Combining process selection and lot sizing models for the production scheduling of electrofused grains Jose Roberto Dale Luche Reinaldo Morabito1 Vitória Pureza Department of Production Engineering Universidade Federal de São Carlos, Brazil 13565-905, São Carlos, SP.
(
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Abstract This work presents optimisation models to support decisions in the production planning and control of the electrofused grain industry. A case study was carried out in a Brazilian company with the aim of helping to increase productivity and improve customer service concerning meeting deadlines. Mixed integer linear programming models combining known models of process selection and single-stage lot sizing are applied for the production scheduling of electrofused grains. Optimising such scheduling is not a simple task mainly because of the scale of the equipment setup times, the diversity of the products and the limitations of the order due dates. A constructive heuristic is also proposed as an alternative solution method, particularly for large sized instances. The results show that the models, as well as the heuristic are capable of producing better solutions than the ones currently used by the company. Key words: production scheduling, process selection, lot sizing, electrofused grains.
1. Introduction This study deal with the Production Planning and Control (PPC) of the electrofused grains industry, which has various production units in the southeast of Brazil. It is based on a case study in a plant in Sao Paulo State that has a monthly production of more than two thousand tons of electrofused grained raw materials. This plant uses large amounts of electric power for electrofused microgrit production, reaching a monthly average of 14 MW/h, which has caused problems regarding the power cut crisis in the country over the last years. This situation has encouraged research in alternative ways to optimise the company’s processes 1
Corresponding author: phone 55-16-33518237, fax 55-16-33518240,
[email protected].
2 and PPC. Various critical decisions should be considered in the PPC, particularly regarding the inventory levels of raw materials and intermediate products, and the production scheduling of furnaces and subsequent processes, like crushing, grinding and classification, depending on the final product demands. These are typical decisions in electrofused grain industries. In this work, mixed integer linear programming models are formulated to help in taking decisions for the production scheduling of electrofused grains. These models combine known models of process selection and single-stage lot sizing so that they can be viewed as lot sizing models which, instead of using “product lots” use “process lots”. Moreover, each lot simultaneously produces a set of products. Optimising such production scheduling is not a simple task mainly because of the scale of the equipment setup times, the diversity of the products and the limitations of the order due dates. Production managers of Brazilian electrofused grain companies find planning the production quite demanding due to the combinatorial nature of the problem, among other aspects. Techniques of mathematical programming seem not to have been effectively applied to such a problem as yet, which could bring many benefits to the process of taking decisions in the PPC of these companies. It is common that a production schedule needs to be modified several times due to urgent customer orders and unexpected equipment failures therefore highlighting the importance of a model capable of generating efficient production programs in a feasible time. For cases where the sizes of the problem instances make the use of exact methods prohibitive, a constructive heuristic is also proposed. This article is organized as follows: in the next section, the production process of electrofused grains is briefly introduced. There is a discussion of the production scheduling problem based on the plant of the company studied, which is also applied to other companies of the sector. In section 3, concepts of the selection process and lot sizing models are combined, focusing on the problem formulation of the case study. The proposed model (MLP - Minimization of Lack of Production), as well as the heuristic procedure intend to minimize the amount of non-produced ordered products, in other words, it consists of minimizing the production shortages and the deadline delays. Variations and extensions of this model are also discussed. In section 4, the computational results from the solution of the models using the modelling language GAMS (with the solver CPLEX), applied to simulated data and current data from the company are presented. These results show that the models are consistent and able to generate better schedules than the ones currently used by the PPC programmers of the
3 company. Finally, in section 5, the conclusions and perspectives for future research are discussed. 2. Problem definition 2.1- Process description The studied factory produces white electrofused aluminium oxide and all products are grains whose sizes vary from some centimetres to a few micrometers. The alloy of alumina and bauxite is made of a reduction process in a liquid phase in furnaces (Higgins furnaces). Each furnace has three electrodes placed among the raw material, which is heated up to a temperature of approximately 2500 ºC. After reduction, the material is put to rest until it becomes solid. Afterwards, it is carried to a cooling area where it stays for over 30 hours to cool. In this factory the yield is carried by travelling cranes to the first crush in a hydraulic crusher. After that, the material undergoes crushing and classification. Figure 1 depicts the basic material flow diagram of this production process. For further information on the electrofused grain processes and the size of particles (grains) and mills adjustments, see e.g. Allen (1990) and Alcoa (2002). stock
crusher
mill
carry
Higgins furnace
travelling cranes
cooling
classification
output
Figure 1 - Material flow diagram in the factory.
4 2.2 – Production system The production system of the company is intermittent and repetitive: changes in the type of products always take place, resulting in a great variety of products. The layout is set as a function of the product: the machines are allocated according to the operation sequence the product is submitted to. Since all products have basically the same sequence of operations, the production system flow pattern is a flow-shop. Different important decisions are involved in the PPC of this factory, mainly concerning the inventory levels of raw materials and intermediate products and to the production scheduling of the furnaces and subsequent processes, like crushing, grinding and classification, all dependent on the final product demands. The classification scheduling of the grains should be made together with the scheduling of furnaces, crushers and mills, however it involves some difficulties. The classification is carried out by a set of shaking sieves, with the aim of separating the grains by size (the size is set by the amount of holes per inch in the sieve), and it can be assembled with different combinations of sieves. There are sieves of different dimensions, for instance, product EC31_120 is a grain that initially goes through a sieve of 100 holes per inch and then stops in a sieve of 120 holes per inch (the number of holes per inch can vary from 5 holes, in a sieve for thick grains, up to 220 in a sieve for minute grains). Table 1 presents the amounts (in kilos/day) of various electrofused grains (rows of the Table), which can be produced from 10 different processes (combining a set of sieves with the adjustment of crushers and mills; see columns 1 to 10). For instance, process 1 is able to produce 2,000 kilos of product EC31_36, 600 kilos of product EC31_120, 500 kilos of product EC31_150, and so on, adding up 27,700 kilos per day (last line of the Table). Notice that the same product can be produced by different assembles of sieves in different amounts. However, once the production line is established and it has chosen the set of sieves to operate, some products from this arrangement might not have been demanded. This is due to the fact that because the products and the amounts of each product to be produced in a day in the selected process are already established, it is common to produce and stock products which have still not been demanded for a long time and even until the end of the planned time. A way of restricting the stocked amount is to use the surplus of medium and thick grains as raw materials for the demanded production of tiny grains. However, this implies in additional production times and costs, as those grains must go back to the mill to be grinded again.
5 Table 1 - Amounts (in kilos/day) of electrofused grains (lines of the Table) produced by processes 1-10 of sieves (columns). PRODUCT EC31_36 EC31_120 EC31_150 EC31_180 EC31_220 EG52_120 EG52_150 EG52_180 EC31R_3-8_1-4 EC31R_3-5_7 EC31R_5-16_4 EC31R_08_F EC31R_08_F1 EC31R_08_F2 EC31R_08_F3 EC31R_08_F4 EC31R_08_F5 EC31R_08_G EC31R_08_G1 EC31R_08_G2 EC31R_08_G3 EC31R_08_G4 EC31R_08_G5 TOTAL
1 2000 600 500 300 300
2000 5000 2000 5000 6000
2 2000
3 2000 600
4 2000 300
300 500 300 300 2000 5000 2000 5000
PROCESS 5 6 2000 2000 600 300 300
300 500
500 200
300 2000 5000 2000 10000
400
300 2000 5000 2000 8000
1000 4000 1000 7000 800
500
2000 5000 2000 7000
2000 1000
500 500 2000 5000 2000 10000 1500 1000
300 2000 5000 2000 10000 1000 1000 1000
10000 800 1000 3600 4800 5000
10000 5000
2000
1000 1000 1000 4000
5000 8000 27400
10
1000 1000 1000
2000
27700
300
9 2000 600 300
1000
900 1000
8 2000
500
1000 500
2000
7 2000 1000
1200 27400
800 27700
7000 500 27000
25700
1000 27500
1000
8000
26100
26100
27000
3. Problem modelling Problem modelling is based on a combination of the models of process selections and lot sizing, as discussed below. 3.1 - Process selection models In these sorts of models, the product demands are established for each period of finite planning, and each product can be produced by different alternative processes. The production costs and the resources used depend on the chosen process. The resources have limits of availability in each period and various products compete for these resources according to each production process. The models should determine how much to produce of each product in each process, resulting in minimizing the production costs or maximizing the profit contribution, subject to the constraints of limited resources and meeting the demand (e.g., Johnson and Montgomery, 1974; Hax and Candea, 1984; Williams, 1993; Graves et al, 1993, Nahmias, 1995).
6 3.2 – Lot sizing models Lot sizing models are related to process selection models and they basically consist of planning the amount of products to be produced in each period of a finite planning horizon, so that the demand is met and a criterion is optimised, for instance, to minimize production costs or maximize the profit contribution (e.g., Johnson and Montgomery, 1974; Hax and Candea, 1984; Askin and Standridge, 1993, Gershwin, 1994, Graves et al, 1993). For Feng and Cheng (1998), solving lot sizing models has been difficult as the industrial environments and practices have become more complex. Experts point out five dimensions of complexity for single-stage lot sizing models: (i) limited availability of multiple resources, (ii) occurrence of many products sharing the same resources, (iii) variable demands period to period, and different periods in planning, (iv) setup times and (v) setup costs to produce a lot of a certain product. When setup times and costs are considered, the lot sizing models become difficult to solve (Bitran and Yanasse, 1982; Maes and Wassenhove, 1988). Various studies can be found in the literature covering models and solution methods for different lot sizing problems, for instance for the capacited lot sizing problem (CLSP) (Bodt et al, 1984; Gunther, 1987; Trigeiro et al, 1989; Maes et al, 1991), the discrete lot sizing and the scheduling problem (DLSP) (Fleishmann, 1990, 1994; Salomon, 1991; Salomon et al, 1997; Eijl and Hoesel, 1997; Bruggeman and Jahnke, 1999), the proportional lot sizing and scheduling problem (PLSP) (Drexl and Haase, 1995), and the continuous setup lot sizing problem (CSLP) (Bitran and Matsuo, 1986; Karmarkar et al, 1987; Matta and Guignard, 1989; Kimms, 1996). Drexl and Kimms (1997) and Karimi et al (2003) present model reviews of various problems of lot sizing, including the general lot sizing and scheduling problem (GLSP). Other recent studies covering lot sizing models are found for example in Armentano et al (1999), Clark and Clark (2000), Haase and Kimms (2000), Meyr (2000, 2002) and Fleszar and Hindi (2004). We are not aware of models in the literature combining decisions of process selection and lot sizing problems, as in the present work. 3.3 – Classification of Grains In this case study, the classification of grains is made by a set of sieves. Production scheduling is not only limited to the choice of sieves to make up the assembly, but it also depends on the crusher and mill programming. Larger amounts of tiny or thick grains define the adjustment (squeeze) of the mills (there are various production curves with different granulometric distributions able to be implemented). Therefore, each process basically
7 consists of adjusting the mills and choosing a set of sieves to be used in a period (e.g., a workday). The thicker the curve region of the grains, the larger is the production in tons. On the other hand, the thinner the curve region of the grains, the smaller the production is. The assembly of sieves defines which are the selected products (produced) whereas the adjustment of mills sets the amount of each product in each period. In every workday, the company wants to accomplish at most one process setup, because the machine setup times are very long. The company wants to avoid the process interchange throughout the day, as a large part of the setup time of the process takes place before the beginning of the first work shift (the shifts do not add up to 24 hours a day). The mix of products to be produced in one day is defined in the process, which is subject to various technological restrictions. For each period, it needs to find the process that best provides the demanded amount of each product, keeping in mind that, on one hand, all of the products in demand will hardly be produced by this process and, on the other hand, other nondemanded products will probably be produced. Grain production scheduling is still conditioned to: a. The planning time generally being one month, where 19 periods are considered, referring to the days in which there is production in the company; b. New customers’ orders may be accepted in a plan, that is, it may be necessary to remake the scheduling to meet urgent demands (inclusively this may happen in the “frozen periods” of the rolling horizon); c. The initial inventory of the products is not considered in the schedules, i.e., it is simply withdrawn from the demanded amount of these products in the first periods, before scheduling; d. It is assumed that the products are produced in a single production stage (independent demand). In this issue, the limitations of machine capacities are considered when developing the production processes, i.e., each production process takes into account the capacity restrictions of the equipment in the production line. 3.4 – Process selection and lot sizing modelling As already mentioned, the models presented here are a combination of process selection and lot sizing models. The PPC of the company barely foresees the daily inventory
8 and shortage costs of products. Moreover, it hardly provides feasible production schedules due to the commonly tight order due dates (dealt with by the sales department) and the increase of weekly and fortnightly customer requests (with smaller demanded amounts) relating to the monthly requests. Thus, due to these difficulties and the long setup times for process change, it is admitted that the problem consists of finding a production schedule that minimizes the lack of production using at most one process per period (i.e., the process does not need to be changed throughout the day). This practice is desirable by the plant managers as the demand requirements are frequently infeasible in practice (due to very strict deadlines dealt with by the sales department). The following model minimizes the lack of production, and makes use of slackness and surplus variables in the demand constraints. Model MLP (Minimizing the Lack of Production): Variables:
x jt : indicates if process j (j = 1, 2, ..., n) is used in period t (t = 1, 2,.., T) fit: slackness of product i in period t eit: surplus of product i in period t
Parameters:
aij: amount of product i (i = 1, 2, ..., m) produced by process j (the model assumes that the production leadtime is the period); dit: demand of product i in period t (the model assumes that the initial product inventory is null – in case it is positive, the demand is reduced appropriately).
Min z =
T
m
t =1 i =1 t
n
f it
aij x jt '+ f it − eit =
(1) t
t = 1,..,T
(2)
t = 1,..,T
(3)
x jt ∈ {0,1}, f it ≥ 0, eit ≥ 0 , j = 1,..,n, i = 1,..,m, t = 1,..,T
(4)
t' =1 j =1 n j =1
x jt ≤ 1,
t' =1
d it ',
i = 1,..,m ,
The objective function (1) minimizes the lack of production of the products in demand. The demand constraint (2) includes the slackness and surplus variables of each product i in each period t (the total amount of a product produced until a certain period, plus a slackness or minus a surplus, should be equal to the accumulated demand until this period). Constraints (3) impose that at most one process is used in each period t and constraints (4),
9 the integrality of variables xjt and the non-negativity of variables fit and eit. Notice that model (1)-(4) can be observed as a lot sizing model that, instead of using “product lots”, uses “process lots” to produce a set of products. An optimum solution of model MLP always fulfils f it eit = 0 . Although the major concern of the PPC programmers is with the lack of production, in cases where the model solution results in z = 0, meaning that all of the order due dates are met in the planning, a slight modification of the model can also minimize the surplus of production (i.e.,
T
m
t =1 i =1
eit )
or the number of periods needed for the demanded production. In the latter, the idea is to produce as soon as possible, meaning that, even though it is possible to postpone the production at a certain period, this is not done due to the opportunity of receiving new orders and being able to produce them during the planning. This can result in carrying out larger stocks due to the chance of being ahead of the production of some products, however maximizing the plant productivity. A way of doing this is simply redefining the objective function (1) as:
Min z = K
T
m
t =1 i =1
f it +
T
n
t =1 j =1
tx jt
(5)
where K in (5) is a sufficiently large number so that the first objective (minimizing the lack of production) rules the second (minimizing the number of periods). The model can also be rewritten so that the value of the objective function z literally corresponds to the number of necessary periods (instead of a penalty function as in (5)):
Min z = K
T
m
t =1 i =1 n j =1
x jt ≥
n j =1
x j , t +1
f it +
T
n
t =1 j =1
x jt t = 1,..,T-1.
(6) (7)
Subject to constraints (2), (3) and (4). Constraint (7) imposes an order in the assignment of variables xjt, in a way that period t+1 is not used without period t. In this case, the solution always applies to only the first periods available for production. Despite the fact that the company does not change processes
10 during the production period, the models above can be modified to consider setup times and inventory costs in order to investigate possible practical solutions that the plant managers are not exploring at the time. For instance, model (1)-(4) can also be extended to consider setup times in a production period as follows: Model MLP with setup times: Additional variables : qjt: fraction of period t (i.e., a production day) producing process j ( 0 ≤ q jt ≤ 1 ) Additional parameters: st: fraction of a period setting up process j, i.e., its setup time ( 0 ≤ st j ≤ 1 );
Min z =
T
m
t =1 i =1 t
n
t' =1 j =1 n j =1
f it
aij q jt '+ f it − eit =
(8) t t' =1
d it ',
i = 1,..,m ,
( st j x jt + q jt ) ≤ 1,
q jt ≤ x jt
j = 1,..,n ,
t = 1,..,T
(9)
t = 1,..,T
(10)
t = 1,..,T
(11)
x jt ∈ {0,1}, q jt ≥ 0, f it ≥ 0, eit ≥ 0 , j = 1,..,n, i = 1,..,m, t = 1,..,T (12)
where the capacity constraints (10) consider the setup times of the processes in a production day and constraints (11) relate to the definition of qjt. Note that, unlike model (1)-(4), model (8)-(12) uses more than one process in a period. The model above can be reformulated to also consider product inventory (either on-hand or backlog inventory) carried out between the periods and the corresponding costs, as: Model MLP with setup times and inventory costs: Alternative variables: I it+ and I it− : on-hand inventory and backlog inventory levels of product i at the end of period t, Additional parameters: hit: on-hand inventory unit cost of product i in period t;
11 Min z = K
T
m
t =1 i =1 n j =1
I it− +
T
m
t =1 i =1
hit I it+
aij q jt + ( I i+,t −1 − I i−,t −1 ) − ( I it+ − I it− ) = d it
(13) i = 1,..,m , t = 1,..,T (14)
Subject to (10) and (11)
x jt ∈ {0,1}, q jt ≥ 0, I it+ ≥ 0, I it− ≥ 0, , j = 1,..,n, i = 1,..,m, t = 1,..,T (15)
where, as in (5), K is a sufficiently large number so that the first objective (minimizing the lack of production) rules the second (minimizing the on-hand inventory costs), and constraints (14) are the inventory balancing equations. Note that the inventory of product i in period t is given by I it = I it+ − I it− and only one of the terms on the right-hand-side of the equation can be positive. The models above can still be extended to deal with problems by combining decisions of lot sizing and scheduling (sequencing) of process lots, such as in the DLSP, PLSP, CSLP and GLSP discussed in Section 2. Other possible aims, namely, minimizing production, setup, inventory, backlogging and lost sales costs, prioritising products by profit contribution, urgent orders and preferential customers, among others, can be dealt with by adaptations in the models above and/or applying goal programming (as e.g. Munhoz and Morabito, 2001). Obviously the application of this technique here involves some difficulties, as the models involve integer (binary) variables. 3.5 – A constructive heuristic The problem instance size may impose limitations to the application of exact methods to the proposed models. For this reason, a heuristic was developed to solve model MLP (model (1)-(4)) in order to deal with larger problems. The heuristic constructs a solution sequentially, one period at a time. The choice of the process of a particular period t is provided by a function that aims to evaluate “how well” each process fulfils the overall due demand of all products from period t to T. Let I´it be the due demand of product i in period t, i.e., the product demand in period t minus its inventory at the end of period t-1 ( I ´it = d it − I i ,t −1 ) (recall that I i ,t −1 < 0 implies in a backorder inventory). Then, for each process j (j = 1.. n), the evaluation function Fjt is defined as:
12 m
T
Fjt =
Min
i =1 t " = t
(aij − I ´it " ) (t"−t + 1) v
,0
(16)
Note that Fjt only comprises terms associated to periods with a lack of production and the impact of each term in the evaluation decreases the farther the period t” is from the current period t. Such impact is controlled by the exogenous parameter v. Although production anticipation is often necessary to meet large demands of subsequent periods, it is also reasonable to expect that the demands of later periods will be fulfilled afterwards with other processes, which are yet to be decided. The p processes with the highest evaluations (set P) are selected for further examination. Each process in P is used as a tentative process for period t. According to this provisional decision, on-hand and backlog inventories and due demands of all products are updated.
The evaluation function is then used to determine tentative processes for the
remaining periods t+1 to T, one period at a time. In each period, only the best evaluated process is considered. Note that the process selection in any period depends on how the decision of the previous period affects on-hand and backlog inventories and due demands. Note also that the larger the exogenous parameter p is, the larger the number of provisional solutions generated. The provisional solution from period t to T that provides the smallest lack of production permanently defines the process for time t. The procedure is then repeated for t = t+1 up to t =T in order to obtain the complete solution. A more formal description of the heuristic is provided by the following steps: 1. Let I ´it = d it of each product i in each period t (i=1..m; t=1…T). 2. For t1=1 to T: 2.1. Compute Fj,t1 (equation (16)) for each process j (j = 1.. n). Make P = the set of the p processes with the highest evaluations. 2.2. For each r ∈ P : 2.2.1. Make tentative_process(t1) = r. 2.2.2. For all products i (i=1..m), update (provisionally) the on-hand ( I i+,t1 ) and backlog ( I i−,t1 ) inventories in period t1 and the due demands I ´it in all remaining periods t1+1 to T. 2.2.3. For t2=t1+1 to T:
13 2.2.3.1.Compute Fj,t2 for each process j (j = 1.. n). Let q = process with the highest evaluation. 2.2.3.2. Make tentative_process(t2)=q. 2.2.3.3. For all products i (i=1..m), update (provisionally) the on-hand ( I i+,t 2 ) and backlog ( I i−,t 2 ) inventories in period t2 and the due demand I ´it in all remaining periods t2+1 to T. 2.2.4. Let f´r,t1,T = lack of production of all products from period t1 to T given tentative_process(t1) = r.
2.3. Select j´ as the process of period t1 for which f´r,t1,T is
minimum
( j´= arg min r∈P { f ´r ,t1,T } ). For all products i (i=1..m), update the on-hand ( I i+,t1 ) and backlog ( I i−,t1 ) inventories in period t1 and the due demands I ´it in all remaining periods t1+1 to T.
4. Computational results The experiments were performed using a microcomputer Pentium IV 3.0 GHz, 2.0 Gb of RAM. The modelling language GAMS 2.0 with the solver CPLEX 7.0 (Brooke et al., 1992) was used to solve the mathematical models. Reviews on modelling languages are found for example in Kao (1998) and Kiup (1993). In all experiments we used the default parameters of CPLEX with a null tolerance for the gap of optimality. The constructive heuristic described in Section 3.5 was implemented in Borland Delphi 7. Preliminary tests suggested that rather than a single pair of values for parameters v and p, a parametric variation within some given ranges should provide a better performance. We used v={2,3,4} and p={3,4,5,6} as standard settings, which means that the solutions are the best from 3x4=12
runs while runtimes consist of the accumulated times of these runs. 4.1. – Experiments with simulated data Initially a fictitious list was created with n=10 arbitrary processes to produce m=15 products in the planning of T=10 periods, with a stipulated demand per period. This simulated situation of a smaller scale than an actual situation allows for a simplified analysis of the functionality and consistence of the models and solutions. The input data are presented in Tables 2 and 3.
14 Table 2: Amount (aij) in kilograms produced in a day of product i using process j. PRODUCT EK8A-16 EK8A_20 EK8A_24 EK8A_30 EK8A_36 EK8A_46 EK8A_54 EK8A_60 EK8A_80 EK8A_100 EK8A_120 EK8A_150 EK8A_180 EK8A_220 EK8A_FFF
1
0 0 0 0 0 0 0 1500 1000 300 300 250 250 250 200
2 1000 1000 0 0 0 0 0 0 1500 300 300 250 250 250 200
3 2000 500 500 500 500 0 0 0 500 0 0 0 0 0 0
4
0 0 0 0 0 1500 800 1500 1500 500 500 0 500 400 0
PROCESS 5 6 500 0 500 0 300 0 300 0 0 0 0 0 0 0 2000 3000 1500 2000 300 700 300 700 200 500 200 500 200 500 300 0
7
0 0 0 0 0 0 0 3000 2500 800 800 700 700 700 600
8
9 1000 1000 200 200 200 200 200 1000 1000 1000 1000 1000 500 400 400
10 700 700 700 0 0 0 0 0 0 0 0 0 0 0 0
8
9 600 0 0 1000 800 0 0 1500 0 3000 0 0 0 2000 0
10 500 1000 0 0 0 0 1500 0 2000 0 1500 1500 1500 0 2000
0 0 0 600 700 700 700 0 0 0 500 300 300 500 500
Table 3: Amount (dit) in kilos demanded from product i in period t. PRODUCT EK8A-16 EK8A_20 EK8A_24 EK8A_30 EK8A_36 EK8A_46 EK8A_54 EK8A_60 EK8A_80 EK8A_100 EK8A_120 EK8A_150 EK8A_180 EK8A_220 EK8A_FFF
1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3
0 0 1000 0 0 1000 0 0 0 0 0 0 0 500 0
4
0 0 0 0 0 0 0 1000 0 0 0 0 0 0 1000
PERIOD 6 0 0 0 0 0 0 300 700 300 0 0 500 0 500 2000 0 0 500 0 0 0 0 0 0 0 0 0 0 0 0 5
7 400 400 0 0 0 0 0 0 0 0 0 0 0 0 0
0 600 500 0 1000 0 0 0 1000 0 0 0 0 0 0
By applying the MLP model (model (1)-(4)) to the data in Tables 2 and 3, the optimal solution of Tables 4 and 5 is obtained with all demands met until the end of the planning. The production schedule uses the 10 periods (Table 4) and results in a total stock of 34,600 kg at the end of this planning (Table 5). In period 4 there is a shortage of 100 kg of product EK8A_FFF that is missed out in Table 5. This is due to the fact that the amount of a product in a period can be produced in the next periods before the end of the planning. Therefore, the lack of the product EK8A_FFF, that took place in period 4, was fulfilled in period 5. Both GAMS/CPLEX and the heuristic described in Section 3.5 required only a few seconds to solve the problem.
15 Table 4: Process j used in each period t by the solution of model MLP (i.e., with xjt = 1) for the data in Tables 2 and 3. Period Process
1 4
2 10
3 5
4 7
5 8
6 9
7 8
8 3
9 9
10 7
Table 5: Total excess or shortage of products provided by the processes of Table 4 for the data of Tables 2 and 3. Product EK8A-16 EK8A_20 EK8A_24 EK8A_30 EK8A_36 EK8A_46 EK8A_54 EK8A_60
Stock 3700 1700 400 400 200 1800 600 7000 Total Stock
Product EK8A_80 EK8A_100 EK8A_120 EK8A_150 EK8A_180 EK8A_220 EK8A_FFF
Stock 7000 1400 3900 2700 2200 1300 300 34600
As mentioned before, most of the setup of the process takes place before the beginning of the first work shift, and the company did not have accurate estimations of the process setup times. For the application of model MLP with setup times (model (8)-(12)) for the data of Tables 2 and 3, we simply consider that: (i) the setup times of all the processes are equal, i.e., stj = st, and (ii) in each period t, the setup of the first process is done before the beginning of
the period. Therefore, constraint (10) of the model is redefined as: n j =1
( st.x jt + q jt ) ≤ 1 + st ,
t = 1,..,T
(17)
where, as before, st is the fraction of the period setting up the process ( 0 ≤ st ≤ 1 ). Note in (17) that if st = 1, only one process is allowed in each period t (as in model MLP), whereas if st < 1, then more than one process may be used in the period. Table 6 presents the total shortage (z) and the maximum number of processes in a period (max) obtained with model MLP with setup times (with constraint (10) replaced by (17)), by varying st = 0, 0.1, 0.2, …, 1. As expected, for st = 1, the solution (total shortage of z = 100) is the same obtained by model MLP.
16 Table 6: Results of model MLP with setup times by varying st = 0, 0.1, 0.2, …, 1 for the data in Tables 2 and 3. st z max
0 0 3
0.1 0 3
0.2 0 3
0.3 0 3
0.4 40 2
0.5 100 1
0.6 100 1
0.7 100 1
0.8 100 1
0.9 100 1
1 100 1
4.2. – Experiments with the company’s data A restricted list of processes used by the company was taken as an initial base for the production scheduling. The reason for this was the huge number of possible production processes that can be implemented in the factory, resulting from the combination of different amounts of sorts of products in a process. Considering that the average daily production is 28 tons, that there are about 50 different products and that the products can be produced initially in amounts of 300 kg (where 5 kg is the minimum increment scale), this would result in an enormous number of processes, making the models computationally intractable. Techniques of column generation (processes) could also be applied here, as for example in the literature of cutting and packing problems (e.g. Dyckhoff and Finke, 1992; Morabito and Arenales, 2000). However, there are additional difficulties to define feasible processes, as a function of the restrictions of the various machines involved in the production line. Due to the enormous number of processes (in the order of thousands), the company firstly provided the 140 most used standard processes (to feed the models). It is known that based only on those 140 processes, it is possible that model MLP is not able to produce better results than the ones programmed by the company’s PPC, since the company’s PPC has more flexibility in the processes´ definition and choice. On the other hand, this first approach can facilitate the PPC programmers to find results more quickly and, eventually, better than the ones obtained without using a computer. In addition, new processes, set by the programmers, can be added to the model processes listed to test their benefits. Therefore, the input data consist initially of n=140 processes (with the setup times already subtracted from the production capacity of each process), m=50 products and T=19 periods (days) in the one-month planning. These data are detailed in Luche (2003). The production scheduling performed by the PPC of the company in this planning resulted in a total shortage of 13,450 kg, which means that it does not meet all of the deadlines. In general, the programmers of the company need many hours (sometimes up to days) to achieve an acceptable production schedule.
17
Model MLP using 140 processes: The MLP model was applied to the data with 140 processes to minimize the lack of production. The obtained result of GAMS/CPLEX (total shortage of z = 42,370 kg in 9 seconds of runtime) is far from the scheduling performed by the company PPC (13,450 kg). This is due to the fact that the model operates with only 140 processes, while the PPC programmers have great flexibility to formulate new processes to meet the demand in a better way. The constructive heuristic found a solution with a total shortage of 42,890 kg.
Model MLP using 159 processes: Nineteen processes were added to the input data, corresponding to the processes used in the company’s PPC scheduling in the one-month planning under consideration (remember that the company scheduling solution does not cover all deadlines). The solution value of the constructive heuristic is 11,125 kg, achieved after 9.5 seconds of runtime, i.e., 17% better than the company solution. Therefore, it is expected that the model would find (in the worst case) at least a solution as good as the one found by the heuristic or the company. In fact, in this simulation a much better result than the one of the company was obtained: total shortage of z = 10,475 kg in 26 seconds of runtime. It is worth pointing out that the difference is relevant (a reduction of 22% in the shortage). In other words, some of the 19 additional processes included in the 140 standard processes are important for a production scheduling with a better chance of meeting deadlines. The experiments of this section show that, in real situations, the quality of the solutions may be very susceptible to the set of processes (available in the input data of the model). In spite of the frequent use of about 140 standard processes by the company, this isolated group may not be capable of providing good production schedules. It may be necessary that other processes, which exploit the demand patterns in each period, should also be considered by the models and the heuristic. This gives hints concerning the importance of the interaction between the PPC programmers and the solution approaches in order to test the benefits resulted from the inclusion of new processes, due to the demands of each planning period. It is worth mentioning that model MLP becomes harder to solve optimally for GAMS/CPLEX (in terms of computer runtimes) as the number of processes (n) and the number of periods (T) increase. For instance, taking this example with n=159 processes, if we double the number of periods (from 19 to 38 periods), using the same product demand of the first 19 periods to the last 19 periods, the runtime required to solve the model increases more than ten times (from 26 to 293 seconds). Conversely, if we halve the number of periods (from 19 to 9 periods), the runtime is only a couple of seconds. Moreover, the model also becomes
18 harder to solve as the product demand increases or is more distributed over the periods, as shown in the next section. Applying model MLP with setup times (with constraint (10) replaced by (17) as before) to this company instance with n=159, the GAMS/CPLEX was unable to find optimal solutions within one hour for st = 0.1, 0.2, …, 1. Note that, despite the correspondence between model MLP and model MLP with setup times for st = 1, the latter is much more difficult to solve than the former. In particular, for st = 0, the optimal solution of model MLP with setup times does not have a shortage (z = 0) and involves up to six processes in the same period. 4.3. – Experiments with randomly generated data To better evaluate the computational performance of model MLP and the heuristic procedure, we carried out experiments with 7 sets of 10 examples randomly generated (a total of 70 examples). The data generation was inspired by the company’s example analysed above. In the first data set, each example contains the same m=50 products, n=159 processes and T=19 periods (days) of the company’s example. For each product, the demands of the periods were added up (some of the periods have no demand), after which the total value was randomly sorted among all periods. In order to avoid periods with a very small demand, if the sorted quantity in a period was less than a pre-defined lower bound, this quantity was transferred to other periods and the period remained with no demand of that product. In the next five data sets, the product demands of data set 1 were simply reduced by 10, 20, 30, 40 and 50 percent, respectively. Conversely, in the last data set, the product demands of data set 1 were incremented by 10 percent. The average results obtained by solving model MLP by GAMS/CPLEX and by the constructive heuristic for each data set are presented in Table 7. The third column in each Table shows the mean computer runtime Time (in seconds) obtained by GAMS/CPLEX to achieve either the optimal solution or a maximum of 3 runtime hours (i.e., a time limit of 10800 seconds). The fourth column presents the average percent deviation (APD) from the model solution and the mean total computer runtime Time (in seconds) required by the heuristic with parameters v and p standard setting. The fifth column shows the heuristic results by applying the extended setting v={2,3,4,5} and p={3,4,5,6,7,8}. For the latter, note that results for each instance are the best of 4x6=24 runs with accumulated runtimes.
19 Table 7: Mean values of the 7 data sets of 10 examples randomly generated with n=159 (total of 70 examples). Data set Demand variation (%) Model MLP Heuristic Heuristic (ext) APD (%) APD (%) Time (s) Time (s) Time (s) 1 0 11.4 8.7 1324
2
-10
3
-20
4
-30
5
-40
6
-50
7
+10
984 337 42 17 8 4648
64
156
11.5
11.5
62
7.9
153
4.9
63
150
6.4
5.6
60
4.7 60
6.3
147
3.3
146
3.7
60
145
14.4
12.6
67
163
Note in data set 1 that the random examples with product demands distributed in all periods are, on average, harder to solve optimally for GAMS/CPLEX (in terms of computer runtimes) than the company’s example with demands more concentrated in a smaller number of periods. Note also in data sets 1-6 that as we reduce the product demands, the examples become, on average, easier to solve optimally (as shown by the decreasing mean runtimes). Conversely, the larger the demand, the harder the problem becomes: optimality could not be proved within the imposed time limit for only one example of data set 1 and for four examples of data set 7, which explains the larger average runtime. In data sets 3-6 there are examples with optimal solution value z = 0 (null shortage), which could be solved by the model using the objective function (5), instead of (1), in order to also minimize the number of periods needed. It should be noted the reasonably good results provided by the heuristic; parameters v and p standard setting (column 4) produced average solution values ranging from 4.7 to 14.4% above the average model solution value. If the extended parameter setting is used (column 5), these ranges are reduced to 3.3% and 12.6% at the extra cost of 90 seconds of runtime. Observe that as the product demands increase, the solution quality given by the heuristic tends to deteriorate, whereas the computer runtime is not particularly sensitive to the variation of the demand.
20
5. Conclusions The results show that the optimization models are able to produce better solutions than the ones currently used by the company. Due to the relatively short computational runtimes using GAMS/CPLEX, such an approach can perform different production scheduling simulations (exploring a number of situations), which provides flexibility and effectiveness for the individuals liable to take decisions. Moreover, the approach also makes the application of rolling techniques easier and allows the production and sales departments of the company to quickly analyse the incorporation of new customers´ orders when planning. As observed in the computational experiments, for the models and the heuristic to find good production schedules (besides the standard processes), the input of new processes that take into account the mix of the demand in each period of the planning horizon is important. However, the complexity of the problem substantially increases as new processes are added to the input data, as shown in the experiments with 140 and 159 processes. Another factor that affects the problem complexity is the increase in the product demand, also illustrated in some experiments. An interesting perspective for future research is the application of column (process) generation techniques in the solution of these models, in a similar way to, for example the models of cutting and packing problems in the literature. However, the procedure of providing a feasible process for the electrofused grain plant involves certain modelling difficulties, due to the technical restrictions of some equipment from the production line. In this research subject, solutions of relaxed models can probably be useful to favour the generation of new processes, as they result in various processes in a single period. The combination of these processes in a single process can be considered to establish a more suitable new process for the period. Regarding the use of models with setup times, carrying out other investigations in the company is in our research agenda aiming at analysing the advantages and disadvantages of the solutions in depth with more than one setup per period (with regards to the solutions with one setup for a period at most), which could justify a change in the company policy in avoiding process interchanges within a workday.
Acknowledgements: The authors would like to thank the two anonymous referees for their useful comments and suggestions, and Alcoa-EMAS (nowadays a member of Treibacher Schleifmittel group in Brazil), in particular Luis A. Camilotti and Marcelo Suster, for supporting and taking part in this research. This research was partially supported by CNPq.
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