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A new formulation for a variant of the Capacitated Lot Sizing and Scheduling Problem with Sequence Dependent setup costs and times: Single machine, multiple products and periods
2010 Management and Control of Production Logistics University of Coimbra, Portugal September 8-10, 2010
A new formulation for a variant of the Capacitated Lot Sizing and Scheduling Problem with Sequence Dependent setup costs and times: Single machine, multiple products and periods I. Chaieb Memmi*, S. Hammami* * ESTI, 45 Rue des Entrepreneurs, Charguia II, 2035 Tunis Carthage, Tunisia (Tel: 216-71-941-579; e-mail: [email protected], [email protected] )
Abstract: We present in this paper a new formulation of the Capacitated Lot sizing and Scheduling Problem with Sequence Dependent set up cost and times (CLSPSD) problem, well known as an NP-Hard problem. We formulate the problem as a non linear model and use it to modify a lot sizing problem described in literature and applied to a real-life problem of a Canadian paper maker where the paper machine represents the bottleneck resource and consequently requires an efficient production planning approach optimizing resource utilization and minimizing production, setup and inventory costs. In these previous works, the dependency of the costs and times with the production sequences is not thoroughly dealt with. We prove that by really treating the problem of determining the production sequences, we handle to considerably reduce the supply chain total cost. This cost includes production costs, inventory holding costs, start up and setup costs. The reduction in the total cost varies from 6 to 52% when setup costs vary from 20% to 200% of inventory holding costs. some other researches having dealt with the lot sizing problems with set up costs dependent on sequences are described. In section three, we describe a real-life problem of a Canadian paper maker where the paper machine represents the bottleneck resource and consequently requires an efficient production planning approach, optimizing resource utilization and minimizing production, setup and inventory costs. This problem has already been treated in (Bouchriha et al., 2007) and we aim to study the profits in total cost that offer the relaxation of some assumptions; in fact, we no more impose that a predetermined production sequence must be maintained, or that all the types of products must be manufactured at each production period. Consequently, we change the model described by the authors in a Capacitated Lot sizing and Scheduling Problem with Sequence Dependent set up cost and times, and evaluate the benefits of this model on the total cost considered.
1. INTRODUCTION A supply chain consists of all parties involved, directly or indirectly, in fulfilling a customer request. The supply chain includes not only the manufacturer and suppliers, but also transporters, warehouses, retailers, and even customers themselves. Within each organization, such as manufacturer, the supply chain includes all functions involved in receiving and filling a customer request. To encounter the fierce competitions encompassed in today’s global market, companies have to well manage their supply chains. A proper management of supply chain systems leads to significantly increase profit and reduce operation costs. A wide variety of models for production planning and inventory management are offered in the literature. Lot sizing problems over a planning horizon make plans that minimize cost while satisfying forecasted external demands. These problems are classified in (Jans and Degraeve, 2004) into two classes: -
-
The forth section presents results and simulations whereas the last one concludes our work and offers some perspectives.
The continuous time-scale, constant demand and infinite time horizon lot sizing problems. The famous Economic Order Quantity model (EOQ) and the Economic Lot Scheduling Problem (ELSP) belong to this category;
Table 1. Lot Sizing Models
Lot Sizing Models
The discrete time scale, dynamic demand and finite time horizon lot sizing models. In general, we refer to this type of planning as dynamic lot sizing.
We are interested in this paper to the second category of lot sizing problems. A second classification of Lot Sizing Models is provided in (Jans and Degraeve, 2007), see Table 1. This classification will be discussed section 2 where 978-3-902661-81-4/10/$20.00 © 2010 IFAC
The single item uncapacitated lot sizing problem (ULS) The Capacitated Lot Sizing Problem (CLSP) Small bucket models Multi-Level Lot Sizing 2. LITERATURE REVIEW
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Let us define in what follows the variants of lot sizing problems mentioned in Table 1.
sequences. From this set and in each period, the best sequence is retained regard to its cost and whether or not each item’s inventory is empty. Well, even if the tailor-made enumeration method of the branch-and-bound type solves problem instances optimally, the important number of assumptions hindrances applying this method in more general cases. In fact, the authors assume that:
The single item uncapacitated lot sizing problem (ULS): In its simplest form, the dynamic lot sizing problem tries to find a production plan that minimizes the total costs of production, set up and inventory for one item over a multiperiod horizon. This plan has to satisfy deterministic demand. The capacitated multi-item lot sizing problem (CLSP) is a typical example of a large bucket model, where many different items can be produced on the same machine in one time period. The production capacity of this machine is limited. Small bucket models differ from the large bucket ones by an assumption made, which imposes that, during each time period; at most one type of item can be produced on the same machine. This is the case for the Continuous Set Up Lot Sizing Problem (CSLP).
-
if the production of an item starts in a period then the inventory of the item must be empty at the end of the previous period,
-
the set up is preserved between two adjacent periods, this means that we must first produce the item i at period t if this item is the last one produced in period (t-1),
-
for each item, at most one lot will be produced in a period.
Due to this last assumption, we could no more optimize the quantities to produce from each item. In fact, optimizing these quantities may imply to produce the quantities demanded over two periods, in only the first period.
The Proportional Lot Sizing and Scheduling Problem (PLSP) relaxes the restriction of allowing production for only one product in each time period. Nevertheless, in the PLSP, at most two different items can be produced in each time period. There is still at most one set up in each period, but the set up from the previous period can be carried over to the next period. The Discrete Lot Sizing and Scheduling Problem (DLSP) differs from the CSLP in that a discrete production policy is assumed, implying that an item must be produced at full capacity.
In (Kovács et al., 2009), the authors develop an approach partly based on the work of (Haase and Kimms, 2000). They address the capacitated lot-sizing and scheduling problem with sequence-dependent setup times and costs (CLSPSD). They show that the complexity of this large-bucket lot-sizing problem originates from the series of implicit sequencing problems that have to be solved for the items produced in each time period. Consequently, they develop an efficient algorithm in order to determine during pre-processing all item sequences that could appear in an optimal solution. A novel MIP formulation of CLSPSD that relies on a compact representation of those sequences by using item-related binary variables is introduced. According to authors, the model developed allows solving in reasonable time instances where the product of the number of items and number of time periods is at most 60–70.
In a Multi-level Lot Sizing problem, the production planning is not only done at the final level for the finished product, but also for the components and subassemblies needed to make this finished product. Production at one level leads to demand for components at a lower level, and at the highest level, production is triggered by market demand. In addition to this set of models described in (Jans and Degraeve, 2007), authors in (Salomon et al., 1997), consider another type of lot sizing problems, called the Discrete Lot sizing and Scheduling Problem with Sequence Dependent set up cost and times (DLSPSD). So, as it is underlined in the designation of this lot sizing problem type, setup costs and times depend on the production sequences retained. To solve this problem, and based on a transformation of the DLSPSD into a Travelling Salesman Problem with Time Windows, an exact solution procedure is developed. The sensitivity of the proposed approach’s performance is analysed for different problem characteristics: problem dimension, inventory holding costs, set-up times and production capacity utilization. Even if the set-up times are not explicitly taken into account in the definition of production capacity utilization, the authors have investigated the relationship between this capacity and the average CPU-time of their algorithm.
As in (Haase and Kimms, 2000), and both using MIP techniques, exact optimisation approaches to CLSPSD have been suggested by (Gupta and Magnusson, 2005). The authors define a MIP that simultaneously makes the lotsizing, and all sequencing decisions within time periods. The developed approach allows finding optimal solutions only for very small instances (three items and three time periods). For larger problems, a heuristic procedure is proposed in the same paper. The authors recommend using this procedure when there are many more products than there are planning periods. From a practical point of view, the procedure comforts firms which need to solve the problem repeatedly for a few periods at a time. In (Bouchriha et al., 2007), a real case study is presented. The authors discuss a specific lot sizing problem and different scenarios where a predetermined production sequence has to be maintained, all types of products have to be produced in each production period, etc. The production is planned under a cyclic manner. For this, a new approach is proposed to first compute the cycle time by assuming constant demand, then,
The authors, in (Haase and Kimms, 2000), deal with lot sizing and scheduling for a single-stage, single-machine production systems where set up costs and times are sequence dependent. They formulate a large-bucket mixed integer programming which is based on the set of efficient 84
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I i0 = K
lot sizes are determined for each product within each cycle in order to satisfy demand. When comparing the different scenarios analysed: fixing a cycle for all products, or the quantity to produce in each cycle, etc., the authors show that the scenario fixing each of the cycle and the quantities is the most expansive. The other scenarios aren’t considerably different in terms of potential gains. Among theses scenarios, the last one is the less constrained, in fact, all the products types have not to be produced in each period, besides, the production sequences have not to be prefixed. Even though these assumptions are removed, this scenario does not rigorously illustrate the CLSPSD problem, we’ll explain more in detail this point of view when describing the mathematical model developed by the authors.
I it ≥ SS i = K
Model (1):
(1.1)
∑ ( k it ρ it + ait Qit ) = C t t = 1,..., T
(1.2)
qit . ρ it ≤ Qit ≤ M . ρ it i = 1,..., n, t = 1,..., T
(1.3)
i =1
(1.7)
Table 3. Model (1) Decision variables (1)
Subject to:
n
Qit ≥ 0, I it ≥ 0 i = 1,..., n t = 1,..., T
Period index (t∈{1, …, T}) Set of products to be manufactured on the machine (IP⊂{1, …, n}) i Product index, i∈IP hit Inventory holding cost for product i ($/ton per period), i∈IP bit Unit production cost ($/ton), i∈IP Kit Setup cost for product I ($), i∈IP M A very large number qit Minimum quantity to produce when a setup occurs (ton) dit Demand of item i that corresponds to period t (ton), i∈IP kit Setup time for item i at period t (hour), i∈IP ait Unit of capacity consumption of item i at period t (hour/ton), i∈IP Ct Production capacity per period (hour machine) SSi Safety stock for product i (ton), i∈IP Ii0 Stock level at the end of the first period (ton), i∈IP, It is a constant (K)
The model described in (Bouchriha et al., 2007) and noted Model (1), determines optimum production batch size and setup for each planning period that minimizes the total inventory, setup and production costs for all products over the entire planning period. Model (1) is a mixed integer program described, using notations in Tables 2 and 3, as follows:
Qit + I it −1 = I it + d it i = 1,..., n, t = 1,..., T
(1.6)
T IP
3.1 Initial Mathematical model
t =1 i =1
ρ it ∈ {0,1} i = 1,..., n t = 1,..., T
Table 2. Model (1) Parameters
3. MATHEMATICAL MODEL
n
(1.5)
Here, we need to underline that constraints (1.3) impose that a minimum quantity has to be produced if a setup occurs. From this Model, we’ll omit in the subsequent paragraph the terms Kit.ρit, from the objective function, keep constraints (1.1), (1.2), (1.4), (1.5), (1.7) and modify constraints (1.3) and (1.6). Our objective is to introduce setup costs incurred each time we switch from one item’s type to the next.
To recap, we simultaneously try to find, at each period of the planning horizon, the finished products quantities to manufacture and the production sequences to adopt. Finding these production sequences lead us to deal simultaneously with two planning levels: tactical and operational.
T
i = 1,..., n t = 1,..., T
(1.4)
The constraints include material balance (1.1), production output capacity of machine (1.2) and constraints requiring setup in periods in which production occurs, constraints (1.3) and (1.6). Constraint (1.4) implies that an initial inventory must be fixed in the planning problem. The authors define this parameter according to the average of the demand. Constraints (1.5) imply that, at the end of each period, the stock level must be at least equal to the safety stock level. Backorders are not allowed, (1.7).
We aim in this paper to continue the works presented in (Bouchriha et al., 2007) and specifically the last scenario described. We then modify the lot sizing problem on a Capacitated Lot sizing and Scheduling Problem with Sequence Dependent set up cost and times, and consider the single machine variant of this problem, multiple products, multiple periods and dynamic demands. Our objective is to really focus on set up costs and times depending on production sequences. The model we develop determines in each production period, the lot sizes and the production sequences. These sequences have not to be respected from one period to another. Moreover, we are not obliged to produce all types of products in each period. The set of items to produce in each period will be dependent on the optimization of the production batches and different costs.
Min ∑ ∑ (hit . I it + bit . Qit + K it . ρ it )
i = 1,..., n
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ρit Binary variables that corresponds to 1 if the product is manufactured at period t and 0 otherwise, i∈IP Qit The quantity of product i to produce ate period t (ton), i∈IP Iit Stock level of product I at the end of the period t (ton), i∈IP
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3.2 Mathematical model for the setup
period (t+1) verifies ∑ xl i t +1 = 0 what means only one
Taking into account setup costs within production periods and between period t and period (t+1) involves combining the following Model to the Model (1) described previously. The notations used here are explained in Tables 4 and 5.
product is manufactured after product l, at the period (t+1).
i
⎛n n ⎞ T −1 n n Min∑ ⎜ ∑ ∑ cijt . xijt ⎟ + ∑ ⎛⎜ ∑ ∑ cklt . y klt ⎞⎟ t =1 ⎝ i =1 j =1, j ≠i ⎠ ⎠ t =1 ⎝ k =1 l =1,k ≠l T
Table 4. Model (2) Parameters
(2)
Subject to:
zit ≤ Qit ∀i ∀t
(2.1)
Qit ≤ zit .M
(2.2)
∀i ∀t
0 ≤ Posit ≤ ∑ zit ∀i ∀t
(2.3)
Posit ≤ zit .M
∀i ∀t
(2.4)
zit ≤ Posit .M
∀i ∀t
(2.5)
zit + z jt ≤ 1 + Pos jt − Posit ∀i ∀j ≠ i ∀t
(2.6)
1 − xijt ≤ z jt − zit + ( Pos jt − Posit ) − 1 ∀i ∀j ≠ i ∀t
(2.7)
i
( z kt + zl t +1 + (1 − ∑ xkit ) + (1 − ∑ xil t +1) − 3) ≤ y klt .M i
i
∀k , l ∈ {1,.., n}, l ≠ k , ∀t = 1,..., T − 1 Qit ≥ 0, I it ≥ 0 ∀i, ∀t z it ∈ {0,1},
(2.9)
Posit ∈ IN , ∀i
xijt ∈ {0,1}, ∀i
∀j ≠ i
(2.8)
∀t
(2.10)
∀t
(2.11)
y klt ∈ {0,1}, ∀k ∀l ≠ k ∀t = 1,..., T − 1
(2.12)
Constraints (2.1) and (2.2) indicate that if we manufacture product i in period t then variables zit equal 1, thus, if amounts Qit are strictly positive then zit equal 1. Relation between variables zit and Posit are determined by constraints (2.3), (2.4) and (2.5). While constraints (2.4) and (2.5) affect 0 to Posit when product i isn’t manufactured at period t, i.e. when zit equal 0, constraint (2.3) determine their domains of variation. These domains are delimited by 0 and the number of products manufactured at period t, i.e. the sum of zit. Constraints (2.6) avoid attributing the same position in the production sequence to two different items. In fact, if we manufacture products i and j at period t, i.e. if zit and zjt equal 1, then (Posjt-Posit) is necessary greater than 1 or lower than (-1). Constraints (2.7) attribute the value 1 to variables xijt when products i and j are manufactured at period t (zit and zjt equal 1) and item’s j position succeeds item’s i position in the production sequence, what means product j is manufactured immediately after product i (Posjt-Posit=1). Finally constraints (2.8) imply that if product k has the last position in the production sequence at period t and product l has the first position in the production sequence at period (t+1) then the binary variable yklt equals 1. The last product of the production sequence at period t verifies ∑ xkit = 0 what
csi Start up costs of product i, i∈IP cij Setup costs incurred when manufacturing item j after item i ($), i, j∈IP ckl Has the same signification that cij, it is used here just to underline the fact that this setup cost is incurred between two distinguished periods ($), k, l∈IP
Table 5. Model (2) Decision variables xijt Binary variables that correspond to 1 if the product j is manufactured immediately after product i at period t and 0 otherwise, i, j∈IP, t∈{1, …, T} yklt Binary variables that correspond to 1 if product k is the last product of the production sequence in period t and l the first product of the production sequence in period t+1, and 0 otherwise, k, l∈IP, (t∈{1, …, T}) zit Binary variables that correspond to 1 if product i is manufactured in period t and 0 otherwise, i∈IP, (t∈{1, …, T}) Posit Integer variables that indicate the position of product i in the production sequence, at period t, i∈IP, (t∈{1, …, T}) si Binary variable that correspond to 1 if product i is the first product manufactured in period 1 and 0 otherwise, i∈IP As we can see in Model (2), due to the absolute values, the mathematical model is a mixed integer non linear program. Solving this model with a commercial solver like LINGO may give local optimum solutions. 2.3 CLSPSD mathematical model The CLSPSD problem is finally formulated as follows: Model (3):
⎛n ⎞ ⎜ ∑ csi . si + ⎟ ⎜ i=1 ⎟ ⎜T ⎛n ⎟ n n ⎞ Min⎜ ∑ ⎜ ∑ (hit . I it + bit . Qit ) + ∑ ∑ cijt . xijt ⎟ + ⎟ i =1 j =1, j ≠i ⎠ ⎟ ⎜ t =1 ⎝ i=1 ⎜ T −1⎛ n n ⎟ ⎞ ⎜ + ∑ ⎜ ∑ ∑ cklt . y klt ⎟ ⎟ ⎠ ⎝ t =1 ⎝ k =1 l =1,k ≠l ⎠
(3)
Subject to:
i
(1.1), (1.2), (1.4), (1.5), and (1.7), (2.1) to (2.12). We have to note that in constraints (1.2), variables ρit are substituted by zit.
means no product is manufactured after product k, at the period t. The first product of the production sequence at
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And:
si ≤ zi1 ∀i
(3.1)
( zi1 + (1 − ∑ xki1) − 1) ≤ si .M
∀i
(3.2)
i
si ∈ {0,1}, ∀i
(3.3)
In the objective function, the first term corresponds to start up costs, the second to inventory costs, the third to productions costs, the forth to setup costs within production periods and the last to setup costs between periods.
these costs. If we observe Model 1, we can obviously notice that when setup costs are more important than holding costs, we’ll prefer producing in few periods and stocking important amounts of products. On the contrary, when setup costs are not very important, we prefer producing just the needed quantities of products in each production period in order to hold small amounts in stocks. When solving Model 1, we obtain the quantities to produce and to have in stocks in each production period, and the set of products to manufacture in each period. However, these sets of products aren’t scheduled and this is due to the fact that the problem of setup costs and times depending on sequences isn’t really dealt with. In our model, we have considered this dependency and we aim by analyzing these two following scenarios to demonstrate how the problem of simultaneously scheduling production and determining lot sizes can modify the amounts of products to plan each period, so the scheduling problem can stage-manage the planning problem.
Constraints (3.1) assign the value 0 to si if product i is not produced in the first period and constraints (3.2) assign the value 1 to si if product i is manufactured in the first period (zi1=1), and this product is the first product manufactured at this period, this means that no product precedes i, so ∑ xki1 = 0 . k
To solve this non linear Integer problem, including O(n2T) variables and O(n2T) constraints, we use the commercial solver LINGO version 12.0.
Scenario 1: setup costs = 20% holding costs Scenario 2: setup costs = 200% holding costs Since the holding costs are unitary, we have estimated total holding costs by multiplying the random variables chosen from the interval [1,10] (Tale 6) by the average demand. Choosing the average demand is due to the fact that in one hand, we have no idea about the average level of stocks; in the other hand the average demand can comfort the two opposite scenarios: important unitary holding costs, low unitary holding costs. In fact, when the inventory holding costs aren’t important, we have tendency to increase the amounts stocked; in opposite, if the holding costs are important we try to reduce as possible the amounts stocked.
4. RESULTS In this section, we focus on two points; first, we compare our results with those published in (Bouchriha et al., 2007), secondly, we analyse the sensitivity of these results to two parameters: inventory costs and setup costs. 4.1 Comparison with the initial model (1) In this paragraph and in order to compare our results to those described in (Bouchriha et al., 2007), we use the illustrative example given Table 6.
The obtained results are given Table 7. If we observe Scenario 2, for Model 1, the way of thinking is, since the holding costs are not very important, we have to produce, if the machine’s capacity allows, all the needed amounts of products and if it is possible, one product type per period (in our example, we have only two periods, this is why P1 and P2 are produced on the same period). Well, by solving simultaneously the lot sizing problem and the scheduling problem, we obtain different production sequences that lead to a decrease in the total cost of the supply chain.
Table 6. The illustrative example T IP dit
2 3 Random number that belongs to the interval [10,200] Ii0 Random number that belongs to the interval [0,40] SSi Random number that belongs to the interval [0,100] Ct 100 000 hours bit Random number that belongs to the interval [1,10] hit Random number that belongs to the interval [1,10] cij and Random number that belongs to intervals defined ckl proportionally to the intervals of hit. Kit Random number that belongs to intervals defined proportionally to the intervals of hit. qit Random number that belongs to the interval [50,150] ait Random number that belongs to the interval [10,20] kit Random number that belongs to the interval [1,5] M 109 Here, we analyse two scenarios: important holding costs and low holding costs. Before, we should explain our interest to
It is the same for scenario 1 where setup costs are not very important, and so we can produce, at each period all the products’ types; this is well illustrated in Model 1. In fact, we produce two types of products in period 1 and three in period 2. Our Model proves that simultaneously solving the lot sizing and scheduling problems don’t systematically lead to the Model’s 1 conclusion. In fact, by producing three types of products on period 1 and only one on period 2 we can divide by two the supply chain total cost. The same reasoning can be applied to study production costs versus setup costs.
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4.2 Analysis of the model’s sensitivity
Table 7. Results of the illustrative example Model 1 1st period - Qit - Iit
(P1,P2) (105 P1,100 P2) (5 P1,40 P2,10 P3)
P3-P2-P1/ (1 P3,265 P2,205 P1) (105 P1,205 P2,11 P3)
2nd period - Qit - Iit
(P1,P2,P3) (100 P1,165 P2,150 P3) (5 P1,5 P2,130 P3)
P3 24 P3 (5 P1,5 P2,5 P3)
Total cost
5102 $
2432 $
In this paragraph, we want to compare our results with those of Model 1 when varying the setup costs from 20% to 200% of the holding costs. Fig.1 shows that, in all cases, we manage to significantly reduce the total cost. This reduction varies from 6 to 52%. Reducing these costs can be better illustrated when we consider more production periods. As in (Haase and Kimms, 2000) and from a practical point of view, our model comforts more firms aiming to solve the CLSPSD problem repeatedly for few periods at a time. 12000 10000
5653 $
0
Total cost for CLSPLD Model
200%
8568 $
2000
160%
Total cost
4000
180%
P1-P2-P3 (100 P1,200 P2,25 P3) (5 P1,5 P2,5 P3)
140%
(P3) (105 P3) (5 P1,5 P2,130 P3)
6000
120%
2 period - Qit - Iit
nd
8000
80%
P2-P1/ (65 P2,105 P1) (5 P1,5 P2,10 P3)
100%
(P1,P2) (205 P1,265 P2) (105 P1,205 P2,10 P3)
60%
1st period - Qit - Iit
20%
Scenario 2
Model 4
40%
Scenario 1
Total Cost for Model I
We have to underline that computational times for all the simulations we have done are about 1 second.
Fig. 1. Comparison of total costs when varying the percentage Setup costs/Holding costs efficient rescheduling opportunities. International Journal of Production Economics, 66, 159-169. Jans, R. and Z. Degraeve (2007). Meta-heuristics for dynamic lot sizing: A review and comparison of solution approaches. European Journal of Operational Research, 177, 1855-1875. Kovács, A., K.N. Browna and A. Tarim (2009). An efficient MIP model for the capacitated lot sizing and scheduling problem with sequence dependent setups.
5. CONCLUSIONS In this paper, we have dealt with the Capacitated Lot sizing and Scheduling Problem with Sequence Dependent set up cost and times. The problem we have treated is characterized by one machine, a dynamic and deterministic demand, multiple product types and few production periods. We formulate the problem as a non linear mixed integer model and compare our results with those described in (Bouchriha et al., 2007). Even though the optimal solution was not found, using the solutions provided by the solver LINGO during the solution process, we manage to reduce the supply chain total cost, this cost includes production costs, inventory holding costs, start and set up costs. This reduction can significantly be amplified when we increase the number of production periods. In future works we’ll make use of Meta heuristics in order to treat a problem with an important number of product types and production periods.
International Journal Economics, 118, 282-291.
of
Production
Salomon, M., M.M. Salomon, L.N. Van Wassenhove, Y.D. Dumas and S. Dauzere-Peres (1997). Solving the discrete lot sizing and scheduling with sequence dependent setup costs and setup times using the Travelling Salesman Problem with time windows.
European Journal of Operational Research, 100, 494-513.
REFERENCES Bouchriha, H., M. Ouhimmou and S. D’Amours (2007). Lot sizing problem on a paper machine under a cyclic production approach. International Journal of Production Economics, 105, 318-328. Gupta, D. and T. Magnusson (2005). The capacitated lot sizing and scheduling problem with sequence dependent setup costs and setup times. Computers & Operations Research, 32, 727-747. Haase K. and A. Kimms (2000). Lot sizing and scheduling with sequence-dependent setup costs and times and
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A new formulation for a variant of the Capacitated Lot Sizing and Scheduling Problem with Sequence Dependent setup costs and times: Single machine, multiple products and periods I. Chaieb Memmi*, S. Hammami* * ESTI, 45 Rue des Entrepreneurs, Charguia II, 2035 Tunis Carthage, Tunisia (Tel: 216-71-941-579; e-mail: [email protected], [email protected] )
Abstract: We present in this paper a new formulation of the Capacitated Lot sizing and Scheduling Problem with Sequence Dependent set up cost and times (CLSPSD) problem, well known as an NP-Hard problem. We formulate the problem as a non linear model and use it to modify a lot sizing problem described in literature and applied to a real-life problem of a Canadian paper maker where the paper machine represents the bottleneck resource and consequently requires an efficient production planning approach optimizing resource utilization and minimizing production, setup and inventory costs. In these previous works, the dependency of the costs and times with the production sequences is not thoroughly dealt with. We prove that by really treating the problem of determining the production sequences, we handle to considerably reduce the supply chain total cost. This cost includes production costs, inventory holding costs, start up and setup costs. The reduction in the total cost varies from 6 to 52% when setup costs vary from 20% to 200% of inventory holding costs. some other researches having dealt with the lot sizing problems with set up costs dependent on sequences are described. In section three, we describe a real-life problem of a Canadian paper maker where the paper machine represents the bottleneck resource and consequently requires an efficient production planning approach, optimizing resource utilization and minimizing production, setup and inventory costs. This problem has already been treated in (Bouchriha et al., 2007) and we aim to study the profits in total cost that offer the relaxation of some assumptions; in fact, we no more impose that a predetermined production sequence must be maintained, or that all the types of products must be manufactured at each production period. Consequently, we change the model described by the authors in a Capacitated Lot sizing and Scheduling Problem with Sequence Dependent set up cost and times, and evaluate the benefits of this model on the total cost considered.
1. INTRODUCTION A supply chain consists of all parties involved, directly or indirectly, in fulfilling a customer request. The supply chain includes not only the manufacturer and suppliers, but also transporters, warehouses, retailers, and even customers themselves. Within each organization, such as manufacturer, the supply chain includes all functions involved in receiving and filling a customer request. To encounter the fierce competitions encompassed in today’s global market, companies have to well manage their supply chains. A proper management of supply chain systems leads to significantly increase profit and reduce operation costs. A wide variety of models for production planning and inventory management are offered in the literature. Lot sizing problems over a planning horizon make plans that minimize cost while satisfying forecasted external demands. These problems are classified in (Jans and Degraeve, 2004) into two classes: -
-
The forth section presents results and simulations whereas the last one concludes our work and offers some perspectives.
The continuous time-scale, constant demand and infinite time horizon lot sizing problems. The famous Economic Order Quantity model (EOQ) and the Economic Lot Scheduling Problem (ELSP) belong to this category;
Table 1. Lot Sizing Models
Lot Sizing Models
The discrete time scale, dynamic demand and finite time horizon lot sizing models. In general, we refer to this type of planning as dynamic lot sizing.
We are interested in this paper to the second category of lot sizing problems. A second classification of Lot Sizing Models is provided in (Jans and Degraeve, 2007), see Table 1. This classification will be discussed section 2 where 978-3-902661-81-4/10/$20.00 © 2010 IFAC
The single item uncapacitated lot sizing problem (ULS) The Capacitated Lot Sizing Problem (CLSP) Small bucket models Multi-Level Lot Sizing 2. LITERATURE REVIEW
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Let us define in what follows the variants of lot sizing problems mentioned in Table 1.
sequences. From this set and in each period, the best sequence is retained regard to its cost and whether or not each item’s inventory is empty. Well, even if the tailor-made enumeration method of the branch-and-bound type solves problem instances optimally, the important number of assumptions hindrances applying this method in more general cases. In fact, the authors assume that:
The single item uncapacitated lot sizing problem (ULS): In its simplest form, the dynamic lot sizing problem tries to find a production plan that minimizes the total costs of production, set up and inventory for one item over a multiperiod horizon. This plan has to satisfy deterministic demand. The capacitated multi-item lot sizing problem (CLSP) is a typical example of a large bucket model, where many different items can be produced on the same machine in one time period. The production capacity of this machine is limited. Small bucket models differ from the large bucket ones by an assumption made, which imposes that, during each time period; at most one type of item can be produced on the same machine. This is the case for the Continuous Set Up Lot Sizing Problem (CSLP).
-
if the production of an item starts in a period then the inventory of the item must be empty at the end of the previous period,
-
the set up is preserved between two adjacent periods, this means that we must first produce the item i at period t if this item is the last one produced in period (t-1),
-
for each item, at most one lot will be produced in a period.
Due to this last assumption, we could no more optimize the quantities to produce from each item. In fact, optimizing these quantities may imply to produce the quantities demanded over two periods, in only the first period.
The Proportional Lot Sizing and Scheduling Problem (PLSP) relaxes the restriction of allowing production for only one product in each time period. Nevertheless, in the PLSP, at most two different items can be produced in each time period. There is still at most one set up in each period, but the set up from the previous period can be carried over to the next period. The Discrete Lot Sizing and Scheduling Problem (DLSP) differs from the CSLP in that a discrete production policy is assumed, implying that an item must be produced at full capacity.
In (Kovács et al., 2009), the authors develop an approach partly based on the work of (Haase and Kimms, 2000). They address the capacitated lot-sizing and scheduling problem with sequence-dependent setup times and costs (CLSPSD). They show that the complexity of this large-bucket lot-sizing problem originates from the series of implicit sequencing problems that have to be solved for the items produced in each time period. Consequently, they develop an efficient algorithm in order to determine during pre-processing all item sequences that could appear in an optimal solution. A novel MIP formulation of CLSPSD that relies on a compact representation of those sequences by using item-related binary variables is introduced. According to authors, the model developed allows solving in reasonable time instances where the product of the number of items and number of time periods is at most 60–70.
In a Multi-level Lot Sizing problem, the production planning is not only done at the final level for the finished product, but also for the components and subassemblies needed to make this finished product. Production at one level leads to demand for components at a lower level, and at the highest level, production is triggered by market demand. In addition to this set of models described in (Jans and Degraeve, 2007), authors in (Salomon et al., 1997), consider another type of lot sizing problems, called the Discrete Lot sizing and Scheduling Problem with Sequence Dependent set up cost and times (DLSPSD). So, as it is underlined in the designation of this lot sizing problem type, setup costs and times depend on the production sequences retained. To solve this problem, and based on a transformation of the DLSPSD into a Travelling Salesman Problem with Time Windows, an exact solution procedure is developed. The sensitivity of the proposed approach’s performance is analysed for different problem characteristics: problem dimension, inventory holding costs, set-up times and production capacity utilization. Even if the set-up times are not explicitly taken into account in the definition of production capacity utilization, the authors have investigated the relationship between this capacity and the average CPU-time of their algorithm.
As in (Haase and Kimms, 2000), and both using MIP techniques, exact optimisation approaches to CLSPSD have been suggested by (Gupta and Magnusson, 2005). The authors define a MIP that simultaneously makes the lotsizing, and all sequencing decisions within time periods. The developed approach allows finding optimal solutions only for very small instances (three items and three time periods). For larger problems, a heuristic procedure is proposed in the same paper. The authors recommend using this procedure when there are many more products than there are planning periods. From a practical point of view, the procedure comforts firms which need to solve the problem repeatedly for a few periods at a time. In (Bouchriha et al., 2007), a real case study is presented. The authors discuss a specific lot sizing problem and different scenarios where a predetermined production sequence has to be maintained, all types of products have to be produced in each production period, etc. The production is planned under a cyclic manner. For this, a new approach is proposed to first compute the cycle time by assuming constant demand, then,
The authors, in (Haase and Kimms, 2000), deal with lot sizing and scheduling for a single-stage, single-machine production systems where set up costs and times are sequence dependent. They formulate a large-bucket mixed integer programming which is based on the set of efficient 84
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I i0 = K
lot sizes are determined for each product within each cycle in order to satisfy demand. When comparing the different scenarios analysed: fixing a cycle for all products, or the quantity to produce in each cycle, etc., the authors show that the scenario fixing each of the cycle and the quantities is the most expansive. The other scenarios aren’t considerably different in terms of potential gains. Among theses scenarios, the last one is the less constrained, in fact, all the products types have not to be produced in each period, besides, the production sequences have not to be prefixed. Even though these assumptions are removed, this scenario does not rigorously illustrate the CLSPSD problem, we’ll explain more in detail this point of view when describing the mathematical model developed by the authors.
I it ≥ SS i = K
Model (1):
(1.1)
∑ ( k it ρ it + ait Qit ) = C t t = 1,..., T
(1.2)
qit . ρ it ≤ Qit ≤ M . ρ it i = 1,..., n, t = 1,..., T
(1.3)
i =1
(1.7)
Table 3. Model (1) Decision variables (1)
Subject to:
n
Qit ≥ 0, I it ≥ 0 i = 1,..., n t = 1,..., T
Period index (t∈{1, …, T}) Set of products to be manufactured on the machine (IP⊂{1, …, n}) i Product index, i∈IP hit Inventory holding cost for product i ($/ton per period), i∈IP bit Unit production cost ($/ton), i∈IP Kit Setup cost for product I ($), i∈IP M A very large number qit Minimum quantity to produce when a setup occurs (ton) dit Demand of item i that corresponds to period t (ton), i∈IP kit Setup time for item i at period t (hour), i∈IP ait Unit of capacity consumption of item i at period t (hour/ton), i∈IP Ct Production capacity per period (hour machine) SSi Safety stock for product i (ton), i∈IP Ii0 Stock level at the end of the first period (ton), i∈IP, It is a constant (K)
The model described in (Bouchriha et al., 2007) and noted Model (1), determines optimum production batch size and setup for each planning period that minimizes the total inventory, setup and production costs for all products over the entire planning period. Model (1) is a mixed integer program described, using notations in Tables 2 and 3, as follows:
Qit + I it −1 = I it + d it i = 1,..., n, t = 1,..., T
(1.6)
T IP
3.1 Initial Mathematical model
t =1 i =1
ρ it ∈ {0,1} i = 1,..., n t = 1,..., T
Table 2. Model (1) Parameters
3. MATHEMATICAL MODEL
n
(1.5)
Here, we need to underline that constraints (1.3) impose that a minimum quantity has to be produced if a setup occurs. From this Model, we’ll omit in the subsequent paragraph the terms Kit.ρit, from the objective function, keep constraints (1.1), (1.2), (1.4), (1.5), (1.7) and modify constraints (1.3) and (1.6). Our objective is to introduce setup costs incurred each time we switch from one item’s type to the next.
To recap, we simultaneously try to find, at each period of the planning horizon, the finished products quantities to manufacture and the production sequences to adopt. Finding these production sequences lead us to deal simultaneously with two planning levels: tactical and operational.
T
i = 1,..., n t = 1,..., T
(1.4)
The constraints include material balance (1.1), production output capacity of machine (1.2) and constraints requiring setup in periods in which production occurs, constraints (1.3) and (1.6). Constraint (1.4) implies that an initial inventory must be fixed in the planning problem. The authors define this parameter according to the average of the demand. Constraints (1.5) imply that, at the end of each period, the stock level must be at least equal to the safety stock level. Backorders are not allowed, (1.7).
We aim in this paper to continue the works presented in (Bouchriha et al., 2007) and specifically the last scenario described. We then modify the lot sizing problem on a Capacitated Lot sizing and Scheduling Problem with Sequence Dependent set up cost and times, and consider the single machine variant of this problem, multiple products, multiple periods and dynamic demands. Our objective is to really focus on set up costs and times depending on production sequences. The model we develop determines in each production period, the lot sizes and the production sequences. These sequences have not to be respected from one period to another. Moreover, we are not obliged to produce all types of products in each period. The set of items to produce in each period will be dependent on the optimization of the production batches and different costs.
Min ∑ ∑ (hit . I it + bit . Qit + K it . ρ it )
i = 1,..., n
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ρit Binary variables that corresponds to 1 if the product is manufactured at period t and 0 otherwise, i∈IP Qit The quantity of product i to produce ate period t (ton), i∈IP Iit Stock level of product I at the end of the period t (ton), i∈IP
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3.2 Mathematical model for the setup
period (t+1) verifies ∑ xl i t +1 = 0 what means only one
Taking into account setup costs within production periods and between period t and period (t+1) involves combining the following Model to the Model (1) described previously. The notations used here are explained in Tables 4 and 5.
product is manufactured after product l, at the period (t+1).
i
⎛n n ⎞ T −1 n n Min∑ ⎜ ∑ ∑ cijt . xijt ⎟ + ∑ ⎛⎜ ∑ ∑ cklt . y klt ⎞⎟ t =1 ⎝ i =1 j =1, j ≠i ⎠ ⎠ t =1 ⎝ k =1 l =1,k ≠l T
Table 4. Model (2) Parameters
(2)
Subject to:
zit ≤ Qit ∀i ∀t
(2.1)
Qit ≤ zit .M
(2.2)
∀i ∀t
0 ≤ Posit ≤ ∑ zit ∀i ∀t
(2.3)
Posit ≤ zit .M
∀i ∀t
(2.4)
zit ≤ Posit .M
∀i ∀t
(2.5)
zit + z jt ≤ 1 + Pos jt − Posit ∀i ∀j ≠ i ∀t
(2.6)
1 − xijt ≤ z jt − zit + ( Pos jt − Posit ) − 1 ∀i ∀j ≠ i ∀t
(2.7)
i
( z kt + zl t +1 + (1 − ∑ xkit ) + (1 − ∑ xil t +1) − 3) ≤ y klt .M i
i
∀k , l ∈ {1,.., n}, l ≠ k , ∀t = 1,..., T − 1 Qit ≥ 0, I it ≥ 0 ∀i, ∀t z it ∈ {0,1},
(2.9)
Posit ∈ IN , ∀i
xijt ∈ {0,1}, ∀i
∀j ≠ i
(2.8)
∀t
(2.10)
∀t
(2.11)
y klt ∈ {0,1}, ∀k ∀l ≠ k ∀t = 1,..., T − 1
(2.12)
Constraints (2.1) and (2.2) indicate that if we manufacture product i in period t then variables zit equal 1, thus, if amounts Qit are strictly positive then zit equal 1. Relation between variables zit and Posit are determined by constraints (2.3), (2.4) and (2.5). While constraints (2.4) and (2.5) affect 0 to Posit when product i isn’t manufactured at period t, i.e. when zit equal 0, constraint (2.3) determine their domains of variation. These domains are delimited by 0 and the number of products manufactured at period t, i.e. the sum of zit. Constraints (2.6) avoid attributing the same position in the production sequence to two different items. In fact, if we manufacture products i and j at period t, i.e. if zit and zjt equal 1, then (Posjt-Posit) is necessary greater than 1 or lower than (-1). Constraints (2.7) attribute the value 1 to variables xijt when products i and j are manufactured at period t (zit and zjt equal 1) and item’s j position succeeds item’s i position in the production sequence, what means product j is manufactured immediately after product i (Posjt-Posit=1). Finally constraints (2.8) imply that if product k has the last position in the production sequence at period t and product l has the first position in the production sequence at period (t+1) then the binary variable yklt equals 1. The last product of the production sequence at period t verifies ∑ xkit = 0 what
csi Start up costs of product i, i∈IP cij Setup costs incurred when manufacturing item j after item i ($), i, j∈IP ckl Has the same signification that cij, it is used here just to underline the fact that this setup cost is incurred between two distinguished periods ($), k, l∈IP
Table 5. Model (2) Decision variables xijt Binary variables that correspond to 1 if the product j is manufactured immediately after product i at period t and 0 otherwise, i, j∈IP, t∈{1, …, T} yklt Binary variables that correspond to 1 if product k is the last product of the production sequence in period t and l the first product of the production sequence in period t+1, and 0 otherwise, k, l∈IP, (t∈{1, …, T}) zit Binary variables that correspond to 1 if product i is manufactured in period t and 0 otherwise, i∈IP, (t∈{1, …, T}) Posit Integer variables that indicate the position of product i in the production sequence, at period t, i∈IP, (t∈{1, …, T}) si Binary variable that correspond to 1 if product i is the first product manufactured in period 1 and 0 otherwise, i∈IP As we can see in Model (2), due to the absolute values, the mathematical model is a mixed integer non linear program. Solving this model with a commercial solver like LINGO may give local optimum solutions. 2.3 CLSPSD mathematical model The CLSPSD problem is finally formulated as follows: Model (3):
⎛n ⎞ ⎜ ∑ csi . si + ⎟ ⎜ i=1 ⎟ ⎜T ⎛n ⎟ n n ⎞ Min⎜ ∑ ⎜ ∑ (hit . I it + bit . Qit ) + ∑ ∑ cijt . xijt ⎟ + ⎟ i =1 j =1, j ≠i ⎠ ⎟ ⎜ t =1 ⎝ i=1 ⎜ T −1⎛ n n ⎟ ⎞ ⎜ + ∑ ⎜ ∑ ∑ cklt . y klt ⎟ ⎟ ⎠ ⎝ t =1 ⎝ k =1 l =1,k ≠l ⎠
(3)
Subject to:
i
(1.1), (1.2), (1.4), (1.5), and (1.7), (2.1) to (2.12). We have to note that in constraints (1.2), variables ρit are substituted by zit.
means no product is manufactured after product k, at the period t. The first product of the production sequence at
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And:
si ≤ zi1 ∀i
(3.1)
( zi1 + (1 − ∑ xki1) − 1) ≤ si .M
∀i
(3.2)
i
si ∈ {0,1}, ∀i
(3.3)
In the objective function, the first term corresponds to start up costs, the second to inventory costs, the third to productions costs, the forth to setup costs within production periods and the last to setup costs between periods.
these costs. If we observe Model 1, we can obviously notice that when setup costs are more important than holding costs, we’ll prefer producing in few periods and stocking important amounts of products. On the contrary, when setup costs are not very important, we prefer producing just the needed quantities of products in each production period in order to hold small amounts in stocks. When solving Model 1, we obtain the quantities to produce and to have in stocks in each production period, and the set of products to manufacture in each period. However, these sets of products aren’t scheduled and this is due to the fact that the problem of setup costs and times depending on sequences isn’t really dealt with. In our model, we have considered this dependency and we aim by analyzing these two following scenarios to demonstrate how the problem of simultaneously scheduling production and determining lot sizes can modify the amounts of products to plan each period, so the scheduling problem can stage-manage the planning problem.
Constraints (3.1) assign the value 0 to si if product i is not produced in the first period and constraints (3.2) assign the value 1 to si if product i is manufactured in the first period (zi1=1), and this product is the first product manufactured at this period, this means that no product precedes i, so ∑ xki1 = 0 . k
To solve this non linear Integer problem, including O(n2T) variables and O(n2T) constraints, we use the commercial solver LINGO version 12.0.
Scenario 1: setup costs = 20% holding costs Scenario 2: setup costs = 200% holding costs Since the holding costs are unitary, we have estimated total holding costs by multiplying the random variables chosen from the interval [1,10] (Tale 6) by the average demand. Choosing the average demand is due to the fact that in one hand, we have no idea about the average level of stocks; in the other hand the average demand can comfort the two opposite scenarios: important unitary holding costs, low unitary holding costs. In fact, when the inventory holding costs aren’t important, we have tendency to increase the amounts stocked; in opposite, if the holding costs are important we try to reduce as possible the amounts stocked.
4. RESULTS In this section, we focus on two points; first, we compare our results with those published in (Bouchriha et al., 2007), secondly, we analyse the sensitivity of these results to two parameters: inventory costs and setup costs. 4.1 Comparison with the initial model (1) In this paragraph and in order to compare our results to those described in (Bouchriha et al., 2007), we use the illustrative example given Table 6.
The obtained results are given Table 7. If we observe Scenario 2, for Model 1, the way of thinking is, since the holding costs are not very important, we have to produce, if the machine’s capacity allows, all the needed amounts of products and if it is possible, one product type per period (in our example, we have only two periods, this is why P1 and P2 are produced on the same period). Well, by solving simultaneously the lot sizing problem and the scheduling problem, we obtain different production sequences that lead to a decrease in the total cost of the supply chain.
Table 6. The illustrative example T IP dit
2 3 Random number that belongs to the interval [10,200] Ii0 Random number that belongs to the interval [0,40] SSi Random number that belongs to the interval [0,100] Ct 100 000 hours bit Random number that belongs to the interval [1,10] hit Random number that belongs to the interval [1,10] cij and Random number that belongs to intervals defined ckl proportionally to the intervals of hit. Kit Random number that belongs to intervals defined proportionally to the intervals of hit. qit Random number that belongs to the interval [50,150] ait Random number that belongs to the interval [10,20] kit Random number that belongs to the interval [1,5] M 109 Here, we analyse two scenarios: important holding costs and low holding costs. Before, we should explain our interest to
It is the same for scenario 1 where setup costs are not very important, and so we can produce, at each period all the products’ types; this is well illustrated in Model 1. In fact, we produce two types of products in period 1 and three in period 2. Our Model proves that simultaneously solving the lot sizing and scheduling problems don’t systematically lead to the Model’s 1 conclusion. In fact, by producing three types of products on period 1 and only one on period 2 we can divide by two the supply chain total cost. The same reasoning can be applied to study production costs versus setup costs.
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4.2 Analysis of the model’s sensitivity
Table 7. Results of the illustrative example Model 1 1st period - Qit - Iit
(P1,P2) (105 P1,100 P2) (5 P1,40 P2,10 P3)
P3-P2-P1/ (1 P3,265 P2,205 P1) (105 P1,205 P2,11 P3)
2nd period - Qit - Iit
(P1,P2,P3) (100 P1,165 P2,150 P3) (5 P1,5 P2,130 P3)
P3 24 P3 (5 P1,5 P2,5 P3)
Total cost
5102 $
2432 $
In this paragraph, we want to compare our results with those of Model 1 when varying the setup costs from 20% to 200% of the holding costs. Fig.1 shows that, in all cases, we manage to significantly reduce the total cost. This reduction varies from 6 to 52%. Reducing these costs can be better illustrated when we consider more production periods. As in (Haase and Kimms, 2000) and from a practical point of view, our model comforts more firms aiming to solve the CLSPSD problem repeatedly for few periods at a time. 12000 10000
5653 $
0
Total cost for CLSPLD Model
200%
8568 $
2000
160%
Total cost
4000
180%
P1-P2-P3 (100 P1,200 P2,25 P3) (5 P1,5 P2,5 P3)
140%
(P3) (105 P3) (5 P1,5 P2,130 P3)
6000
120%
2 period - Qit - Iit
nd
8000
80%
P2-P1/ (65 P2,105 P1) (5 P1,5 P2,10 P3)
100%
(P1,P2) (205 P1,265 P2) (105 P1,205 P2,10 P3)
60%
1st period - Qit - Iit
20%
Scenario 2
Model 4
40%
Scenario 1
Total Cost for Model I
We have to underline that computational times for all the simulations we have done are about 1 second.
Fig. 1. Comparison of total costs when varying the percentage Setup costs/Holding costs efficient rescheduling opportunities. International Journal of Production Economics, 66, 159-169. Jans, R. and Z. Degraeve (2007). Meta-heuristics for dynamic lot sizing: A review and comparison of solution approaches. European Journal of Operational Research, 177, 1855-1875. Kovács, A., K.N. Browna and A. Tarim (2009). An efficient MIP model for the capacitated lot sizing and scheduling problem with sequence dependent setups.
5. CONCLUSIONS In this paper, we have dealt with the Capacitated Lot sizing and Scheduling Problem with Sequence Dependent set up cost and times. The problem we have treated is characterized by one machine, a dynamic and deterministic demand, multiple product types and few production periods. We formulate the problem as a non linear mixed integer model and compare our results with those described in (Bouchriha et al., 2007). Even though the optimal solution was not found, using the solutions provided by the solver LINGO during the solution process, we manage to reduce the supply chain total cost, this cost includes production costs, inventory holding costs, start and set up costs. This reduction can significantly be amplified when we increase the number of production periods. In future works we’ll make use of Meta heuristics in order to treat a problem with an important number of product types and production periods.
International Journal Economics, 118, 282-291.
of
Production
Salomon, M., M.M. Salomon, L.N. Van Wassenhove, Y.D. Dumas and S. Dauzere-Peres (1997). Solving the discrete lot sizing and scheduling with sequence dependent setup costs and setup times using the Travelling Salesman Problem with time windows.
European Journal of Operational Research, 100, 494-513.
REFERENCES Bouchriha, H., M. Ouhimmou and S. D’Amours (2007). Lot sizing problem on a paper machine under a cyclic production approach. International Journal of Production Economics, 105, 318-328. Gupta, D. and T. Magnusson (2005). The capacitated lot sizing and scheduling problem with sequence dependent setup costs and setup times. Computers & Operations Research, 32, 727-747. Haase K. and A. Kimms (2000). Lot sizing and scheduling with sequence-dependent setup costs and times and
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