Lot-Sizing and Furnace Scheduling in Small Foundries

: A lot-sizing and scheduling problem prevalent in small market-driven foundries is studied. There are two related decision levels: (1) the furnace scheduling of metal alloy production, and (2) moulding machine planning which specifies the type and size of production lots. A mixed integer programming (MIP) formulation of the problem is proposed, but is impractical to solve in reasonable computing time for non-small instances. As a result, a faster relax-and-fix (RF) approach is developed that can also be used on a rolling horizon basis where only immediate-term schedules are implemented. As well as a MIP method to solve the basic RF approach, three variants of a local search method are also developed and tested using instances based on the literature. Finally, foundry-based tests with a real order book resulted in a very substantial reduction of delivery delays and finished inventory, better use of capacity, and much faster schedule definition compared to the foundry’s own practice.


Introduction
Foundries are common in every region of Brazil, producing many types of products, ranging from simple items for domestic use to sophisticated parts for the automobile and machine tool industries. According to the Brazilian Foundry Association [1], the sector is growing rapidly, having produced over 12,000 tonnes a day (October/2005) and directly employing about 60,000 people.
Foundries can be classified as either captive or market-driven. The former tend to be part of large companies, such as car manufacturers, that totally absorb the foundry production, composed of large quantities of a small number of parts with relatively stable demand. On the other hand, market-driven foundries are generally small or medium sized companies that nevertheless produce a huge range of items with vastly varying demand. While captive foundries use a small number of different metal alloys, market-driven foundries need to work with a wide variety due to the diversity of their clients. This paper focuses on the problem of planning and scheduling production in small market-driven foundries.
Sounderpandian et al. [13] and Gravel et al. [14] also studied to large foundries, but dealt specifically with the problem of job sequencing on machines. Most papers on this topic are concerned with production scheduling in large steel plants. By their nature, such plants do not have moulding sections since they produce steel sheets, of varying sizes and types, in rolling machines with cylinders that are configured according to the sheet specifications. Tang et al. [15] published a review of research on production planning and scheduling in steel production, while a variety of papers have studied the practical problems found in large steel plants, including [16][17][18][19][20][21][22][23].
In practice, the problem is tackled sequentially. The lot sizing of foundry products is carried out first, taking into account their demand and due dates. The production of alloys is then scheduled as a function of the lot sizes of the end products. However, this approach can result in a poor furnace schedule given the lack of two-way linkage between the lot sizing of end products and the scheduling of alloys. In this paper, we propose an optimization model where the two problems are solved in an integrated manner. The model put forward in this paper is closely related to the General Lot Sizing and Scheduling Problem (GLSP) and its extensions [8,[24][25][26][27][28]. The GLSP schedules multiple products on a single machine and allows many setups in each single 'large-bucket' time period. This paper adapts the GLSP to include backlogs and product-group setups, with an emphasis on rolling horizon use.
speed and solution quality for this kind of model as an important research area. Recent research by Gupta and Magnusson [29] into capacitated lot sizing with sequence-dependent setup costs states that it is still difficult to obtain near-optimal solutions for industrial-size problems. Dillenberger et al. [30] formulated a lot sequencing and sizing model with representation of sequence-independent setup times on multiple machines. The resulting mixed integer programming (MIP) model is difficult to solve optimally for large realistic problems, and so the authors resorted to the fix-andrelax method (Beraldi et al. [31]), more widely known as relax-and-fix (Wolsey [32], Kelly and Mann [33]). The current paper uses a similar basic approach. Dillenberger et al. [30] illustrated the viability and value of the relax-and-fix method, applying it to sizeable real problems from three IBM plants, and obtaining acceptable solutions in reasonable computing time, even on the slower machines of the 1990s. Section 2 describes the lot sequencing and sizing in the context of found while section 3 presents the results MIP model. An initial attempt to optimally solve this model using advanced optimisation software was not successful. To try to overcome this, section 4 presents rolling horizon solution strategy, but the software still failed to find good solutions within reasonable computational time. The relax-and-fix heuristic approach was then applied on a rolling horizon basis in section 5, giving good results. Relax-and-fix involves the solution of a period-by-period sequence of partially-relaxed MIPs, each one with just a reduced set of binary variables whose number is small enough to obtain good solutions. As the horizon rolls forward in time over the periods, each set of binary variables is permanently fixed at their solution values.
For comparison, three methods involving neighbourhood search on the reduced set of binary variables were developed, namely, descent heuristic, diminishing neighbourhood search and simulated annealing. Several authors have already explored a similar approach, among them Kuik et al. [34] who used simulated annealing and tabu search on a lot-sizing problem with sequenceindependent setups, Teghem et al. [35] who employed linear programming within a simulated annealing for a combinatorial production planning problem, and Fleischman and Meyr [26] who applied threshold accepting search on the general lot sizing problem (GLSP) with sequenceindependent setup costs (but not times), denoted GLSPST. Similarly, Meyr [27] developed an efficient algorithm for the GLSPST that uses multiple runs on a local search for lot sequencing with linear programming (LP) dual reoptimization for rapid lot-sizing.
Computational tests comparing all the methods are presented and analysed in section 6, including in-foundry comparisons with practiced schedules.

Problem Definition
A key process in a foundry is the transformation of ore and scrap metal into alloys with specified levels of carbon, silicon, zinc, etc, that determine properties such as brittleness and resistance to corrosion. The alloy, still in a liquid state, is then poured into moulds, normally made of sand and resin where it cools to produce final items. These two processes of alloy production and item moulding must be jointly scheduled. After the cast items have been cooled, they are deburred and made available for delivery.
In small foundries, only one furnace is usually operating at any point in time, so that just a single alloy can be produced in each time period. This is different from large foundries where several furnaces can be in operation simultaneously, enabling the production of several alloys in the same period [12]. Furthermore, in small foundries, the preparation of sand moulds is a manual process that is carried out as soon as the next day's production schedule is specified and is not a production bottleneck. Rather, the furnace is the production bottleneck. This contrasts with automated foundries, where multiple moulding machines of varying capacity and efficiency have to be scheduled. In this case, the bottleneck could be either mould preparation or the furnaces, depending on item orders and schedules. Figure 1 illustrates the main activities in a small market-driven foundry. Clients randomly and spontaneously submit orders specifying the item type, quantity and alloy. The Production Planning department negotiates due dates with the client, often agreeing unachievable dates that result in delivery delays and the possible loss of future orders from the client. Thus the minimisation of delays is one of the principal concerns of the foundry company.
The Production Planning department specifies which items should be produced during the next few days, advises the Moulding section which moulds to prepare in advance, and determines the specific alloys to be melted. The scheduling is guided by due dates in the following days as well as by delayed orders. Excessive changes of alloys are undesirable, and so setups are considered in the model.
The manufacturing system has the following characteristics and assumptions: • An alloy is generally used in several products, but a product is made from just one alloy.
• The output weight of an alloy is equal to the total gross weight of the products in which the alloy is used.
• A product cannot be manufactured in a given time period unless the alloy which it is made from is also processed in that period. Processed alloys cannot be held over to the next period. • In each time period only one alloy can be processed on the furnace.
• A setup changeover from one alloy to another consumes capacity time in a manner that is independent of the sequence in which the alloys are processed.
• All products have a demand over a planning horizon that would be met if capacity were sufficient.
However, delays will occur if capacity is tight, and so backlogs must be represented in the model. • The objective is to schedule lot sizes and to sequence setups in order to minimize a penaltyweighted sum of product backlogs, finished inventories and setup changeovers.

Mathematical Model
We now propose a mixed integer programme (MIP) to model the problem described above.
The following notation is used: Furthermore, consider the following definitions from the General Lot-Sizing and Scheduling Problem (GLSP) model [8,[24][25][26][27][28]: First sub-period in period t (F 1 = 1) The total number of sub-periods over the planning horizon We can now formulate the following MIP model: subject to: The first part of the objective function (1)  h and s k to explore alternative production schedules. Such exploration does not generally give the efficient surface [36], but can provide enough scenarios to guide the decion maker.
Constraints (2) balance inventories, delays, demands and production of items for every item in each period. Constraints (3) not only keep production within the furnace capacity, but also ensure that only items of the same alloy are produced in a particular furnace loading. As k n y and k n1 y − are 8 both binary variables, constraints (4) and the objective (1) force the continuous variables k n z to be equal 1 if there is a changeover to alloy k or to equal 0 otherwise. Along with constraints (7), the k n z variables assume just 0 or 1 values, even if the x in and k n y variables are not integer optimal (for example, at nodes during a branch-and-bound search). Constraints (5) and (6) ensure that there is only a single furnace loading in each sub-period.
In captive foundries, lot sizes tend to be large since there are just a few types of items (generally components of standard products) which have a large stable demand, so that the integrality of x in can usually be relaxed. However, in contrast, the integrality condition is necessary for small market-driven foundries with their many small orders. This feature is not taken into account in most lot sizing models [8].
Constraints (9)  Model (1)-(9) shares some similarities with the GLSP model [8,[24][25][26][27][28]. Sequencing decisions are implicitly determined by the furnace setup variables k n y , but differently from the GLSP: • a sub-period covers the set of products of a given alloy rather than a single product; • the sub-periods in our model represent the time it takes to process a furnace load, and so have predetermined lengths, whereas in the GLSP the duration of a particular small time bucket is a decision outcome; • the number of sub-periods per period is fixed, being equal to the number of furnace loads that can be processed per period, although it could be any predetermined number, as in the GLSP.
Thus the GLSP's small and large bucket concepts are both present in the model (1)-(9), since during each period only one type of alloy (an intermediate item which must be used in the period) can be produced together with multiple ordered items. This type of lot sizing and scheduling problems is found in many other applications such as the production of soft drinks or tomato sauce [37].
Depending on the number of items and periods, lot-sizing MIP models are often very large in practice so that even advanced solvers such as Cplex 7.1 [38] are unable to identify probablyoptimal solutions in acceptable computational time. Trying to solve the model (1)-(9) with realistic data using MIP solver Cplex 7.1 on a Pentium III 500 MHz with 512 MB of RAM, the default branch-and-cut (B&C) search ran out of memory, achieving only poor solutions.
However, it is generally not worthwhile to invest a lot of computing time in the search for an exact optimal solution, given that input data are often imprecise in small foundries and in manufacturing in general. A more useful outcome is a quickly-obtained solution of good quality.
Delay penalties are usually subjective estimates and the order book is changeable, being updated daily, so that a theoretically optimal solution to model (1)-(9) will almost surely not be the best in practice and should be used as a guide rather than a command. As a result, the exact model may be relaxed to an easier one that includes integer variables only for the first immediate periods (where decisions scarcely change), after which the order book is updated, and the model applied to the next immediate periods, and so on. Such a rolling horizon strategy of approaching the problem is widely used in practice. The use of a rolling horizon is not only a useful practical approach (since it takes account of daily changes to the order book), but it works as a very good heuristic strategy to solve a problem even if the order book is not changed, as we will see in the computational experiments where fixed parameters are used.
Several authors have pursued this approach. Clark [39] and Stadtler [40] showed that this flexible approach can handle large multi-level MRP-type problems over long planning horizons with sequence-independent (Stadtler) and sequence-dependent (Clark) setup times. Suerie and Stadtler [41] used the same approach tested on smaller problems with a tight reformulation and valid inequalities providing very good fast solutions. The relax-and-fix method as implemented in the current paper fits well into rolling horizon usage, as will be shown below.

Rolling Horizon Model
To illustrate how the Rolling Horizon strategy works, suppose each period t is a workday, as in the foundry that motivated this study. Consider a planning horizon of T = 5 workdays of which only the first day (t = 1) will be scheduled in detail. This is achieved by dividing the first day into L =η 1 =10 sub-periods, as up to L furnace loadings can be processed each day. The remaining days t = 2,....,5 have just one sub-period each (η 2 = η 3 = η 4 = η 5 = 1). Thus F 1 = 1; L 1 = 10; F 2 = L 2 = 11; i.e., there are η = 14 sub-periods n (as illustrated in Figure   2). The variables k n y for the larger sub-periods n = F 2 ,...,F 5 are then redefined as "the number of loadings using material k produced in sub-period n".
Only the scheduled decisions relative to the η 1 = 10 sub-periods of day 1 are actually implemented. The decisions for the remaining 4 days are used only to evaluate the impact of future available capacity, i.e., to identify a provisional production plan in order to have advance warning of possible production backlogs and be able to act accordingly. Under standard rolling horizon practice, the model is reapplied one period later covering periods t = 2, …, T+1 with updated demand data over the rolled-forward T-period horizon, then over periods t = 3, …, T+2, and so on, using fresh demand forecasts [42].
To reduce problem complexity and solution time, the integer x in variables are relaxed for sub-periods n = F 2 ,...,F T , given that these variables' decisions are never in fact implemented. The k n y variables for sub-periods n = F 2 ,...,L T could also have been relaxed, but initial computational experiments indicated that they should remain integer in order to improve future capacity evaluation.
These modifications result in the following model for rolling horizon use, denominated RH: Model RH: subject to: Constraints (3) are now replaced by (12) for the first period and (13) for the remaining periods.
concepts for rolling horizon use. The large-bucket time period used for scheduling a whole day, for example, is split into several small-bucket scheduling time periods (for instance, 10 loadings of materials). Days 2 to 5 are, temporarily, indivisible large buckets with production of multiple materials.

Solution Methods
Model RH is much smaller than the model (1)-(9), but still not small enough to be solved optimally with realistic data using the MIP solver Cplex 7.1 within acceptable computing time.
However, it is possible to sub-optimally solve the model using the relax-and-fix method. This The problem in step 1 is solved using one of the four methods (RF, DH, DN and SA) described in sections 5.1 to 5.4 below. The problem in step 2 can be optimally solved in a few seconds with the Cplex MIP solver, since a binary variable k n y which is fixed to 1 implies, by constraints (12) and (15), that x in = 0 for all i∉S(k), i.e., products that do not use material k are not manufactured in subperiod n, thus eliminating many integer variables and constraints. Consequently, the solution methods developed in the rest of this paper focus on step 1.

Basic Relax-and-Fix Method (RF):
The basic approach to solving step 1 of the relax-and-fix method simply uses the incumbent solution that results from running the Cplex MIP solver for 3, 6 and 12 minutes respectively for small, medium and large problems (as defined in Table 1 of section 4.1). This method is denoted RF.

Descent Heuristic (DH)
The basic RF method is dependent on a MIP solver for both MIP problems in the two steps that arise in the relax and fix procedure above. The first problem is to solve (and then fix) just the first day's binary variables. The second problem, in step 2, is to try to find an optimal solution for variables k n y (n = F 2 , ..., L T ) and x in (n = F 1 , ..., L 1 ).
To solve the first MIP, a local search descent heuristic (DH) [43][44][45] can be used to Three alternative ways of obtaining a starting solution for the descent heuristic were at first considered: 1. For n = 1,...,η 1 , choose the value of v n to be k with the probability given by |S(k)| / N where |S(k)| is the size (cardinality) of the set S(k). Thus, the more products that can be made from k, the more likely it is that k will be selected; 2. Run a MIP solver for a few minutes to obtain an initial heuristic solution; 3. For n = 1,...,η 1 , choose the value of v n to be k, uniformly sampled from {1, ..., K}.
However, after initial tests the first way was selected and the other two were discarded.
The search stops after 1000 iterations, which was found to be sufficient to obtain a good solution in acceptable computing time. This number of iterations was fixed as the stopping criterion in order to fairly compare the three types of heuristic.
In order to determine neighbouring solutions it is necessary to specify how the vector v is changed, which defines decisions to the first η 1 sub-periods. Just one sub-period has its value changed (in the Diminishing Neighbourhood Heuristic more than one can be changed in the beginnig, but ends with just one changing). The neighbourhood move implemented slightly biases the selection towards more widely-used alloys, and was adopted after initial testing showed its positive impact. Two alternative procedures were used to randomly choose n*, the sub-period to change the material k: 1. With 90% probability: The value of n* is uniformly sampled from the set {1,...,η 1 }.
2. With 10% probability: Let k = v n be the material currently produced in a given sub-period n. We want that the more products S(k) made from material k, the less the chance of selecting sub-period n. So, sample the value of k with probability (N-|S(k)|) / (N(K*-1)) where K* is the number of different materials to be produced in period 1. The value of n* is then uniformly sampled from those sub-periods in which material k is produced.
Once n* is chosen, a new key material k is selected in one of two ways: Researchers have proposed many ways to improve the descent heuristics performance, including [43]- [51]. In this paper, we use two strategies: Diminishing Neighbourhood search method and Simulated Annealing, with the same basic parameters as the local search described above.

Diminishing Neighbourhood (DN) Search
This method adapts the local search described in the previous section, beginning with a large neighbourhood to encourage diversity and then gradually diminishing its size so as to increasingly

Simulated Annealing (SA)
Simulated Annealing is a variant of the local search descent heuristic that tries to avoid getting trapped at a local optimum by permitting worsening moves away with probability: where Temp is a gradually-cooling "temperature" and ?ofv the amount by which the new move worsens the objective function value. As the search progresses, the best solution encountered is recorded.
Previous computational tests indicated that the following parameters produce, in general, the best results. The starting temperature Temp start is a function of the initial solution ( [48] where µ = 0.6 and θ = 0.9 indicate that a solution which is 60% worse than the current one has 90% probability of acceptance at the start of the search. 50 iterations were allowed in order to reach equilibrium at a given temperature before cooling but only 10 iterations after a solution was accepted, even when worse. Each time the temperature was cooled in this way, it was reduced by 5%. In addition, each time a worse solution was accepted, the temperature was again cooled (slightly) as follows:  (24) where ofv(S) is the objective function value for the previous solution. The worse the accepted solution, the greater the reduction in temperature, thus making the acceptance of future worse solutions less likely from then on.

Computational Experiments
The computational experiments were divided into two parts. In order to evaluate the method in different situations, the first part used randomly generated data based on modified intervals based on [25]. The second part made use of an order book from a real-world foundry, so that the method's outcome could be compared with schedules used in practice.
Before describing the data generation, consider following parameter definitions: A value α i = 0 means that the item i due date is day 1, and α i < 0 means that item i is not delayed when the planning begins. Suppose that an item i is already delayed by α i > 0 at the beginning of the planning. Then, at the end of period t, t = 1,…,T the item's delay will be α i +t. If orders for an item have different due dates, then this item will be considered as two distinct items in order to have two different delay penalty values. Although this could considerably increase the model size, in practice it will do so only a little at most, since in small market-driven foundries only a few items are doubled ordered within the one-week planning horizon. Note that this doubling of items will not create false setups (as it would in classical lot sizing models with item setups) since the doubled items belong to the same set S(k), as in constraints (12).
It might be tempting to explicitly prohibit delays in the model, but as these are frequently alerted and, if necessary, the scheduler could increase the value of parameter β i for that item or renegotiate the item's due-date.

Generation of Test Data
Previous experience [42] indicates that certain parameters may affect solution quality, namely: • problem size (N,K) • size of setup penalty st k

• tightness of capacity Cap
Larger problems, bigger setup penalties, and tighter capacity are each expected a priori to adversely affect solution quality and computing time, but may do so in different degrees for each solution method. The uniform sampling intervals used to randomly generate the test data were based on those in [25] and are shown in Table 1. Though not encompassing every possible situation, the values are sufficiently typical to be confident that the test results will point to generally applicable conclusions.
The furnace capacity was generated as follows: first calculate the resources needed to exactly produce the total item demand over the planning horizon (in this case 5 periods, i.e., 50 furnace loadings); then add the total setup time needed if the furnaces were setup just once for each alloy; finally divide by the number of furnace loads, i.e., 50. Hence: Thus in Table 1  The parameters (N, K), s k and Cap were varied in a 3-factor experimental design. Each factorial combination was generated 10 times, using a different random seed each time, resulting in a total of 3×2×4×10 = 240 instances.

Solution Quality
In this section, we first analyse the quality of the solutions obtained by the relax-and-fix (RF) method described in section 5.1. We then compare the results from the three neighbourhood search approaches and the basic RF method. Finally we consider computing times.

Evaluation of the RF method
In order to evaluate the performance of the RF method, we used the solutions and lower bounds obtained by the Cplex 7.1 solver applied to model (1)-(9) in a general purpose branch-and-cut search.
Note that a solution to model RH is not a solution to model (1)-(9), as just the first day's loadings are scheduled and actually implemented, whereas the other days are planned only approximately. However, the application T times of model RH, starting consecutively at periods 1, 2, …, 5, with an always-shortening horizon (T = 5,4,3,2,1), will provide a feasible solution to model (1)-(9), enabling a comparison of results, similar to the internally rolling schedule in Stadtler [40].
The final value of the objective function is gradually accumulated over the T applications of model RH, each of which contributes its period 1 part: of expression (10), i.e., excluding the part for periods 2 onwards: ∑ ∑ = = Table 2 shows the variation of the mean objective value compared to the lower bounds supplied by Cplex after running for one hour. The variation is calculated as: where the method is either Cplex or RF.
As noted in section 3, attempts at solving model (1)- (9) to simultaneously schedule all furnace loadings over the whole 5-day horizon resulted in mediocre solutions. The Cplex incumbent solution, after 1 hour, was on average 22.26% worst than the lower bound. Even if more time was allowed (10 hours for some instances) the Cplex solutions were still poor. On the other hand, running the RF method for 3, 6, and 12 minutes for the small, medium and large problems respectively resulted in better solutions for all three cases that were on average 8.6% worse than the Cplex lower bound. Irrespective of the method, Table 2 shows that, while this gap tends to grow with problem size, there is no clear relationship with tightness of capacity or size of setup penalty.

Evaluation of the DH, DN and SA methods
After evaluating the quality of the RF method, we now compare the three neighbourhood search methods (DH, DN and SA) and the basic RF method using model RH.
The basic RF method of section 5.1 used the MIP incumbent solution found within the time limits. Table 3 shows the percentage of instances for which this method found the MIP optimal solution. Observe that for small problems (N = 10, K = 2) the method easily found optimal solutions to all of the MIPs, but the percentage of optimal solutions found reduced as problem size increased. Again, there is no clear relationship with tightness of capacity or size of setup penalty.
Overall the three heuristics performed nearly as well as the basic RF method, the best being SA, followed by DN. The DH heuristic converged rapidly to a local optimum while the DN and SA ones took many more iterations to achieve their best solution.
Note in Table 4 that the performance of the DH, DN and SA heuristic relative to the basic RF method improved as problem size increased. For small and medium problems, the DH method had the worst mean performance (4.36% and 3.38%, respectively), followed by DN (1.93% and 2.72%),   variables had been previously fixed, leaving only the non-zero integer x in values to be optimised.

Evaluation in a Small Foundry
The methods were also tested on real world instances at a small foundry that used a 5-day In other words, the DH schedule reduced item-day delays by 85% and kg-day delays by 40%, a very substantial gain in efficiency.
The DH schedule used less capacity (93.1% usage) than the foundry's manual schedule (98.7%) and produced a wider variety of types (34 alloys in opposition to 28). Note from Tables 5 and 6 that the 5-day totals of the DH backlogs and stocks are substantially lower, i.e., respectively 27% and 90% by weight, and 15% and 71% by number of items. This reflects the foundry's concern to maximize utilization of capacity, a policy which tends to prioritize of larger lots, resulting in the repeated postponement or early production of orders, thus creating larger backlogs and inventories as can be seen in Tables 5 and 6.
Excluding time for input data, the DH method generated a schedule in ten minutes, compared to the two days (16 working hours) of elapsed time that it took to specify the manual schedule.
This improved scheduling permits better negotiation of delivery dates (with simulation, for example), reducing promises of impossible deadlines. It also avoids an additional problem that frequently occurs in the foundry (and which the proposed method easily resolves), namely the overutilization of the furnaces in an attempt to reduce delays. Such overloading is physically possible but unadvisable as smelting takes longer, the furnace internal coating deteriorates sooner, and more energy is consumed, thus increasing production costs.
A further advantage for the foundry is that, when the order book is updated daily, rescheduling of the following days is easily carried out, allowing the inclusion of new orders.
In this paper a mixed integer linear programme (MIP) was proposed to model the production planning and scheduling in small foundries. The model integrated lot sizing and scheduling on a single capacitated machine (furnace) in a production environment where key alloys are first produced and subsequently transformed into a number of ordered items made from just one type of alloy. The foundry's main concern was to minimize delays. The inventory of alloys was forbidden, and at any time only one type of alloy could be produced. The setups for changing alloys were sequence-independent. There were neither setup times nor costs between the production of final products. Lot sizes of final items were assumed to be integers and backorders were allowed.
It was not possible to optimally solve the overall model within viable computing time, even using an advanced heavy-duty MIP solver. In order to efficiently but approximately solve the model, a rolling horizon approach and associated relax-and-fix procedure was developed. Four solution methods were proposed using a basic relax-and-fix (RF) approach and three variants of neighbourhood search. These four methods were tested with 240 generated instances based on Haase and Kimms (2000), and showed that the relax-and-fix approach provides a good compromise between speed and quality of solution, that local search is faster with slightly worse solutions, and that, for the data used, simulated annealing generally resulted in better solutions than the other two local search variants.
Tests were also carried out with a real-world instance from a foundry. The results showed that not only do the methods help small foundries to considerably reduce delays, but also that the improved schedules are generated in a very small fraction of the time of those created manually in the foundry. Ongoing efforts are continuing with the foundry to obtain better quality data with the aim of making a more precise comparison between the schedules currently used in practice and those output by the proposed method. In parallel, other small foundries in the region are being sought for further case comparisons. To facilitate such collaboration, a tool with a visual interface is being developed to output schedules and alloy mixes, within a wider objective of providing production planning and scheduling software for small and medium-sized foundries.
* Acknowledgements: The authors would like to thank the referees for their valuable reviews and helpful suggestions, and the National Research Council (CNPq), Brazil, for financial support.