Dynamic Lot Sizing Model for Stochastic Inventory Factors - Assignment Example

Running Header: Dynamic Lot Sizing Model for Stochastic Inventory Factors Student’s Name Lecturer Course Title Date Table of Contents Table of Contents 2 1.0. Introduction 3 1.1. Efficiency-Based Lot Sizing Problems 3 1.2. Capacitated Lot Sizing Problem 7 1.3. Multi-Level Capacitated Lot Sizing Problems 10 1.4. Solving Lot Sizing Issues in Remanufacturing 11 1.5. Carbon Emission Constraints in Dynamic Sizing Lot 13 1.6. Conclusion 16 1.7. References 18 Literature Review 1.0. Introduction Dynamic lot sizing is a topic that has widely been researched by various researchers in the world. One of the main aspects that have been widely considered in these researches involves identifying and solving various problems encountered in dynamic lot sizing in different situations. This paper evaluates various researches on dynamic lot sizing. The paper classifies these researches based on the problems the researcher focused on solving. Articles solving same kind of problems are classified together. These classifications include efficiency-based lot sizing problems, capacitated lot sizing problem, multi-level capacitated lot sizing problems, economic lot sizing problems, solving lot sizing issues in remanufacturing, and carbon emission constraints in dynamic sizing lot. The paper then gives a brief conclusion, based on what is deduced from this review. 1.1. Efficiency-Based Lot Sizing Problems A number of researchers conducted diverse studies on solving various dynamic lot sizing problems in unclassifiable situation. These include Tunc et al (2016), Rossi, Kilic & Tarim (2015), Carvalho et al. (2016), Onal et al. (2015), Onal (2016), Telha and Vyve (2016), Kian, Gurler & Berk (2014), and Hwang and Kang (2016). Although they employed diverse measures, they all focused on enhancing the efficiency of their system to reduce on production cost and maximize on the outcome. The models classified in this group are quite diverse though there are some that have slight similarities. For instance, Tunc et al and Rossi, Kilic & Tarim employed unified mixed integer programming modelling technique in the development of their models to resolve the identified problems. Nevertheless, their approaches on other aspects were quite different. Tunc et al and Kian, Gurler & Berk focused on a similar problem, though at different levels where Tunc et al majored on concave issues, Kian, Gurler & Berk focused on convex issues. There is also a great level of similarity between Onal and Onal et al work, where Onal appears as an advancement of the work of Onal et al. Apart from Tunc et al and Rossi, Kilic & Tarim, each of other articles in this group employed different techniques to develop their solutions. Carvalho et al. (2016) employed new lagrangian heuristic to create a solution, while Hwang and Kang (2016) employed geometric method. Tunc et al., Chen, Tunc et al and Rossi, Kilic & Tarim, and Rossi Kilic & Tarim, demonstrated the efficiency of their proposed model by comparing them with the previously presented models in the literature. The authors provide a number of advantages to demonstrate their effectiveness and also employ a computational to achieve the same. They employed mathematical computational models to justify their proposition. On the contrary, Carvalho et al employed computational test to justify the model efficiency, while the Hwang and Kang model was not tested. Thus, the efficiency of the two models over the existing models is unknown. This makes the reliability of the two models low or unclear. Nevertheless, it is clear that Carvalho et al model is functional, while the operational ability of Hwang and Kang is unknown. Tunc et al (2016) investigated stochastic lot sizing issue experienced with piecewise linear concave costs of ordering. On the contrary, Kian, Gurler & Berk (2014), focused on issues in dynamic lot sizing with regard to polynormial-type convex production and setup costs of zero. Kian, Gurler & Berk (2014) work was therefore considerably complex compared to Tunc et al. work, and it therefore used mixed integer nonlinear programming formulation and dynamic programming formulation to develop the new model, Tunc et al on the other hand employed the (R, S) policy and formulate its mixed integer programming. This variation can be explain by slight difference in the researchers focus such that when Kian, Gurler & Berk focused on convex economic production function, Tunc et al. focused on piecewise linear concave costs of ordering. In this regard, Kian, Gurler & Berkwork was more complex compared to that of Tunc et al. Nevertheless, they both employed the concept of derivation to come up with models to address their identified problems in the dynamic lot sizing. Onal et al. (2015) and Onal (2016) focused their research on resolving problem with selling of perishable items. However, the two employed did focus on attaining the same goal. Onal et al. focused on determining the best form of perishable items allocation, while Onal developed an algorithm to attain the most effective form of distributing and procuring perishable goods. In this regard, they both employed different technology to attain their goals. Onal et al simulated different cases of allocation, while Onal derived his model using complex polynomial time equation. Nevertheless, Onal et al. research outcome demonstrated that the issue of economic lot sizing with perishable items can be handled by use of polynomial time, based on the four assignment techniques, where there are no capacities for procurement. In this regard, Onal was employing the Onal et al proposed technique to develop the most feasible model. It can therefore be concluded that Onal work was an advancement of Onal et al work. The two papers demonstrated that establishing optimal transfer and procurement plan is NP-hard, and thus, it offers polynomial time addressable special cases. The other three researches were quite unique in their approach. Carvalho et al. (2016) centred on lot sizing of firms containing multiple plants, where every plant contains limited planning horizon split into periods. The researchers therefore focused on developing a new lagrangian heuristic solution, which outperforms all previous techniques employed to handle such problems. Although there are other researchers in this section that focused on creating heuristic solutions such Kian, Gurler & Berkwork, only Carvalho et al. considered employing lagrangian technique which is extensively used in solving capacitated lot sizing issues rather than general cases. Hwang and Kang (2016) on the other hand focused on the problem of lot-sizing with backlogging in stepwise costs of transportation. Inventory is backlogged or carried over in a trade-off with transportation and production set-up costs. Particularly, inventory is the chief source for demand consolidation over time to augment the delivery Full-Truck-Load. The researchers propose that there are no hypothetical production motives that result to an essential property for delivery of Less-Than-Load. In Addition, the cargo of Less-Than-Load does not have any unit carried over in the backlogged or previous period for the following period. The problem is solved in two phases where geometric method to pre-process suitable functional Full-Truck-Load delivery values and in the next phase a residual zoning algorithm containing Full-Truck-Load and Less-Than-Load delivery is offered. This was quite a unique approach which focused more on simulation, rather than others that centred on derivation. Similar others researchers in this category, Telha and Vyve (2016) focused on derived a more efficient estimation equation to be used in economic lot sizing problem solving. The most important aspect of this derivation is that it did not focus on particular economic issues, but it is a general application that can be tried in any situation. However, the applied situation must have progressive function of setup cost, and requires considering the time aspect in production. The main similarity between Telha and Vyve and other articles in this section such as Tunc et al is that, Telha and Vyve model is purely based on derivation which is guided by employing various assumptions. The efficiency of the developed algorithm is also tested for efficiency as it is recorded in most of the analysed models. 1.2. Capacitated Lot Sizing Problem Capacitated lot sizing problem is another aspect of dynamic lot sizing that is extensively investigated. This issue was investigated by Zhang (2015), Fiorotto, De-Araujo & Jans (2015), Hellion, Mangione & Penz (2014), and Boonmee and Sethanan (2016). Both Xiao et al. (2015) and Fiorotto, De-Araujo & Jans (2015) focused on parallel-machine capacitated lot sizing and scheduling issues with preference constraints, machine eligibility, time windows and set up time for sequence dependent. While Fiorotto, De-Araujo & Jans (2015) employed Dantzing-Wolfe decomposition to effectively reformulate the issue, Xiao et al created mixed integer programming model to solve the issue. Nevertheless, the two adopted a similar advanced procedure where the main issue in Xiao et al. is decomposed into small problems and single-machine scheduling set of sub-problem, by decomposition of Lagrangian. Similarly, the master issue in Fiorotto, De-Araujo & Jans is solved by employing two solution techniques, which integrated Dantzing-Wolfe and Langrangian relaxation decomposition in form of a hybrid. The only main difference between the two is that Fiorotto, De-Araujo & Jans. research employed flow constraint as connecting constraints, with the aim of obtaining lower bounds of high quality. Founded on production quantities transfers a basic heuristic is employed to create feasible solutions. Calculation by use of literature data is presented. The results demonstrate that the methods of hybrid create least bounds of competitive upper bounds and excellent quality. Xiao et al. on the other hand integrated annealing algorithm which focused on looking for improved solution in the construction stage feasibility. The calculation experiments in Xiao et al. demonstrated that the hybrid proposed algorithm performs better than heuristic algorithms. Boonmee and Sethanan (2016), Zhang (2015) and Xiao et al. (2015) use mixed integer programming model in solving capacitated lot sizing issues that each focused on. However, when Zhang employed non-static programming-founded algorithm to address this issue in polynomial time, Xiao et al used Lagrangian decomposition, and Boonmee and Sethanan used different structural algorithms to resolve their problem. Zhang (2015) used this model to resolve the problem of capacitated lot sizing with outsourcing, where the capacity of production is constant and contains uncapacitated outsourcing. In every case, the demand was contrasted by both outsourcing and production. Boonmee and Sethanan (2016) on the other had used mixed integer programming to resolve multi-level capacitated lot sizing tool of computation and scheduling issues in the planning of hen eggs production focusing on cost minimization. A mixed integer programming model in this case was created to address small-size issues and Particle Swarm Optimization for large-size problems. However, the need to address multiple social terms of learning in Boonmee and Sethanan resulted to further application of gbest, lbest and nbest social structures and local procedure of search to decide the novel pullet and chick allocation sequence. Re-order and re-initialization strategies were employed to improve the chance of efficient solution during the searching. Nevertheless, mixed integer programming was the basic method to enhance successful development of the best solution for capacitated lot sizing problems. This conclusion is made considering that even Fiorotto, De-Araujo & Jans employed mixed integer programming model at the end of their research to test on the efficiency of their new model. Thus, mixed integer programming can be regarded as a basic model for development of capacitated lot sizing problems solution. Unlike other articles discussed under capacitated lot sizing Hellion, Mangione & Penz (2014) demonstrated unique features in their research. Their technique was not found in any other paper, but had similar focus as Xiao et al. and Fiorotto et al. However, Hellion, Mangione & Penz focused on single-item capacitated lot sizing rather than parallel-machine capacitated lot sizing. Hellion, Mangione & Penz (2014) focused on single-item capacitated lot sizing issue with concave storage and production cost, considering dynamic time windows and minimum quantity of order. The paper modelled frequency limitations on the lots production by non-static time windows. Between two consecutive lots of production, there were at R periods at most and Q periods at least. The research presented the most efficient algorithm O((T − Q)2 (R−Q)T 4/Q3 ), that is bounded by O(T7). Four of the researches conducted under capacitated lot sizing issues that include Hellion et al., Boonmee and Sethanan, Fiorotto, De-Araujo & Jans and Xiao et al. measured the effectiveness of their developed algorithms or models. However, apart from Fiorotto, De-Araujo & Jans and Xiao et al. that employed mixed integer programming software model to test the efficiency of their models, others employed unique techniques. Boonmee and Sethanan model effectiveness was measures by use of heuristic algorithm acquired by contrasting their solutions and solution of relative improvement with regard to gbest, lbest and nbest social and Particle Swarm Optimization algorithms. Hellion, Mangione & Penz (2014) employed polynomial time computational technique, where computations were used to demonstrate the efficiency of the proposed algorithm. 1.3. Multi-Level Capacitated Lot Sizing Problems Chen (2015) and Almeder et al. (2015) focused on problems facing dynamic multi-level capacitated lot sizing problems. Despite focusing on a similar topic, the two employed diverse techniques to solve the identified issues. Chen (2015) proposes a novel fix-and-optimized technique to address two multi-level capacitated lot sizing problems, where mixed integer programming was also employed, while Almeder et al. (2015) use modelling and the synchronization to resolve the problem. The only other common aspect between the two established the researchers tested the efficiency of the new models by comparing their performance with the existing multi-level capacitated lot sizing problems model. Mathematical computation was used to show the efficiency excellence of the proposed solution in both cases. However, Chen also used mixed integer programming for model testing over other existing models. Chen (2015) two multi-level capacitated lot sizing problems contained set up carryover and the multi-level capacitated lot sizing problems with no set up carryover. The technique iteratively solved a sub-problems series of the mixed integer programming lot sizing problem model. Every sub-problem re-optimizes a binary decision variables subset determined, founded on the binary variables interrelatedness in the model constraints, while fixing other binary variables values. Founded on the fix-and-optimized, the paper also developed the multi-level capacitated lot sizing problems variable neighbourhood search technique with no setup carryover. This further enhanced the solution attained by the fix-and-optimized through search space diversification. Benchmark instances numerical experiments demonstrated that both the variable neighbourhood search and fix-and-optimized techniques can acquire an improved solution for more instances contrasted with the fix-and-optimized technique suggested in previous literature. Almeder et al. (2015) conducted an intensive literature study to demonstrate the inefficiency of multi-level capacitated lot sizing problems in capturing precedence relations and resources requirements, and later develop an integrated model that is more efficient compared to the traditional multi-level capacitated lot sizing problems models. The model either offered infeasible plans of production or costly needless inventory plans, based on the lead time postulation. Almeder et al. handled the issue by clearly modelling the two factors and the synchronization of products batches using the multi-level scheduling formulation and lot sizing. Almeder et al integrated two models of multi-level capacitated lot sizing problems that include a model permitting lot streaming and another one regarding batch production. 1.4. Solving Lot Sizing Issues in Remanufacturing Lot sizing problems in remanufacturing is another aspect that has highly attracted the attention of researchers in this topic. The topic has been addressed by Cunha and Melo (2016), Baki et al. (2014), Pineyro and Viera (2014), Parsopoulos Konstantaras & Skouri (2015), Sifaleras and Konstantaras (2015a) and Sifaleras and Konstantaras (2015b). Both Cunha and Melo (2016) and Baki et al. used the ordinary mixed-integer programming solver to develop their solutions. The performance efficiency test of the two model created by Cunha and Melo and Baki et al. also used the mixed integer programming model, where the new models performance was tested over the existing models. Others employed different techniques, although the techniques employed by Sifaleras and Konstantaras (2015a) and Sifaleras and Konstantaras (2015b) were closely related. Sifaleras and Konstantaras (2015a) employed a heuristic optimal heuristic algorithm known as General Variable Neighbourhood Descent Heuristic Search, while Sifaleras and Konstantaras (2015b) used of meta-heuristic algorithm of Variable Neighbourhood Search. The two also focused on the same research which involved resolving multi-production dynamic lot size in product recovery and returns that normally occur in reverse logistic. Similar to Sifaleras and Konstantaras (2015b), Parsopoulos, Konstantaras & Skouri (2015) also employed meta-heuristic algorithm to investigate differential evolution. The performance of algorithms developed by Sifaleras and Konstantaras (2015a), Sifaleras and Konstantaras (2015b) and Parsopoulos, Konstantaras & Skouri were all evaluated and contrasted with a state-of-the-art mechanism, and with past evaluated meta-heuristic algorithms. They were all found to outperform Gurobi optimizer state-of-the-art solution Cunha and Melo (2016) considered remanufacturing as an essential technique to lower environmental degradation in the industrialized goods production. This process should involve employment of production systems that handle reutilization of returned materials that include reverse logistics. Cunha and Melo regard a production planning issue emanating from the reverse logistic context referred as economic lot sizing remanufacturing. In this situation, the single item deterministic demand over a limited horizon of time must be satisfied. This can be conducted either from remanufactured or newly produced items, and the objective consists in lowering the total cost of production. The research presents a strengthened Wagener and an extended multi-commodity formulation that employs priori addition of newly described liable inequalities in the original variable space. The researchers also suggested a new dynamic heuristic solution founded on the cost to spontaneously establish the partial version size of Wagner-in based formulation. The results of the computation demonstrate that the new partial wagner-in founded formulation in the size spontaneously established in a heuristic manner perform better than all other tested techniques. This include the most excellent performing formulation of shortest path accessible in the literature, after considering the quantity of cases solved to justify optimality by use of an ordinary mixed integer programming solver. This new technique permitted over 96 per cent optimality of the evaluated lot sizing remanufacturing cases, with different setups, including a number of cases which could not be otherwise solved. Pineyro and Viera (2014) is a short note where corrections are done on the main paper the authors published earlier. Unlike all other papers under this section that intensively provide analysis of various aspects of solving remanufacturing problem, this note only provide a correction to the previous work. Thus, it is not very useful for this research. It could have been very useful if the original paper was used. 1.5. Carbon Emission Constraints in Dynamic Sizing Lot Carbon emission is another aspect that has highly been investigated with regard to dynamic sizing lot. According to Purohit et al. (2016, p. 654), firms across the globe are highly pushed by their main stakeholders who include customers and governments to reduce carbon emission created while conducting their business. In this regards a number researchers have considered investigating on various production aspects that are associated with carbon emission issues in dynamic lot sizing. The aspect of carbon emission constraints in the dynamic lot sizing is reviewed by four articles that include Purohit et al. (2016), Purohit, Choudhary & Shankar (2015), Wang and Choi (2015) and Absi et al. (2016). Although Purohit, Choudhary & Shankar 2015 and Purohit et al. 2016 seems similar in many ways, the two other researches; Wang and Choi and Absi et al. are unique and thus, unrelated. Both Purohit et al. (2016) and Purohit, Choudhary & Shankar (2015) investigated the problem of inventory lot-sizing under a dynamic stochastic demand situation, with cycle service and emission level limitations, considering carbon trade and cap regulatory mechanism. They both examined impacts of system- and emission-related factors on performance of supply chain. The main difference between the two is that Purohit et al. (2016) employed the mixed integer linear programming model to establish the impacts of system related, product related, and emission parameters features on the performance of supply chain, while Purohit, Choudhary & Shankar (2015), did the same by creating an integer linear programming model. The two methods involved the use of broad computational experiments to address general form of business environments. In their investigation, the researchers demonstrated that the level of cycle service and variation demand coefficient has essential effects on emission and total cost, irrespective of the demand variability level, while the effect of the pattern of product’s demand is only essential at only lower demand variability level. The outcomes also demonstrated that increasing carbon price value lowers total cost, total inventory, and total emission. They also showed emission reduction scope by augmenting carbon price at higher cycle service levels and variation demand coefficient. The two researches are very similar, such that the only difference is the employment of different models and despite of this variation, the two researches managed to make the same conclusion. This is a clear indication that the two models are highly reliable in performing this form of experiments Wang and Choi (2015) investigated the carbon management under the trading mechanism of the carbon emission. This is done for optimization of planning lot sizing production in stochastic make-to-order production, with the aim of maximizing the wealth of shareholder. Unlike the two previous researches; Purohit Choudhary & Shankar (2015) and Purohit et al. (2016) that only focus on the investors’ economic benefits, Wang and Choi research is more concerned on both investors’ economic benefits and environmental effects related with production planning. Unlike Purohit et al. and Purohit, Choudhary & Shankar researches that are conducted by use of integer linear programming model, Wang and Choi research is based on numerical experiments research technique. The experiments demonstrated the essential influence of caps, pricing, and trading of carbon emission on the lot sizing dynamic decisions policy. Their research outcome highlighted the crucial carbon management roles in production planning, for attaining both economic and environmental benefits. Absi et al.(2016) research is a continuation of the researchers previous work; published in 2013, where four forms of carbon emission constraints were proposed in multi-mode dynamic lot sizing. These forms include rolling, global, cumulative, and periodic carbon emission constraints. The carbon emission constraint is regarded to describe an upper limit on the mean emission in every product (Absi et al., 2016, p. 849). The researchers focus on single-item lot sizing instead of multiple-mode lot sizing. The research extends the constraint analysis to the realistic fixed carbon emission case related with each mode, as well as its carbon emission unit. Various problems structural properties are presented which assist in making a generalized view that, focusing on fixed carbon emission creates NP-hand problem. The research presents a number of dominant properties which assist in the proposition of two non-static programming algorithms. The first algorithm is polynomial which is considered during fixed number of modes, and it resolves issues only when parameters of carbon emission are stationary. The second algorithm is pseudo-polynomial, and it addresses single-item lot sizing problem, containing fixed and periodic carbon emission constraints issues, with stationary parameters of carbon emission and costs, demonstrating that the investigated issue is in weak sense NP-hard. This research is considerably different from the previous discussed studies focusing on carbon emission. It is highly complex and its research objective is highly different from the previously reviewed researches. 1.6. Conclusion In conclusion, the review noticed a number of commonly addressed issues which include carbon emission lot sizing, capacitated lot sizing, economic lot sizing and multi-level lot sizing issues in different situation. The reviewed researches demonstrate that there are various techniques employed to handle different dynamic lot sizing issues, based on different situations. Different researchers have tried to develop different algorithms or models with higher efficiency than the already existing ones. Some researchers have also tried to improve their own models with time. Among all employed models and algorithms mixed integer programming, polynomial time and integrated Lagrangian heuristic techniques are the most commonly employed techniques in development of these solutions. However, mixed integer programming formulation is highly twisted to accommodate the research requirements. When some use its basic concepts, others try to use modified part of it which include unified, linear, and non-linear. Mixed integer programming has been extensively used not just in models development but also in model testing. It can be cited as the highly used formulation method in solving dynamic lot sizing problems. Most researchers have employed computational test to verify the efficiency of their models, while many others have considered employing these test to proof the model optimality by comparing the new models performance with existing models. Generally, the researchers have demonstrated a continuous improvement of dynamic lot sizing problem solving model in various situation, which is a clear indication that there is no optimal solution to these problems. A new and more improved model will always be produced based on change of technology and identification of limitations in the current models. 1.7. References Absi, N., Dauzere-Peres, S., Kedad-Sidhoum, S., Penz, B & Rapine, C, 2016, “The single-item green lot-sizing problem with fixed carbon emissions,” European Journal of Operational Research, vol. 248, pp. 846-855. Almeder, C., Klabjan, D., Traxler, R & Almada-Lobo, B, 2015, “Lead time considerations for the multi-level capacitated lot-sizing problem,” European Journal of Operational Research, vol. 241, pp. 727-738. Baki, M. F., Chaouch, B. A & Abdul-Kader, W, 2014, “A heuristic solution procedure for the dynamic lot sizing problem with remanufacturing and product recover,” Computer & Operations Research, vol. 43, pp. 225-236. Boonmee, A & Sethanan, K, 2016, “A GLNPSO for multi-level capacitated lot-sizing and scheduling problem in the poultry industry,” European journal of Operational Research, vol.250, pp. 652-665. Carvolho, D. M & Nascimento, M.C. V, 2016, “Largrangian heuristics for the capacitated multi-plant lot sizing problem with multiple periods and items,” Computer & Operations Research, vol. 71, pp. 137-148. Chen, H, 2015, “Fix-and-optimized and variable neighborhood search approaches for multi-level capacitated lot sizing problems,” Omega, vol. 56, pp. 25-36. Cunha, J. O & Melo, R. A, 2016, “A computational comparison of formulations for the economic lot-sizing with remanufacturing,” Computers & Industrial Engineering, vol. 92, pp. 72-81. Fiorotto, D. J., De-Araujo, S. A & Jans, R, 2015, “Hybrid methods for lot sizing on parallel machines,” Computers & Operations Research, vol. 63, pp. 136-148. Hellion, B., Mangione, F & Penz, B, 2014, “A polynomial time algorithm for the single-item lot sizing problem capacities, minimum order quantities and dynamic time windows,” Operations Research Letters, Vol. 42, pp. 500-504. Hwang, H-C & Kang, J, 2016, “Two-phase algorithm for the lot-sizing problem with backlogging for stepwise transportation cost without speculative motives,” Omega, vol. 50, pp. 238-250. Kian, R., Gurler, U & Berk, E, 2014, “The dynamic lot-sizing problem with convex economic production costs and setups,” International Journal Production Economics, vol. 155, pp. 361-379. Onal, M, 2016, “The two-level economic lot sizing problem with perishable items,” Operations Research Letters. doi.org/10.1016/j.orl.2016.03.017 Onal, M., Romeiin, H. E., Sapra, A & Heuvel, W. V. D, 2015, “The economic lot-sizing problem with perishable items and consumption order preference,” European Journal of Operational Research, vol. 244, pp. 881-891. Parsopoulos, K. E., Konstantaras, I & Skouri, K, 2015, “Metaheuristic optimization for the single-item dynamic lot sizing problem with returns and remanufacturing,” Computers & Industrial Engineering, vol. 83, pp. 307-315. Pineyro, P & Viera, O, 2014, “Note on “the economic lot-sizing problem with remanufacturing the one-way substitution”,” International Journal of Production Economics, vol. 156, pp. 167-168. Purohit, A. K., Choudhary, D & Shankar, R, 2015, “Inventory lot-sizing under dynamic stochastic demand with carbon emission constraints,” Procedia – Social and Behavioral Sciences, vol.189. vol. 193-197. Purohit, A. K., Shankar, R., Dey, P. K, & Choudhary, A, 2016, “Non-stationary stochastic inventory lot-sizing with emission and service level constraints in a carbon cap-and-trade system,” Journal of Cleaner Production, vol. 113, pp. 654-661. Rossi, R., Kilic, O. A & Tarim, S. A, 2015, “Piecewise linear approximations for the static-dynamic uncertainty strategy in stochastic lot-sizing,” Omega, vol. 50, pp.126-140. Sifaleras, A & Konstantaras, I, 2015a, “General variable neighborhood search for the multi-product dynamic lot sizing problem in closed-loop supply chain,” Electronic Notes in Discrete Mathematics, vol. 47, pp. 69-76. Sifaleras, A & Konstantaras, L, 2015b, “Variable neighborhood descent heuristic for solving reverse logistics multi-item dynamic lot-sizing problems,” Computers & Operations Research, doi.org/10.1016/j.cor.2015.10.004 Telha, C & Vyve, M.V, 2016, “Efficient approximation schemes for economic lot-sizing in continues time,” Discrete Optimization, vol. 20, pp. 23-39. Tunc, H., Kilic, O. A., Tarim, S. A & Eksioglu, B, 2016, “The stochastic lot sizing problem with piecewise linear concave ordering costs,” Computers & Operations Research, vol. 65, pp. 104-110. Wang, X. J & Choi, S. H, 2015, “Stochastic lot sizing manufacturing under the ETS system for maximization of shareholder wealth,” European Journal of Operational Research, vol. 246, pp. 66-75. Xiao, J., Yang, H., Zhang, C., Zheng, L & Gupta, J. N. D, 2015, “A hybrid lagrangian-simulated annealing-based heuristic for the parallel-machine capacitated lot-sizing and scheduling problem with sequence-dependent setup times,” Computers & Operations Research, vol.63, pp. 72-82. Zhang, M, 2015, “Capacitated lot-sizing problem with outsourcing,” Operations Research Letters, vol.43, pp. 479-48. Read More