برنامه ریزی تولید منابع محدود در صنایع غذایی نیمه آماده

The resource-constrained production planning problem in semicontinuous multiproduct food industries is addressed. In particular, the case of yogurt production, a representative food process, in a real-life dairy facility is studied in detail. The problem in question is mainly focused on the packing stage, whereas timing and capacity constraints are imposed with respect to the batch stage to ensure the generation of feasible production plans. A novel mixed discrete/continuous-time mixed-integer linear programming model, based on the definition of families of products, is proposed. Timing and sequencing decisions are taken for product families rather than for products; thus, reducing significantly the model size. Additionally, material balances are realized for every particular product, permitting the detailed optimization of inventory and operating costs. Packing units operate in parallel and share resources. Qualitative as well as quantitative objectives are considered. Several industrial case studies, including also some unexpected events scenarios, have been solved to optimality.

The theme of production planning for the process industries has received significant attention in the past 20 years. Initially, from the early 1990s to the early 2000s, this was due to the resurgence in interest in flexible processing either as a means of ensuring responsiveness or adapting to the trends in process industries towards lower volume, higher value-added materials in the developed economies (Shah, 1998). More recently, the topic has received a new impetus as enterprises attempt to optimize their overall supply chains in response to competitive pressures or to take advantage of recent relaxations in restrictions on global trade. The production planning problem at a single site is usually concerned with meeting fairly specific production requirements. Customer orders, stock imperatives or higher-level supply chain or long-term planning would usually set these. It is concerned with the allocation over time of scarce resources between competing activities to meet these requirements in an efficient fashion. The key components of the resulting resource-constrained planning problem are resources, tasks and time. The resources need not be limited to processing equipment items, but may include material storage equipment, transportation equipment (intra- and inter-plant), operators, utilities (e.g., steam, electricity, and cooling water), auxiliary devices and so on. The tasks typically comprise processing operations (e.g., reaction, separation, blending, and packing) as well as other activities which change the nature of materials, and other resources such as transportation, quality control, cleaning, and changeovers. There are both external and internal elements to the time component. The external element arises out of the need to co-ordinate manufacturing and inventory with expected product liftings or demands, as well as scheduled raw material receipts and even service outages. The internal element relates to executing the tasks in an appropriate sequence and at right times, taking account of the external time events and resource availabilities. Overall, this arrangement of tasks over time and the assignment of appropriate resources to the tasks in a resource-constrained framework must be performed in an efficient fashion, which implies the optimization, as far as possible, of some objective. Typical objectives include the minimization of total cost or maximization of profit, maximization of customer satisfaction, minimization of deviation from target performance (Shah, 1998). Mathematical programming techniques, especially Mixed-Integer Linear Programming (MILP) because of its rigorousness, flexibility and extensive modeling capability, have become one of the most widely explored methods for process planning and scheduling problems (Floudas & Lin, 2005). The application of mathematical programming approaches implies the development of a mathematical model and an optimization algorithm. Most approaches aim to develop models that are of a standard form (from linear programming models for refinery planning to mixed-integer non-linear programming models for multipurpose batch plant scheduling). These may then be solved by standard software or specialized algorithms that take account of the problem structure. A critical feature of mathematical programming approaches is the representation of the time horizon. This is because activities interact through the use of resources and therefore the discontinuities in the overall resource utilization profiles must be tracked over time; to be compared with resource availabilities to ensure feasibility. The complexity arises because these discontinuities (unlike discontinuities in availabilities) are functions of any schedule proposed and are not known in advance. Excellent reviews on the optimal scheduling for the process industries can be found in Kallrath (2002) and Méndez, Cerdá, Grossmann, Harjunkoski, and Fahl (2006). The literature in the field of production scheduling and planning of food processing industries is rather poor. Entrup, Günther, Van Beek, Grunow, and Seiler (2005) presented three different MILP model formulations, which employ a combination of a discrete and a continuous time representation, for scheduling and planning problems in the packing stage of stirred yogurt production. They accounted for shelf life issues and fermentation capacity limitations. However, product changeover times and production costs were ignored. The latter makes the proposed models more appropriate to cope with planning rather than scheduling problems, where products changeovers details are crucial. The data set used to demonstrate the practical applicability of their models consisted of 30 products based on 11 recipes that could be processed on 4 packing lines. They reported near-optimal solutions within reasonable computational time for the case study solved. Marinelli, Nenni, and Sforza (2007) addressed the planning problem of 17 products in 5 parallel packing machines, which share resources, in a packing line producing yogurt. Their optimization goal was the minimization of inventory, production and machines setup cost. Sequence-dependent changeover costs and times were not considered. The authors presented a discrete mathematical planning model which failed to obtain the optimal solution of the real application in an acceptable computation time. Thus, they proposed a two-stage heuristic for obtaining near-optimal solutions for the problem under study. Recently, Kopanos, Puigjaner, and Georgiadis (2010) developed a mixed discrete/continuous-time MILP formulation for the simultaneous production scheduling and lot-sizing in yogurt production lines sharing common resources (e.g., fruit-mixers). Although the problem was mainly focused on the packing stage, timing and capacity constraints of the fermentation stage were also included in the model. The overall formulation takes into account sequence-dependent changeover times and costs, production overtimes as well as typical daily production shutdown and setup times due to hygienic requirements. Despite the significant literature body in the broad area of process planning and scheduling very few contributions model resource (e.g., labor) constraints in addition to the classical production constraints. Roughly speaking, resources could be mainly classified into non-renewable and renewable. Non-renewable resources do not recover their capacity after the completion of the tasks that consumed them. For instance, raw materials and intermediate products can be considered as non-renewable resources. On the other hand, renewable resources recover their capacity after the completion of the tasks that used them. Renewable resources like manpower are called discrete renewable resources, while resources such as utilities (e.g., electricity, vapor, and cooling water) are usually referred as continuous renewable resources. Pinto and Grossmann (1997) presented a MILP sequential approach based on a slot-based continuous-time representation that extended a former mathematical formulation for unconstrained multistage batch plants (Pinto & Grossmann, 1995). As the number of binary variables and big-M constraints substantially increased, the general MILP resource-constrained model became almost computationally unsolvable. Consequently, they developed a problem solution methodology that combined a branch-and-bound MILP algorithm with disjunctive programming. Slot-based representations were also presented by Lamba and Karimi (2002) and Lim and Karimi (2003) to tackle semicontinuous scheduling problems of single-stage parallel production lines with resource constraints. Lamba and Karimi, 2002 used identical slots across all processors while Lim and Karimi (2003) employed asynchronous slots. Since the underlying idea of an asynchronous slot is similar to the unit-specific time event, checkpoints for resource utilization were placed at the start of each slot. Additional variables and constraints should be included to establish the slot relative positions. Méndez and Cerdá (2002a) developed a precedence-based MILP model that independently handles unit allocation and task sequencing decisions through different sets of binary variables. Sequencing variables allowed to order the tasks allocated either to the same equipment unit or to another discrete resource. In this way, an important saving in binary variables was achieved. Afterwards, Méndez and Cerdá (2002b) reported a more general MILP formulation to deal with both continuous and discrete finite renewable resources. Each continuous resource was divided into a discrete number of subsources or pieces that were assigned to tasks through new allocation variables. Then, sequencing variables were still used to ordering tasks allocated to the same discrete or continuous resource item. The maximum number of pieces into which a continuous renewable can be divided was a model parameter, while each piece capacity was a non-negative variable selected by the model. However, the proposed resource representation may sometimes exclude the problem optimum from the feasible space and, consequently, optimality was not guaranteed. It should be pointed out that the models of Méndez and Cerdá (2002a) and Méndez and Cerdá (2002b) can potentially lead to overestimation of utility levels. Sundaramoorthy, Maravelias, and Prasad (2009) proposed a discrete-time MILP model for the simultaneous batching and scheduling in multiproduct multistage processes under utility (e.g., cooling water, steam, and electricity) constraints. Since different tasks often share the limited utilities at the same time, they used a common time-grid approach. Further, the proposed method handles the batching decisions (i.e., the number and sizes of batches) seamlessly without the usage of explicit batch-selection variables. Finally, they introduce a new class of inventory variables and constraints, in order to preserve batch identity in storage vessels. Marchetti and Cerdá (2009) presented a general precedence-based MILP framework to the short-term scheduling of multistage batch plants that accounted for sequence-dependent changeover times, intermediate due dates and limited availability of discrete and continuous renewable resources. Their formulation relied on a continuous-time formulation based on the general precedence notion that uses different sets of binary variables to handle allocation and sequencing decisions. To avoid resource overloading, additional constraints in terms of sequencing variables and a new set of 0–1 overlapping variables were presented. Finally, preordering rules can be added in their MILP model.