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Lot-sizing with non-stationary cumulative capacities
Operations Research Letters 35 (2007) 549 – 557
Operations Research Letters www.elsevier.com/locate/orl
Lot-sizing with non-stationary cumulative capacities F. Zeynep Sargut, H. Edwin Romeijn∗ Department of Industrial and Systems Engineering, University of Florida, P.O. Box 116595, Gainesville, FL 32611-6595 USA Received 14 January 2006; accepted 7 September 2006 Available online 30 October 2006
Abstract We study a new class of capacitated economic lot-sizing problems. We show that the problem is NP-hard in general and derive a fully polynomial-time approximation algorithm under mild conditions on the cost functions. Furthermore, we develop a polynomial-time algorithm for the case where all cost functions are concave. © 2006 Elsevier B.V. All rights reserved. Keywords: Capacitated lot-sizing; Fully polynomial-time approximation scheme; Dynamic programming
1. Introduction In this paper we study a new dynamic lot-sizing problem which we will refer to as the lot-sizing problem with cumulative capacities (LSP-CC). In traditional capacitated lot-sizing models, the quantity produced in each period is limited by some capacity, but any capacity remaining at the end of a period is essentially lost. In contrast, our new model applies to settings where any remaining capacity is transferred to the next production period. This means that the cumulative quantity that we can produce up to and including period t is constrained by the cumulative capacity up to and including period t.
∗ Corresponding author. Tel.: +1 352 392 3088;
fax: +1 352 392 2537. E-mail address: [email protected]fl.edu (H.E. Romeijn). 0167-6377/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2006.09.004
This problem may occur if the capacity constraints do not correspond to a perishable production capacity, such as time, but instead to a physical capacity in the form of a monetary budget or raw materials. The problem also appears in certain settings where the planning of procurement, production, and inventory holding of raw materials and final product is integrated. This two-level lot-sizing model has been studied extensively for the case where all cost functions are concave and capacities are absent (see, e.g., [15]), or procurement is subject to stationary capacities (see, e.g., [11,12]). Although in the presence of general procurement capacities the problem is known to be NP-hard even when all cost functions are concave (see [7]), Kaminsky and Simchi-Levi [12] derive certain conditions on the cost functions under which the problem is solvable in polynomial time. If the total cost of procurement and inventory of raw materials are exogenous to the model, the two-level problem reduces the
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LSP-CC. We study this problem under both general and concave cost functions. In Section 2, we formally introduce the LSP-CC. Then in Section 3, we prove that the problem is NPhard for general cost functions and provide a fully polynomial time approximation scheme (FPTAS) in case all cost functions are nondecreasing. In Section 4 we then develop a dynamic programming approach that solves the problem in polynomial time when all cost functions are concave. Finally, in Section 5 we briefly discuss an extension where backlogging is allowed. 2. Model formulation
• xt denotes the quantity produced in period t with associated cost function ct (t = 1, . . . , T ); • It denotes the inventory level at the production level at the end of period t with associated cost function ht (t = 1, . . . , T ). Without loss of generality, all cost functions are assumed to be equal to zero when their argument is zero. For convenience, we will define cumulative demands and capacities as dst = t=s+1 d (0 s < t T ) and C0t = t=1 C (1t T ). Then we can formulate the single-level LSP-CC as follows: minimize
(ct (xt ) + ht (It ))
t=1
subject to xt + It−1 = dt + It , t = 1, ..., T , t x C0t , = 1, ..., T , =1
xt , It 0,
• We show that the problem is NP-hard under general cost functions and develop a FPTAS for the case where all cost functions are nondecreasing. • We develop a dynamic programming algorithm that solves the problem in O(T 4 ) time when all cost functions are concave. • We extend these results to allow for backlogging.
3. General cost functions
We consider a single-level system in which the goal is to satisfy a deterministic and dynamic sequence of demands over a finite and discrete planning horizon at minimum cost through production and inventory. We denote the planning horizon by T . The demand faced in period t is denoted by dt (t = 1, . . . , T ). In period t (t = 1, . . . , T ), the procurement capacity is given by the total unused capacity through period t − 1 plus an additional capacity of Ct units that becomes available in period t. Our decision variables and associated cost functions are given below.
T
Note that we may, without loss of generality, assume that the initial inventory level I0 = 0. Our contributions in this paper with respect to this new lot-sizing problem are the following:
t = 1, ..., T .
(LSP-CC)
We next turn to an analysis of the LSP-CC with general cost functions. In particular, we will show that this problem is NP-hard in general. In addition, we provide a FPTAS for the LSP-CC with nondecreasing cost functions, i.e., an algorithm that finds a feasible solution with relative error no more than ε > 0 in an amount of time that is polynomial in the input size of the problem and 1/ε. 3.1. Proof of NP-hardness In this section, we will show that LSP-CC with zero holding cost is NP-hard by reducing the subset sum problem to an instance of LSP-CC. The SUBSET SUM problem is defined as follows (see [8]): SUBSET SUM: Given positive integers a1 , a2 , ..., aT and A, does there exist a set S ⊆ {1, ..., T } such that i∈S ai = A? The following theorem shows that the LSP-CC is NP-hard in general by reducing the SUBSET SUM problem to it. Theorem 3.1. The LPS-CC is NP-hard. Proof. Consider an instance of the SUBSET SUM problem. We then define the following instance of the LSP-CC: Ct = at for t = 1, ..., T , 0 for t = 1, ..., T − 1, dt = A for t = T ,
F.Z. Sargut, H.E. Romeijn / Operations Research Letters 35 (2007) 549 – 557 ct(x) a1+ ... +at+1
at+1 at
at
a1+ ... +at ↑
x
Fig. 1. Production cost function in period t.
ct (x) =
⎧ 0 ⎪ ⎪ ⎨1 +
at −1 at x
⎪ ⎪ ⎩x + 1
ht (It ) = 0
if x = 0, if 0 < x at for t = 1, ..., T , t a , if at < x =1
if It 0 for t = 1, ..., T .
The production cost function is illustrated in Fig. 1. First observe that, since the only non-zero demand is in period T , each production quantity will be used to satisfy the demand in that period and thus the cumulative production over the planning horizon is equal to A, i.e., T=1 x = A. Moreover, the production cost in period t satisfies ct (x) = x if x ∈ {0, at } and ct (x) > x otherwise. This implies that the cost of any production plan with production in any period t not equal to either 0 or at exceeds A. In other words, if the LSPCC has a solution with cost A the set of periods in which production takes place provides a solution to the SUBSET SUM problem. On the other hand, if the SUBSET SUM problem has a solution, say S, producing xt = at in all periods t ∈ S provides a solution to the LSP-CC with cost A. 3.2. A fully polynomial time approximation scheme Although the LSP-CC is NP-hard for general cost functions, we will show in this section that a FPTAS exists when all cost functions are nondecreasing and all problem data (i.e., demands, capacities, and costs) are integral. In particular, we will draw heavily on the paper by van Hoesel and Wagelmans [10] and modify the FPTAS that they developed for the standard capacitated lot sizing problem (CLSP) to our problem.
551
Following van Hoesel and Wagelmans [10], we start by introducing a non-traditional dynamic programming formulation of the problem that runs in pseudopolynomial time. Let B be any integer upper bound on the optimal solution value, say z∗ , to the LSP-CC. For t = 1, ..., T and b = 0, 1, ..., B, we then define Ft (b) to be the maximal ending inventory in period t (It ) that can be achieved with a budget of b in the first t periods, i.e., the total production and inventory holding costs in periods 1, . . . , t do not exceed b. Since for t = 1 the cumulative capacity constraint is equal to the traditional capacity constraint we immediately obtain that F1 (b) =
max
d1 x1 C1
{x1 − d1 |c1 (x1 ) + h1 (x1 − d1 ) b}
for b = 0, . . . , B. A recursion for the values Ft (b) can then be derived by, for t = 2, . . . , T , allocating a budget of 0 a b to periods 1, . . . , t − 1 and a budget b − a for period t, and optimizing over the value of a. For the CLSP, van Hoesel and Wagelmans [10] show that, for a given value of a, (i) if it is possible to feasibly extend a production plan for periods 1, . . . , t − 1 that ends with the maximal inventory level Ft−1 (a) to period t it is optimal to do so; (ii) otherwise, attention may be restricted to xt = 0 and dt It−1 < Ft−1 (a). We will show that these two properties also hold for the LSP-CC. First, note that the flow balance constraints imply that t−1 =1 x = d0,t−1 + It−1 so that the cumulative capacity constraint in period t can be written as xt C0t −
t−1
x = C0t − d0,t−1 − It−1 .
=1
Moreover, since the demand in period t needs to be satisfied we need to have xt max{0, dt − It−1 }. The next proposition now shows that properties (i) and (ii) hold for the LSP-CC. Proposition 3.2. If there exists max{0, dt −Ft−1 (a)} xt C0t − d0,t−1 − Ft−1 (a) such that ct (xt ) + ht (Ft−1 (a) + xt − dt ) b − a
(1)
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then we only need to consider production plans with It−1 = Ft−1 (a). Otherwise, we only need to consider production plans with xt = 0 and dt It−1 < Ft−1 (a). Proof. Suppose we have a feasible solution with It−1 < Ft−1 (a) and xt > 0. Then if we increase It−1 and decrease xt by the same amount while retaining nonnegativity of xt condition (1) as well as the bound constraint on xt will still be satisfied. Moreover, the resulting value of It remains unchanged. Therefore, we can without loss of optimality in the dynamic programming recursion increase It−1 and decrease xt until either It−1 becomes equal to Ft−1 (a) or xt becomes equal to zero, whichever happens earlier. In the former case, the first claim follows. In the latter case, the second claim follows. As a result of the Proposition 3.2 the following dynamic programming recursion is obtained Ft (b) = max 0a b ⎧ maxmax{0,dt −Ft−1 (a)} xt C0t −d0,t−1 −Ft−1 (a) ⎪ ⎨ {Ft−1 (a) + xt − dt |ct (xt ) max +ht (Ft−1 (a) + xt − dt ) b − a}, ⎪ ⎩ max0 It 0 to achieve a solution to with value at most (1 + ε)z∗ in polynomial time, yielding the desired FPTAS. It remains to be shown that a cost bound B with the desired property can be found in polynomial time. Once again we will modify the approach of van Hoesel and Wagelmans [10] to apply to the LSP-CC. The idea of their algorithm is to find the smallest value L with the property that there exists a feasible solution to the problem such that all cost functions contribute at most L to the total cost. Hence, such a feasible solution has cost at most 2T L. Clearly, in any optimal solution of the original problem, each cost function contributes no
more than z∗ so that L z∗ . This implies that B =2T L is an upper bound on z∗ such that B 2T z∗ . To show that L can be found in polynomial time, we first 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 . Given this value, we define for each period t the following upper bounds on the production and inventory level: x¯t ≡ max{x : 0 x C0t , ct (x) } and I¯t ≡ max{I : I 0, ht (I ) }. These bounds can be determined using binary search in a total of O(T log(CT )) time. It is easy to see that a feasible solution in which each cost function contributes at most exists if and only if there exists a feasible solution that satisfies the above upper and lower bounds on the production and inventory levels. We can check this as follows: let mt denote the largest ending inventory in period t achievable by a production plan for the first t periods that satisfies all upper and lower bounds. In particular, we have m1 = min{x¯1 − d1 , I¯1 } and mt = min{mt−1 + min{C0t − d0,t−1 − mt−1 , x¯t } − dt , I¯t }, t = 2, . . . , T . If mT 0 then there exist a solution in which each cost function contributes at most . Now noting that L is bounded from above by maxt=1,...,T max{ct (C0t ), ht (dtT )} the value L can be determined in polynomial time using binary search.
4. Concave costs 4.1. Introduction The LSP-CC may be formulated as a minimum cost network flow problem as illustrated in Fig. 2. Nodes (D, 2), ..., (D, T ) are demand nodes, where (D, t) represents the demand in period t, dt . Nodes (C, 1), . . . , (C, T ) are transshipment nodes and node (C, T + 1) is a supply node having supply equal to the cumulative demand over the planning horizon, i.e., its demand is equal to the negative thereof: −d0T . As in traditional lot-sizing models, an arc from node (D, t) to node (D, t + 1) (t = 1, . . . , T − 1) represents the amount held in inventory at the end of period t, It , and
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553
Fig. 2. Network flow representation of LSP-CC.
the cost function of the flow on that arc is ht . (Note that, without loss of optimality, the inventory at the end of period T is equal to zero.) In addition, an arc from node (C, t) to node (D, t) represents the amount produced in period t, xt , and the cost function of the flow on that arc is ct . Finally, an arc from node (C, t +1) to node (C, t) (t = 1, . . . , T ) represents the cumulative production amount in periods 1, . . . , t. An arc of this form has no cost but a finite capacity, C0t . (There are no capacities on the other arcs.) Production and inventory cost functions are commonly concave, representing the economies of scale that are often found due to the presence of, for example, fixed setup costs for production.When the functions ht and ct (t =1, . . . , T ) are concave, the LSP-CC thus becomes a minimum concave cost network flow problem. Despite the fact that an extreme point optimal solution to such problems exists, they are generally NP-hard (see [3]). However, polynomial-time algorithms have been found for many lot-sizing problems with concave costs by employing the structure of extreme point solutions for these problems. In particular, it is well-known that the standard uncapacitated economic lot-sizing model is solvable in O(T log T ) time (see [1,5,14]). Under stationary (timeinvariant) production capacities this model is solvable in O(T 3 ) time when inventory costs are linear (see [9]) and in O(T 4 ) time with general concave inventory cost functions (see [6]). On the other hand, the economic lot-sizing model with nonstationary (timevarying) capacities is NP-hard, even if the production cost functions have a fixed-charge structure and the holding cost functions are linear (see [7]). In the remainder of this section we will develop a polynomial-
time algorithm for the LSP-CC under nonstationary capacities. 4.2. Solution approach As mentioned above, minimum concave cost network flow problems have an extreme point optimal solution. Extreme point solutions to minimum cost network flow problems have the property that there does not exist a cycle formed by free arcs, i.e., arcs whose flow is strictly between its lower bound (usually 0) and upper bound (see, e.g., [2]). In the remainder of this section we will assume for simplicity that all demands are strictly positive. However, all results can be easily extended to the general case where periods with zero demand are allowed. Consider the set of free arcs in an extreme point solution to the network flow formulation of the LSPCC. Since all inventory and production arcs are uncapacitated, when we ignore the cumulative production arcs the set of remaining free arcs decomposes into disjoint connected and acyclic components. Each of these components is characterized by a consecutive set of demand periods, say t + 1, . . . , (where 0 t < T ), with the property that It = 0, Is > 0, I = 0.
s = t + 1, ..., − 1,
We will refer to a component satisfying these properties as a subplan, denoted by (t, ). As we will show later, the cost structure of the LSP-CC allows us to compute the minimum cost of supplying the demands in any subplan independently of the other subplans.
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This is perhaps surprising since the subplans seem to be intimately related through the (cumulative) production capacities. Now denote the cost of subplan (t, ), i.e., the minimum cost of supplying the demands in that subplan, by t and let F (t) be the minimum total cost of satisfying the demands in periods t + 1, ..., T , so that the optimal solution value of the LSP-CC is given by F (0). This then immediately leads to the following dynamic programming formulation of the LSP-CC: F (t) =
min
=t+1,...,T
{t + F ()}
for t = 1, ..., T − 1 and F (T ) = 0. This recursion is, of course, very similar to the recursion obtained for the economic lotsizing problem without capacities or with stationary capacities. It is easy to see that, if the subplan costs t were given, the LSP-CC could be solved in O(T 2 ) time. However, since there are O(T 2 ) subplans the computation of the subplan costs can be expected to be the bottleneck operation and the challenge is therefore to develop efficient algorithms for finding these costs. 4.3. Computing the subplan costs It is easy to see that all cumulative production arcs carry a positive flow that is possibly equal to their capacity. Now first note that it is easy to determine the flow on the cumulative production arcs that connect a pair of subplans in an extreme point solution. In particular, if (t, ) is a subplan, we know that the flow from node (C, t + 1) to node (C, t), i.e., the flow from the current subplan to past subplans, should be equal to d0t since It = 0 by the definition of a subplan, regardless of the actual subplans present in the solution up to time t. Moreover, since the cost on the cumulative production arcs is costless, it follows that we can view the subplan (t, ) in isolation by redefining the capacity of the cumulative production arc from ≡C −d node (C, s + 1) to node (C, s) to be Cts 0s 0t for s = t + 1, . . . , . (For convenience, we also define Ctt ≡ 0.) Furthermore, we know that, in an extreme point solution, there cannot be a cycle formed by free arcs alone. We conclude that between each pair of production periods within a given subplan there must be at least one cumulative production arc that is at capac-
ity. However, since the capacities of the cumulative production arcs are nondecreasing over time, we may assume without loss of generality that the first cumulative production arc between two production periods in a subplan is at capacity. The following theorem characterizes the solutions that we need to consider for a given subplan. Theorem 4.1. Consider a subplan (t, ) and let s be a production period in that subplan. Furthermore, let sˆ be the previous production period in the subplan (where sˆ = t if s is the first production period). The production quantity in period s is then equal to: • dt − Ctsˆ if s is the last production period in the subplan (which is only feasible if d0 C0s ); − C otherwise. • Cts t sˆ Proof. Let us first consider the case where period s is the last production period in the subplan. Then the quantity produced in period s should bring the total production quantity in the subplan to dt . Since the cumulative production quantity up to the previous production period sˆ was binding, the quantity that remains . (Note that if to be produced in period s is dt − C0ˆ s = C = 0 by definition so that sˆ = t we have that C0ˆ 0t s the production quantity is dt .) This is only feasible if the total demand in the subplan can be produced by = C − d or, equivalently, period s, i.e., if dt Cts 0s 0t if d0 C0s . If s is not the last production period in the subplan, we use the fact that we only need to consider extreme point solutions. Therefore, the cumulative production arc entering nodes (C, sˆ ) and (C, s) are at full capacity. Therefore, the quantity produced in period s should be equal to the difference between the two capacities, − C . (Note that if sˆ = t we obtain that production Cts t sˆ .) in period s is equal to Cts Fig. 3 illustrates an extreme point solution with two subplans and the associated arc flows. We will next formulate the problem of computing the minimum cost of a subplan (t, ) as a dynamic programming problem. To this end, we define states of the form (s, sˆ , b) where t sˆ s + 1 and b ∈ {0, 1}. The first element, s, denotes the current period while the second element, sˆ denotes the previous production period in the subplan. The third element
F.Z. Sargut, H.E. Romeijn / Operations Research Letters 35 (2007) 549 – 557
555
Fig. 3. Extreme point solution with two subplans and the associated arc flows.
indicates whether production has been completed for the subplan (b = 1) or not (b = 0). The source node is (t, t, 0) while the sink node is ( + 1, + 1, 1). For each state (s, sˆ , b) with t s − 1 there are up to three potential decisions to be made: (i) do not produce in period s + 1, (ii) produce in period s + 1 but do not complete all production, and (iii) complete all production in period s + 1. This means that from state (s, sˆ , 0) the following states can be reached: • (s + 1, sˆ , 0): this means that we do not produce in period s + 1. The cost of this decision is equal to hs+1 (Ctsˆ − dt,s+1 ) since the production in the previous production period sˆ was up to capacity. Note that this decision is only feasible if the demand in period s+1 can be satisfied using past production, i.e., if Ctsˆ dt,s+1 . • (s + 1, s + 1, 0): this means that we produce up to capacity in period s + 1. The cost of this decision is equal to cs+1 (Ct,s+1 −dt,s+1 ). −Ctsˆ )+hs+1 (Ct,s+1 Note that this decision is only feasible if the demand in period s+1 can be satisfied using production up to period s + 1, i.e., if Ct,s+1 dt,s+1 (since otherwise ending inventory in period s +1 would be negative). Moreover, we only need to consider this decision if Ct,s+1 < dt (since otherwise period s + 1 should be the last production period in the subplan). • (s + 1, s + 1, 1): this means that we make period s + 1 the last production period in the subplan. The cost of this decision is equal to cs+1 (dt − Ctsˆ ) + hs+1 (ds+1, ). Note that this decision is only feasible if dt Ct,s+1 , so that all demand of the subplan can indeed be satisfied using production up to time s + 1.
From state (s, sˆ , 1) with t s − 1, on the other hand, the only state that can be reached is • (s + 1, sˆ , 1): this means that we do not produce in period s + 1. The cost of this decision is equal to hs+1 (ds+1, ). From states ( − 1, sˆ , b), we can only go to states (, sˆ , 1) and (, , 1). Since the period is the latest period that the last production can occur. Finally, from state (, sˆ , 1) (which means that we have satisfied all demands in the subplan with production up to period sˆ ) we can reach state ( + 1, + 1, 1) at no cost. We are now ready to prove the main result of this section: Theorem 4.2. The LSP-CC can be solved in O(T 4 ) time. Proof. We first calculate and store the values dt for 1 t < T in O(T 2 ) time as well as the values C0t for 1 t T in O(T ) time. The dynamic programming formulation for computing the costs of a single subplan has O(T 2 ) states and a constant number of decisions per state. Since, using the information obtained in the preprocessing step, the costs of all decisions can be found in constant time, the cost of a single subplan can be found in O(T 2 ) time. The time required to find all subplan costs is therefore O(T 4 ). Since the high level dynamic program that employs the subplan costs runs in O(T 2 ) time, the LSP-CC with concave costs can be solved in O(T 4 ) time. 5. Allowing for backlogging In this section, we will generalize the LSP-CC to allow for backlogging. It immediately follows that this
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generalization is NP-hard in general. In addition, the results of Section 3.2 and van Hoesel and Wagelmans [10] can be used to develop a FPTAS for this generalization under the additional assumption that the production and backlogging cost functions are concave (but the inventory holding cost functions are only required to be monotone). If the production and backlogging cost functions are monotone but not concave we can instead modify the two very recently developed FPTASs for the economic lot-sizing problem that are presented in the papers by Chubanov et al. [4] and Ng et al. [13] in a straightforward manner by appropriately modifying the feasible production quantities in a given period with respect to the cumulative rather than the more traditional individual period capacities. However, note that the running time of these procedures (when expressed as a function of T and the approximation factor only) are O(T 11 /6 ) and O(T 7 log T /4 ), respectively, as opposed to O(T 3 log2 T +T 3 log T /2 ) for the former. In the remainder of this section we show that the dynamic programming algorithm developed in Section 4 can be modified to yield an O(T 4 ) algorithm for the LSP-CC with backlogging. We let ut be the quantity backlogged in period t with associated cost function bt (t = 1, . . . , T ). We assume these backlogging cost functions are concave and, without loss of generality, equal to zero at zero. In that case, the mathematical programming formulation of the problem becomes: minimize
T
(ct (xt ) + ht (It ) + bt (ut ))
t=1
subject to It−1 +xt +ut = dt +It +ut−1 , t = 1, ..., T , t
x C0t ,
= 1, ..., T ,
xt , It , ut 0, uT = 0,
t = 1, ..., T ,
=1
where we have assumed without loss of generality that I0 = u0 = 0. We can again view each extreme point solution as decomposed into a number of subplans, albeit with a slight modification of the definition of a subplan. In
particular, we now define a subplan (t, ) to be the consecutive set of demand periods t + 1, . . . , with the property that It = 0 and ut = 0, Is > 0 or us > 0, s = t + 1, ..., − 1. I = 0 and u = 0. The costs of the subplans can now be determined using a similar dynamic programming approach as for the case without backlogging. The only modification is that, whereas a decision was earlier deemed infeasible if it led to a negative inventory level, such a decision is now feasible where the cost of negative inventory is interpreted as the cost of the corresponding backlogged amount. We therefore have the following result. Theorem 5.1. The LSP-CC with backlogging can be solved in O(T 4 ) time. Acknowledgement The work of the second author was supported by the National Science Foundation under Grant no. DMI0355533. References [1] A. Aggarwal, J.K. Park, Improved algorithms for economic lot-size problems, Oper. Res. 41 (3) (1993) 549–571. [2] R.K. Ahuja, T.L. Magnanti, J.B. Orlin, Network Flows: Theory, Algorithms, and Applications, Prentice-Hall, Englewood Cliffs, NJ, 1993. [3] G.R. Bitran, H.H. Yanasse, Computational complexity of the capacitated lot size problem, Manage. Sci. 18 (1982) 12–20. [4] S. Chubanov, M. Kovalyov, E. Pesch, An FPTAS for a singleitem capacitated economic lot-sizing problem with general cost structures, Math. Programm. 106 (2006) 453–466. [5] A. Federgruen, M. Tzur, A simple forward algorithm to solve general dynamic lot sizing models with n periods in O(n log n) or O(n), Manage. Sci. 37 (1991) 909–925. [6] M. Florian, M. Klein, Deterministic production planning with concave costs and capacity constraints, Manage. Sci. 18 (1971) 12–20. [7] M. Florian, J.K. Lenstra, A.H.G. Rinnooy Kan, Deterministic production planning: algorithms and complexity, Manage. Sci. 26 (7) (1980) 669–679. [8] M.R. Garey, D.S. Johnson, Computers and Intractability, W.H. Freeman and Company, New York, NY, 1979. [9] C.P.M. van Hoesel, A.P.M. Wagelmans, An O(T 3 ) algorithm for the economic lot-sizing problem with constant capacities, Manage. Sci. 42 (1) (1996) 142–150.
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Operations Research Letters www.elsevier.com/locate/orl
Lot-sizing with non-stationary cumulative capacities F. Zeynep Sargut, H. Edwin Romeijn∗ Department of Industrial and Systems Engineering, University of Florida, P.O. Box 116595, Gainesville, FL 32611-6595 USA Received 14 January 2006; accepted 7 September 2006 Available online 30 October 2006
Abstract We study a new class of capacitated economic lot-sizing problems. We show that the problem is NP-hard in general and derive a fully polynomial-time approximation algorithm under mild conditions on the cost functions. Furthermore, we develop a polynomial-time algorithm for the case where all cost functions are concave. © 2006 Elsevier B.V. All rights reserved. Keywords: Capacitated lot-sizing; Fully polynomial-time approximation scheme; Dynamic programming
1. Introduction In this paper we study a new dynamic lot-sizing problem which we will refer to as the lot-sizing problem with cumulative capacities (LSP-CC). In traditional capacitated lot-sizing models, the quantity produced in each period is limited by some capacity, but any capacity remaining at the end of a period is essentially lost. In contrast, our new model applies to settings where any remaining capacity is transferred to the next production period. This means that the cumulative quantity that we can produce up to and including period t is constrained by the cumulative capacity up to and including period t.
∗ Corresponding author. Tel.: +1 352 392 3088;
fax: +1 352 392 2537. E-mail address: [email protected]fl.edu (H.E. Romeijn). 0167-6377/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2006.09.004
This problem may occur if the capacity constraints do not correspond to a perishable production capacity, such as time, but instead to a physical capacity in the form of a monetary budget or raw materials. The problem also appears in certain settings where the planning of procurement, production, and inventory holding of raw materials and final product is integrated. This two-level lot-sizing model has been studied extensively for the case where all cost functions are concave and capacities are absent (see, e.g., [15]), or procurement is subject to stationary capacities (see, e.g., [11,12]). Although in the presence of general procurement capacities the problem is known to be NP-hard even when all cost functions are concave (see [7]), Kaminsky and Simchi-Levi [12] derive certain conditions on the cost functions under which the problem is solvable in polynomial time. If the total cost of procurement and inventory of raw materials are exogenous to the model, the two-level problem reduces the
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LSP-CC. We study this problem under both general and concave cost functions. In Section 2, we formally introduce the LSP-CC. Then in Section 3, we prove that the problem is NPhard for general cost functions and provide a fully polynomial time approximation scheme (FPTAS) in case all cost functions are nondecreasing. In Section 4 we then develop a dynamic programming approach that solves the problem in polynomial time when all cost functions are concave. Finally, in Section 5 we briefly discuss an extension where backlogging is allowed. 2. Model formulation
• xt denotes the quantity produced in period t with associated cost function ct (t = 1, . . . , T ); • It denotes the inventory level at the production level at the end of period t with associated cost function ht (t = 1, . . . , T ). Without loss of generality, all cost functions are assumed to be equal to zero when their argument is zero. For convenience, we will define cumulative demands and capacities as dst = t=s+1 d (0 s < t T ) and C0t = t=1 C (1t T ). Then we can formulate the single-level LSP-CC as follows: minimize
(ct (xt ) + ht (It ))
t=1
subject to xt + It−1 = dt + It , t = 1, ..., T , t x C0t , = 1, ..., T , =1
xt , It 0,
• We show that the problem is NP-hard under general cost functions and develop a FPTAS for the case where all cost functions are nondecreasing. • We develop a dynamic programming algorithm that solves the problem in O(T 4 ) time when all cost functions are concave. • We extend these results to allow for backlogging.
3. General cost functions
We consider a single-level system in which the goal is to satisfy a deterministic and dynamic sequence of demands over a finite and discrete planning horizon at minimum cost through production and inventory. We denote the planning horizon by T . The demand faced in period t is denoted by dt (t = 1, . . . , T ). In period t (t = 1, . . . , T ), the procurement capacity is given by the total unused capacity through period t − 1 plus an additional capacity of Ct units that becomes available in period t. Our decision variables and associated cost functions are given below.
T
Note that we may, without loss of generality, assume that the initial inventory level I0 = 0. Our contributions in this paper with respect to this new lot-sizing problem are the following:
t = 1, ..., T .
(LSP-CC)
We next turn to an analysis of the LSP-CC with general cost functions. In particular, we will show that this problem is NP-hard in general. In addition, we provide a FPTAS for the LSP-CC with nondecreasing cost functions, i.e., an algorithm that finds a feasible solution with relative error no more than ε > 0 in an amount of time that is polynomial in the input size of the problem and 1/ε. 3.1. Proof of NP-hardness In this section, we will show that LSP-CC with zero holding cost is NP-hard by reducing the subset sum problem to an instance of LSP-CC. The SUBSET SUM problem is defined as follows (see [8]): SUBSET SUM: Given positive integers a1 , a2 , ..., aT and A, does there exist a set S ⊆ {1, ..., T } such that i∈S ai = A? The following theorem shows that the LSP-CC is NP-hard in general by reducing the SUBSET SUM problem to it. Theorem 3.1. The LPS-CC is NP-hard. Proof. Consider an instance of the SUBSET SUM problem. We then define the following instance of the LSP-CC: Ct = at for t = 1, ..., T , 0 for t = 1, ..., T − 1, dt = A for t = T ,
F.Z. Sargut, H.E. Romeijn / Operations Research Letters 35 (2007) 549 – 557 ct(x) a1+ ... +at+1
at+1 at
at
a1+ ... +at ↑
x
Fig. 1. Production cost function in period t.
ct (x) =
⎧ 0 ⎪ ⎪ ⎨1 +
at −1 at x
⎪ ⎪ ⎩x + 1
ht (It ) = 0
if x = 0, if 0 < x at for t = 1, ..., T , t a , if at < x =1
if It 0 for t = 1, ..., T .
The production cost function is illustrated in Fig. 1. First observe that, since the only non-zero demand is in period T , each production quantity will be used to satisfy the demand in that period and thus the cumulative production over the planning horizon is equal to A, i.e., T=1 x = A. Moreover, the production cost in period t satisfies ct (x) = x if x ∈ {0, at } and ct (x) > x otherwise. This implies that the cost of any production plan with production in any period t not equal to either 0 or at exceeds A. In other words, if the LSPCC has a solution with cost A the set of periods in which production takes place provides a solution to the SUBSET SUM problem. On the other hand, if the SUBSET SUM problem has a solution, say S, producing xt = at in all periods t ∈ S provides a solution to the LSP-CC with cost A. 3.2. A fully polynomial time approximation scheme Although the LSP-CC is NP-hard for general cost functions, we will show in this section that a FPTAS exists when all cost functions are nondecreasing and all problem data (i.e., demands, capacities, and costs) are integral. In particular, we will draw heavily on the paper by van Hoesel and Wagelmans [10] and modify the FPTAS that they developed for the standard capacitated lot sizing problem (CLSP) to our problem.
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Following van Hoesel and Wagelmans [10], we start by introducing a non-traditional dynamic programming formulation of the problem that runs in pseudopolynomial time. Let B be any integer upper bound on the optimal solution value, say z∗ , to the LSP-CC. For t = 1, ..., T and b = 0, 1, ..., B, we then define Ft (b) to be the maximal ending inventory in period t (It ) that can be achieved with a budget of b in the first t periods, i.e., the total production and inventory holding costs in periods 1, . . . , t do not exceed b. Since for t = 1 the cumulative capacity constraint is equal to the traditional capacity constraint we immediately obtain that F1 (b) =
max
d1 x1 C1
{x1 − d1 |c1 (x1 ) + h1 (x1 − d1 ) b}
for b = 0, . . . , B. A recursion for the values Ft (b) can then be derived by, for t = 2, . . . , T , allocating a budget of 0 a b to periods 1, . . . , t − 1 and a budget b − a for period t, and optimizing over the value of a. For the CLSP, van Hoesel and Wagelmans [10] show that, for a given value of a, (i) if it is possible to feasibly extend a production plan for periods 1, . . . , t − 1 that ends with the maximal inventory level Ft−1 (a) to period t it is optimal to do so; (ii) otherwise, attention may be restricted to xt = 0 and dt It−1 < Ft−1 (a). We will show that these two properties also hold for the LSP-CC. First, note that the flow balance constraints imply that t−1 =1 x = d0,t−1 + It−1 so that the cumulative capacity constraint in period t can be written as xt C0t −
t−1
x = C0t − d0,t−1 − It−1 .
=1
Moreover, since the demand in period t needs to be satisfied we need to have xt max{0, dt − It−1 }. The next proposition now shows that properties (i) and (ii) hold for the LSP-CC. Proposition 3.2. If there exists max{0, dt −Ft−1 (a)} xt C0t − d0,t−1 − Ft−1 (a) such that ct (xt ) + ht (Ft−1 (a) + xt − dt ) b − a
(1)
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then we only need to consider production plans with It−1 = Ft−1 (a). Otherwise, we only need to consider production plans with xt = 0 and dt It−1 < Ft−1 (a). Proof. Suppose we have a feasible solution with It−1 < Ft−1 (a) and xt > 0. Then if we increase It−1 and decrease xt by the same amount while retaining nonnegativity of xt condition (1) as well as the bound constraint on xt will still be satisfied. Moreover, the resulting value of It remains unchanged. Therefore, we can without loss of optimality in the dynamic programming recursion increase It−1 and decrease xt until either It−1 becomes equal to Ft−1 (a) or xt becomes equal to zero, whichever happens earlier. In the former case, the first claim follows. In the latter case, the second claim follows. As a result of the Proposition 3.2 the following dynamic programming recursion is obtained Ft (b) = max 0a b ⎧ maxmax{0,dt −Ft−1 (a)} xt C0t −d0,t−1 −Ft−1 (a) ⎪ ⎨ {Ft−1 (a) + xt − dt |ct (xt ) max +ht (Ft−1 (a) + xt − dt ) b − a}, ⎪ ⎩ max0 It
more than z∗ so that L z∗ . This implies that B =2T L is an upper bound on z∗ such that B 2T z∗ . To show that L can be found in polynomial time, we first 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 . Given this value, we define for each period t the following upper bounds on the production and inventory level: x¯t ≡ max{x : 0 x C0t , ct (x) } and I¯t ≡ max{I : I 0, ht (I ) }. These bounds can be determined using binary search in a total of O(T log(CT )) time. It is easy to see that a feasible solution in which each cost function contributes at most exists if and only if there exists a feasible solution that satisfies the above upper and lower bounds on the production and inventory levels. We can check this as follows: let mt denote the largest ending inventory in period t achievable by a production plan for the first t periods that satisfies all upper and lower bounds. In particular, we have m1 = min{x¯1 − d1 , I¯1 } and mt = min{mt−1 + min{C0t − d0,t−1 − mt−1 , x¯t } − dt , I¯t }, t = 2, . . . , T . If mT 0 then there exist a solution in which each cost function contributes at most . Now noting that L is bounded from above by maxt=1,...,T max{ct (C0t ), ht (dtT )} the value L can be determined in polynomial time using binary search.
4. Concave costs 4.1. Introduction The LSP-CC may be formulated as a minimum cost network flow problem as illustrated in Fig. 2. Nodes (D, 2), ..., (D, T ) are demand nodes, where (D, t) represents the demand in period t, dt . Nodes (C, 1), . . . , (C, T ) are transshipment nodes and node (C, T + 1) is a supply node having supply equal to the cumulative demand over the planning horizon, i.e., its demand is equal to the negative thereof: −d0T . As in traditional lot-sizing models, an arc from node (D, t) to node (D, t + 1) (t = 1, . . . , T − 1) represents the amount held in inventory at the end of period t, It , and
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553
Fig. 2. Network flow representation of LSP-CC.
the cost function of the flow on that arc is ht . (Note that, without loss of optimality, the inventory at the end of period T is equal to zero.) In addition, an arc from node (C, t) to node (D, t) represents the amount produced in period t, xt , and the cost function of the flow on that arc is ct . Finally, an arc from node (C, t +1) to node (C, t) (t = 1, . . . , T ) represents the cumulative production amount in periods 1, . . . , t. An arc of this form has no cost but a finite capacity, C0t . (There are no capacities on the other arcs.) Production and inventory cost functions are commonly concave, representing the economies of scale that are often found due to the presence of, for example, fixed setup costs for production.When the functions ht and ct (t =1, . . . , T ) are concave, the LSP-CC thus becomes a minimum concave cost network flow problem. Despite the fact that an extreme point optimal solution to such problems exists, they are generally NP-hard (see [3]). However, polynomial-time algorithms have been found for many lot-sizing problems with concave costs by employing the structure of extreme point solutions for these problems. In particular, it is well-known that the standard uncapacitated economic lot-sizing model is solvable in O(T log T ) time (see [1,5,14]). Under stationary (timeinvariant) production capacities this model is solvable in O(T 3 ) time when inventory costs are linear (see [9]) and in O(T 4 ) time with general concave inventory cost functions (see [6]). On the other hand, the economic lot-sizing model with nonstationary (timevarying) capacities is NP-hard, even if the production cost functions have a fixed-charge structure and the holding cost functions are linear (see [7]). In the remainder of this section we will develop a polynomial-
time algorithm for the LSP-CC under nonstationary capacities. 4.2. Solution approach As mentioned above, minimum concave cost network flow problems have an extreme point optimal solution. Extreme point solutions to minimum cost network flow problems have the property that there does not exist a cycle formed by free arcs, i.e., arcs whose flow is strictly between its lower bound (usually 0) and upper bound (see, e.g., [2]). In the remainder of this section we will assume for simplicity that all demands are strictly positive. However, all results can be easily extended to the general case where periods with zero demand are allowed. Consider the set of free arcs in an extreme point solution to the network flow formulation of the LSPCC. Since all inventory and production arcs are uncapacitated, when we ignore the cumulative production arcs the set of remaining free arcs decomposes into disjoint connected and acyclic components. Each of these components is characterized by a consecutive set of demand periods, say t + 1, . . . , (where 0 t < T ), with the property that It = 0, Is > 0, I = 0.
s = t + 1, ..., − 1,
We will refer to a component satisfying these properties as a subplan, denoted by (t, ). As we will show later, the cost structure of the LSP-CC allows us to compute the minimum cost of supplying the demands in any subplan independently of the other subplans.
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This is perhaps surprising since the subplans seem to be intimately related through the (cumulative) production capacities. Now denote the cost of subplan (t, ), i.e., the minimum cost of supplying the demands in that subplan, by t and let F (t) be the minimum total cost of satisfying the demands in periods t + 1, ..., T , so that the optimal solution value of the LSP-CC is given by F (0). This then immediately leads to the following dynamic programming formulation of the LSP-CC: F (t) =
min
=t+1,...,T
{t + F ()}
for t = 1, ..., T − 1 and F (T ) = 0. This recursion is, of course, very similar to the recursion obtained for the economic lotsizing problem without capacities or with stationary capacities. It is easy to see that, if the subplan costs t were given, the LSP-CC could be solved in O(T 2 ) time. However, since there are O(T 2 ) subplans the computation of the subplan costs can be expected to be the bottleneck operation and the challenge is therefore to develop efficient algorithms for finding these costs. 4.3. Computing the subplan costs It is easy to see that all cumulative production arcs carry a positive flow that is possibly equal to their capacity. Now first note that it is easy to determine the flow on the cumulative production arcs that connect a pair of subplans in an extreme point solution. In particular, if (t, ) is a subplan, we know that the flow from node (C, t + 1) to node (C, t), i.e., the flow from the current subplan to past subplans, should be equal to d0t since It = 0 by the definition of a subplan, regardless of the actual subplans present in the solution up to time t. Moreover, since the cost on the cumulative production arcs is costless, it follows that we can view the subplan (t, ) in isolation by redefining the capacity of the cumulative production arc from ≡C −d node (C, s + 1) to node (C, s) to be Cts 0s 0t for s = t + 1, . . . , . (For convenience, we also define Ctt ≡ 0.) Furthermore, we know that, in an extreme point solution, there cannot be a cycle formed by free arcs alone. We conclude that between each pair of production periods within a given subplan there must be at least one cumulative production arc that is at capac-
ity. However, since the capacities of the cumulative production arcs are nondecreasing over time, we may assume without loss of generality that the first cumulative production arc between two production periods in a subplan is at capacity. The following theorem characterizes the solutions that we need to consider for a given subplan. Theorem 4.1. Consider a subplan (t, ) and let s be a production period in that subplan. Furthermore, let sˆ be the previous production period in the subplan (where sˆ = t if s is the first production period). The production quantity in period s is then equal to: • dt − Ctsˆ if s is the last production period in the subplan (which is only feasible if d0 C0s ); − C otherwise. • Cts t sˆ Proof. Let us first consider the case where period s is the last production period in the subplan. Then the quantity produced in period s should bring the total production quantity in the subplan to dt . Since the cumulative production quantity up to the previous production period sˆ was binding, the quantity that remains . (Note that if to be produced in period s is dt − C0ˆ s = C = 0 by definition so that sˆ = t we have that C0ˆ 0t s the production quantity is dt .) This is only feasible if the total demand in the subplan can be produced by = C − d or, equivalently, period s, i.e., if dt Cts 0s 0t if d0 C0s . If s is not the last production period in the subplan, we use the fact that we only need to consider extreme point solutions. Therefore, the cumulative production arc entering nodes (C, sˆ ) and (C, s) are at full capacity. Therefore, the quantity produced in period s should be equal to the difference between the two capacities, − C . (Note that if sˆ = t we obtain that production Cts t sˆ .) in period s is equal to Cts Fig. 3 illustrates an extreme point solution with two subplans and the associated arc flows. We will next formulate the problem of computing the minimum cost of a subplan (t, ) as a dynamic programming problem. To this end, we define states of the form (s, sˆ , b) where t sˆ s + 1 and b ∈ {0, 1}. The first element, s, denotes the current period while the second element, sˆ denotes the previous production period in the subplan. The third element
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555
Fig. 3. Extreme point solution with two subplans and the associated arc flows.
indicates whether production has been completed for the subplan (b = 1) or not (b = 0). The source node is (t, t, 0) while the sink node is ( + 1, + 1, 1). For each state (s, sˆ , b) with t s − 1 there are up to three potential decisions to be made: (i) do not produce in period s + 1, (ii) produce in period s + 1 but do not complete all production, and (iii) complete all production in period s + 1. This means that from state (s, sˆ , 0) the following states can be reached: • (s + 1, sˆ , 0): this means that we do not produce in period s + 1. The cost of this decision is equal to hs+1 (Ctsˆ − dt,s+1 ) since the production in the previous production period sˆ was up to capacity. Note that this decision is only feasible if the demand in period s+1 can be satisfied using past production, i.e., if Ctsˆ dt,s+1 . • (s + 1, s + 1, 0): this means that we produce up to capacity in period s + 1. The cost of this decision is equal to cs+1 (Ct,s+1 −dt,s+1 ). −Ctsˆ )+hs+1 (Ct,s+1 Note that this decision is only feasible if the demand in period s+1 can be satisfied using production up to period s + 1, i.e., if Ct,s+1 dt,s+1 (since otherwise ending inventory in period s +1 would be negative). Moreover, we only need to consider this decision if Ct,s+1 < dt (since otherwise period s + 1 should be the last production period in the subplan). • (s + 1, s + 1, 1): this means that we make period s + 1 the last production period in the subplan. The cost of this decision is equal to cs+1 (dt − Ctsˆ ) + hs+1 (ds+1, ). Note that this decision is only feasible if dt Ct,s+1 , so that all demand of the subplan can indeed be satisfied using production up to time s + 1.
From state (s, sˆ , 1) with t s − 1, on the other hand, the only state that can be reached is • (s + 1, sˆ , 1): this means that we do not produce in period s + 1. The cost of this decision is equal to hs+1 (ds+1, ). From states ( − 1, sˆ , b), we can only go to states (, sˆ , 1) and (, , 1). Since the period is the latest period that the last production can occur. Finally, from state (, sˆ , 1) (which means that we have satisfied all demands in the subplan with production up to period sˆ ) we can reach state ( + 1, + 1, 1) at no cost. We are now ready to prove the main result of this section: Theorem 4.2. The LSP-CC can be solved in O(T 4 ) time. Proof. We first calculate and store the values dt for 1 t < T in O(T 2 ) time as well as the values C0t for 1 t T in O(T ) time. The dynamic programming formulation for computing the costs of a single subplan has O(T 2 ) states and a constant number of decisions per state. Since, using the information obtained in the preprocessing step, the costs of all decisions can be found in constant time, the cost of a single subplan can be found in O(T 2 ) time. The time required to find all subplan costs is therefore O(T 4 ). Since the high level dynamic program that employs the subplan costs runs in O(T 2 ) time, the LSP-CC with concave costs can be solved in O(T 4 ) time. 5. Allowing for backlogging In this section, we will generalize the LSP-CC to allow for backlogging. It immediately follows that this
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generalization is NP-hard in general. In addition, the results of Section 3.2 and van Hoesel and Wagelmans [10] can be used to develop a FPTAS for this generalization under the additional assumption that the production and backlogging cost functions are concave (but the inventory holding cost functions are only required to be monotone). If the production and backlogging cost functions are monotone but not concave we can instead modify the two very recently developed FPTASs for the economic lot-sizing problem that are presented in the papers by Chubanov et al. [4] and Ng et al. [13] in a straightforward manner by appropriately modifying the feasible production quantities in a given period with respect to the cumulative rather than the more traditional individual period capacities. However, note that the running time of these procedures (when expressed as a function of T and the approximation factor only) are O(T 11 /6 ) and O(T 7 log T /4 ), respectively, as opposed to O(T 3 log2 T +T 3 log T /2 ) for the former. In the remainder of this section we show that the dynamic programming algorithm developed in Section 4 can be modified to yield an O(T 4 ) algorithm for the LSP-CC with backlogging. We let ut be the quantity backlogged in period t with associated cost function bt (t = 1, . . . , T ). We assume these backlogging cost functions are concave and, without loss of generality, equal to zero at zero. In that case, the mathematical programming formulation of the problem becomes: minimize
T
(ct (xt ) + ht (It ) + bt (ut ))
t=1
subject to It−1 +xt +ut = dt +It +ut−1 , t = 1, ..., T , t
x C0t ,
= 1, ..., T ,
xt , It , ut 0, uT = 0,
t = 1, ..., T ,
=1
where we have assumed without loss of generality that I0 = u0 = 0. We can again view each extreme point solution as decomposed into a number of subplans, albeit with a slight modification of the definition of a subplan. In
particular, we now define a subplan (t, ) to be the consecutive set of demand periods t + 1, . . . , with the property that It = 0 and ut = 0, Is > 0 or us > 0, s = t + 1, ..., − 1. I = 0 and u = 0. The costs of the subplans can now be determined using a similar dynamic programming approach as for the case without backlogging. The only modification is that, whereas a decision was earlier deemed infeasible if it led to a negative inventory level, such a decision is now feasible where the cost of negative inventory is interpreted as the cost of the corresponding backlogged amount. We therefore have the following result. Theorem 5.1. The LSP-CC with backlogging can be solved in O(T 4 ) time. Acknowledgement The work of the second author was supported by the National Science Foundation under Grant no. DMI0355533. References [1] A. Aggarwal, J.K. Park, Improved algorithms for economic lot-size problems, Oper. Res. 41 (3) (1993) 549–571. [2] R.K. Ahuja, T.L. Magnanti, J.B. Orlin, Network Flows: Theory, Algorithms, and Applications, Prentice-Hall, Englewood Cliffs, NJ, 1993. [3] G.R. Bitran, H.H. Yanasse, Computational complexity of the capacitated lot size problem, Manage. Sci. 18 (1982) 12–20. [4] S. Chubanov, M. Kovalyov, E. Pesch, An FPTAS for a singleitem capacitated economic lot-sizing problem with general cost structures, Math. Programm. 106 (2006) 453–466. [5] A. Federgruen, M. Tzur, A simple forward algorithm to solve general dynamic lot sizing models with n periods in O(n log n) or O(n), Manage. Sci. 37 (1991) 909–925. [6] M. Florian, M. Klein, Deterministic production planning with concave costs and capacity constraints, Manage. Sci. 18 (1971) 12–20. [7] M. Florian, J.K. Lenstra, A.H.G. Rinnooy Kan, Deterministic production planning: algorithms and complexity, Manage. Sci. 26 (7) (1980) 669–679. [8] M.R. Garey, D.S. Johnson, Computers and Intractability, W.H. Freeman and Company, New York, NY, 1979. [9] C.P.M. van Hoesel, A.P.M. Wagelmans, An O(T 3 ) algorithm for the economic lot-sizing problem with constant capacities, Manage. Sci. 42 (1) (1996) 142–150.
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