Ind. Eng. Chem. Res. 1997, 36, 2691-2700
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Planning and Scheduling of Parallel Semicontinuous Processes. 1. Production Planning Conor M. McDonald* and Iftekhar A. Karimi Advanced Process Control and Optimization, Central Research and Development, E. I. DuPont de Nemours, Experimental Station, Wilmington, Delaware 19880-0101
In this two-part series of papers, production planning and scheduling models are developed for the case of semicontinuous processes which are assumed to comprise several facilities in distinct geographical locations, each potentially containing multiple parallel units. The models developed are deterministic in nature and are formulated as mixed-integer linear programs (MIP’s). Part 1 deals with multiperiod midterm planning models where sourcing considerations are important, given that products can be manufactured at several facilities, often on different continents. Optimal allocation of assets to production tasks in order to satisfy the fluctuating demands of the global marketplace over an extended horizon is the main goal in this kind of model. Plan performance is assessed relative to an objective function involving maximization of earnings and minimization of production, inventory, and transportation costs. In these types of models, the actual timing and sequencing of production campaigns is not determined. For this purpose, part 2 discusses the application of two novel short-term scheduling formulations to a singlestage, multiproduct, multiprocessor facility. The goal is to minimize the production, inventory, and transition costs for a single facility. These are continuous time formulations in the sense that they allow production events to occur at any point over the scheduling horizon while retaining the ability to assess costs at discrete points in time, as in the classic multiperiod formulations. Thus, they overcome the restrictions imposed by the use of time boundaries in multiperiod models. For purposes of illustration, a limited number of examples are presented which are modifications of real industrial problems. 1. Introduction Given an increasingly competitive global marketplace, and the unwillingness to commit to large investments in new plants, large chemical companies have become more focused on improving the way in which they plan and operate their facilities. Manufacturing flexibility, customer responsiveness, lower operating costs, and reduced investments in inventory have become the battle cries of upper management. However, these goals are often in conflict. Given the complexity of large chemical operations, spreadsheet planning approaches quickly become inadequate, and more rigorous methods that determine how to optimally utilize resources are required. This work concentrates on the operation of large-scale continuous processors that operate in parallel. Multiple products are typically manufactured on these units so that semicontinuous operation is required. An obvious example is the manufacture of several grades of polymer on the same extruder. These processors can be collected in one facility, and there can be several such facilities which are globally dispersed. This environment provides a challenging framework for the planning and scheduling decision process. Roughly, planning can be considered as the task of optimally allocating resources to tasks over a relatively long time horizon, while scheduling implies making exact timing decisions in response to the marketplace over a much shorter time frame. This work concentrates on deterministic mathematical programming models to attack both these classes of problems. Production planning and scheduling models for continuous plants have received considerably more atten* Author to whom all correpondence should be addressed. E-mail address:
[email protected]. S0888-5885(96)00901-3 CCC: $14.00
tion in the field of operations research than in chemical engineering. Multiperiod models are commonly employed for planning purposes, where the goal is the optimal utilization of the manufacturing asset base in the face of fluctuating demand. Bitran and Hax (1984) present a practically oriented review of the use of mathematical programming in production planning. Linear and dynamic programming approaches to solving aggregated lot-sizing models are discussed. These multiperiod models are usually mixed-integer linear programs (MIP’s) and are therefore computationally intensive for the large-scale problems of practical interest. One approach to circumvent this drawback is to reformulate these problems through the employment of variable disaggregation. For example, the classic single product lot-sizing problem (an MIP) can be solved as an LP by a number of reformulations as described by Nemhauser and Wolsey (1988). Their common features are that they contain many more variables than the original formulations, but the linear programming relaxations are much tighter. One chemical engineering paper of direct relevance in this area is that of Sahinidis and Grossmann (1991b) who use a generic multiperiod model as a basis for investigating the impact of various variable disaggregation strategies to improve the performance of large-scale MIP’s. Another related paper is that of Sahinidis and Grossmann (1991c) who develop a multiperiod investment model for the allocation of resources and capacity additions in dedicated and flexible plants over a long time horizon. Norton and Grossmann (1994) extend their model by allowing alternate forms of flexibility and illustrate their approach with an industrial problem. In relation to scheduling problems, research has focused on the economic lot scheduling problem, which involves minimizing inventory and setup costs for a multiproduct facility (see Elmaghraby (1978) for a © 1997 American Chemical Society
2692 Ind. Eng. Chem. Res., Vol. 36, No. 7, 1997
review). Often, the assumptions invoked are too restrictive; these include consideration of infinite horizons and constant demand rates (see Carreno (1990) for example). Variations have been proposed as in the development for multiple processors (Carreno, 1990) or time varying lot sizes (Dobson, 1987). Geoffrion and Graves (1976) develop a model for parallel continuous lines, formulating it as a quadratic assignment problem. Short-term scheduling has received much less attention for semicontinuous operations in the chemical engineering literature, where most of the effort has been expended on the area of batch processing involving the manufacture of premium-value products in low volume. However, this excludes the large and important class of chemical semicontinuous operations which are under consideration in this work. One exception is the paper of Sahinidis and Grossmann (1991a) who formulate the cyclic scheduling problem for continuous parallel facilities as a mixed-integer nonlinear program (MINLP). Inventory, production, and transition costs as well as profit contribution constitute the objective function. A constant demand rate is assumed which means that the discretization of the time horizon can be avoided. The optimal cycle time is an output of the model. A key element of the work is the development of the concept of time slots which can be variable in length and within which exactly one product is made. This slot formulation allows variable length campaigns but cannot be used for the case of dynamic demand with due dates, where it is necessary to establish inventory levels at discrete points in time. Pinto and Grossmann (1994) extend the formulation of Sahinidis and Grossmann (1991a) to the multistage case. The same assumptions are made, but intermediate storage is allowed between stages which leads to nondifferentiabilities in the mass balance equations. Special methods of dealing with this occurrence are presented. Kondili et al. (1993a) present a general state-task MIP formulation for the scheduling of batch operations. The state-task framework was originally developed for batch processes but has been extended both conceptually and in range of application (Pantelides, 1993). A key feature is the uniform discretization of time (UDM), and the length of the smallest time slice has a major effect on the size and performance of the MIP. These formulations share many of the same characteristics as the multiperiod models but with much smaller time slices. Given the large number of time periods that are required for a typical industrial problem, the method is computationally intensive, limiting the opportunity for its application. Zentner et al. (1994) discuss and compare features of uniform discretization models (UDM) and nonuniform continuous models (NUCM). In the latter case, the only time discretization allowed is for defined fixed points where model parameter levels may change. They explore techniques for integrating features of these different modeling approaches for the various levels encountered in a generally stated planning and scheduling problem. Kondili et al. (1993b) develop a planning model for multiproduct continuous plants which allows production events to lie anywhere along the time horizon demarcated with a prespecified set of time boundaries. Formulations for the problem of generating optimal sequences given a matrix of sequence dependent setup costs have been developed, and they usually have some variant of the traveling salesman problem embedded in them. Pekny et al. (1990) present a parallel algorithm for scheduling in the face of sequence-dependent costs, but due dates are not considered. Miller et al. (1993)
describe an application of such techniques to an industrial problem where various heuristic approaches were tested in addition to the exact methods. Tandon et al. (1995) use simulated annealing to minimize a tardiness penalty function but do not provide any comparison to exact methods. The main focus of this work is to provide methods of determining optimal production plans and schedules for a fixed set of manufacturing assets which operate semicontinuously. The emphasis is therefore concentrated on obtaining more cost-effective ways of running existing facilities to satisfy marketplace demands. Methods used are divided into two general classes: (i) midterm planning models and (ii) short-term scheduling models. The midterm models typically consider horizons ranging from 1 to 3 years, while the short-term horizons are from 2 to 6 months, although these time frames may well vary. The midterm models focus on the total manufacturing asset base, given that products can be made in more than one facility and that there may exist complex interfacility material flows, often across continents. The customer base will typically be global. The goal is then to obtain the optimal sourcing structure, balancing the costs of production, inventory, and transportation. Multiperiod models are very common in this arena. The scheduling models incorporate very different goals. In this case, the driving force is often optimal operation of a single facility which can contain multiple processors. Goals are satisfaction of customer demand while meeting inventory targets and minimizing costs. In this case, many more real world constraints must be considered, such as sequencing and minimum run lengths. A conventional multiperiod model will not be able to satisfy these constraints, and a more sophisticated model is required. In part 1 of this series, a generic multiperiod model for midterm planning is presented. Some of the drawbacks in relation to a formulation such as this are described, in addition to methods that can be used to avoid such problems. Part 2 (following paper in this issue) presents two novel formulations for the shortterm scheduling problem. The fundamental contribution in this area has been to combine the features of a continuous time formulation (the so-called slot formulations) which escape the restrictions of time boundaries and the standard multiperiod formulations which have serious drawbacks when applied to shorter term scheduling problems. 2. Midterm Planning Models Many chemical ventures feature complex global supply chains: a large number of products are sold worldwide and are manufactured at multiple facilities spread across continents. Often, internally generated demand exists so that intermediate products manufactured at one or more facilities are used by the same venture at several other facilities, complicating the picture even further. Given the unpredictable nature of the demands placed on these manufacturing sites, creating optimal plans and schedules consistently is an extremely challenging tast. This paper develops models to aid the global supply chain organization to run its business more efficiently and cost-effectively. These models consider existing capacity only, and the potential to install green field capacity is not incorporated. However, expansion of capacity at an existing site is easily handled by the models. Planning ranges are typically as long as the length of time into the future that demand information of
Ind. Eng. Chem. Res., Vol. 36, No. 7, 1997 2693
Figure 1. Simplified supply chain.
sufficient quality is available. This is typically in the range of 1-3 years. Demand is assumed to be deterministic in nature and is given in terms of due dates which correspond to the end of planning periods such as months or quarters. The demands can therefore vary by product, but the due dates do not. In order to provide a buffer against the uncertainty associated with the demand, safety stock is kept. The amount of safety stock that is desired to be in inventory at the end of a planning time period is the target safety stock level, and this is assumed to be a supplied deterministic quantity. This amount is usually derived from statistical considerations and is dependent on the variance in demand and how often the product is made. The statement of the general problem for planning purposes is given as follows: Given is a fixed set of facilities located in distinct geographical locations, with each site containing one or more semicontinuous multiproduct single-stage processors, that provide products to a given number of external customers as well as supplying intermediates internally. A time horizon divided into a number of time periods is also given with demands due at the end of each time period. Then, the goal is to determine the sourcing or production plan over the full horizon which optimizes a specified performance function which can include some or all of the following criteria: maximizing net income; minimizing production, inventory and transportation costs, and costs due to missed orders. The outputs of such a model are therefore the allocation of assets over time to achieve the desired goals, along with inventory and shortfall levels and internetwork flows. The sequencing and exact timing of production events are not considered. 2.1. Notation. The following sets will be used throughout this paper. The set of products in the system is denoted by the index set I ≡ {i}. This set can be classified in three categories, (1) raw materials (denoted by the set IRM) (2) intermediate products (denoted as IIP) (3) finished products (denoted as IFP), so that I ) {IRM ∪ IIP ∪ IFP}. Note that an intermediate product can also belong to the set of finished products. The set of machines is denoted as J ≡ {j}, and the set of facilities where these processors are located is denoted as S ≡ {s}. The set of customers is denoted as C ≡ {c}, while the set of time periods is denoted as T ≡ {t}. 2.2. Development of a Planning Model. Before describing the model, the basic structure of the supply chain network is described. For the purposes of the development in this paper, two layers of the supply chain network will be considered: (1) manufacturing facilities and (2) customers. Commonly, a layer of distribution warehouses might also need to be considered, but in what follows it is assumed that the production facilities can feed the customer nodes directly, so that the warehouse layer is ignored and inventory is carried at the production facilities. An
Figure 2. Sample facility with multiple processors.
example is shown in Figure 1, where there are three facilities (s1, s2, s3) feeding four customer regions (c1...c4). Note that s1 and s3 manufacture both finished products and intermediates from raw materials. The intermediate products are shipped to s2 where they are converted into additional finished products which are subsequently sold to customers. The facilities in the supply chain can comprise varying degrees of complexity. A site may contain one singlestage processor (e.g. a large reactor) so that the facility s conforms exactly to this one unit. A sample topological representation of a multiprocessor facility (labeled s1) is shown in Figure 2. It is composed of four units or processors (j1...j4), arranged in two contiguous machine blocks, with raw material RM1 feeding the first block of three machines, and a second raw material RM2 feeding machine j4. There are many examples of these kinds of parallel semicontinuous operations, including extruders, packaging lines, blending operations, dryers, filters, spinning machines, and grinding mills. These single-stage units are capable of continuous operation, but they operate semicontinuously due to the multiproduct demand placed on them. This leads to the classic economic lot-sizing problem, where longer running times for products lead to high inventory and safety stock levels, while yielding higher capacity utilization and incurring lower transition costs. The optimal balancing of inventory and transition costs is the main goal in this case. In conclusion, there are two major goals in developing optimization models for a multifacility, multiprocessor manufacturing environment: (i) optimization of the total supply chain network from procurement through production to distribution; (ii) optimization of the operation of the individual units within the production nodes of the supply chain. In the experience of the authors, the class of problems just described is more common than the equivalent planning and scheduling problem for batch processes, even though this latter area has received far more attention in the chemical engineering literature. Variables. The first step is to define the variables required to model production, consumption of raw and intermediate materials, supply of finished products to customers, interfacility shipments, inventory levels,
2694 Ind. Eng. Chem. Res., Vol. 36, No. 7, 1997
shortage levels, and safety stock levels. These are now defined in index notation: Pijst: production amount of i ∈ I\IRM on processor j at facility s in time period t RLijst: corresponding run length of product i ∈ I\IRM on processor j at facility s in time period t Cist: consumption of raw material or intermediate i ∈ I\IFP at a facility s in time period t Iist: inventory level for i ∈ I\IRM at the end of time period t at location s Sisct: supply of finished product i ∈ IFP from facility s to customer c in time period t σiss′t: flow of intermediate product i ∈ IIP from facility s to s′ in time period t Iict : amount of shortage of finished product i ∈ IFP for customer c in time period t ∆ Iist : deviation below safety stock target for product i ∈ I at location s in time period t Cost Coefficients. The cost components of a global chemical business derive from production, carrying inventory, inability to fulfill orders, purchasing raw materials, and transporting materials (which may include import duties and other foreign transaction costs). The associated cost parameters are defined as follows: hist: inventory cost for holding a unit of product i in inventory at facility s for the duration of the time period t µic: revenue per unit of product i ∈ IFP sold to customer c pis: price of raw material i ∈ IRM at facility s ζis: penalty for dipping below safety target of product i at a site s νijs: variable cost to produce a unit of product i ∈ I\IRM on processor j at site s tss′/tsc: transportation cost to move a unit of product from site s to site s′ or customer c Note that these costs can be easily defined over the time periods t if such data are available. For example, the inventory cost coefficients could be increased for the time period corresponding to fiscal year-end, when increased emphasis is placed on reducing the inventory component of working capital. The inventory and shortage penalties are assessed at the end of a time period t and are calculated as simple accumulations. They are not obtained by an integration over the time period. This second approach, though more accurate, is not applied because it requires precise timing and sequencing information which is not available from the results of the midterm planning model. In addition, it leads to nonlinearities in the resulting model. It is also assumed that backorders can be satisfied so that if a shortage does occur, then the associated demand can be fulfilled in the future, although a penalty is assessed for these delayed deliveries. General Data. In addition to the cost data required, other parameters relating to manufacturing and customers are needed: Rijst: effective rate for product i on processor j at facility s in time period t (it includes adjustment to the rate relating to efficiency, utility, and/or yield as defined for a particular manufacturing environment) βi′is: the yield-adjusted amount of raw or intermediate i ∈ I\IFP that must be consumed to produce a unit of i′ ∈ I\IRM at facility s Hjst: amount of time available for production on processor j at s during time period t dict: demand for finished product i for customer c due at end of time period t
L Iist : safety stock target for product i at facility s in time period t Iis0: inventory of product i held in facility s at start of planning horizon 2.3. Midterm Planning Model. The generic (LP) midterm planning model, denoted as formulation M, is now presented. In the objective function, R represents the revenues arising from the sale of finished products, and C represents the total costs arising from production and distribution of these products.
objective function: max R - C or min C
∑ µicSisct
R)
i,s,c,t
C)
pisCist + ∑histIist + ∑ tscSisct + ∑ νijsPijst + ∑ i,s,t i,s,t i,s,c,t ζisI∆ist + ∑µicI∑ tss′σiss′t + ∑ ict i,s,s′,t i,s,t i,c,t
i,j,s,t
manufacturing constraints: ∀ i ∈ I\IRM
Pijst ) RijstRLijst
(1)
∑i RLijst e Hjst ∑
Cist )
∑j Pi′jst
∀ i ∈ I\IFP
βi′is
i′3βi′is*0
Cist )
(2)
σis′st ∀ i ∈ IIP ∑ s′
(3) (4)
supply chain constraints: Iist ) Iis(t-1) +
σiss′t ∑j Pijst - ∑ s′ ∑c Sisct
Iict g Iic(t-1) + dict -
∑s Sisct
∑ Sisct′ e t′et ∑dict′
∀ i ∈ I\IRM (5) ∀ i ∈ IFP
∀t∈T
(6) (7)
s,t′et
I∆ist g IList - Iist
(8)
lower bound constraints: ∆ Pijst, RLijst, Cist, Iist, Iict, σiss′t, Iist g 0
upper bound constraints: Pijst e RijstHjst; RLijst e Hjst; Sisct e
∑dict′
t′et
Iict e
∑dict′; t′et
I∆ist e IList
The objective function can be of different forms. The most common one would be the maximization of contribution, defined as revenues netted of costs. Another objective is the minimization of cost without regard to income. Note that the model must include the inventory and shortage penalties as a bare minimum; otherwise meaningless solutions will be obtained. Inventory penalties are assessed as simple linear functions (histIist) which is exact for a single product if it is produced for the duration of a time period and supply is withdrawn continuously over the time period to satisfy demand at
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the end of the time period. This is more accurate for longer time periods. Other linear approximations can be used such as assessing the cost on the basis of what is produced during a time period. This is done in part 2 (following paper in this issue), where the supply can be considered as being instantaneously withdrawn at the end of the time period to satisfy demand at that same point. Equation 1 simply relates the production amount to the run length through the rate. Equation 2 specifies that the total amount of time spent producing cannot exceed the time available for production. Scheduled downtime and an approximate estimate for downtime due to transitions is factored into Hjst. In a short-term scheduling model, this approximation would not be adequate. Equation 3 models the consumption of raw or intermediate materials using the bills of material. For raw materials, the consumption variable is simply the amount that must be purchased from an external supplier. It is assumed here that the raw materials are available on demand, although this could easily be modified to include lower and upper bound restrictions. For intermediate products consumed at a site s, supply must come from production at that same site, or it must be shipped from some other site s′. Equation 4 forces all material shipped to a facility s to be consumed there in the same time period. Thus, inventory is only held where a product is manufactured, eliminating redundant network material flows. Equation 5 represents the fundamental material flow constraint. Inventory held at the end of period t equals that held at the end of the previous time period (t - 1) plus production during t minus outflow of intermediates to other plants and shipments of finished products to customers. Finished products are obviously not shipped between facilities (unless they are also intermediate products). They are stored in inventory at the facility where they are manufactured, or they are supplied from that facility to a customer. Intermediate products can be shipped between facilities (and supplied to customers if they are also sold as finished product). Customer shortfalls are the cumulative differences between demand and supply. Equation 6 indicates that shortfalls in supply carry from one period to the next. Obviously, shortages will be zero when supply fulfills demand. Equation 7 allows supply from a current time period to fulfill orders from a previous time period, subject to the upper bound of cumulative total demand until that period. Equation 8 shows that if the inventory level exceeds the safety target, then the safety stock ∆ shortage defaults to zero due to the positivity of Iist ; otherwise it is equal to the deviation from the target L level, whose maximum level is Iist . Inventory Costs. At this point, it is instructive to examine the manner in which inventory costs are modeled. Consider a single product without regard for multiple time periods. The inventory carrying cost is h, the safety stock penalty is ζ, and the shortfall penalty is µ, with all costs positive and expressed per unit. Figure 3 plots the inventory level versus associated cost with the slopes of the following three regions marked: (I) I < 0; (II) 0 e I e IL; (III) I > IL. The safety stock target IL is the desired level of inventory which is held to buffer against unpredicted contingencies. Any buildup over this level is excess inventory; negative inventory implies shortages. Some conditions must hold if this cost function is to yield inventory levels that drive toward the correct optimum of the safety stock level: (1) (-µ) < (h - ζ) < h; (2) h < ζ. Condition 1 ensures
Figure 3. Inventory costs.
that a convex piecewise linear function is obtained as in Figure 3. Note that the slope in region I is negative so that the cost in region I increases with decreasing I. Condition 2 ensures that the slope in region II is also negative. so that the optimum value of inventory resides at the target safety stock level, IL. If 0 < h - ζ < h, then a convex function is also obtained, but with the optimum now driven toward zero rather than IL, an undesired result. In formulation M, note that the positive and negative inventory levels are modeled as separate variables (I and I- respectively). I- therefore takes µ as a positive coefficient. 2.4. Minimum Run Length Constraints. Formulation M is adequate for situations where a facility can run products for any length of time without incurring significant changeover costs. However, in the case of large chemical facilities, this assumption is unrealistic. For reasons of operating efficiency, some form of minimum run lengths, denoted as MRLijs, must be imposed. In this case, formulation M, which is an LP, cannot be used. In order to gain a better understanding of these restrictions, some general properties can be deduced by examining the relationship between minimum run lengths and lengths of the individual time periods. Define NCijst as an estimate of the number of campaigns of product i that can be run within a time period t at site s:
NCijst )
Hjst MRLijs
Depending on the value of NCijst different constraints need to be appended to formulation M. These can be divided into three general classes, as defined below. Case 1: NCijst . 1. If the minimum run length of a product is much less than the length of the time period, then one has only to ensure that campaigns are of the required length. These constraints are easily formulated by defining
{
1
Yijst )
if product i is manufactured on processor j at s in period t otherwise
0
and then writing
RLijst - HjstYijst e 0 RLijst - MRLijsYijst g 0
(9)
This ensures that RLijst ) 0 if Yijst ) 0 and that MRLijs e RLijst e Hjst if Yijst ) 1. It is also possible to introduce a fixed-charge component into the objective function with Yijst, if such data are available. While these binary
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constraints are easily generated, they are difficult to deal with, corresponding to a lot-sizing problem with the addition of the minimum run length constraint. Thus, they are a very challenging set of constraints, especially for large problems, featuring large (LP) relaxation gaps. Case 2: NCijst ≈ 1. In this case, including eqs 9 in the formulation will not be adequate. For example, if the length of the campaign is of the same order as the horizon time for a particular period, then, because all campaigns must occur within a time period, there may well be some slack time which will not be utilized. Consider campaigns that run for 80% of a time period on the average: the remaining 20% of capacity will have to remain unused in order to satisfy the minimum run length constraint of eqs 9. To circumvent this problem, run lengths are allowed to “spill over” into the following time period. First, a new binary variable needs to be defined:
{
1
YSijst )
0
if a campaign of i spills over from t to t + 1 on unit j at site s otherwise
Then, the following set of equations can be used to formulate the new run length constraints:
RLijst - HjstYijst e 0
(10)
RLijst - MRLijsYijst + MRLijs[YSijs(t-1) + YSijst] g 0 (11) [RLijst + RLijs(t+1)] - MRLijsYSijst g 0
∀ t < |t| (12)
campaigns can span multiple time periods and sequencing can be considered explicitly. Case 3: NCijst , 1. For this case, campaign lengths extend beyond the length of the time periods. Usually, this implies that the time periods are of a very short duration and a campaign can extend over multiple time periods. Therefore, this corresponds to the case of shortterm scheduling. The short-term scheduling models to be discussed in part 2 fully determine the sequencing while dissolving the restrictions imposed by the time period boundaries. 2.5. Additional Considerations. The previous section presented a basic formulation for a general class of midterm planning problems. There are a number of additional issues that arise in applying this basic formulation to an industrial problem, and these are now discussed. There are many additional issues and constraints that can be considered, but a representative sample of the more important ones has been selected. Supply Constraints. 1. Customer Qualification. Customers often place stringent demands on the consistency of supplied products, and therefore, even though multiple facilities can manufacture the same product, sourcing must take place from one site. For a different facility to become a supplier, a customer must be qualified to receive product from this new facility. One way of modeling this situation is by defining a binary variable as follows:
{
1
Yisc )
0
if product i is supplied from facility s to customer c otherwise
Then the constraint becomes
YSijst e Yijst
(13)
∑i YSijst e 1
(14)
∑t Sisct - ∑t dictYisc e 0
(15)
∑s Yisc e nic
YSijst + YSijs(t+1) e 1
∀ t < |t|
Equation 10 is required as in case 1. Equation 11 can relax the standard minimum run length constraint as given by eqs 9 for period t if YSijst ) 1 or if YSijs(t-1) ) 1, noting that a campaign can be split across two time periods where each individual run length is less than the minimum run length, but their sum is not. If YSijst ) 1, then eq 12 ensures that the combined run lengths in two consecutive time periods exceed the minimum run length. Equation 13 ensures that only if a campaign is running in the current period can it extend into the next one. Equation 14 shows that obviously only one campaign can spill over per time period t, although the possibility of none spilling over is permitted. Equation 15 forbids campaign spillover occurring in two consecutive time periods. Note that eqs 12 and 15 are not required for the last time period. The above formulation allows portions (which can be less than the minimum run length) of a single campaign to be in two adjacent periods. This overcomes a major limitation when minimum run lengths are not substantially less than the time period length in question. Therefore, if a product run does spill over to the next period, t + 1, then it clearly must be the last product to run in the current period t and the first to run in t + 1. The sequencing is decided for this particular product. However, no specification in relation to sequencing the other products is made, and this decision lies in the domain of short-term scheduling. Note that this approach will be generalized in part 2 so that portions of
where nic (often equal to 1) is the maximum number of facilities which can source product i to customer c. This constraint can be constructed for subsets of the time horizon, so that the sourcing is allowed to switch from one facility to another at various points in time. In some situations, there may well be cost information in relation to qualifying customers, and these fixed charge terms can also be included in the objective function. 2. Customer Supply Constraint. Another constraint is in relation to the minimum fraction of demand for a product i that a customer c should receive from facility s, defined as Fisc. Then,
Sisct g Fiscdict This type of constraint is typically used for situations where the demands of established customers are satisfied from the same facility as before, while allowing new demand to be satisfied from a different facility. 3. Internal Supply Constraint. Similar constraints can also be constructed for internal sourcing. Define ωissj as the fraction of the total internal shipments to sj of product i which must come from site s. Then,
σis′sjt ∑ s′
σissjt g ωissj
Ind. Eng. Chem. Res., Vol. 36, No. 7, 1997 2697
ωissj ) 1 is a common scenario in order to ensure that a facility can rely on a consistent feed of intermediate product. Time Lags. Given the geographical reach of the supply chain network, there will be significant time lags in transporting material between nodes, and these affect the shipment and supply variables. Assuming that all time periods are of equal length, Lsc and Lss′ are defined as the transportation lags that are integer multiples of the time period lengths. This adjustment neglects inaccuracies due to lags that are not integer multiples of the time periods, a reasonable assumption for midterm planning. Equations 4 and 6 are then recast as
Cist )
σis′s(t-L ∑ s′
Iict g Iic(t-1) + dict -
s′s)
∑s Sisc(t-L ) sc
Note that additional inventory penalties are also assessed in the objective function for the duration of the transportation lag. Product Families. Families of products are a common occurrence so that transition times and costs between products of the same family are negligible, while family transitions incur significant changeover costs. Adjustment of the formulation for this scenario is simple. F ≡ {f} represents the set of families, and Λif is defined as the cross-set indicating that product i is a member of family f. Then, only the minimum run length constraints require modification and are written for case 1 as
RLfjst - HjstYfjst e 0 RLfjst - MRLfjsYfjst g 0 RLfjst )
∑ RLijst
i∈Λif
Extension to case 2 is obvious. The fixed-charge term in the objective function is also modified to be expressed in terms of family campaigns. The manufacturing and mass flow constraints still need to be written over the set of products. 3. Examples Some representative examples are now described. These share characteristics similar to problems encountered in industry. The typical industrial problem will however be larger in size than those supplied here and will have many more constraints which can be considered extraneous for the purposes of this work. The number of facilities can be anywhere from two to twenty, and the number of products in the supply chain network could range from ten to several thousand. Usually, special techniques have to be invoked to solve these larger problems. This will generally involve decomposition of the problem into contiguous subgroups based on the problem data. For example, for multifacility problems with large amounts of interfacility flow, the formulation could be solved initially without regard to minimum run length constraints to decide the optimal sourcing structure as long as this does not lead to unrealizable schedules. This will often be the case as long as it is ensured that the manufacture of amounts considerably less than the minimum run length is forbidden.
Table 1. Family Information for Example 1 F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11
Rf (mass/time)
MRLf (time)
FCf
0.85 0.42 0.88 0.44 0.82 0.77 0.97 0.82 0.97 0.70 0.71
22 60 29 3 31 33 19 22 19 50 50
4.4 6.2 6.2 0.2 6.0 6.0 4.2 4.2 4.0 9.0 8.8
3.1. Example 1. This first example is for a pair of facilities producing a total of 34 products. Each facility contains a single processor so the set J is superfluous. The first facility s1 manufactures 23 products (labeled I1 through I23), and there are 11 product families (F1 through F11) whose rates, minimum run lengths, and fixed-charge costs are given in Table 1. The second facility s2 is dependent on the first and manufactures 11 products, labeled I24 through I34, with each product at s2 requiring 1 unit of the first product of each family of s1. This means that 1 unit of P1 is required to make 1 unit of P24, 1 unit of P4 to make 1 unit of P25, and so on. There are 12 periods (each of 1 month) in the planning horizon with demand due at the end of each period. The dependent facility s2 is assumed not to be capacity constrained. The demand for the 11 products at s2 is derived as 50% of the demand for the products they consume from the supplying facility s1. ζi ) 2hi is assumed in all examples. I0i ) 0, µi ) 5, hi ) 0.28, and νi ) 1 for facility s2. The target safety stock levels for facility s2 are assumed to equal the average monthly demand. Table 2 supplies the product to family relationship for s1 along with beginning inventory (I0i ), safety stock target level (ILi ), inventory cost (hi), variable production cost (νi), shortage cost (µi), the demand (d1...d12), and the uptime in each time period (Ht). Note that hi is the cost of carrying one unit of inventory over the full planning horizon, so that the cost to carry for one period is hit ) hi/12. Clearly, this example corresponds to case 1 for facility s1 with NCit . 1. There are no manufacturing constraints for facility s2. The problem was solved using OSL accessed via GAMS (Brooke et al., 1988). The family campaigns are shown in Table 3. The campaigns by product are given in Table 4 for family F1 and all the products in facility s2. The total cost of the plan is $13 454. The relaxed MIP had an objective function of $13 203, and the relative gap between the final integer solution and the lower bound was 1.6%. A limit of 100 cpu s was set. Facility s1 is capacity constrained, with no unused capacity. Shortages amounted to 14% of total demand at s1 and 18% at facility s2. Deviation from safety stock targets was significant with the percentage achievement of safety stock targets at 64% for s1 and 62% for facility s2. Clearly, strategies for the debottlenecking of facility s1 need to be implemented. Expansion of the existing facility can be accurately incorporated into the current model. However, for addition of new equipment involving major capital outlay, separate models would need to be developed; this kind of model is outside the scope of the current research. 3.2. Example 2. This example features a single facility with 7 parallel processors that manufacture 14 products. There are 4 time periods, the first two lasting 15 days each, and the second two lasting 30 days each. The minimum run length for all products on all machines is 10 days. The uptime data in hours are
2698 Ind. Eng. Chem. Res., Vol. 36, No. 7, 1997 Table 2. Product Information for Example 1 F
I
I0i
ILi
hi
νi
µi
d1
d2
d3
d4
d5
d6
d7
d8
d9
d10
d11
d12
F1
I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14 I15 I16 I17 I18 I19 I20 I21 I22 I23
110 16 35 25 55 0 100 30 65 65 8 0 42 0 100 6 85 92 18 6 110 0 15
15 12 1 1 8 3 40 14 4 16 1 4 3 1 19 5 41 27 5 49 24 5 4
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.22 0.22 0.24 0.24 0.24 0.24 0.24 0.22 0.24 0.24 0.22 0.22 0.26 0.26 0.25 0.25
0.88 0.88 0.88 0.88 0.88 0.88 0.88 0.78 0.78 0.86 0.86 0.86 0.86 0.84 0.80 0.84 0.84 0.80 0.80 0.92 0.92 0.90 0.90
4.4 4.4 4.4 4.4 4.4 4.4 4.4 3.9 3.9 4.3 4.3 4.3 4.3 4.2 4.0 4.2 4.2 4.0 4.0 4.6 4.6 4.5 4.5
14 10 1 0 7 2 32 14 4 15 1 3 3 0 18 5 48 25 3 58 26 5 5 320
15 11 1 1 8 2 33 15 3 16 0 2 3 0 19 4 50 30 5 54 24 4 6 320
17 11 1 0 9 2 35 16 4 15 1 4 3 0 18 5 48 25 6 52 23 4 2 280
14 11 1 0 8 1 34 14 2 15 1 3 3 0 18 4 48 24 3 50 23 4 2 320
14 12 0 0 8 3 45 14 4 14 0 2 3 0 18 3 48 31 5 50 22 4 1 320
14 15 1 0 8 2 40 14 3 15 1 4 3 0 17 4 35 25 4 48 24 4 3 320
18 10 1 0 11 2 41 14 4 15 0 3 2 1 17 5 36 25 5 48 23 4 4 320
15 12 1 0 5 3 45 15 3 15 1 4 3 0 22 5 35 26 4 45 24 4 4 130
10 12 2 1 7 2 42 10 4 16 0 2 3 0 19 5 35 31 4 42 23 4 4 320
16 11 1 0 7 2 43 14 4 15 1 4 3 0 19 5 35 29 4 44 22 4 2 150
14 12 1 0 7 2 42 14 4 15 0 3 3 0 21 4 35 24 5 44 25 4 4 320
14 12 1 0 7 2 41 14 4 15 1 5 3 0 20 4 33 29 4 42 23 4 3 290
F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 Ht
Table 3. Campaigns for Example 1 at Facility s1 RLft F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11
t1
t2
t3
t4
t5
48.2
t6
t7
85.9
29.0
127.8 31.8
74.5
t8
t9
t10
t11
t12
41.2
31.8
28.7
51.3
38.8
51.1
52.3
51.1
5
53.1
31.0
21.6 100.5 42.3
20.6 49.5 49.5
106.8
59.8
65.4
48.0
45.5 35.4 158.1 52.1
58.5 67.0 94.7
197.2
19.0 141.4
31.5 64.3 255.7
92.8 75.9
19.6 60.4 52.6 56.5
44.5 44.3 181.7
19.6 49.5
50.5
Table 4. Run Lengths from Example 1 for Family F1 and Facility s2 RLit I1 I2 I3 I24 I25 I26 I27 I28 I29 I30 I31 I32 I33 I34
t1
t2
t3
t4
t5
t6
2.4 45.9 15.0 1.0 3.0 15.0 16.0 4.0
8.0 1.0 1.0 8.0 8.0 1.0
6.0 27.0 54.0
2.0 1 27.0
t7
t8
t9
t10
t11
t12
27.1 14.1
17.6 14.1
28.2 0.4
24.7 26.6
24.7 14.1
8.0
5.0
8.0
7.0
7.0
3.0
1.0
1.0
1.0
1.0
7.6 2.0
8.4
16.0
8.0
5.0 12.0
2.0 15.0
56.5 29.4 9.0
6.0
1.0
1.0 14.0 7.0 2.0 1.0 5.0
8.0 2.0 16.0 26.0
15.0
1.0
1.0 1.0 2.0 27.4 49.0
Table 5. Uptime for Machines in Example 2
16.0 2.0 2.0 14.6
3.0 13.0
3.0 16.0 21.0
47.0
15.0
Table 6. Product Data for Example 2
Hjt
t1
t2
t3
t4
I
I0i
ILi
hi
µi
d1
d2
d3
d4
J1 J2 J3 J4 J5 J6 J7
360 360 360. 360 360 360 360
360 360 260 360 360 360 360
720 720 520 720 720 720 720
720 720 720 720 320 720 420
I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14
10700 1060 1080 1080 20030 1080 1080 1080 1080 1080 1080 2060 1080 1080
2000 1060 1080 1080 2000 1080 1080 1080 1080 1080 1080 2060 1080 1080
1.30 1.23 1.38 1.38 1.76 1.38 1.38 1.38 1.38 1.38 1.38 2.00 1.38 1.38
5.1 4.8 5.4 5.4 6.9 5.4 5.4 5.4 5.4 5.4 5.4 8.1 5.4 5.4
16400 1550 18500 3900 19750 10400 19725 7550 750 7850 6975 2900 7550 2500
16400 1550 18500 3900 19750 10400 19725 7550 750 7850 6975 2900 7550 2500
5600 3100 41400 8300 38900 18400 40600 14200 1300 16500 9900 6300 14200 5000
60000 3100 32600 7300 40100 23200 38300 16000 1700 14900 18000 5300 16000 5000
supplied in Table 5. This example conforms to case 2, where the minimum run length is of the same order of magnitude as the time periods. In the original problem, there were additional time periods of longer length but these are extraneous for purposes of illustration of this example. The problem data are supplied in Table 6, which contains the starting inventory level (I0i ), the safety stock target levels (ILi ), the holding cost hi, the
profit margin (µi), and the demand for the time periods (dit). Note that the holding cost is expressed in dollars per 1000 mass per day, so that hit varies for the time
Ind. Eng. Chem. Res., Vol. 36, No. 7, 1997 2699
387
function of $1 077 000, and the final relative gap between the integer solution and the lower bound was 1.8%. It is possible to obtain the optimal solution by subdividing the machines into their independent machine subgroups (this strategy will be discussed further in part 2). If this is done, then the optimal production plan obtained is given in Table 9. It is worthwhile to comment on the performance of the MIP solvers used in the examples. Firstly, the LP solver of CPLEX 4.0 is superior to that of OSL. However, performance for an individual MIP is very much dependent on the characteristics of a particular problem. For example 1, OSL outperformed CPLEX 4.0 with OSL consistently generating a much better solution in much lower computational time. The best MIP solution generated by CPLEX 4.0 was 25% from the lower bound, even after allowing it to run for 50 000 iterations. However, for example 2, CPLEX 4.0 performed better. This is confirmation of the anecdotal evidence that the performance of these two MIP solvers is problem dependent, and extensive testing on both of them should be performed. Ideally, CPLEX 4.0 is the solver of choice because of the speed of its LP solver which can lead to computational savings for MIP problems that take a large number of iterations. The examples were run on an IBM RS6000/3CT.
320
4. Conclusions
278 442 30 80 310
In this paper, a general multiperiod supply chain production planning model has been presented. The complications arising from the incorporation of minimum run length constraints were explored in order to illuminate the efficacy of these models for midterm planning problems. It was seen that the multiperiod model becomes inadequate when the time scale of a planning period is much less than the time scale of an individual production event. This supplies a natural stepping stone to part 2 of this series, where a continuous time formulation is developed for the short-term scheduling problem.
Table 7. Product Rate Data for Example 2 Rij
J1
I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I11 I14
J2
J3
J4
22 28
115
J5
J6
J7
12
12
124 100 150 25
99 50 43
102 132 12
12 50
23
94
Table 8. Production Plan for Example 2 (All Machines Included) total cost: J
I
t1
J1
I3 I6 I11 I4 I10 I1 I2 I7 I14 I5 I12 I5 I12 I8 I9 I13
298 62
J2 J3 J4 J5 J6 J7
28 240 360 332
$1 145 000
t2 ‚‚‚ ‚‚‚
‚‚‚
290 70 360
‚‚‚
120
‚‚‚
240
178 106 212 260 240 67 53 118 242 360
‚‚‚
t4 263
76 520
‚‚‚ ‚‚‚
298 62
t3 334 240 134
...
410 187 435 285 480 240 436
240 ‚‚‚
‚‚‚
284
164 720
‚‚‚
Table 9. Optimal Production Plan for Example 2 total cost: J
I
t1
J1
I3 I6 I11 I4 I10 I1 I2 I7 I14 I5 I5 I12 I8 I9 I13
298 62
J2 J3 J4 J5 J6 J7
28 240 360 305 360 290 70 120 240
$1 143 000
t2 ‚‚‚ ‚‚‚
‚‚‚
‚‚‚ ‚‚‚
178 106 212 260 27 93 240 360 118 242 298 62
‚‚‚
‚‚‚
‚‚‚ ‚‚‚
‚‚‚
t3
t4
334 21 134
263 219 77
76 520 213 410 720 195 525 436 284
‚‚‚
‚‚‚
‚‚‚
164 720
Literature Cited
387
‚‚‚
320 278 442 30 80 310
‚‚‚
periods. The rates for the products on the machines is given in Table 7. The results are supplied in Table 8. The dots between run lengths indicate the spreading of a campaign from one period to the next. For example, production of I6 on machine J1 runs for 62 h at the end of period t1 and continues into period t2 for another 178 h. The run lengths in both of these time periods are less than the minimum run length. Note that the solution supplied in Table 8 completely specifies the sequence for all machines simply because there is no more than one campaign that is not less than the minimum run length in each time period. For example, the schedule for J7 is derived as I13 f I8 f I13 f I8 f I13 f I9. The total cost of the plan is $1 145 000. CPLEX 4.0 was used to solve this problem, and a limit of 100 cpu s was set. The reported solution was obtained after 75 cpu s with 7800 nodes explored. The relaxed MIP had an objective
Bitran, G. R.; Hax, A. C. The role of mathematical programming in production planning. In Production Planning and Scheduling: Mathematical Programming Applications; Lawrence, K. D., Zanakis, S. H., Eds.; Industrial Engineering and Management Press: Atlanta, GA, 1984; p 21. Brooke, A.; Kendrick, D.; Meeraus, A. GAMS: A User’s Guide. Scientific Press: Palo Alto, CA, 1988. Carreno, J. J. Economic lot scheduling for multiple products on parallel identical processors. Mgmt. Sci. 1990, 36 (3), 348. Dobson, G. The economic lot scheduling problem: Achieving feasibility using time-varying lot sizes. Oper. Res. 1987, 35 (5), 764. Elmaghraby, S. E. The economic lot scheduling (ELSP): Review and extensions. Mgmt. Sci. 1978, 24 (6), 587. Geoffrion, A. M.; Graves, G. W. Scheduling parallel production lines with changeover costs: Practical application of a quadratic assignment/LP approach. Oper. Res. 1976, 24 (4), 595. Kondili, E.; Pantelides, C. C.; Sargent, R. W. H. A general algorithm for short-term scheduling of batch operations - I. MILP formulation. Comput. Chem. Eng. 1993a, 17 (2), 211. Kondili, E.; Shah, N.; Pantelides, C. C. Production planning for the rational use of energy in multiproduct continuous plants. Comput. Chem. Eng. 1993b, 17, S123. Miller, D. L.; Singh, H.; Rogers, K. A. A modular system for scheduling chemical plant production. In Proceedings of FOCAPO II; Rippin, D. W. T., Hale, J. C., Davis, J. F., Eds.; Cache Publications: Austin, TX, 1993; p 355. Nemhauser, G. L.; Wolsey, L. A. Integer and Combinatorial Optimization; John Wiley & Sons: New York, 1988. Norton, L. C.; Grossmann, I. E. Strategic planning model for complete process flexibility. Ind. Eng. Chem. Res. 1994, 33, 69.
2700 Ind. Eng. Chem. Res., Vol. 36, No. 7, 1997 Pantelides, C. C. Unified frameworks for optimal process planning and scheduling. In Proceedings of FOCAPO II; Rippin, D. W. T., Hale, J. C., Davis, J. F., Eds.; Cache Publications: Austin, TX, 1993; p 253. Pekny, J. F.; Miller, D. L.; McRae, G. J. An exact parallel algorithm for when production costs depend on consecutive states. Comput. Chem. Eng. 1990, 14 (9), 1009. Pinto, J. M.; Grossmann, I. E. Optimal cyclic scheduling of multistage continuous multiproduct plants. Comput. Chem. Eng. 1994, 18 (9), 797. Sahinidis, N. V.; Grossmann, I. E. MINLP model for cyclic multiproduct scheduling on continuous parallel lines. Comput. Chem. Eng. 1991a, 15 (2), 85. Sahinidis, N. V.; Grossmann, I. E. Reformulation of multiperiod MILP models for planning and scheduling of chemical processes. Comput. Chem. Eng. 1991b, 15 (4), 255. Sahinidis, N. V.; Grossmann, I. E. Multiperiod investment model for processing networks with dedicated and flexible plants. Ind. Eng. Chem. Res. 1991c, 30, 1165.
Tandon, M.; Cummings, P. T.; LeVan, M. D. Scheduling of multiple products on parallel units with tardiness penalties using simulated annealing. Comput. Chem. Eng. 1995, 19 (10), 1069. Zentner, M. G.; Pekny, J. F.; Reklaitis, G. V.; Gupta, J. Practical considerations in using model-based optimization for the scheduling and planning of batch/semicontinuous processes. J. Process Control 1994, 4 (4), 259.
Received for review December 23, 1996 Revised manuscript received April 21, 1997 Accepted April 22, 1997X IE960901+
X Abstract published in Advance ACS Abstracts, June 15, 1997.
