SOLUTION APPROACHES FOR THE SOFT DRINK INTEGRATED PRODUCTION LOT SIZING AND SCHEDULING PROBLEM

Transcription

1 SOLUTON APPROACHES FOR THE SOFT DRNK NTEGRATED PRODUCTON LOT SZNG AND SCHEDULNG PROBLEM Deiseara Ferreira, Reinaldo Morabito Universidade Federal de São Carlos Departaento de Engenharia de Produção São Carlos, SP Brazil deise@dep.ufscar.br, orabito@power.ufscar.br Socorro Rangel UNESP São Paulo State University R: Cristovão Colobo- 2265; São José do Rio Preto, SP Brazil socorro@dcce.ibilce.unesp.br Abstract n this paper we present a ixed integer-prograing odel that integrates production lot sizing and scheduling decisions of beverage plants with sequence-dependent set up costs and ties. The odel considers that the industrial process produces soft drink bottles in different flavours and sizes, and it is carried out in two production stages: liquid preparation (Stage ) and bottling (Stage ). The odel also takes into account that the production bottleneck ay alternate between Stages and, and a synchrony of the production between these stages is required. A relaxation approach and several strategies of the relax-and-fix heuristic are proposed to solve the odel. Coputational tests with instances generated based on real data fro a Brazilian soft drink plant are also presented. The results show that the solution approaches are capable of producing better solutions than those used by the copany. Key words: ixed integer prograing, lot-scheduling, soft drink industry, relax-and-fix heuristic. 1 ntroduction The soft beverage industry consists of copanies which produce, arket, package, sell and deliver non-alcoholic beverages, including carbonated soft drinks, to custoers (Montag, 2005). While the sector has any sall local copanies, at least three large systes aintain the highest degree of international brand recognition. The credit quality of a global beverage copany is dependent on the relationships between the concentrated producer,

2 which owns the brand, and its bottlers. The concentrated producer owns the tradeark and prootes the brand advertising which is aied at the end-consuers. The bottlers aintain the physical bottling and distribution infrastructures. They also retain and cultivate long-ter relationships with the retailers that are not easily replaced. The success of the concentrate producer is largely linked to the presence of bottlers that understand the local arket and culture. Given an environent of intense copetition, changing consuer deands and rising costs, the soft drink sector is faced with two ajor challenges: aintaining product relevance, brand strength and arket position, as well as balancing pricing strategies with volue growth to achieve sustained growth and stable or iproving efficiency and profitability. Soft beverages are widely popular in any arkets and deand is generally stable given that they are low cost, repeat purchase ites. They are consued by a wide range of social and econoic groups, although per capita consuption is higher in soe arkets. Brazil is one of the largest producers of carbonated soft drinks with ore than 800 plants, and a consuer arket of ore than 13 billion litres a year, which is the third in the world. However, the national per capita consuption is around 65 litres, a odest nuber when copared to other arkets. According to the Brazilian association of soft drink beverages (ABR, 2007), the stabilization of the currency in the 1990 s contributed to a significant increase in the Brazilian production, rising fro a production of 5.9 billion litres in 1991, to 9.14 billion in 1995, and billion in Over the last years the rate has reained stable producing billion in 2006, which is 4.75% higher than in The diversity of products offered to consuers, the scale of plants and the coplexity of odern filling lines require the adoption of optiization-based progras to generate efficient production plans. The production lot sizing and scheduling proble is present in the Brazilian soft drink production, as well as in other industrial processes such as, e.g., foundry (Araujo et al., 2006), anial nutrition (Toso et al., 2007) and electro fused grains (Luche et al., 2007). n general, in the industry practice, the lot sizing and scheduling proble is solved separately, that is, first the production lot sizes are defined and afterwards the production schedules. However, it is often of interest to integrate the two probles in the decision process. These probles, either integrated or not, have been studied by several authors (e.g.; Fleischann, 1990; Pinedo, 1995; Drexl and Kis, 1997; Meyr, 2000; Karii et al., 2003). Production lot sizing and scheduling probles can be very difficult depending on the restrictions which have to be et and the cobinatorial structure (classified in general as NPhard optiization probles, e.g. Bitran and Yanasse, 1982; Meyr, 2002). The production 2

3 anager needs to consider product deands, achine capacities, raw aterial availabilities, set up ties, aong other aspects. Mixed integer prograing odels for the production planning of beverages have been proposed in the literature. For instance, Clark (2003) studies the production lot sizing proble of a canning line at a drinks anufacturer, considering non-sequence-dependent setup ties of the canning line to changeover between canned products if no change of liquid is involved, and extra fixed setup ties if a change of liquid is involved. Toledo et al. (2007a, 2007b) proposes a two-stage ulti-achine odel that integrates the lot sizing and scheduling decisions, and takes into account sequence-dependent setup ties and the synchrony between the production stages. Solution ethods based on heuristics and etaheuristics are also proposed. n this paper we propose a ixed integer-prograing odel that integrates the lot sizing and scheduling decisions which considers the synchrony between the stages of the liquid flavour preparation and liquid bottling of a Brazilian soft drink plant. t is sipler than the odel presented in Toledo et al. (2007a), as it considers ore siplifying assuptions. Both odels are based on the GLSP (General Lot sizing and Scheduling Proble) proposed in Fleishann and Meyr (1997). n GLSP the planning horizon is divided into T acroperiods. t is a big bucket odel and to obtain the order at which the ites will be produced, each acro-period is divided into a nuber of icro-periods, and only one ite can be produced in each icro-period. The reaining part of this paper is organized as follows. n the next section we briefly describe the soft drink production process. The proposed optiization odel is described in Section 3. n Section 4 we present the solution approaches based on a relaxation algorith and different strategies of the relax-and-fix heuristic. Results of the coputational tests are given in Section 5 and final rearks in Section 6. 2 The soft-drink production process As entioned, soft drinks of different flavours and bottles types (disposable and recycled) are produced in two ain stages: liquid flavour preparation (Stage ) and liquid bottling (Stage ). n Stage, the liquid flavour (flavour concentrated or syrup plus water) is prepared in tanks of different capacities. Two different liquid flavours cannot be prepared in the sae tank at the sae tie, and for technical reasons the tank ust be epty before a new lot of liquid flavour can be prepared in that tank. A iniu quantity of liquid flavour ust 3

4 be prepared in order to assure liquid hoogeneity. To properly ix the necessary ingredients, the tank propeller ust be copletely covered. n Stage, the liquid flavours are bottled in the filling lines. A filling line is ade up of a conveyor belt and achines that wash the bottles, fill the with a cobination of liquid flavour and water (carbonated or noncarbonated) and then seal, label and pack the. The production is carried out in the order described above. f for any reason it is necessary to reove a bottle fro the conveyor belt, it will be done at the end of the production process, before packaging. There is only one entry for the bottles in the filling line. For convenience sake, we treat each filling line as a single achine. Each filling line can receive liquid flavour fro only one tank at a tie, no atter the nuber of available tanks. However, a tank can supply a liquid flavour for several filling lines siultaneously if they are bottling soft drinks of the sae flavour. n the scheatic representation of the production process given in Figure 1, all the M filling lines receive water fro the sae source, and liquid flavour fro only one tank at a tie. Note that, in this exaple, tank N is assigned to supply liquid flavour for the filling lines k and M siultaneously, but these filling lines only receive liquid flavour fro this tank. Liquid flavor preparation Bottling Filling line 1 Water Tank 1. Tank t.. Filling line k Wash bottles fill label seal package Pallet. nventory Tank N Filling line M Figure 1. Soft drink production process. The filling lines are initially adjusted to produce soft drinks of a given flavour in a given bottle size. At each changeover of liquid flavour in the tanks and/or soft drink in the filling lines, a set up tie (cleaning and/or achine adjustents) sequence dependent is necessary. For exaple, if a soft drink of a diet flavour has been prepared and a noral one is 4

5 to be prepared next, the cleaning tie is saller than the other way round. n the tanks, a changeover tie is necessary even if the sae liquid flavour is to be prepared next. The production planning involves then the lot size and production sequence definition at each period of the planning horizon. Although, in general, the production planning is ade to eet a pre-specified soft drink deand, it is coon the arrival of urgent product deands that should be et at once. Besides the sequence dependent changeover costs and ties, another iportant factor to be considered in the production planning of soft drink plants is the synchrony between Stages and. f the synchrony is not taken into account, a production schedule ay becoe unfeasible in practice. The liquid flavour in a given tank cannot be sent to the filling lines, unless they are ready to initiate the bottling process. n the sae way, if the necessary liquid flavour is not ready, the filling line ust wait for its preparation. Figures 2 and 3 show the production planning for three types of soft drinks (represented by the nubers 1-3) produced fro two types of liquid flavours (represented by the letters a and b). However, in Figure 2 there is no synchrony between the two stages. The rectangles represent the lot size and the spaces between the the changeover tie fro one soft drink to another in the filling line, or the changeover tie fro one liquid flavour to another in the tank. n Figure 3, the black rectangles represent the waiting tie to synchronize the production of the stages. Note in Figure 2 that at the beginning of the planning horizon, the tank needs soe tie in order to prepare for liquid flavour a. However, the filling line has already begun bottling soft drink 1. n the first changeover (fro liquid flavour a to b and soft drink 1 to 2), although the changeover tie is the sae in both stages, the filling line is ahead of the tank since it did not wait for the tank to be ready in the previous period. n the next changeover, the bottling process is initiated before the second lot of liquid flavour b is ready (this type of changeover occurs if ore than one full tank capacity of liquid flavour is necessary given the soft drink lot size). The following soft drink produced needs the sae liquid flavour as the previous one, but is of a different bottle size. n this case the changeover tie in the tank is saller than the changeover tie in the filling line. The tank should have waited for the filling line set up before releasing the liquid flavour. f the waiting ties are considered in both stages, the production schedule should be as shown in Figure 3. Note that, after synchronizing the two stages, the necessary production capacity is higher than the available one and the proposed schedule is not viable. Therefore, the synchrony between the two stages ust be taken into account while the lot size and schedule is being planned. 5

6 Tank a b b b Line Available capacity Figure 2 Non-synchronized schedule Tank a b b b Line Available capacity Figure 3 Synchronized schedule The production process described above is coon for the three soft drink plants visited in this research and differs ainly by the nuber of final products, the production capacities (tanks and filling lines nubers) and the degree of coputer software utilization during the production planning. Plant A (of large scale) produces ore than 100 products characterized by the bottle type and flavours (for exaple, a soft drink of 600 l and orange flavour is one product, while a soft drink of 2l and the sae flavour is another). t has nine tanks and seven filling lines of different capacities. Plant B (of ediu scale) produces 48 products and has seven tanks and three filling lines, all of different capacities. The third one, Plant C (of sall scale), has only two filling lines, one to produce soft drinks using glass bottles and the other using plastic bottles. The latter can produce 27 products and has several allocated tanks. The synchrony between the two stages does not need to be considered in the case of Plant C, since the production bottleneck is always in the bottling stage. Plant A has soe coputer software that integrates diverse departents (e.g., sales, production planning and control, logistics). The production planning is done with the help of three pieces of different software. The first one is used to define an initial lot size considering the filling line capacities, the second one is used to adjust the lot sizes considering the tank capacities, and the third one is used to define the production schedule. Various anual adjustents are still necessary when considering other constraints, such as resource availabilities, achine aintenance and urgent product deands. The production planning of 6

7 Plants B and C are anual and only spreadsheets and databases are used to help the decision process. The arket growth potential together with the increasing nubers of new products and the copetition for the arket share has posed several challenges to the soft drinks producers. There is a great concern to iprove the production and process anaging. The optiization odel proposed in the next section is useful to develop specific software that helps the decision process and the analysis of different production situations. 3 - Model developent The production planning proble for the soft drink plant described in the previous section can be basically stated as follows. Define the lot size and lot schedule of the products taking into account the synchrony between the two production stages, the product deands and the capacities of the tanks and filling lines, iniizing the overall production costs. The ixed integer optiization odel presented in this section considers the synchrony between the production stages and integrates the lot sizing and scheduling decisions as well as the odel in Toledo et al. (2007a). However, as pointed out in Section 1, it differs fro the latter in three ajor aspects. A siplification of the proble is considered supposing that each filling line, thereafter called achine, has a dedicated tank. Each tank can be filled, in turn, with all the liquid flavours needed by the associated achine. The planning horizon is divided into T periods (acro-periods). t is a big bucket odel and to obtain the order at which the products are produced, each acro-period is divided into a nuber of icro-periods. The total nuber of icro-periods is defined by the user, but should be set as the axiu nuber of setups in each acro-period. The icro-period size is flexible and its length is defined by the odel, since it depends on the lot sizes of the liquid flavour or the product. The total nuber of icro-periods in both production stages is the sae and only one liquid flavour or product can be produced in each icro-period. The synchrony between the two stages is considered using continuous variables, instead of binary ones, as described below. The odel size is defined by (J, M, F, T, N) representing respectively the nuber of soft-drink products (ites), the nuber of achines (both filling lines and tanks), the nuber of liquid flavours, the nuber of periods or acro-periods and the nuber of icro-periods (i.e. nuber of setups in each period). Let ( i, j,, k, l, t, s) be the index set defined as: i, j {1.. J}, { 1.. M}, k, l {1.. F}, t { 1.. T} and s { 1.. N}. Consider also, that the following sets and data are known: 7

8 Sets S t = set of icro-periods in period t; P t = first icro-period of period t; λ = set of achines that can produce ite j; j α = set of ites that can be produced on achine ; β = set of liquid flavours that can be produced on tank ; γ l = set of ites that can be produced on achine and need liquid flavour l. The data and variables described below with superscript relate to Stage (tank) and with superscript relate to Stage (bottling): Data d jt = deand for ite j in period t; h j = (non-negative) inventory cost for ite j; g j = (non-negative) backorder cost for ite j; s = changeover cost fro liquid flavour k to l; kl s ij = changeover cost fro ite i to j; b kl = changeover tie fro liquid flavour k to l; b ij = changeover tie fro ite i to j; a j = production tie for one lot of ite j; K = total capacity of tank, in litres of liquid; K t = total tie capacity in achine in period t; r jl = quantity of liquid flavour l necessary for the production of one lot of ite j; = nitial inventory for ite j; + j0 Variables: + = inventory for ite j at the end of period t; jt jt = backorder for ite j at the end of period t; x jt = production quantity in achine of ite j in icro-period s; 1 if the tank is setup for liquid flavour l in icro-period s y ls = 0 otherwise y js z kls z ijs 1 if the achine is setup for ite j in icro-period s = 0 otherwise 1 if there is changeover in tank fro liquid flavour k to l in icro-period s = 0 otherwise 1 if there is changeover in achine fro ite i to j in icro-period s = 0 otherwise To include the synchrony between the two production stages in the odel, another set of variables is necessary. As discussed in Section 2, a achine ust wait until the liquid flavour is ready in the tank. The set of continuous variables, v 0, coputes this waiting tie for each achine in each icro-period s. The waiting tie is equal to the difference between the tank changeover tie and the achine changeover tie. That is: s 8

9 , = 1... M, s = 1... N. v b z b z s kl kls ij ijs k β l β i α j α f the achine changeover tie fro ite i to j is greater than the tank changeover tie fro liquid flavour k to l, the waiting variable is zero and only the achine changeover tie is considered in the associated capacity constraint. Otherwise, the total waiting tie of the acro-period is taken into account. The Two-Stage Multi-Machine lot-scheduling odel (P2SMM) is then: (1) Subject to: Stage (Tank) (2) r x jl js j γ l j γ l J T M N M N + j jt + j jt + kl kls + ij ijs j= 1 t= 1 = 1 s= 1 k β l β = 1 s= 1 i α j α Min Z= ( h g ) s z s z K yls, = 1... M, l β, s = 1,..., N; (3) r x jl js q ls y ls, = 1... M, l β, s = 1,..., N; (4) y l s 1) y l β = 1 M ( ls,..., ; l β t = 1... T, s S { P}; t t (5) z kls yk ( s 1) + yls 1 = 1,..., M ; k, l β; s = 2,..., N; (6) zkls yj( s 1) + yls 1 = 1,..., M ; k, l β; t = 2,..., T; s = Pt ; j γ k (7) zkl1 yl1 = 1,..., M, l β ; k β 1 kls,..., M ; k β l β (8) z 1 Stage (bottling) (9) + j(t 1) + jt + λ j s S t x js = + jt = t = 1... T, s S t ; + j(t 1) + d jt, j = 1,..., J, t = 1,..., T; (10) a x + b z + v K, = 1,..., M ; t = 1... T; j js ij ijs s t j α s St i α j α s St s St (11), = 1... M, s = 1... N. v b z b z s kl kls ij ijs k β l β i α j α K (12) xjs yjs, = 1,..., M ; j α ; t = 1... T, s S t ; a t j (13) = 1, = 1... M, s = 1... N; (14) y js j α zijs y + i( s 1) js y -1, = 1,..., M ; i, j α ; s = 2,..., N; 9

10 (15) zij1 yj1 = 1,..., M ; j α ; i α (16) (17) z ijs i α j α 1, + jt, jt 0, j = 1,..., J, t = 1,..., T ; = 1,..., M, i e j α, k e l β, t = x js, 1,..., T, v s, s St. z ijs, = 1,..., M, s = 1,..., N ; zkls 0 ; y js, y = 0 / 1, ls The objective function (1) is to iniize the total su of product inventory, product backorder, achine changeover and tank changeover costs. n Stage, the deand for liquid flavour l is coputed in ters of the production variables. That is, the deand for liquid flavour l in each tank in each icro-period s is given by r lj x js. Therefore, the constraints (2) together with constraints (3) guarantee that if tank is setup for liquid flavour l in icro-period s ( y = 1), there will be a production of liquid flavour l (between the ls iniu quantity necessary for liquid hoogeny and the tank axiu capacity). Constraints (4) force the idle icro-periods to happen at the end of the associated acroperiod. Constraints (5) control the liquid flavours changeover. Note that the tank setup does not hold fro one acro-period to another if the setup variable in the last icro-period is zero. Therefore, constraints (6) are needed to count the changeover between acro-periods. Note also that the setup variables in Stage indicate which liquid flavour was prepared in the last non-idle icro-period of each acro-period. Constraints (7) count the first changeover of each tank. Constraints (8) ensure that there is at ost one changeover in each tank in each icro-period s. n Stage, constraints (9) represent the flow conservation constraints for each ite in each acro-period. Since the production variable is defined for each icro-period, to obtain the total production of ite j in a given acro-period t, it is necessary to calculate the associated production variables over all achines where it can be produced ( λ ) and icro-periods ( s St j γ l ) of acro-period t. Constraints (10) represent the achine capacity in each acro-period. Note here the inclusion of the waiting variable, v s, to ensure that the lot schedule is feasible. The waiting tie in achine in each icro-period s is coputed by constraints (11) as explained above. Constraints (12) ensure that there is a production of ite j only if the associated setup variable is set to one, and constraints (13) refer to a single ode production in each icro-period s. Note that the production ay not occur although the achine is always ready to produce an ite. Constraints (14) count the changeover in each j 10

11 achine in each icro-period s. Constraints (15) count the first changeover of each achine and constraints (16) ensure that there is at ost one changeover in each achine in each icro-period s. Finally, constraints (17) define the variables doain. Note that the changeover variables z kls and z ijs are continuous. Constraints (5) and (14) together with the optiization sense (iniization) ensure that these variables will take only 0 or 1 values. The above odel can be easily adapted to represent the particularities of the different soft drinks copanies studied. n the coputational tests described in Section 5, the instances were generated based on data collected fro Plant A. When defining the soft drinks lot size and schedule, this plant considers that the product inventories in a given period ust be sufficient to cover the product deands in the next period. To have a fair coparison between odel P2SMM and the copany solutions, a new set of constraints should be included in the odel: (18) d ( +1), j = 1,..., J, t = 2,..., T jt j t Plant A has various tanks and filling lines (achines) with different capacities. Soe tanks (and achines) are allocated to produce only a given subset of flavours (and ites), whereas others can produce any one. There is a single liquid flavour ( l = p ) whose deand is by far superior to the others. While ost of the liquid flavours have deands around 20,000 units per period, flavour p has a deand of around 150,000 units. t also has a high setup tie and cost. Therefore, in Plant A there is a tank which is copletely allocated to continuously produce flavour p. That is, whenever a achine is ready to produce an ite that uses this flavour, the tank is also ready to release it. To represent this situation in odel P2SMM, the changeover tie in Stage fro any flavour k to flavour p, and fro flavour p to any flavour k, is set to zero ( b b = 0 ). Although there is no need to setup the tanks for flavour p, there = kp pk is still a need to setup the achines when ites that need this flavour are produced. However, the tank setup variable, y ps, cannot be set to zero, since when this is done, constraints (2) together with (3) ipose that the production of any ite that uses this flavour is zero (if y ps = 0 then x = 0, for j γ p ). Therefore, the tank capacity (constraints (2) and (3)) for js l = p is dropped. 11

12 4. Approaches to solve the odel P2SMM The solution of practical instances of odel P2SMM using the exact ethods included in standard software such as CPLEX (log, 2006) was not satisfactory (see Section 5). This indicated the need to develop specific solution strategies, which are described next. 4.1 The relaxation approach n soe soft drink plants the liquid preparation in Stage generally does not represent a bottleneck for the production process. That is, the tank capacities are large enough to ensure that, whenever a achine needs a liquid of a given flavour, it will be ready to be released. Therefore, there is no need to control either the changeover in the tanks or the synchrony between the two production stages. Only the iniu tank capacity constraints to ensure the liquid hoogeny is necessary in Stage. This situation was explored as a solution approach to odel P2SMM. The Relaxation Approach (RA) is based on the idea that once the production decision is taken in Stage, the decision for Stage is easily taken. n the first step, the lot size and schedule of the ites in the achines are decided using a one stage odel obtained fro odel P2SMM by dropping the changeover variables of Stage, z kls by constraints:, and the waiting variables of Stage, v s. The constraints (10) are replaced (10a) a x + b z K, = 1... M, t = 1... T ; j js ij ijs t j α s St i α j α s St and the objective function is redefined as: (1a) J T M N + j jt j jt ij ijs j= 1 t= 1 = 1 s= 1 i α j α. Min Z = ( h + g ) + s z The sets of constraints ((2), (3), (9), (10a), (11)-(17)) and the objective function (1a) define a One-Stage Multi-Machine lot-scheduling odel (P1SMM). An adjustent of the solution of this odel ight be necessary to take into account the synchrony between the two stages. This can be obtained by odel P2SMM with the setup variables fixed according to the solution of odel P1SMM. That is, if ite j is produced, then the tank and the achine ust be setup. This procedure is outlined in Figure 4, where σ j is the liquid flavour necessary to produce ite j. f the odels in Algorith RA are solved by a standard software (e.g. the branch and cut ethod in CPLEX) and their optial solution are not achieved in a pre- 12

13 defined aount of tie, the branch and cut execution is stopped and the best solution is considered. Algorith RA Step1 Solve the P1SMM odel. Step 2 f P1SMM is feasible then for = 1... M, j α s = 1... N do f x > 0 then js Fix the setup variables of odel P2SMM according to: y js = 1 and yls = 1, l σ j endfor endif endfor Step 3 Solve the P2SMM odel obtained in Step 2. Figure 4 Algorith RA 4.2 Relax-and-fix strategies n the relax-and-fix heuristic, the integer variable set is partitioned into disjunctive sets, Q i, i = 1,... P. At each iteration, the variables of only one of these sets are defined as integers, while the variables of the others are relaxed and defined as continuous ones. The resulting sub-odel is then solved. A sub-set of the integer variables of the sub-odel are fixed at their current values and the process is repeated for all the sets. Besides the variable set partition, criteria to fix the variables ust also be defined before applying the procedure. The ain feature of this heuristic is the solution of sub-odels that are saller, and possibly easier to solve, than the original one. The partition of variables and the criteria used to fix the variables have a strong connection with the degree of the sub-odel difficulty (Wolsey, 1998). The relax-and-fix heuristic can be described as the RF algorith outlined in Figure 5. Algorith RF nitialization Define a partition of the integer variables set into P subsets, Q i, i = 1,... P and a criterion to fix variables. for k = 1... P do Step 1 Relax the variables in the set Q j, j = k +1,..., P. Solve the resulting sub-odel. Step 2 f the sub-odel given in Step 1 is unfeasible, stop. Otherwise, fix the variables in set Q k to their current values. Endfor Figure 5 Algorith RF 13

14 The relax-and-fix heuristic has been largely used as a ethod to obtain good prial bounds (feasible solutions) for hard ixed-integer progras either on its own or in hybrid algoriths. n the usual relax-and-fix strategy, the variables are grouped by periods (acroperiods) and only the integer variables are fixed at each iteration - the heuristic iterations nuber is thus the nuber of periods (Dillenberger et al., 1994). Various strategies to partition the set of binary variables are explored in Escudero and Saleron (2005). Toledo (2005) uses a relax-and-fix heuristic to solve the soft drinks lot scheduling proble described in Toledo et al. (2007a). The criterion used is to first fix the binary variables in Stage, then the ones in Stage, in a backward fashion. The relax-and-fix heuristic has also been used in cobination with etaheuristics. Pedroso and Kubo (2005) presents a hybrid tabu search procedure in which the relax-and-fix heuristic is used either to initialize a solution or to coplete partial solutions. The hybrid approach is applied to solve a big bucket lot-sizing proble with setup and backlog costs. At each iteration of the relax-and-fix heuristic, only the variables of a given period that concern a single product are fixed. The ain advantage of this strategy is to solve saller sub-odels since soe ono-period, ono-achine ulti-ites lot sizing probles are hard to solve. Other relax-and-fix strategies also fix continuous variables (Federgruen et al., 2004). The P2SMM odel proposed in Section 3 presents various possibilities to build sets Q, i = 1... P. The setup and changeover variables are indexed by stages, achines, ites and i periods. These indexes are explored when defining various variable partition strategies. Different criteria were proposed to fix variables. For exaple, after solving a sub-odel in a given iteration, it is possible to fix only the binary variables associated to non-zero production variables. Table 1 shows 15 relax-and-fix strategies, divided into three groups. Group 1 has five strategies (G1.1 G1.5), Group 2 has seven ( G2.7), and Group 3 has three. The first colun shows the strategy nae (Strat.), the second and third coluns show the criteria used for the partition (Part.) and fixing (Fix) the variables, respectively. The variables presented in Table 1 are the sae ones used to describe the odel P2SMM, however, for siplicity sake, soe of their indexes were oitted. 14

15 Table 1 Relax-and-fix Strategies Strat. Part. Fix G1.1 Period y, y G1.2 Period y, y, z, z G1.3 Period y, z, y, z, x G1.4 Period y, z, y, z i.th.p.* G1.5 Period G3.1 Micro-period y, y G3.2 One icro-period per period y, y First and last icro-period of each G3.3 period y, y * i.th.p.: if there is production The first five strategies (G1.1 G1.5) use the usual criteria of partitioning the variables according to periods (acro-period). They differ fro each other by the criteria used to fix the variables in a given iteration. These criteria are based on the idea that the subodels should have a diension that favours the decision process. The objective was to evaluate the influence of the variables in the sub-odel solution. Take for exaple strategies G1.1 and G1.2. Note that when the variable y, z, y, z i.th.p. with re-evaluation Machine/Period y, y G2.2 Stage then Stage y, y G2.3 Stage then Stage y, y G2.4 Period/ Stage then Stage y, y G2.5 Period/ Stage then Stage y, y G2.6 Machine/Period/ Stage and Machine/Period/ Stage Machine/Period/ Stage and G2.7 Machine/Period/ Stage y, y y, z, y, z i.th.p. with re-evaluation y is fixed to one, it only assures that the achine is prepared, i.e. it does not indicate if the ite will be produced or not. Variable besides defining that the tank will be prepared, also states that there will be production of an ite between the tank capacities (constraints (2) and (3) in the P2SMM odel). Strategy G1.2 is as flexible as G1.1, however, it also fixes the continuous changeover variables ( z, z ). This reduces the sub-odel size both in ters of variables and constraint nubers. The latter y, 15

16 ight be reduced by the pre-processing routines usually included in optiization packages. The lot size is only fixed when variable x is fixed as in strategy G1.3. Strategy G1.4 is ore flexible than the previous ones since the variables are fixed only if there is production (x > 0). Besides allowing a redefinition of the lot sizes, this strategy also allows any ite to be produced in idle periods at future iterations. An attept to iprove this strategy is done by reevaluating the idle icro-periods in previous iterations which did not have any variables fixed. n case they are no longer idle, fix the associated variables. n this strategy, G1.5, at each iteration, the sub-odels becoe less flexible than the ones in strategy G1.4, but ight becoe saller as ore variables are fixed. Group 2 strategies explore the ulti-achine and the two-stage structure of the odel to partition the set of variables. The objective was to evaluate the influence of each achine (stage) in the solution of the sub-odels. Strategy is siilar to strategy G1.1, but considers both the achine and period index to partition the variable set. The setup variables of both stages ( y and y ) are fixed at each iteration. All the other strategies in this group consider one stage at a tie. n Strategies G2.2 and G2.3, the partition is done according to the production stage, therefore the order to fix the variables changes accordingly. First, the variables fro Stage (or Stage ), then the ones in Stage (Stage ). n strategies G2.4 and G2.5, a reduction in the sub-odel size of strategies G2.2 and G2.3 is obtained by including the variable period in the partition criteria. A further reduction in the sub-odel size is obtained in strategy G2.6 by, besides considering the variable period and stage, also considers the achines in the partition. The criterion used to fix the variables in strategies -G2.6 is the sae as the one used in strategy G1.1, i.e. only the binary variables ( y, y ) are fixed at each iteration. Strategy G2.7 cobines the sub-odels size of strategy G2.6 with the flexibility to fix the variables of strategy G1.5. Besides the binary variables, the continuous variables ( z, z ) are also considered to be fixed when there is production ( x > 0). n this strategy, the idle icro-periods in previous iterations which did not have any fixed variables are also re-evaluated for further variable fixing. Note that only strategies, G2.6 and G2.7 consider the achine index in the partition criteria. The criterion used to partition the variables set in the Group 3 strategies is the icroperiods. This criterion iplies a higher nuber of iterations, but saller sub-odels. For exaple, if strategy G1.1 is applied to odel P2SMM, there are T iterations and sub-odels with 2 S t ( α + β ) binary variables each, and if strategy G3.1 is applied, there are N iterations and sub-odels with ( α + β ) binary variables each. However, fixing a 16

17 variable of a given icro-period also reduces the heuristic flexibility. The results obtained testing strategies G3.1-G3.3 were not good copared to the ones in Groups 1 and 2. For soe instances, they even failed to produce feasible solutions. Therefore, these results were not included in the coputational tests described in the next section. 5 Coputational Tests n this section we present and analyze the coputational results of the solution approaches presented in Section 4, applied to solve real instances of the production planning proble of Plant A. The P2SMM odel (Section 3) and the RA and RF algoriths (Section 4) were coded in the AMPL odelling language (Fourer et al., 1983). The P2SMM odel and the sub-odels necessary in the RA and RF algoriths were solved using the optiization syste CPLEX version 10.0 (log, 2006). At each iteration of algoriths RA and RF, a ixed integer optiization odel needs to be solved. Although generally sipler than odel P2SMM, these odels are still difficult to solve. Therefore, if they were not solved to optiality in a pre-specified aount of tie, their execution was stopped and the best solution was analysed. The runs were executed using an ntel Pentiu 4, 1.0 GB de RAM, 3.2 GHz. 4.1 The test bed Several visits to Plant A were ade in order to understand its production processes and to collect the necessary data to siulate its lot sizing and scheduling proble. Data such as product deands, changeover ties in both stages, tank and achine capacities, aong others, were collected during the visits. Several interviews with workers fro different departents of the plant were conducted in order to obtain the appropriate inforation. For reasons of confidentiality, soe values were odified to protect interests of the copany. The data was used to generate 15 instances. Each period of the planning horizon refers to one production week (five days). The first instance (P1) was generated based on data related to two achines that can produce ites in coon. The first one (achine 1) can produce 23 ites and the second one (achine 2) only 10 out of these. That is, there are 13 ites that can be produced on any one of these two achines. Eighteen different flavours are necessary to produce this set of ites. Machine 1 was available for four working days per week (total of 5,760 inutes per week) and achine 2 for six working days (total of 8,640 inutes per week). t was estiated that 17

18 the tank could have up to five changeovers per day. Taking the average working days (5 days) of the achine, it is possible to have up to 25 changeovers per week. The product deand data for instance P1 is associated to three weeks. Therefore, this instance has three acroperiods (3 weeks) with a total of 75 icro-periods (25 per acro-period). The production scheduling for this instance was provided by Plant A, aking the coparison with the solution approaches tested possible. Four other instances (P2-P5) were generated by odifying part of the data used in instance P1. The inventory costs were doubled (P2), the changeover costs were halved (P3), the total deand was redistributed aong the periods (P4), and the achines capacities were reduced (P5). These odifications are detailed in Table 2. The reaining ten instances (P6-P15) were generated using product deand data related to a period of 30 weeks. Each one of these instances is associated to three consecutive weeks. nstances P6, P7, P14 and P15 are associated to periods of higher deands when copared to P8-P13. Except for the deands, all the other paraeters used to generate these instances were the sae ones used to generate instance P1. More details of the procedure used to generate instances P1-P15 and their coplete data are found in Ferreira (2006). All the 15 instances of odels P2SMM and P1SMM (used in Step 1 of algorith RA) have the sae diensions, as shown in Table 3. Note that the nuber of variables and constraints of the odels are relatively large. nstance P1 P2 P3 P4 P5 Table 2. Modifications in instance P1 to generate P2-P5 Modifications Plant A data nventory costs of P1 doubled. Backorder cost of P1 doubled. The total deand of P1 was randoly redistributed aong the periods. The achines capacities were reduced. Table 3. nstances diensions P1-P15 Model Variables Binary variables Constrains P2SMM P1SMM Results The coputational experients were divided into three parts. First, all the solution approaches were applied to solve instance P1 and copared to the copany solution. nstances P1-P5 were used to evaluate the solution strategies (algoriths RA and RF). The 18

19 best solution strategies were then used to solve instances P6-P15. To siulate Plant A practice, the total execution tie for solving each instance of the P2SMM odel by CPLEX, algorith RA and algorith RF was set to four hours. n general, the soft drink lot sizing and scheduling are prepared three to four days in advance and, according to the production anager, it takes around four hours to be copleted. Therefore, this tie liit is considered acceptable for the decisions involved. All the relax-and-fix strategies presented in Table 1 were applied to solve odels P2SMM and P1SMM (Step 1 of algorith RA). To have a fair coparison, the total execution tie was divided as follows. The axiu execution tie for algorith RF when applied to solve the P2SMM odel was set to three hours. The relax-and-fix solution was then used as a first integer solution in CPLEX, which was executed for another hour. n the RA algorith, algorith RF was executed for at ost three hours and only to solve the odel in Step 1. The odel in Step 3 was then solved by CPLEX with a axiu of one-hour execution tie (total of four hours to execute algorith RA). Results for instance P1 Six CPLEX strategies were used to solve the P1 instance of odel P2SMM and the P1 instances of the sub-odels in Steps 1 and 3 of algorith RA. The CPLEX syste allows for the adjustent of various paraeters of its branch and cut algorith. n this test we copared the perforance of the subroutines to find prial bounds (heuristics), the pre-processing subroutines, and the subroutines to obtain dual liits (cutting planes generation). The syste default paraeters autoatically decide the use or not of these subroutines. The sybol in Table 4 indicates which one of the above subroutines was activated or deactivated. For exaple, the use of the CPLEX strategy run5 eans that the instance was solved by the CPLEX default paraeters, except for the intensive use of the CPLEX heuristic and the deactivation of the cutting plane generation subroutines. Table 4. Runs with different CPLEX Paraeters Nae Default ntensive use of CPLEX Heuristics on Cutting generation - off planes Preprocessing - off run1 run2 run3 run4 run5 19

20 run6 The total cost values (in thousands of onetary units) and the corresponding integer gap for each one of the six CPLEX strategies applied to solve odel P2SMM and the odels associated to algorith RA are shown in Table 5. As discussed in Section 4.1, instance P1 is the only one of the proble sets for which we are aware of the corresponding production schedule used by Plant A. This copany solution eets all product deands with no delays, yielding a total cost of in thousands of onetary units. Coparing this solution with the ones given in Table 5, we note that all the six CPLEX strategies supplied solution values worse than the copany one. The solution values given by algorith RA are better that odel P2SMM for all but one CPLEX strategy (run2). However, the associated gap values are too high ( 98.0% for P2SMM and 97.6% for RA), which indicates that the linear relaxations of the odels are very weak and the total execution tie was not enough for CPLEX to find good prial bounds. The best three strategies for P2SMM were run2, run6 and run1, and for RA they were run3, run4 and run6. Aong the best strategies, only two (runs 4 and 6) ake intensive use of the CPLEX heuristics. Moreover, the variation of the gap value aong the first three strategies (run1-run3) was between 97.6% and 98.5%, while aong the last three the gap value was between 97.7% and 98.6% for both P2SMM and RA ( 0.9%). These results indicated the need to further explore the odel structure to develop specific solution strategies and justify the need for algoriths RA and RF. Table 5. Results for different CPLEX strategies nstance P1 Est. P2SMM GAP (%) RA GAP (%) (Z) (Z) run run run run run run The solutions obtained with the intensive use of the CPLEX heuristic (run4-run6) were not strongly better than the ones given by the other three strategies. Therefore, in the reaining part of the tests, the CPLEX was used with the paraeters defined in run1-run3. All the relax-and-fix strategies given in Table 1 were cobined with the CPLEX strategies and applied to solve the P1 instance of odel P2SMM and the P1 instance of the sub-odel 20

21 in Step 1 of the RA algorith. Due to space liitation, only the solution values that are better than the copany one are presented (all the other results can be found in Ferreira, 2006). The P2SMM allows for backorders and since the solution value given by Plant A does not include backorder costs, a version of the proble without backorders was also solved. Table 6 suarizes all the runs executed presenting the best solution values (total cost in thousands of onetary units) with and without backorders (Z and Znb, respectively). The first colun in Table 6 indicates the solution approach, the second one shows the CPLEX and the relax-andfix cobination (CPLEX_RF), the third colun the Z value, the fourth and sixth colun the percentage by which the given solution is better (or worse) than the copany s one. The fifth colun presents the Znb value. The best solution values are highlighted in bold. Table 6. Best solution values nstance P1. Approach Strategy Z Percentage(% Percentag Znb ) e run1_g P2SMM run3_ run1_g run1_g run1_g run1_ run1_g RA run2_g run2_g run2_ run2_g run3_ run3_g Observe in Table 6 that the RA algorith cobined with run2_ presented the best solution value when backorders are allowed. When backorders are not allowed, the RA algorith is still better than the direct solution of P2SMM. However, it is 26.3 % better than the copany s one when cobined with run1_. t is interesting to note that the inventory and changeover costs in this solution are both saller than the ones in the solution fro the copany. n general, the RA algorith provided ore copetitive solutions, even when backorders are allowed. All together, there were 20 solutions that were better than the copany s one, seven without backorders. Therefore, the proposed odel and solution approaches are copetitive with the industrial practice. Nevertheless, the associated gap is still high, above 96%. 21

22 Results for instances P1-P5 All the solution strategies used to solve instance P1 were also applied to solve instances P2-P5. However, only the best results are presented in Table 7. Below each solution value (Z, in thousands of onetary units), the used relax-and-fix strategy is shown. Note that since only the tank capacity constraints are included in Stage of P1SMM odel (used in Step 1 of the RA algorith), the relax-and-fix strategy G2.7 had to be adapted and renaed as G2.8 in order to be used in RA. The best solution value for each instance is highlighted in bold. Aong the 15 relax-and-fix strategies proposed in section 4.2, four strategies provided the best solution values. For P2SMM the best relax-and-fix strategies were, G2.6 and G2.7, and for RA the best ones were and G2.8. n general, for P2SMM, the relax-and-fix strategies provided best results when cobined with run1, while for RA the best relax-and-fix results were obtained with run2 (three out of five instances for both approaches). Although the results are ixed, it is possible to point out the best cobination CPLEX_RF for both P2SMM and RA, which are respectively run1_g2.7 and run2_. Results for instances P6-P15 Considering the results in Table 7, the P6-P15 instances were solved using only the cobination run1_g2.7 for P2SMM and run2_ for RA. Table 8 presents, for each instance, the solution value (Z, in thousands of onetary units) and the percentage (pw) by which the solution is considered worse when copared to the one given by the other approach. The best solution for each instance is considered to be 0% worse. The RA algorith gave the best solution value for nine out of the ten instances solved, which are up to 50.1% better than the ones given by P2SMM. The average solution values are also the saller ones. The results confir the superiority of the RA algorith when copared to the direct solution of the P2SMM odel instances. However, it is worth entioning that, for all instances P1-P15, the gap between the best feasible solutions of odel P2SMM (found within the tie liit) and its linear relaxation solution is over 90%. This fact suggests that there is scope for further research exploring specific solution approaches. Plant A did not provide the corresponding production lot sizes and schedules to these instances. Therefore, it was not possible to copare the solution values obtained by the proposed approaches used to solve instances P6-P15 to the copany s ones. 22

23 P2SMM RA Table 7. Best solutions values for instances P1-P5 nstances run1 run2 run3 P G G P G G2.7 P G P G G2.6 P G G G2.6 Average P G2.8 P G2.8 P G2.8 P P Average Table 8. Solution values for instances P6-P15 - Z (pw%) Noe P2SMM run1_g2.7 RA run2_ P (20.7%) (0.0%) P (13.9%) (0.0%) P (10.5%) (0.0%) P (39.0%) (0.0%) P (9.3%) (0.0%) P (50.1%) (0.0%) P (0.0%) (12.6%) P (40.6%) (0.0%) P (23.6%) (0.0%) P (33.5%) (0.0%) average Final Rearks n this paper a Two Stage Multi-Machine lot-scheduling odel is presented to solve the production-planning proble of soft drink plants. The proposed odel is useful to represent the proble and showed the liitations of the available general-purpose 23