1 ç =1 c çèè + T 2P T ç =1 log c ç =æ 2 of the cost functions, as discussed in the next section. 5 Special cases and extensions The model for which we have developed the approximation scheme in the preceding sections, is quite general. On one hand, stronger results can be obtained for interesting special cases. On the other hand, our results can be extended to even more general capacitated lot sizing problems encountered in the literature. 5.1 No backlogging In our exposition, we have only used the concavity of the production cost functions to evaluate è2è in Subsection 2.2 and the ærst expression in è5è in Subsection 2.3 eæciently. To be more precise, the assumption is used to deal eæciently with the possibility of backlogging. Hence, in case backlogging is not allowed, it is not necessary to assume that the production cost functions are concave. Therefore, wehave the following result. Theorem 11 If backlogging is not allowed, the approximation scheme is still correct if the production cost functions are only nonídecreasing and not concave. 14

15 5.2 Pseudoílinear cost functions An important special case is the one in which all cost functions are pseudoílinear, as is often assumed in the literature èsee, for instance, Baker et al. 1978; Lambrecht and Vander Eecken 1978; Bitran and Yanasse 1982; Chung and Lin 1988; Chung, Flynn and Lin 1994; Chen, Hearn and Lee 1994aè. In the appendix we show that in this case the dynamic programming algorithm can be adapted to run in OèTBè time. Also, the simple polynomial heuristic of Section 3.2 runs in OèT log U è time, because each of the bounds çc t ;u t and v t can now each be calculated analytically in constant time. Hence, a straightforward version of the approximation scheme runs in OèT log U + T 3 =æè time. Using Kovalyov's complexity improvement idea, we obtain the following result. Theorem 12 If all cost functions are pseudoílinear, then the fully polynomial approximation scheme runs in OèT log U + T 2 log T + T 2 =æè time. 5.3 Piecewise concave or convex cost functions Love è1973è and Swoveland è1975è consider the problem in which the cost functions are piecewise concave èsee also Chen, Hearn and Lee 1994bè. Let us ærst discuss how our approximation scheme should be adapted if the backlogging cost functions are piecewise concave èand noníincreasingè instead of simply concave. Our DP algorithm is only aæected with respect to the evaluation of è2è and the ærst expression in è5è, since these are the only steps in the algorithm where concavity is used. If the backlogging cost function of period 1 consists of n 1 concave pieces, it is easily seen that evaluating è2è can be done by performing at most n 1 binary searches, instead of just one. The evaluation of the ærst expression in è5è can be adapted in a similar way. Hence, if each backlogging function consists of at most n concave pieces, then the complexity of the dynamic programming algorithm, as given in Theorem 8, is increased by at most a factor n. The following result is now obvious. Theorem 13 If the backlogging cost functions are piecewise concave and the number of pieces is polynomially bounded in the size of the problem instance, then there exists a fully polynomial approximation scheme. Also note that lower and upperbounds on the inventory levels can easily be incorporated in our approximation scheme, since these bounds can be modeled by deæning the holdingíbacklogging costs to be inænite outside the feasible range. Now suppose that the production cost functions are piecewise concave and monotone. Again we only have to discuss how this aæects the evaluation of è2è and the 15

16 ærst expression in è5è. Let us consider the latter. If p t èx t è is concave on some interval ëx l ;x u ëçë0;c t ë, then for any a, 0çaçb,wehave max x l çx tçx uff t,1èaè+x t,fd t j p t èx t è+h t èf t,1 èaè+x t,d t èçb,ag= max 0çæx tçx u,x lff t,1èaè+x l +æx t,d t j p t èx l +æx t è+h t èf t,1 èaè+x l +æx t,d t è ç b,ag. It is obvious that the value of æx t which maximizes this expression can again be found by binary search. Hence, to evaluate the ærst expression of è5è, it suæces to perform a number of binary searches which is at most the number of concave pieces of p t èx t è. A similar remark holds for the evaluation of è2è. This implies the following result. Theorem 14 If the production cost functions are piecewise concave and the number of pieces is polynomially bounded in the size of the problem instance, then there exists a fully polynomial approximation scheme. Finally, it is worth mentioning that Veinott è1964è and Erenguc and Aksoy è1990è consider models in which the cost functions are èpiecewiseè convex instead of concave. We just note that if both the backlogging and production cost functions are piecewise convex èand monotoneè, our fully polynomial approximation scheme can be applied, since we can still use binary search toevaluate è2è and the ærst expression in è5è eæciently. 5.4 Startíup and reservations costs Karmarkar, Kekre and Kekre è1987è have introduced the dynamic lotísizing problem with startíup and reservation costs. In this model a startíup cost S t is incurred if the production facility is switched on in period t, and a separate reservation cost R t is charged for keeping the facility on whether or not it is used for production. These costs are incurred in addition to the the production cost p t èx t è. To handle this cost structure, the DP algorithm should be modiæed. For t = 1;2;...;T and b = 0;1;...;B, we deæne F t èbè as before. Furthermore, Ft 0èbè is deæned as the maximum value of I t which can be achieved by production in the ærst t periods if the total cost is at most b and the production facility is oæ in period t. Finally, we deæne Ft 1 èbè as the maximum value of I t achievable in the ærst t periods if the total cost is at most b and the production facility is on in period t. Hence, F t èbè = maxfft 0èbè;F1èbèg. t Let us assume that there is no production in period 0. Then, we have, for b = 0; 1;...;B, F 0 1èbè= è,d 1 if h 1 è,d 1 è ç b,1 otherwise 16

17 F 1 1 èbè = max fx 1, d 1 j S 1 + R 1 + p 1 èx 1 è+h 1 èx 1,d 1 èçbg 0çx 1 çc 1 The latter formula can be evaluated analogously to è2è. Let us now consider the recursion formulas for t ç 2. The formula for Ft 0èbè, i.e., x t = 0, is trivial: F 0 t èbè = maxfi t ç 0 j9a2f0;1;...;bg: I t ç F t,1 èaè,d t ; h t èi t èç b,ag This recursion formula can be evaluated in a similar way as the second expression in è5è. Furthermore, we have Ftèbè 1 = max max 0çaçb 8é max ff 0 èaè+x t,1 t,d t j p t èx t è+h t èf 0 èaè+x t,1 t,d t èçb,a,s t,r t g; 0çxtçct 9é= é: maxfi t ç 0 j9a2f0;1;...;bg: I t éf 0 t,1èaè,d t ; h t èi t èç b,a,s t,r t g; max fft,1èaè+x 1 t,d t j p t èx t è+h t èft,1èaè+x 1 t,d t èçb,a,r t g; 0çxtçct maxfi t ç 0 j9a2f0;1;...;bg: I t éf 1 t,1 èaè,d t; h t èi t èç b,a,r t g Of course, this recursion formula resembles è5è. Its correctness is based on properties similar to those stated in Propositions 6 and 7, which can be proven analogously. The only diæerence is that we have to distinguish between the two possible states of the production facility in period t, 1. Eæcient evaluation of è6è can be done analogously to the evaluation of è5è. It follows that the model with startíup and reservation costs can be solved by a dynamic programming algorithm based on the above formulas with complexity OèB 2P T t=1 log c t +B P T t=1 logèp t,1 ç =1 c çèè. Because the simple polynomial approximation algorithm described in Subsection 3.2 can trivially be adapted to incorporate startíup and reservation costs èdistinguish again between the two possible states in every period and deæne corresponding variables and parametersè, we have the following result. Theorem 15 If there are startíup and reservation costs in addition to the usual production costs, then there exists a fully polynomial approximation scheme. é; è6è 6 Concluding remarks We have developed the ærst fully polynomial approximation schemes for singleíitem capacitated lotísizing problems, where the error is measured in the usual way, i.e., as 17

18 the relative deviation form optimality. To the best of our knowledge, even polynomial approximation methods which produce solutions with a relative error bounded by a constant were previously unknown. We have shown that our approach is applicable to many singleíitem capacitated lotísizing models encountered in the literature. The most important idea in our the approximation schemes is the nonítrivial ëdual" DP formulation in combination with budget scaling. A similar approach may result in approximations schemes for problems which are closely related to singleíitem capacitated lotísizing problems, such as certain NPíhard location and network design problems on trees èsee, for instance, Flippo et al. 1996è and NPíhard variants of the discrete lotísizing and scheduling problem èsalomon et al. 1991è. It is unlikely, however, that our results can be extended to fairly general multiíitem capacitated economic lotísizing problems, since these are known to be strongly NPíhard èchen and Thizy 1990è. Appendix: Pseudo-linear cost functions In this appendix, the cost functions are assumed to be of the following form for t = 1;...;T: p t èx t è= h t èi t è= è 0 if x t =0 f t +r t x t if 0 éx t çc t 8é é: 0 if It =0 e t +s t I t if I t é 0 g t, q t I t if I t é 0 where f t ;r t ;e t ;s t ;g t and q t are nonínegative integers. We will show that in this case the complexity of the dynamic programming algorithm can be reduced. Consider the following expression, which is part of recursion è5è. max max 0çaçb 0çxtçct ff t,1 èaè+x t,d t j p t èx t è+h t èf t,1 èaè+x t,d t èçb,ag è7è As before, we would like to evaluate this expression for every b 2f0;1;...;Bg. To do this eæciently, we will no longer consider these expressions for each value of b separately, but we will exploit the fact that for consecutive values of b the expressions are closely related. Our main result will be an OèTBè bound on the total computational eæort to evaluate è7è for all b 2f0;1;...;Bg and all t 2f1;2;...;Tg, instead of the 18

19 OèB 2P T t=1 log c t + B P T t=1 logèp t,1 ç =1 c çèè bound, which was proved for the general case in Section 2. To start the exposition, we rewrite è7è in terms of I t, which results in the following maximization problem. max 0çaçb f I t j F t,1 èaè, d t ç I t ç F t,1 èaè+c t,d t ; a+p t èi t,f t,1 èaè+d t è+h t èi t èçbg We split the maximization problem above into four èpossibly overlappingè subproblems corresponding to the following cases: èiè I t = 0, èiiè no production, i.e., I t = F t,1 èaè, d t, èiiiè I t é 0 and positive production, and èivè I t é 0 and positive production. We will solve these subproblems independently of each other. However, each subproblem is considered for all b 2f0;1;...;Bg simultaneously. We will show that the total computational eæort to solve a subproblem for all b 2f0;1;...;Bg together is OèBè. Subproblem èiè Since the value of I t is æxed, this is essentially a feasibility problem. If the feasible region is noníempty for a certain value of b, then it is also feasible for larger values of b. Hence, the problem boils down to ænding the smallest value of b for which the feasible region is noníempty. This is done by considering b in order of increasing value and keeping track of min f a + p tè,f t,1 èaè+d t èjf t,1 èaè,d t ç0çf t,1 èaè+c t,d t g 0çaçb è8è As soon as è8è is smaller than b, wehave found the smallest value for which the feasible region is noníempty. Otherwise, we proceed with the next value of b. Since è8è can be updated in constant time when the value of b is increased by 1, it follows that it takes in total OèBè time to solve subproblem èiè for all b 2f0;1;...;Bg. Subproblem èiiè The problems are of the following form. max f F t,1èaè, d t j a + h t èf t,1 èaè, d t è ç bg 0çaçb è9è To solve these subproblems eæciently,we consider them in order of increasing value of b. We ærst determine a B, which is deæned as the largest a for which a+h t èf t,1 èaè,d t è ç B. Since F t,1 èaè is nonídecreasing in a, the optimal value of è9è for b = B is F t,1 èa B è, d t. Next we determine the largest a for which a + h t èf t,1 èaè, d t è ç B, 1. Clearly, we can do this by considering a in decreasing order, starting from a B until we reach the desired value. This gives us the optimal value of è9è for b = B, 1, and so 19

20 on. The total computational eæort of this procedure is easily seen to be OèBè. Subproblem èiiiè We now consider the case in which both x t and I t are positive. Substituting the speciæc cost functions, the corresponding problems can be written as max 0çaçb maxf I t j maxf1;f t,1 èaè+1,d t gçi t çf t,1 èaè+c t,d t ; f t +e t +r t d t +èr t +s t èi t,r t F t,1 èaè+açbg è10è Let a l be the smallest value of a with F t,1 èaè+c t,d t ç1. Clearly, values of aéa l can be ignored. If r t + s t = 0, then it is optimal to take I t = F t,1 èaè+c t,d t for all a ç a l. In this case we can use a similar approach as for Subproblem èiiè. Therefore, we assume r t + s t é 0 from now on. Consider for any a 2fa l ;a l +1;...;Bg the maximization problem maxf I t j maxf1;f t,1 èaè+1,d t gçi t çf t,1 èaè+c t,d t ; f t +e t +r t d t +èr t +s t èi t,r t F t,1 èaè+açbg è11è Of course, the optimal value of this problem depends on the value of b. In particular, the feasible region of the maximization problem is empty ifbis less than b l èaè ç f t + e t + r t d t +èr t +s t è maxf1;f t,1 èaè+1,d t g,r t F t,1 èaè+a. On the other hand, if b is larger than b u èaè ç f t + e t +èr t +s t èc t,s t d t +s t F t,1 èaè+a, then the constraint involving b is redundant and it is optimal to take I t equal to its simple upperbound. For values of b from b l èaè tob u èaè, the constraint involving b is binding. Hence, for each value of a, we have the following optimal solution of è11è: 8é,1 if béb l èaè I t = é: H 1 èa; bè çb 1 rt+st èb,f t,e t,r t d t +r t F t,1 èaè,aèc H 2 èaè ç F t,1 èaè+c t,d t if b l èaè ç b ç b u èaè if b ç b u èaè+1 For any value of b 2fa l ;a l +1;...;Bg,we can now rewrite è10è as max è maxf H 1 èa; bè j a l ç a ç b; b l èaè ç b ç b u èaèg; maxf H 2 èaè j a l ç a ç b; b u èaè+1çbg Our approach will be to determine the values maxf H 1 èa; bè j a l ç a ç b; b l èaè ç b ç b u èaèg for all b 2fa l ;a l +1;...;Bg, and independently the values maxf H 2 èaè j a l ç a ç b; b u èaè+1ç bg, b2fa l ;a l +1;...;Bg. In order to do this eæciently,we will use the facts stated in the following three propositions. è 20

21 Proposition 16 The value b u èaè is strictly increasing in a. Proof. The statement follows immediately from the deænition of b u èaè and the fact that F t,1 èaè is nonídecreasing in a. 2 Proposition 17 Suppose that for some ça 2fa l ;a l +1;...;B,1g it holds that b l èçaè ç b l èça +1è, then maxf H 1 èa; bè j a l ç a ç b; b l èaè ç b ç b u èaèg = maxf H 1 èa; bè j a l ç a ç b; b l èaè ç b ç b u èaè; a6=çag. Proof. Because of Proposition 16, we have that ëb l èçaè;b u èçaèë ç ëb l èça +1è;b u èça + 1èë. Therefore, it suæces to show that H 1 èça +1;bèçH 1 èça; bè for all b 2 ëb l èçaè;b u èçaèë. The inequality b l èçaè ç b l èça + 1è immediately implies maxf1;f t,1 èçaè+1,d t g,r t F t,1 èçaè+çaç maxf1;f t,1 èça +1è+1,d t g,r t F t,1 èça +1è+ça+1 or, equivalently, r t F t,1 èça +1è,r t F t,1 èçaè, 1 ç maxf1;f t,1 èça +1è+1,d t g,maxf1;f t,1 èçaè+1,d t gç0 Since H 1 èça+1;bè,h 1 èça; bè = 1 rt+st èr tf t,1 èça+1è,r t F t,1 èçaè,1è for any b 2 ëb l èçaè;b u èçaèë, the desired result now follows. 2 Proposition 18 For all a 2fa l ;a l +1;...;Bg it holds that b l èaè ç a. Proof. b l èaè = f t + e t + r t d t +èr t +s t è maxf1;f t,1 èaè+1,d t g,r t F t,1 èaè+a ç f t +e t +r t d t +r t èf t,1 èaè+1,d t è+s t,r t F t,1 èaè+a = f t +e t +r t +s t +a ç a 2 Theorem 19 The values maxf H 1 èa; bè j a l ç a ç b; b l èaè ç b ç b u èaèg and maxf H 2 èaè j a l ç a ç b; b u èaè+1çbgcan be computed for b = a l ;a l +1;...;B in a total computational eæort which is OèBè. 21

22 Proof. Let us ærst focus on the computation of the values maxf H 2 èaè j a l ç a ç b; b u èaè +1 ç bg, b = a l ;a l +1;...;B. Let a u be the largest value of a for which b u èaè +1 ç B. From b u èa u è ç b l èa u è and Proposition 18 it follows that béa u if b ç b u èaè + 1. Because H 2 èaè =F t,1 èaè+c t,d t is nonídecreasing in a, we can now conclude maxf H 2 èaè j a l ç a ç b; b u èaè+1çbg=h 2 èa u è for b u èa u è+1çbçb. Analogously, we can prove maxf H 2 èaè j a l ç a ç b; b u èaè+1çbg=h 2 èa u,1è for b u èa u,1è+1 ç b ç b u èa u è, and so on. Hence, the procedure boils down to determining for all b 2fa l ;a l +1;...;Bg the largest value b u èaè + 1 which is less than or equal to b. This can easily be done in OèBè time. Let us now consider the computation of maxf H 1 èa; bè j a l ç a ç b; b l èaè ç b ç b u èaèg, b = a l ;a l +1;...;B. Because of Proposition 17 any value ça for which b l èçaè ç b l èça +1è may be ignored while determining these maxima. This implies that it suæces to consider the subsequence A of a l ;a l +1;...;B deæned by the property that a 2 A if and only if there does not exist any a 0 2fa+1;a+2;...;Bg with b l èa 0 è ç b l èaè. Note that this means that both b l èaè and b u èaè are strictly increasing for increasing a 2 A. Also note that A can be constructed in OèBè time. Now deæne for every b 2fa l ;a l +1;...;Bg the possibly empty subset Sèbè of elements of A as follows. If Sèbè =fa 1 ;a 2 ;...;a m g then 1. a 1 is the smallest a 2 A for which b l èaè ç b ç b u èaè, 2. a i, i =2;3;...;m, is the smallest a 2 A for which a i éa i,1,b l èa i èçbçb u èa i è and H 1 èa i ;bè éh 1 èa i,1 ;bè. If Sèbè is empty, then clearly maxf H 1 èa; bè j a l ç a ç b; b l èaè ç b ç b u èaèg =,1. If Sèbè is noníempty, then we have the properties a 1 éa 2 é...éa m and H 1 èa 1 ;bè é H 1 èa 2 ;bè é... éh 1 èa m ;bè. Because of Proposition 18, we know that a i ç b for all i =1;2;...;m. It is now easily veriæed that H 1 èa m ;bè = maxf H 1 èa; bè j a l ç a ç b; b l èaè ç b ç b u èaèg. Besides the fact that we immediately obtain the value maxf H 1 èa; bè j a l ç a ç b; b l èaè ç b ç b u èaèg, there is another reason for keeping track ofsèbè. If for a certain value of b avalue a 0 2 A with b l èa 0 è ç b ç b u èa 0 è is not in Sèbè, then it is not in Sèb 0 è for any b 0 ébwith b l èa 0 è ç b 0 ç b u èa 0 è. This follows from the fact that there exists an a i 2 A with a i éa 0,b l èa i èçbçb u èa i è and H 1 èa i ;bè ç H 1 èa 0 ;bè. Because b l èa i è éb l èa 0 èçb 0 ébçb u èa i è, it holds that b l èa i è ç b 0 ç b u èa i è. Moreover, H 1 èa i ;bèç H 1 èa 0 ;bè implies H 1 èa i ;b 0 èç H 1 èa 0 ;b 0 è. Hence, a 0 is not in Sèb 0 è. We will consider b in order of decreasing value. The elements of subset SèBè can trivially be found in OèBè time. To achieve this complexity bound for all b 2fa l ;a l +1;...;Bg together, we represent the subsets by a list in which the elements are stored 22