Sensitivity of multi-product two-stage economic lotsizing models and their dependency on change-over and product cost ratio s

Transcription

1 Sensitivity two stage EOQ model 1 Sensitivity of multi-roduct two-stage economic lotsizing models and their deendency on change-over and roduct cost ratio s Frank Van den broecke, El-Houssaine Aghezzaf, Hendrik Van Landeghem Deartment of Industrial Management Ghent University, Ghent, Belgium Abstract This study considers the roduction and inventory management roblem of a two-stage semirocess roduction system. In case both roduction stages are hysically connected it is obvious that materials are forced to flow. The economic lotsize deends on the holding cost of the endroduct and the combined change-over cost of both roduction stages. On the other hand this 'flow sho' is forced to roduce at the seed of the slowest stage. The benefit of this aroach is the low amount of Work In rocess inventory. When on the other hand, the involved stages are hysically disconnected, a stock of intermediates acts as a decouling oint. Tyically for the semi-rocess industry are high change-over costs for the rocess oriented first stage, which results in large lotsize differences for the different roduction stages. Using the stock of intermediates as a decouling oint avoids the comlexity of synchronising oerations but is an additional reason to augment the intermediate stock osition. The disadvantage of this model is the high amount of Work-In-rocess inventory. This aer rooses the 'synchronised lanning model' realising a global otimum instead of the combination of two locally otimised settings. The mathematical model roves (for a two-stage single-roduct setting) that the otimal two-stage roduction frequency corresonds with the single EOQ solution for the first stage. A sensitivity study reveals, within these two-stage lotsizing models, the economical cost deendency on roduct and change-over cost ratio s. The urose of this aer is to understand under which conditions the joined setu or the two-stage individual eoq model remain close to the otimal model. Numerical examles rove that the conclusions about the otimal settings remain valid when extending the model to a two-stage multi-roduct setting. The research reveals that two-stage individually otimized EOQ lotsizing should only be used when the end-roduct stage has a high added value and small change-over costs, comared to the first stage. hysically connected oerations should be used when the endroduct stage has a small added value and low change-over costs, or high added value and large change-over costs comared to the first roduction stage. The aer concludes with suggesting a ractical common cycle aroach to tackle a two-stage multi-roduct roduction and inventory management roblem. The common cycle aroach brings the benefit of a reetitive and redictable roduction schedule. Keywords: Economical Lotsizing, Two-stage roduction system, Cyclical roduction Scheduling, Economic Lot Scheduling roblem (ELS).

2 1. Introduction This research aer focuses on the semi-rocess industry and more secific on the question of economical lotsizing decisions for a two-stage roduction setting. Within such a Make-To-Stock environment management faces following dilemmas: In case different roduction stages are hysically connected it is obvious that materials are forced to flow. The economic lotsize deends on the holding cost of the end-roduct and the combined change-over cost of both roduction stages. (joined setu model). On the other hand this 'flow sho' is forced to roduce at the seed of the slowest stage. Furthermore a breakdown of one of the involved stages causes an interrut for the comlete roduction rocess. The benefit of this aroach is the low amount of Work-In-rocess inventory. When on the other hand, the involved stages are hysically disconnected, a stock of intermediates acts as a decouling oint. Tyically for the semi-rocess industry set-u costs for the rocess oriented art are high, which results in large lotsize differences for the different roduction stages (two-stage individual eoq model). Based on these economical lotsize considerations, the size of the intermediate stock is higher than the buffer stock needed to feed the second stage rocess. Most often stage one and two are different deartments or organisations, both trying to 'locally' otimise their oerations. Using the stock of intermediates as a decouling oint avoids the comlexity of synchronising oerations but is an additional reason to augment the intermediate stock osition. The disadvantage of this model is the high amount of Work-In-rocess inventory. This aer rooses the 'synchronised lanning model' bringing a global otimum instead of the combination of two locally otimised settings. In order to understand the dynamics of the system this research defines the (mathematically) otimal roduction frequencies at both roduction stages (otimal two-stage lotsizing model). The main benefit of this aroach, still minimising the inventory and set-u cost function, is the ability to reduce the amount of Work In rocess inventories. A sensitivity study should reveal, within these two-stage lotsizing models, the economical cost deendency on roduct and change-over cost ratio s. The urose of this aer is to understand under which conditions the joined setu or the two-stage individual eoq model remain close to the otimal model. This aer has following structure. Section 1 introduces the roblem.. Section 2 highlights on revious research considering multi-echelon lotsizing models. Section 3 describes the studied two-stage lotsizing models, introduces the synchronised lanning model and erforms, within a single-roduct setting, the sensitivity study for different setu and holding cost ratio s. Section 4 extends the single roduct model to a multi-roduct model, having one first stage roduct and multile second stage roducts. Section 5 suggests a ractical cyclical lanning aroach to tackle a two-stage multi-roduct roduction and inventory management roblem. Section 6 concludes. ISIR2010.docx 2

3 2. revious Research ioneering work on multi-echelon lot size models has been done by Crowston and Wagner (1973). Under EOQ like assumtions of constant continuous demand, instantaneous roduction, zero lead times, infinite lanning horizon, fixed setu costs and linear holding costs they rove that for a multi-stage assembly system, the otimal lotsize at each facility is an integer multile of the lotsize at the successor facility. In order to synchronise oerations, researchers have been interested in determining cyclical roduction schedules for multile roducts over an infinite lanning horizon, minimising the sum of setu and inventory holding costs. Two techniques, assuming static demand, are based on cyclic scheduling and solve the multi-roduct Economic Lot Scheduling roblem (ELS) within a caacity constrained environment. The 'common cycle' aroach, attributed to Hanssmann (1962) solves the model mathematically and determines the otimal length of a roduction cycle where each roduct is roduced at the same common base frequency. A generalisation of this model towards a multi-stage environment is described by El-Najdawi (1993). The 'cyclic scheduling' aroach is an extension of the common cycle model allowing schedules in which each roduct may be roduced more than once during the base cycle time. This aroach is described as the 'basic eriod aroach'. The roduction frequency of each roduct is exressed as an integer multile of a common basic eriod. The fact that the common cycle method is more restricted (allowing only frequency factors equal to one for each involved roduct) exlains why the common cycle (analytical) solution is an uer bound for the ELS roblem (Doll & Whybark 1973). The Economic Lot Scheduling roblem solution, allowing different frequency factors, results in a lan with lower inventory and set-u costs. The set of indeendent, EOQ based, solutions on the other hand reresents the lower bound. In the early seventies a number of authors (Doll & Whybark(1973), Goyal(1973)) resented heuristic aroaches to solve the ELS roblem. Doll & Whybark (1973) resent a heuristic solution and solve the multi-roduct single-stage lotsizing roblem by determining for each roduct a roduction frequency equal to an integer multile of a basic eriod. Elmaghraby (1978) reviews the different techniques for solving the ELS roblem, and indicates the ability of the roosed heuristics to resent a good and accetable solution. All comared heuristics (Doll& Whybark, Goyal) are suerior in their erformance to the analytical aroaches. This is mainly due to the restrictive assumtions made in these analytical models. Davis (1995) rooses a decomosition two-hase aroach to solve the ELS roblem and sequence the roosed lotsizes. This two-hase rocess, combining a heuristic rocedure and mixed integer rogramming, resolves the combinatorial difficulties of jointly determining roduction quantities, frequencies and rocessing sequences. The first hase defines the basic eriod and roduction frequencies. The scheduling sequence is determined by the second hase rocess. The benefits of common reetitive roduction lan for multile items are numerous. The redictability of the schedule allows synchronisation between different roduction stages. In ISIR2010.docx 3

4 situations where the caacity resource constraint is tight a cyclical model, evenly sreading the load on oerations, reduces the ossibility of eak demands arriving all at once and claiming the use of the caacity constrained resource. Simulation results suggest that the fixed cycle aroach, shows greater imrovement over the indeendent cycle aroach in caacity constrained situations (Güder and Zydiak 2000). The imlied cyclical material coordination asects result in a more robust lanning stability (Van Donselaer et al 2000). When extending this model towards a multi-stage rocess, Viswanathan and ilani (2001) demonstrate that using common relenishment eriods can establish a co-ordinated inventory olicy and brings substantial savings for the total system, in comarison with the indeendent cycle aroac These savings aear when the involved set-u costs are above a certain threshold level. The rocess with the largest set-u cost dictates the behaviour of the otential savings. When on the other hand set-u costs are of low imortance a flow roduction should be more adequate. Van den broecke et al (2008), describe a ractical alication within a hotograhic film roducing comany, imlementing a near-otimal solution for the multi-stage, multi-roduct caacitated lot-sizing roblem by rolling out a cyclical roduction lan. 3. Single-roduct two-stage model Within the two-stage single roduct setting, D corresonds with the annual demand. The first stage uses a lotsize Q = D/M, where M equals the annual roduction frequency. The second stage roduction lotsize corresonds with q = Q/k where k is the frequency difference factor between first and second roduction stage. On an annual basis the first and second stage count resectively M and M.k roduction runs. 3.1 Otimal two-stage lotsizing model First stage Second stage Fig 1 : Otimal two-stage lotsizing model Using the model s convention (fig. 1) the second stage lotsize corresonds with q = D/(M.k). Assuming constant demand the average end-roduct stock corresonds with D/(2.M.k). Within the synchronized lanning model, in reaction on the first stage delivery of D/M volume of intermediate roduct, the second stage rocess immediately converts q = D/(M.k) units into endroduct. Based on this synchronisation ush concet, the average intermediate stock corresonds with (Q-q)/2 = [M D/(M.k)]/2 = [D/(2.M)](1-1/k). ISIR2010.docx 4

5 Within the EOQ logic S and s stand for setu costs within first and second roduction stage. and a are roduct costs of both stages and h stands for the inventory holding cost ercentage. Total annual setu and inventory cost (TC) equals : D 1 D D D TC M. S (1 ) s. k. M. M. S s. k. M.. ( ) 2M k 2. k. M 2. M 2. k. M Taking the derivatives to both M and k equal to zero defines the osition of the otimum. Derivative to k : 1 D s. M ( ).. ( ) 0 k² 2M ( ) k² 2. s. M ² 1 ( ) k M 2. s ( ) k with M 2. s Derivative to M : ( ) S s. k 0 2. M ² 2. k. M ² M ( ) S s. 0 M 2. M ² 2.. M ² M ( ) S s. 0 M 2. M ² 2.. M ² ( ) M ². S M.( s. ) M ². S 0 (*) 2 M ² 2. S M 2. S ( ) s. s 2. s. ( ) 2 a ( ) 2. sa ( ). s 2. ( ) ( ) ( ) s s. ( ) 2 s ( ) 2 0 (*) Relacing σ with its value eliminates the first grade term. The mathematical calculus roves that the first stage otimal roduction rate M for the two-stage system equals the single-stage EOQ (first stage) roduction frequency. The otimal values of M and k corresond with : M 2. S 1 ( ) 2. S ( ) ( ). S k.. M 2. s 2. s. s ISIR2010.docx 5

6 Within the sensitivity study of the cost function for various ratios of S/s (setu cost ratio) and / (roduct cost ratio), this aer assumes a first stage setu cost of 5528 euro and a first stage roduct cost of 1 euro/sqm. The annual demand equals 17,244 million sqm. These data corresond with the actual arameter settings for a medical X-ray filmtye (LT2B tye) within a hotograhic film roducing comany. The inventory holding ercentage is set at 16%. Table 1 and 2 illustrate the different settings of setu and roduct cost ratio s and the corresonding cost function. For the highlighted examle, the ratio (-)/ = 2 imlies a second stage roduct cost of = 3 euro/sqm and the ratio S/s = 1.5 imlies a second stage setu cost of s = 3685 euro. Within this examle the otimal two-stage lotsizing model corresonds with M = , k = resulting in a total cost function equal to euro. Table 1 illustrates for various settings of and s, the resulting (otimal) k values. Table 2 lists the corresonding total cost function. Since values of k < 1 are not allowed the otimal total cost is only resented for valid k conditions. k S/s (-)/ Table 1 : k-values for the otimal two-stage lotsizing model TC S/s (-)/ Table 2 : Total within the otimal two-stage lotsizing model ISIR2010.docx 6

7 3.2 Joined setu EOQ model First stage Second stage Fig 2 : Joined setu EOQ model Within the Joined setu EOQ model, both stages have the same lotsize Q and roduce at the same rate (imlying k = 1). Since both stages are hysically connected there is no intermediate stock and the economical roduction lotsize is controlled by the end-roduct holding cost and the sum of the first and second stage setu costs. The lotsize and total annual cost formulate as : Q 2. ( S s) D M k 1 Q 2.( S s) TC M.( S s) 2. M Within the sensitivity analysis, total cost is calculated for various settings of s and (table 3). The comarison of the joined setu total cost function with the otimal total cost only takes lace for the valid (s, ) settings of the otimal two-stage lotsizing model (table 4). deviations to the otimal model can be quite substantial (> 20 %) esecially in the case of high added value by the second roduction stage (high value for (-)/) and large setu costs within the first roduction stage (high value for S/s). TC S/s (-)/ Table 3 : Total within the joined setu EOQ model TC/TC* S/s (-)/ Table 4 : comarison between Joined Setu and Otimal EOQ model ISIR2010.docx 7

8 3.3 Two-stage individual EOQ model First stage Second stage Within the two stage individual EOQ model lotsizes for both stages are determined indeendently, each otimizing their local inventory and change-over costs. Since lotsizes are not synchronised this results in an intermediate stock, acting as a decouling oint. Based on the traditional EOQ logic the lotsizes and total annual cost formulate as : Q 2. S M D Q Fig 3 : Two-stage individual EOQ model 2. S q a 2. s a a Q ka qa 2. S..2. s a a S. a sa. D 1 D TC M. S (1 ) sa. ka. M. a 2M ka 2. ka. M Within the sensitivity analysis, total cost is calculated for various s a and a settings (table 5). The cost function of the individually otimised EOQ model remains close to global otimum, when setu costs in both stages have the same magnitude (S/s a 1) and a high added value in second stage (high value for ( a -)/)). The last condition favours to kee material as long a ossible in intermediate stock osition which corresonds to a decouled situation. On the condition of large differences in setu ratios (high value of S/s a ) and small roduct cost differences (low value for ( a -)/) the deviations towards otimal cost raise from 3 to 5% (table 6). TC S/s (-)/ Table 5 : Total within the two stage individual EOQ model TC/TC* S/s (-)/ ISIR2010.docx 8

9 Table 6 : comarison between Two stage individual and Otimal EOQ model ISIR2010.docx 9

10 3.4 Overview and sensitivity study results Table 7 summarizes for the X-ray filmtye examle, the arameter settings and resulting total costs, of the highlighted examles (see table 1 to 6) within the three studied lotsizing models. The sensitivity study reveals that two stage individually otimized EOQ lotsizing should only be used when the end-roduct stage has a high added value and small setu costs, comared to the first stage (see table 6). The joined setu model, resulting in hysically connected oerations, should only be used when the end-roduct stage has a small added value and low change-over costs, comared to the first stage or in the oosite condition of a high end-roduct added value and high setu cost comared to the first roduction stage (see table 4). Model M k a TC difference Otimal two-stage lotsizing model Joined setu model Two-stage individual EOQ model Table 7 : X-ray filmtye, model results (single roduct) 4. First stage (single-roduct), second stage (multi-roduct) model 4.1 Otimal two-stage lotsizing model (first stage= single-roduct, second stage= multi-roduct) When extending the single-roduct towards the multi-roduct model, the first stage (singleroduct) intermediate is exloded to a number of end-roducts, clustered into three end-roduct clusters (A,B and C). See table 8 for the detailed arametersettings of the X-ray filmtye. The end-roduct clustered lotsize q a stands for the sum of n a lotsizes within the A end-roduct cluster. The cost of n a setus corresonds with s a. The frequency difference factor k a determines the second stage roduction frequency equal to k a.m, with M equal to the first stage roduction frequency. The annual demand for the intermediate roduct is the sum of the three end-roduct cluster demands with D = d a + d b + d c. The intermediate roduct lotsize Q equals D/M and the end-roduct clustered lotsizes corresond resectively with q a =d a /(k a.m), q b =d b /(k b.m), q c =d c /(k c.m). Coating Tye Annual Demand Set-u roduct Clus Ter Nr of roducts Annual Demand Set-u Cluster Set-u (s a ) roduct (D) (S) () (n a ) (d a ) ( a ) LT2B A B C Sum Table 8 : X-ray filmtye, multi-roduct arametersettings ISIR2010.docx 10

11 Within this logic the total cost function describes as : da d TC M. S M.( sa. ka sb. kb sc. kc) ( D 2. M ka k b b dc ) kc 2. h M da. ( ka a db. kb b dc. c ) kc Using the same logic as within section 3.1 and utting the derivatives to resectively M, k a, k b and k c equal to zero, leads to 4 equations resulting into following otimal values. The rovided examle of the X-ray filmtye calculates into M = , k a = , k b = and k c = The total cost corresonds with euro. M 2. S k a d a. ( a ). S. sa k b d b. ( b ). S. sb k c d c. ( c ). S. sc 4.2 Joined setu model (first stage = single-roduct, second stage = multi-roduct) Within the Joined setu model, both stages roduce at the same rate (imlies k a = k b = kc = 1). Since both stages are hysically connected there is no intermediate stock. (cfr D-d a /k a - d b /k b -d c /k c = D-d a -d b - d c = 0) The economical roduction lotsize is controlled by the end-roduct holding cost and the sum of the first and second stage change-over costs. The rovided examle of the X- ray coating tye calculates into M = , k a = k b = k c = 1. The total cost corresonds with euro. The economical lotsizes and frequency factors formulate as : ( a. da b. db c. dc) M k a 1 k b 1 k c 1 2.( S sa sb sc) 4.3 Two-stage individual EOQ model (first stage = single-roduct, second stage = multi-roduct) Within the Two stage individual EOQ model lotsizes for both stages are determined indeendently, each otimizing their local inventory and change-over costs. At the first roduction stage this corresonds with one EOQ lotsize and roduction rate. At the second stage, lotsizes and roduction rates are determined for each end-roduct cluster. Since roduction rates are not synchronised this results in an intermediate stock, acting as a decouling oint. Based on the traditional EOQ logic the lotsizes and total annual cost formulate as described in table 9. For the X-ray filmtye examle total cost corresonds with euro. 1 da db dc h da. a TC M. S ( D ) ( sa. ka sb. kb sc. kc). M.( 2M ka kb kc 2. M ka Coating Tye Annual Demand sqm (D) EOQ lotsize EOQ Frequency (M) Clus Ter Nr of roducts Annual Demand db. kb EOQ Lotsize dc. c ) kc ISIR2010.docx 11 b EOQ Frequency EOQ Freq Factor k i (Q) (n a ) (d a ) q i m i LT2B A B C Table 9 : X-ray filmtye, two-stage individual EOQ model (first hase = single roduct, second hase = multi-roduct)

12 4.4 Comarison of results (first stage = single-roduct, second stage = multi-roduct) Table 10 summarizes for the X-ray filmtye examle, the resulting roduction frequencies and total costs, of the three comared models for a first stage (single-roduct) and second stage (multi-roduct) setting. Model M k a k b k c TC Difference Otimal two-stage lotsizing model Joined setu model Two stage individual EOQ model Table 10 : X-ray filmtye examle, model results (first hase = single roduct, second hase = multi-roduct) 5. Two-stage multi-roduct model When extending the model to a two-stage multi-roduct setting, the first stage and second stage have resectively M i and M i.k ai, M i.k bi, M i.k ci roduction frequencies. The intermediate roducts are referenced with the index i, and a,b,c reresent their end-roduct clusters at the second roduction stage. The numerical examle consists of 5 medical film tyes. The first intermediate roduct LT2B corresonds with the X-ray examle used within the revious sections. Table 11 lists the arameter settings of the studied two-stage multi-roduct model. Coating Tye Annual Demand 1000 sqm (D i ) Setu Є ro duct Є/sqm ( i ) Clus Ter Nr of roducts (n ai ),(n bi ) (n ci ) Annual Demand 1000 sqm (d ai ),(d bi ), (d ci ) Cluster Setu Є (s ai ),(s bi ),(s ci ) roduct Є/sqm ( ai ),( bi ),( ci ) (S i ) 1 LT2B A B C CX1X A B C CXDGU A B C CXDGG A B C CXDGL A B C Table 11 : filmtyes examle, model arameters (two-stage, multi-roduct) ISIR2010.docx 12

13 5.1 Otimal two-stage lotsizing model Based on the results on the single-roduct model (section 3), the otimal first stage roduction frequency corresonds with the single-stage economical solution. For multile roducts lotsize decisions within a roduction stage, the Doll & Whybark heuristic delivers a still unbeaten nearotimal solution for the ELS (Economic Lot Scheduling) roblem and brings the advantage of a cyclical reetitive roduction schedule. For the testcase this results in a common cycle reeated (M base ) times a year. Four coating tyes are roduced twice (K 1 = K 2 =K 3 =K 4 = 2), The fifth filmtye is coated only once (K 5 = 1) within the common cycle. The frequency factors for the second stage (k ai, k bi, k ci ) are determined based on the formula resented in section 4.1. Table 12 lists the results. In Intermed. M base K M k a k b k c TC dex roduct 1 LT2B CX1X CXDGU CXDGG CXDGL Table 12 : Otimal two-stage lotsizing model, roduction frequencies Within a oerational lanning system the roosed cycle length and roduction frequencies should become rounded to more ractical integer values. (M = 6.59 corresonds with a common cycle length of 1 / 6.59 year = 7.79 weeks 8 weeks). 5.2 Joined setu model Corresonding with the logic from revious sections joined setu corresonds with an equal roduction frequency for all roducts at all stages. When defining this common cycle frequency, the model considers the end-roduct holding cost and sum of all setu costs. See table 13. M i ( dai. 2 i ai d ( Si s ai bi. s bi bi d ci sci). ci) Index Intermediate M k a k b k c TC roduct 1 LT2B CX1X CXDGU CXDGG CXDGL Table 13 : Joined setu EOQ model ISIR2010.docx 13

14 5.3 Two-stage individual EOQ model At every stage and for every roduct(cluster) the otimal roduction frequency is determined by the standard EQO logic. Table 14 lists the results and the corresonding total cost. Index Intermediate M k a k b k c TC roduct 1 LT2B CX1X CXDGU CXDGG CXDGL Table 14 : Two-stage individual EOQ model, roduction frequencies. 5.4 Comarison of results Model Otimal twostage lotsizing model Joined setu model Two-stage individual EOQ Model First stage (Doll & Whybark rocedure) o Determine base frequency M base o Determine M i =K i.m base for each intermediate roduct Second stage o Determine k ai, k bi, k ci for each end-roduct cluster First stage & second stage o Determine common M for all intermediate roducts o For all end-roduct clusters k ai = k bi = k ci = 1 First stage o Determine otimal Mi for each intermediate roduct based on EOQ logic Second stage o Determine k ai, k bi, k ci for each end-roduct cluster based on standard EOQ logic Total Diffe rence Conclusions Table 15 : Comarison of model results This aer rooses for a two-stage multi-roduct setting, the 'synchronised lanning model' realising a global otimum instead of the combination of two locally otimised settings. Within the simlified single-roduct case, the mathematical model roves (for a two-stage singleroduct setting) that the otimal two-stage roduction frequency corresonds with the single EOQ solution for the first stage. A sensitivity study reveals, within these two-stage lotsizing models, the economical cost deendency on roduct and change-over cost ratio s. ISIR2010.docx 14

15 The research reveals that two stage individually otimized EOQ lotsizing should only be used when the end-roduct stage has a high added value and small change-over costs, comared to the first stage. hysically connected oerations (joined setu model) should be used when the endroduct stage has a small added value and low change-over costs, or high added value and large change-over costs comared to the first roduction stage. Numerical examles rove that for a ractical case, the conclusions about the otimal settings remain valid when extending the model to a two-stage multi-roduct setting. The suggested common cycle aroach, based on the economic lot scheduling roblem (ELS) is a ractical aroach to tackle a two-stage multi-roduct roduction and inventory management roblem. Within a oerational lanning system the roosed cycle length and roduction frequencies should be rounded to more ractical integer values. References 1. Crowston, W.B., Wagner, M.H., 1973, Dynamic Lot Size models for Multi-Stage Assembly Systems, Management Science, 20(1), Setember Davis, S.G., 1995, An imroved algorithm for solving the economic lotsize roblem (ELS), International Journal of roduction Research, 33(4), Doll, C.L., Whybark, C., 1973, An iterative rocedure for the single-machine multiroduct lot scheduling roblem, Management science, 20(1), Elmaghraby, S.E., 1978, The economic lot scheduling roblem (els) review and extensions, Management science, 24, El-Najdawi, M., Kleindorfer,.R., 1993, Common cycle lot-size scheduling for multiroduct, multi-stage roduction, Management Science 39(7), July 1993, Goyal, S.K., 1973, Scheduling a Multi-roduct Single-Machine system, Oerational Research quarterly, 24(2), Güder, F., Zydiak, J.L., 2000, Fixed cycle ordering olicies for caacitated multile item inventory systems with quantity discounts, Comuters and industrial engineering, 38, Hanssmann, F., 1962, Oerations Research in roduction and Inventory Control, John Wiley & Sons, Viswanathan, S., ilani, R., 2001, Coordinating suly chain inventories through common relenishment eochs, Euroean Journal of oerational researc 129, Van den broecke, F., Aghezzaf, E.H., Van Landeghem, H., 2008, "Imlementing a nearotimal solution for the multi-stage, multi-roduct caacitated lot-sizing roblem by rolling out a cyclical roduction lan", International journal of roduction economics, 112(1), Van Donselaar, K., Van Den Nieuwenhof, J., Visschers, J., 2000, "The imact of material co-ordination concets on lanning stability in suly chains", International journal of roduction economics, 68, ISIR2010.docx 15