Optimal Production Scheduling and Lot-Sizing in ... - ACS Publications

Ind. Eng. Chem. Res. 2010, 49, 701–718

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Optimal Production Scheduling and Lot-Sizing in Dairy Plants: The Yogurt Production Line Georgios M. Kopanos,† Luis Puigjaner,† and Michael C. Georgiadis*,‡ Department of Chemical Engineering, UniVersitat Polite`cnica de Catalunya, ETSEIB, AV. Diagonal 647, 08028 Barcelona, Spain, and Department of Engineering Informatics & Telecommunications, UniVersity of Western Macedonia, Karamanli & Lygeris, 50100 Kozani, Greece

The lot-sizing and production scheduling problem in a multiproduct yogurt production line of a real-life dairy plant is addressed in this work. A new mixed discrete/continuous-time mixed-integer linear programming model, based on the definition of families of products, is proposed. The problem under question is mainly focused on the packaging stage, whereas timing and capacity constraints are imposed with respect to the pasteurization/homogenization and fermentation stage. Packaging units operate in parallel and share common resources. Sequence-dependent times and costs are explicitly taken into account and optimized by the proposed framework. Several scenarios for a large-scale dairy plant have been solved to optimality using the proposed model. Production bottlenecks are revealed, and several retrofit design options are proposed to enhance the production capacity and flexibility of the plant. 1. Introduction Most work undertaken in the area of chemical production scheduling relies on mathematical programming and generic process representations such as the state task network (STN)1 or the resource task network (RTN).2 Both representations have proven to be invaluable in describing multiproduct, multipurpose production facilities of arbitrary structure. These processes do not fall under the usual flow-shop or job-shop operations research, because the number of jobs (tasks) to be executed is not known a priori, processing times can depend on job size, and jobs are linked to each other through material balance constraints and intermediate storage requirements. STN- and RTN-based scheduling formulations can be roughly classified into discrete-time and continuous-time models. A common feature of discrete-time models is the discretization of the time horizon into equal-length intervals, the base time of which is chosen according to the smallest processing time of the particular problem instance at hand. Depending on the time horizon of interest and the resolution of the time grid employed, this approach can become computationally prohibitive for most realistic manufacturing processes. Continuous-time formulations can in principle be used to alleviate some of the computational problems incurred by discrete-time-indexed formulations, and relevant models are also available in the process systems engineering (PSE) literature. However, their computational performance is usually affected by large integrality gaps, commonly defined as the relative difference between the best possible integer solution and the best integer solution found. A plethora of contributions addressing production scheduling problems can be found in the literature of the operational research and PSE communities. Excellent recent reviews covering the short-term batch and continuous process scheduling can be found in Meńdez et al.3 and Floudas and Lin.4 However, the use of optimization-based techniques for scheduling dairy plants is still in its infancy. This fact can mainly be attributed to the complex production recipes, the large number of products * To whom correspondence should be addressed. E-mail: mgeorg@ otenet.gr. † Universitat Polite`cnica de Catalunya. ‡ University of Western Macedonia.

to be produced under tight operating and quality constraints, and the use of mixed-batch and semicontinuous production modes. Entrup et al.5 presented three different mixed-integer linear programming (MILP) model formulations that employ a combination of discrete- and continuous-time representations, for scheduling and planning problems in the packaging stage of stirred yogurt production. The authors 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 details of product changeovers 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 four packaging lines. Authors reported near-optimal solutions within a reasonable computational time for the case study solved. Marinelli et al.6 addressed the planning problem of 17 products in 5 parallel packaging machines that share resources in a packaging line producing yogurt. Their optimization goal was the minimization of inventory, production, and machine setup costs. Sequence-dependent costs and times were not considered. The authors presented a discrete mathematical planning model that 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. Doganis and Sarimveis7 studied the scheduling problem at a yogurt packaging line of a dairy company in Greece. Their objective was to optimally schedule two (or three) parallel conjoined (coupled) packaging machines over a 5-day production horizon in order to meet the weekly demand for 25 different products. Each of the identical machines could produce any of the 25 products. Product changeover times and costs were considered, and total demand satisfaction was imposed. Simultaneous packaging of multiple products was not allowed because the parallel machines shared the same feeding line. Both the latter restriction and the limited number of products considered greatly simplified the problem under question. The apparent reduction of changeover times was transformed into additional

10.1021/ie901013k  2010 American Chemical Society Published on Web 12/04/2009

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Ind. Eng. Chem. Res., Vol. 49, No. 2, 2010

machine idle time. Finally, potential limitations of the fermentation stage were completely ignored. Yogurt production could be considered as a particular case of a batch or semicontinuous production process. The PSE community has addressed and studied these types of production processes during the past 20 years. One of the main features of batch processes is that large numbers of products are produced from a few initial product recipes. The same description holds for yogurt production. Thus, final yogurt products may differ in at least one of the following features: (i) fermentation recipe type origin, (ii) total cup weight, (iii) number of cups per piece, (iv) labeling depending on customer destination, (v) flavors, and (vi) packaging cup type (material, shape, etc.). Packing rates can vary significantly from one product to another. The short shelf life of yogurt does not favor a “make-tostock” production policy, as product inventory has a finite storage time. Therefore, yogurt production is performed in a “make-to-order” environment. In the open literature, a production environment where a continuous production stage is followed by a packaging stage is called “make-and-pack” production.8 Lot-sizing and scheduling constitute the major challenges in this type of production environment. There are two approaches to deal with this kind of problem: (i) the sequential approach, wherein the lot-sizing problem is solved first, followed by the scheduling problem (two-stage procedure), and (ii) a holistic approach according to which the lot-sizing and scheduling problems are solved simultaneously (single-stage procedure). In this work, a new MILP model is proposed for the simultaneous lot-sizing and production scheduling problem in a multiproduct yogurt production line of a dairy plant in Greece. The proposed MILP model uses a mixed discrete/continuous time representation, in which the days of the scheduling horizon are modeled with a discrete-time representation whereas, within each production day, a continuous-time representation is adopted. The problem under consideration is mainly focused on the packaging stage, although it considers timing and capacity constraints with respect to the fermentation stage. Packaging units operate in parallel and share common resources. Sequencedependent times and costs are explicitly taken into account and optimized by the proposed mathematical framework. Typical daily production line shutdown and setup times are modeled to account for hygienic requirements. Production overtimes are allowed as part of the company’s policy. The remainder of this article is organized as follows: Section 2 describes the yogurt production process and is followed by the mathematical model in section 3. Section 4 discusses several cases of a large-scale industrial yogurt production line. Finally, section 5 summarizes this work and the relative merits of the proposed approach. 2. Description of Yogurt Production Processes Many different types of yogurt products are produced worldwide. A scheme of classification that separates all types of yogurt products into four categories based on the physical characteristics (state) of the product can be found in Figure 1. Yogurt products are classified as (i) yogurts (liquid/viscous phase), (ii) concentrated/strained yogurts (semisolid phase), (iii) frozen yogurts (solid phase), and (iv) dried yogurts (powder phase). In concrete terms, yogurt is subdivided into different groups based on (i) legal standards to classify the product on the basis of chemical composition or fat content (full, semiskimmed/ medium, or skimmed/low fat); (ii) the physical nature of the

Figure 1. Yogurt product classification.

Figure 2. Yogurt production process (except for set yogurt).

product (i.e., set, stirred, or fluid/drinking, where the latter is considered stirred yogurt of low viscosity); (iii) flavors (plain/ natural, fruit, or flavored, where the latter two types are normally sweetened); and (iv) postfermentation processing (vitamin addition or heat treatment).9 The two main yogurt product types are set and stirred yogurt.10 Both types are subsequently subjected to cooling and packaging. Additionally, fruit and nuts can be added to stirred yogurt where applicable. The main difference between these two yogurt types is that set yogurt first passes from the packaging lines and afterward is fermented in the final retail container. Figure 2 illustrates the main processing steps for producing stirred yogurt. A brief description of these stages follows. Daily Milk Collection. Milk is collected on a daily basis from dairy farms. The main components of milk are water (mainly), fat, protein, lactose, and minerals. It is worth mentioning that the chemical composition of fresh milk varies from day to day within any particular breed depending on various factors. The stage of lactation, age, and breed of the cow; milking intervals; season of the year and climate temperature; breeding policy; nutrition; and hormones are some of these factors. Preliminary Treatment of Milk Base. The removal of contaminants from the fresh milk to comply with hygiene standards and ensure a better final product quality is of great importance during and/or after milk collection. The main


contaminants are cellular material from the udder of the cow, straws, leaves, hair, seeds, and soil. Cloth filters and centrifugal clarification constitute the most commonly used fresh milk filtration methods. Transportation to Dairy Plant. Fresh milk transportation from dairy farms to the dairy production plant is usually carried out in road tankers with cooled containers (at about 5 °C). In some cases, rail tankers or churns can be used. Standardization. After the fresh milk is delivered to the dairy plant, two types of standardization take place in order to enhance the quality of the final product: (1) The fat content in the milk is standardized. Because the fat content in milk can vary, it is necessary to standardize the milk in order to meet the current compositional standards for yogurt. This standardization can be done by (i) removing part of the fat content from milk, (ii) mixing full cream milk with skimmed milk, and/or (iii) adding cream to full-fat milk or skimmed milk. (2) The solids-not-fat content in the milk is standardized. Addition of milk powder (whole milk powder, skimmed milk powder, whey, buttermilk, casein powder, nonmilk proteins, etc.) in order to adjust protein content and thus overcome the seasonal variation in the protein content in milk. Storage stability is also improved by adding milk powder. In some countries, the fortification of yogurt milk with powder is not allowed; therefore, other methods are employed to increase the solids level. To continue with standardization, binding of water and an increment in viscosity are achieved by adding stabilizers and/or emulsifiers. In most countries, stabilizers are governed by legislative regulation. Sweetening compounds are normally added during the manufacture of fruit/flavored yogurt. Addition of miscellaneous compounds, such as penicillinase additives and preservatives, to achieve specific objectives can also take place. Homogenization and Heat Treatment. After its standardization, milk goes to the homogenizer where large fat globules are separated to ones of smaller diameter. As a consequence, both the creaming effect of the milk fat and the tendency of the fat globules to coalesce or clump are reduced. The homogenization phase contributes to (i) a whiter and more attractive milk color, (ii) an improved mouthfeel of the product, and (iii) an increased milk viscosity. After homogenization, the milk is heated for a short amount of time in order to eliminate pathogens and other undesirable microorganisms. Fermentation and Culture Addition. The heat-treated milk is delivered to multipurpose fermentation tanks, where starter cultures (Streptococcus thermophilus and Lactobacillus delbrueckii) are added to incubate the mix. The fermentation time depends on the temperature, the final product type, and the concentration of the starter cultures in the mix. Therefore, fermentation time can vary significantly. Note that set yogurt is fermented after the packaging stage, in contrast to the other yogurt types (stirred, fruit, etc.). Packaging and Flavoring. In this stage, flavoring, filling, and packaging take place. For fruit yogurts, the flavoring is usually done by the mixing of fruit ingredients with the fermented milk through continuous fruit-mixers in the filling and packaging phase. The other yogurt types skip the flavoring stage. Filling and packaging are performed in parallel packaging machines that can pack many different type of final products, depending on the cup size, the cup type, labeling, yogurt type, and so on. The packaging stage constitutes the bottleneck of a yogurt production facility mainly because of (i) the low packaging rates compared to the flow rates of the previous stages and (ii) the batch nature of the process. Thus, a good scheduling strategy, in the packaging stage, constitutes a critical task in

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the daily operation of a typical dairy plant. The need for cleaning and/or sterilizing activities between the packaging of any two different final products highly complicates the packaging problem. Sequence-dependent changeover times and important sequence-dependent changeover costs due to cleaning requirements should be taken into account. Cold Storage and Quality Control. The products of the packaging stage are placed in cooling storage containers at a temperature below 10 °C, where starter cultures show limited growth. A cooling storage period of 2-5 days is usually required in order to achieve the final stability of the coagulum and preserve a high final product quality. Quality control is also realized at this phase. Distribution to Customers. A dairy plant could distribute final products to customers in different ways, mainly depending on the importance and the customer location. Important customers and local market customers (situated close to the dairy plant) are provided with final products by the firm’s refrigerated trucks. Large international clients realize the transportation of the final products with their own trucks. Finally, the transportation of final products to minor clients, outside of the local market, is usually assigned to third-part logistics companies. It is worth pointing out that vehicles used for transporting yogurt should comply with special recommendations, given that inappropriate refrigeration and/or high shaking of the yogurt can lead to a reduction in viscosity and whey syneresis, and thus to quality deterioration. 3. Mathematical Formulation Problem Statement. In this work, we address the lot-sizing and scheduling problem in a yogurt production line of a multiproduct dairy plant. Packaging units operate in parallel and share common resources such as fruit-mixer equipment units. In the current work, all data were assumed to be deterministic. Some of these data (such as product demand quantities and due dates) can vary significantly during the time horizon of interest. It is beyond the scope of this work to consider demand uncertainty, but uncertainty issues will be explicitly addressed in a future contribution. The problem addressed in this work is formally stated as follows: Given: (1) the number and type of fermentation recipes; (2) the number and type of products; (3) the number of packaging lines operating in parallel; (4) the products assigned to fermentation recipe set and to packaging line suitability; (5) the scheduling time horizon (usually 1 week); (6) the daily plant setup and shutdown time; (7) the fermentation times for every fermentation recipe; (8) the packaging rate for every product at each packaging line; (9) the minimum (depending on pasteurization and fermentation stages) and maximum packaging sizes; (10) the changeover times and costs (including sterilization and cleaning processes) for any pair of products in each packaging line; (11) the forbidden packaging sequences; (12) the costs regarding packaging units operating costs (labor costs are included), storage of final products, preparation of fermentation recipes, and overtimes; (13) the initial inventory for each product; (14) the target inventory for each product; and (15) the product demands and due dates.

704


Determine: (1) the allocation of product families to packaging lines; (2) the sequencing of product families in each packaging line; (3) the inventory level for every product family in every scheduling time interval; and (4) the production run length and starting and completion time for every product family. So that a typical economic objective function representing total costs is minimized. Conceptual Model Design. Scheduling problems are usually NP-hard and no standard solution techniques are available.11 The resolution of real-life scheduling problems by exact methods, such as mathematical programming, is extremely computationally expensive. Thus, in order to moderate the computational burden a product grouping method could be used by exploring the main processing characteristics of the products. Production scheduling in dairy plants typically deals with a large number of products. Fortunately, many products illustrate similar processing characteristics. Therefore, products that share the same processing characteristics could be treated as a product family group. Thus, the scheduling problem being solved focuses on product families rather than on each product separately. The use of product families significantly reduces the size of the underlying mathematical model and, thus, the necessary computational effort without sacrificing any feasibility constraint. In the proposed approach products belong to the same product family if and only if: (i) they come from the same fermentation recipe, (ii) there is no sequence-dependent changeover time among them, (iii) they share the same processing (packaging) rate. When changing the production between two products that are not based on the same recipe, it is always necessary to perform changeover cleaning and/or sterilizing operations. In dairy plants, a “natural” sequence of products often exists (e.g., from the lower taste to the stronger or from the brighter color to the darker) thus the relative sequence of products within a product family is known a priory. Therefore, when changing the production between two products of the same product family, the cleaning and sterilizing can be neglected. Hence, in dairy plants not only the relative sequence of products belonging to the same product family may be fixed but also the relative sequence of product families in each packaging line. In that case, different product families are enumerated according to their relative position within the day. Product Families Demand. The demand demfn for product family f is calculated by aggregating the product demands demcup pn for all products p ∈ FPf that belong to this family. Parameter weightpcup corresponds to the cup weight of product p. Note that demcup pn corresponds to production targets for every product. The initial product inventory has been already subtracted from demcup pn by the logistics department. demfn )

∑

cup demcup pn weightp

∀f, n

(1)

p∈FPf

Constraints. In the proposed mathematical formulation, constraints have been grouped according to the type of decision (assignment, timing, sequencing, etc.) on which they are imposed. It should be emphasized that the proposed model is a crossbreed between a continuous-time model and a discretetime model. More specifically, a continuous-time representation is incorporated within each production day of the production week horizon, which is modeled with a discrete-time grid using a number of time periods (see Figure 4 below).

pack of product Timing Constraints. The packaging time Tfjn pack family f at packaging line j ∈ J equals the packaged amount pack of product family f at the same packaging line divided by Qfjn of product family f on packaging line the packaging rate ratepack fj j ∈ Jpack. This value is greater than a minimum packaging time packmin packmax tfjn and lower than a maximum packaging time tfjn , 12 according to constraint 2. According to Soman et al., in case of high capacity utilization, as in the food industry, the production rate cannot be reduced because of quality problems. is considered fixed in the proposed approach. Thus, ratepack fj

min

tpack Yfjn e Tpack fjn fjn )

Qpack fjn ratepack fj

max

e tpack Yfjn fjn ∀f, j ∈ (Jpack ∩ FJf), n

(2)

Constraints 3 and 4 impose lower and upper bounds, respectively, on each product family completion time Cfjn. Thus, the completion time has to be greater than the daily plant setup for time, setupjn, plus the minimum fermentation time tferm f preparing the fermentation recipe for producing family product pack plus the changeover time sdf ′fj f, plus the packaging time Tfjn for changing the production to family f ′. )Yfjn + Tpack Cfjn g (setupjn + tferm f fjn +

∑

sdf 'fjXf 'fjn

f ′*f,f ′∈JFj

∀f, j ∈ FJf, n

(3)

Constraint 4 ensures that the completion time, Cfjn, of product family f is smaller than the production time horizon horjn minus the daily plant shutdown time shutdownjn. Production line shutdown is realized on a daily basis, as a typical production policy to guarantee the high quality of the final products and to comply with hygienic standards. Cfjn e (horjn - shutdownjn)Yfjn

∀f, j ∈ FJf, n

(4)

Timing and Sequencing Constraints. Constraint 5 guarantees that the starting time of a product family f ′ that follows another product family f on a packaging line j ∈ Jpack at period n (i.e., Xff ′jn ) 1) is greater than the completion time of product family f, Cfjn, plus the necessary changeover time sdff ′j between these product families. Cfjn + sdff 'j e Cf 'jn - Tpack f 'jn + horjn(1 - Xff ′jn) ∀f, f ′ * f, j ∈ (Jpack ∩ FJf ∩ FJf ′), n

(5)

Allocation and Sequencing Constraints. Constraints 6 and 7 state that, if a product family f is allocated to packaging unit j ∈ Jpack at period n (i.e., Yfjn ) 1), at most one product family f ′ is processed before and/or after it, respectively.

∑

Xf 'fjn e Yfjn

∀f, j ∈ (Jpack ∩ FJf), n

(6)

∑

Xff 'jn e Yfjn


(7)

f ′*f,f ′∈JFj

f ′*f,f ′∈JFj

The packaging unit j ∈ Jpack is used in period n (i.e., YJjn ) 1) if at least one product family f is assigned at period n (i.e., Yfjn ) 1). It is noted that no lower bound for the binary variable YJjn is necessary because a cost term, related to unit utilization, is included in the objective function, thus forcing YJjn to zero. YJjn g Yfjn


(8)

Constraint 9 states that the total number of active sequencing binary variables Xff ′jn plus the unit utilization binary variable YJjn should be equal to the total number of active allocation


binary variables Yfjn in a packaging unit j ∈ J at period n. For instance, if three products families are assigned to a unit j, then two sequencing variables will be active. pack

∑ ∑

Xff 'jn + YJjn )

f∈JFj f ′*f,f ′∈JFj

∑Y

fjn

∀j ∈ Jpack, n

f∈JFj

(9) Fermentation Stage Constraints. Constraints on both the fermentation and pasteurization stages must be included in the mathematical model in order to guarantee the feasibility of the production schedule in yogurt production lines. Constraint 10 states that the cumulative packaged quantity of product families f ∈ RFr that come from the same fermentation recipe r should be greater than the minimum produced fermentation recipe amount in the pasteurization and fermentation stages and lower than the maximum production capacity QRmax QRmin r r . QRrmaxYRrn

g

∑

∑

Qpack fjn

g

QRrminYRrn

∀r, n

YRrn e

∑

∑

Constraint 11 guarantees that a fermentation recipe r is produced at period n (i.e., YRrn ) 1), if at least one product family f ∈ RFr is packaged in a packaging unit j ∈ Jpack at the same period n (i.e., Yfjn ) 1). YRrn g

∑

Yfjn

∀r, f ∈ RFr, n

(11)

j∈(Jpack∩FJf)

Constraint 12 ensures that, if no product family f ∈ RFr is packaged in any packaging unit j ∈ Jpack at period n, then a fermentation recipe r is not produced over the same period (i.e., YRrn ) 0). This constraint could be omitted if a recipe cost term is considered in the objective function.

Figure 3. Yogurt production line layout.

∀r, n

(12)

f∈RFr j∈(Jpack∩FJf)

In addition, minimum packaging run limitations QFmin and/or fj might exist for any maximum packaging run limitations QFmax fj product family f. Constraint 13 forces the packaged amount Qpack fjn of a product family f to be greater than its corresponding and lower than its maximum minimum packaging run QFmin fj packaging run QFmax fj . pack min QFmax fj Yfjn g Qfjn g QFfj Yfjn

∀f, j ∈ (Jpack ∩ FJf), n (13)

Tightening Constraints. To reduce the computational effort, constraint 14 may further tighten the mathematical formulation pack of by imposing an upper bound on the packaging time Tfjn product family f on packaging line j ∈ Jpack at period n. Tpack fjn +

∑

sdff 'jXff 'jn e (horjn - shutdownjn - setupjn)Yfjn

f ′*f,f ′∈JFj


f∈RFr j∈(Jpack∩FJf)

(10)

Yfjn

705

(14)

Mass Balance Constraints. In the fresh food industry, backordering is not allowed, as the unsatisfied demand is lost. Therefore, the maximum produced quantity for every product family f and total demand satisfaction are imposed by constraints 15 and 16, respectively. The total product family quantity produced at period n should not exceed the cumulative demand for the same product family f for all periods n′ equal to or greater than the actual period n and equal to or lower than the last period N, as constraint 15 states. Thus, the storage of dairy products is avoided because they are perishable and the product quality depends on its shelf life.

∑

j∈(Jpack∩FJf)

N

Qpack fjn e

∑ dem

fn'

n'gn

∀f, n

(15)

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In dairy plants, the satisfaction of customer demands is of great importance. The inability to satisfy customer demands on time results in a loss of competitive advantage, a loss of market share, customer disappointment, a loss of customers, and so on. Constraint 16 in tandem with constraint 15 forces a total demand satisfaction.

∑ ∑Q

pack fjn

j∈(Jpack∩FJf)

∑ dem

g

n

∀f

fn

(16)

n

Product family inventories Stfn are observed by constraints 17 and 18. The inventory Stfn of product family f is the sum of the previous period inventory, Stfn-1, and the total produced quantity minus the product family demand, demfn, at the current period n.

∑

Stfn g Stfn-1 +

Qpack fjn - demfn

∀f, n > 1

j∈(Jpack∩FJf)

(17) Stfn g

∑

Qpack fjn - demfn

∀f, n ) 1

(18)

j∈(Jpack∩FJf)

Constraint 19 is added to the mathematical formulation if safety , are desired. product family safety stocks, Stfn Stfn g

∀f, n

Stsafety fn

(19)

If product-family-dependent storage limitations exist, then constraint 20 is used. Otherwise, constraint 21 could be included to account for the total plant storage capacity. ∀f, n

Stfn e Stmax f

∑ St

fn

∀n

e Stplant

(20) (21)

f

Share Common Resources Constraints. In the yogurt production line under study, packaging units j ∈ Jpack share fruitmixer equipment units j ∈ Jmix for the production of fruitflavored yogurt products f ∈ Ffruit (see Figure 3). Therefore, common resource constraints 22-26 are incorporated into the proposed mathematical framework. The decision variables of the fruit-mixer unit, Cfjn, Tmix fjn , and Yfjn, are correlated with those of the packaging units through constraints 22-24 since fruitmixers and packaging units are operating simultaneously during the production of flavored yogurt. Cfjn ) Cfj'n

∀f ∈ F

, j ∈ (J

fruit

pack

∩ FJf),

j' ∈ (Jmix ∩ FJJfj), n Tpack fjn

)

Tmix fj'n

∀f ∈ F

, j ∈ (J

fruit

pack

Yfjn ) Yfj'n

(22)

(23)

∀f ∈ Ffruit, j ∈ (Jpack ∩ FJf), j' ∈ (Jmix ∩ FJJfj), n

Figure 4. Production scheduling horizon.

recipe

process type

tfferm

R01 R02 R03 R04 R05 R06 R07 R08 R09 R10 R11 R12

fermentation fermentation fermentation fermentation fermentation fermentation fermentation cooling cooling cooling fermentation fermentation

4.75 4.50 8.25 7.75 5.25 7.25 8.75 1.50 1.50 1.50 8.75 8.75

Finally, feasible sequencing of product families in the fruitmixer equipment units j ∈ Jmix is ensured through the timing constraints 25 and 26. Cfjn + sdff 'j e Cf 'jn - Tmix f 'jn + horjn(1 - Xff'jn) + horjn(2 - Yfjn - Yf 'jn)

(24)

∀f ∈ Ffruit, f ′ ∈ Ffruit,

j ∈ (Jmix ∩ FJf ∩ FJf ′), n: f ′ < f

(25)

Cf 'jn + sdf 'fj e Cfjn - Tmix fjn + horjnXff 'jn + horjn(2 - Yfjn - Yf 'jn) ∀f ∈ Ffruit, f ′ ∈ Ffruit, j ∈ (Jmix ∩ FJf ∩ FJf ′), n: f ′< f

(26)

Objective Function. The objective function to be minimized includes several cost-related factors such as (i) inventory costs, (ii) operating costs, (iii) fermentation recipe preparation costs, (iv) unit utilization costs, and (v) product family changeover costs, as follows: minimize

inventory costs hold fn Stfn + n operating costs

∑ ∑ Cost f

∑ ∑ ∑ Cost

oper pack fjn T fjn

f

j∈(Jpack∩FJf)

fermentation recipe costs rec rn YRrn r n

∑ ∑ Cost

∑∑ f

∩ FJf),

j' ∈ (Jmix ∩ FJJfj), n j

Table 1. Minimum before Packaging Stage Times (h)

+

+

n

unit utilization costs unit jn YJjn pack n j∈J

∑ ∑ Cost

+

changeover costs

∑

f ′*f j∈(Jpack∩FJf ∩FJf ′)

∑ Cost

change ff 'jn Xff 'jn

(27)

n

In a dairy plant, the final yogurt products are kept refrigerated, thus resulting in significant inventory costs, which should be considered in the optimization procedure. Usually, inventory costs also include shelf-life-dependent costs. Roughly speaking, yogurts with long shelf lives exhibit lower inventory costs than short-shelf-life yogurts; thus, products belonging to the same product family were assumed


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Table 2. Product Family Sequences in a Production Day Per Packaging Line (NFM, FM Cases) pkg unit J1 J2 J3 J4

product family relative sequence F20 F12 F01 F23

F21 F11 F02

w w w

w w w

F22 F19 F03

F18 F05

w w

w w

F13 F04

w w

F14 F08

F15 F09

w w

w w

F16 F10

w w

F17 F06

w

F07

Table 3. Main Data for Products and Product Families product family final products product information fermentation recipe packaging unit rate (cups/h) weight (kg) rate (kg/h) inventory cost (€/kg) F01 F02 F03 F04 F05 F06 F07 F08 F09 F10 F11 F12 F13 F14 F15 F16 F17 F18 F19 F20 F21 F22 F23

P01-P05 P06-P08 P09-P11 P12-P16 P17-P20 P21, P22 P23 P24, P25 P26, P27 P28, P29 P30-P33 P34, P35 P36-P38 P39-P41 P42-P44 P45-P47 P48-P50 P51, P52 P53, P54 P55-P60 P61-P71 P72, P73 P74-P76 P77-P79 P80-P82 P83, P84 P85-P89 P90-P93

set set set

R08 R09 R10 R01 R02 R02 R07 R07 R07 R11 R11 R11 R11 R11 R01 R02 R03 R04 R04 R11 R11 R07 R06 R05 R06 R06 R06 R12

fruit fruit fruit fruit fruit

fruit fruit

fruit fruit fruit fruit fruit

J3 J3 J3 J3 J3 J3 J3 J3 J3 J3 J3 J3 J3 J3 J2 J2 J2 J2 J2 J2 J2 J2 J2 J1 J1 J1 J1 J4

2850 2850 2850 2850 2850 8550 3375 6750 3000 6000 4000 3000 9000 6800 2200 2200 2200 2200 2300 2300 1760 2100 2100 420 93 225 450 4600

0.600 0.600 0.600 0.600 0.600 0.200 0.400 0.200 0.380 0.400 0.600 0.450 0.150 0.125 1.000 1.000 1.000 1.000 0.500 0.500 0.750 1.000 1.000 5.000 30.000 10.000 5.000 0.150

1710 1710 1710 1710 1710 1710 1350 1350 1140 2400 2400 1350 1350 850 2200 2200 2200 2200 1150 1150 1320 2100 2100 2100 2790 2250 2250 690

7.50 6.75 6.75 7.50 7.50 7.50 0.45 0.45 0.45 1.50 1.50 0.45 0.45 1.80 3.60 3.60 2.10 3.60 3.60 1.80 0.60 0.45 0.45 3.60 0.45 0.45 0.45 1.80

Table 4. Changeover Times between Product Families sdff ′j (h) product family pkg product unit family F01 F02 F03 F04 F05 F06 F07 F08 F09 F10 F11 F12 F13 F14 F15 F16 F17 F18 F19 F20 F21 F22 F23 J3 J3 J3 J3 J3 J3 J3 J3 J3 J3 J2 J2 J2 J2 J2 J2 J2 J2 J2 J1 J1 J1 J4 a

F01 F02 F03 F04 F05 F06 F07 F08 F09 F10 F11 F12 F13 F14 F15 F16 F17 F18 F19 F20 F21 F22 F23

-a FSb FS FS FS FS FS FS FS FS -

0.00 FS FS FS FS FS FS FS FS -

0.00 0.00 FS FS FS FS FS FS FS -

0.50 0.50 0.50 0.25 FS FS FS FS FS -

0.50 0.50 0.50 FS FS FS FS FS FS -

0.50 0.50 0.50 1.00 0.50 FS 2.00 2.00 2.00 -

0.50 0.50 0.50 1.50 0.75 0.50 2.00 2.00 2.00 -

1.00 1.00 1.00 0.50 1.00 FS FS FS FS -

1.50 1.50 1.50 0.75 1.50 FS FS 0.50 FS -

1.50 1.50 1.50 0.75 1.50 FS FS 0.50 -

0.50 FS FS FS FS FS FS FS -

FS FS FS FS FS FS FS FS -

1.00 0.50 FS FS FS FS 0.50 0.50 -

1.00 0.50 0.50 FS FS FS 0.50 0.50 -

1.50 1.00 1.00 0.50 FS FS 1.00 1.00 -

2.00 2.00 2.00 2.00 1.50 FS 2.00 2.00 -

2.00 2.00 2.00 2.00 2.00 0.50 2.00 2.00 -

1.00 0.50 FS FS FS FS FS 0.50 -

1.00 0.50 FS FS FS FS FS FS -

FS FS -

0.50 FS -

1.00 0.50 -

-

Impossible processing sequence. b Forbidden processing sequence.

to have the same holding costs. Operating costs mainly include labor and utilities costs plus costs due to material losses. The fermentation recipe cost accounts for all costs associated with the preparation of every fermentation recipe.

The unit utilization cost basically represents the costs of shutdown cleaning operations plus the initial unit setup cost. Finally, sequence-dependent changeover costs include the cleaning and/or sterilization costs.

708


Table 5. Changeover Costs between Product Families Costffchange (103 €) ′jn product family pkg product unit family F01 F02 F03 F04 F05 J3 J3 J3 J3 J3 J3 J3 J3 J3 J3 J2 J2 J2 J2 J2 J2 J2 J2 J2 J1 J1 J1 J4 a


FSb FS FS FS FS FS FS FS FS a

0.50 FS FS FS FS FS FS FS FS -

0.50 0.50 FS FS FS FS FS FS FS -

1.20 1.20 1.20 0.75 FS FS FS FS FS -

1.20 1.20 1.20 FS FS FS FS FS FS -

F06

F07

F08

F09

F10

F11 F12 F13 F14

F15

F16

F17

F18 F19 F20 F21 F22 F23

1.20 1.20 1.20 4.00 1.20 FS 15.00 15.00 15.00 -

1.20 1.20 1.20 10.00 1.50 1.20 15.00 15.00 15.00 -

4.00 4.00 4.00 1.20 4.00 FS FS FS FS -

10.00 10.00 10.00 1.50 10.00 FS FS 1.20 FS -

10.00 10.00 10.00 1.50 10.00 FS FS 1.20 -

1.20 FS FS FS FS FS FS FS -

10.00 4.00 4.00 1.20 FS FS 4.00 4.00 -

15.00 15.00 15.00 15.00 10.00 FS 15.00 15.00 -

15.00 15.00 15.00 15.00 15.00 1.20 15.00 15.00 -

4.00 1.20 FS FS FS FS FS 1.20 -

FS FS FS FS FS FS FS FS -

4.00 1.20 FS FS FS FS 1.20 1.20 -

4.00 1.20 1.20 FS FS FS 1.20 1.20 -

4.00 1.20 FS FS FS FS FS FS -

FS FS -

1.20 FS -

4.00 1.20 -

-


Table 6. Product Family Packaging Rates ratefj (kg/h) after Manifold Investment (FM&M Case)

Table 8. Additional Changeover Times between Product Families sdff ′j (h) for the FM&M Case

packaging unit

product family

J1


J2

J3

pkg unit

J4

1710 1710 1710 1710 1710 1350 1140 2400 1350 850

J4 J4 J4 J4 J4 1150

a

2700 1350 850

product family

product family

F06

F08

F09

F10

F23

F06 F08 F09 F10 F23

2.00 2.00 2.00 FS

b

FS 0.50 FS FS

FS 0.50 0.50 FS

2.00 2.00 2.00 2.00 -

a

FS FS FS FS


Table 9. Additional Changeover Costs between Product Families (103 €) for the FM&M Case Costffchange ′jn pkg unit

2200 2200 2200 2200 1150 1150 1320 2100 2100

F06

F06 F08 F09 F10 F23

15.00 15.00 15.00 FS

J4 J4 J4 J4 J4 a

2100 2790 2250

product family

product family

a

F08

F09

F10

F23

b

FS 0.50 FS FS

FS 0.50 0.50 FS

15.00 15.00 15.00 15.00 -

FS FS FS FS


solved by the proposed MILP model. It is pointed out that real data have been modified slightly due to confidentiality issues. Yogurt Production Line Description. The production line produces set, stirred, and/or flavored yogurt. It is noted that flavored yogurt is stirred yogurt with additional fruit (or other type) flavor. Thus, flavored yogurt production should pass through fruit-mixer equipment in order for the addition and mixing of fruit substances to be performed. The yogurt production line consists of (i) a set of cooling tanks (set yogurt), (ii) a set of fermentation tanks (stirred and flavored yogurt),

690

It is worth mentioning that, because full demand satisfaction is imposed, the minimization of total costs is identical to the maximization of total profit. 4. Case Studies In this section, first, the yogurt production line under question is described in detail, and then, several real-world cases are

Table 7. Product Family Sequences in a Production Day Per Packaging Line after Manifold Investment (FM&M Case) pkg unit J1 J2 J3 J4

product family relative sequence F20 F12 F01 F08

w w w w

F21 F11 F02 F09

w w w w

F22 F19 F03 F10

w w w

F18 F05 F06

w w w

F13 F04 F23

w w

F14 F08

w w

F15 F09

w w

F16 F10

w w

F17 F06

w

F07


709

Table 10. Case Study I: Product Demand (Cups) Per Production Day product

n1

n2

P01 P02 P03 P04 P05 P06 P07 P08 P09 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 P20 P21 P22 P23 P24 P25 P26 P27 P28 P29 P30 P31 P32 P33 P34 P35 P36 P37 P38 P39 P40 P41 P42 P43 P44 P45 P46 P47

3115

n3

n4

n5

product P48 P49 P50 P51 P52 P53 P54 P55 P56 P57 P58 P59 P60 P61 P62 P63 P64 P65 P66 P67 P68 P69 P70 P71 P72 P73 P74 P75 P76 P77 P78 P79 P80 P81 P82 P83 P84 P85 P86 P87 P88 P89 P90 P91 P92 P93

2190 4416 14001 5480 5888 11241 4000 1888 1229 715 1560 1215 4416 6341 3715 2319 2592 6138 6480 1620 1380 1318 2193 1274 1671 1325 3312 3312 682 801 249 1057 2472 2472 26496 8531 4717 4093 5743 1172 2700 807 219 1578 518 690

n1

n2

n3

n4

n5 2132 5495 830 9380 1272 2386 315

1782 316 316 1188 1188 316 108 162 753 219 648 1162 648 468 1296 544 219 1099 683

240 40 1071 117 57 1848 195 960 1160 200 710 290 200 518 6600 1442 850 10140 5410 2000 1000

Table 11. Case Study I: Computational Resultsa model

objective function

equations

continuous variables

binary variables

nodes

CPU time (s)

NFM FM FM&M

239648 221502 213793

3607 3194 3558

1358 1232 1386

1050 875 973

29506 33 25

22.25 0.38 0.44

a

Solved in GAMS 22.8, CPLEX 11, on a Dell Inspiron 1520, 2.0 GHz, 2 GB of RAM.

Table 12. Case Study I: Economic Comparative Results (€) model

total cost

inventory cost

operating cost

recipe cost

unit cost

changeover cost

NFM FM FM&M

239648 221502 213793

117415 103127 98140

64933 61425 61403

15250 14500 15000

18000 16000 18000

24050 26450 21250

(iii) four packaging units, and (iv) two fruit-mixer equipment units. The main yogurt production line layout is illustrated in Figure 3. The short-term scheduling time horizon for yogurt production is usually 1 week.13 Regular production is performed from Monday through Friday. Overtime is permitted on Sunday and/ or on Saturday (see Figure 4). Period production time is equal

to 24 h. Daily scheduled plant cleaning operations, shutdownjn, last for 2 h, and the period before the start of the fermentation stage (including pasteurization, homogenization, etc.), reflecting the total plant setup time setupjn, equals to 3 h. Product demand data are packaging stage production targets, and they are provided from the logistics department of the plant. Demand quantities and due dates are based on product orders, from Sunday

710


Figure 5. Case I: Cost (€) comparative analysis. Table 13. Case Study I: Production Schedule for the NFM Case (kg) production periods (days)

product family

pkg unit

n0

n1

n2

n3

n4

n5


J3 J3 J3 J3 J3 J3 J3 J3 J3 J3 J2 J2 J2 J2 J2 J2 J2 J2 J2 J1 J1 J1 J4

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4895 0 0

1869 0 0 1665 1391 0 1138 0 8186 0 0 0 0 0 0 0 4264 2022 0 0 0 19170 0

1314 0 0 0 0 0 0 2650 0 0 0 0 0 0 0 0 407 0 178 0 31923 0 0

0 0 0 0 3683 0 0 0 0 0 0 0 1343 0 0 0 0 0 0 0 32783 0 0

2650 6245 2299 2650 1555 1221 0 1673 0 1376 3726 0 0 0 0 2553 0 0 0 0 0 21443 0

11689 6432 0 6034 4488 0 0 0 0 0 0 2786 7114 10652 1351 0 0 0 1050 10500 0 17438 2783

4895

39706

36471

37808

47390

82315

total daily production

fruit

Figure 6. Case I: Comparison of inventory costs (€) per production period.

through Tuesday of the following week, as well as on forecasts. Demand due dates are given for packaged final products (subtracking the necessary final cold storage and quality control time, which varies between 2 and 5 days). For instance, a product order with a 4-day minimum cold storage time that should be shipped on next Tuesday will have its due date of Friday. Table 1 lists the minimum fermentation times (stirred yogurt) and the minimum cooling times (set yogurt) for each product family f. Table 2 illustrates the relative sequence of product families in a production day. Table 3 provides the main data for all products and product families. In particular, in Table 3, the following information is summarized: (i) the products to

product family set FPf, (ii) the set of flavored yogurts F , (iii) the fermentation recipe origin set RFr of each product family, (iv) the available packaging units set FJf to process each product family, (v) the product and product family packaging rates, (vi) the product cup weight, and (vii) the associated product family inventory cost Costhold fn . The minimum produced quantity of any fermentation recipe QRmin r , due to pasteurization and fermentation stage operability issues, is 1200 kg. The minimum production amount for any product family in any packaging unit QFmin fj is 150 kg. Minimum and maximum packaging times are not imposed. Sequence-dependent changeover times and costs are given in Tables 4 and 5, respectively. Variable operating costs Costoper fjn for packaging product families mainly include labor and utility costs. This is equal to 500 €/h for each packaging unit and

Ind. Eng. Chem. Res., Vol. 49, No. 2, 2010 Table 14. Case Study I: Production Schedule for the FM Case (kg)

Table 16. Case Study I: Inventory Profile for the NFM Case (kg)

production periods (days)

production periods (days)

product family

pkg unit

n0

n1

n2

n3

n4

n5



0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1869 0 0 1665 1391 0 1138 0 8186 0 0 0 0 0 0 0 0 2022 0 0 4035 14175 0

1314 0 0 0 0 0 0 2650 0 0 0 0 0 0 0 0 2673 0 178 0 32783 0 0

0 0 0 0 3683 0 0 0 0 0 0 0 1343 0 0 0 0 0 0 0 32783 0 0

2650 6245 2299 2650 1555 1221 0 1673 0 1376 3726 0 0 0 0 2553 1997 0 0 0 0 26438 0

11689 6432 0 6034 4488 0 0 0 0 0 0 2786 7114 10652 1351 0 0 0 1050 10500 0 17438 2783

total daily production 0 34482 39597 37808 54382 82315 Table 15. Case Study I: Production Schedule for the FM&M Case (kg) production periods (days) product family

pkg unit

n0

n1

n2

n3

n4

n5

F01 F02 F03 F04 F05 F06

J3 J3 J3 J3 J3 J3 J4 J3 J3 J4 J3 J4 J3 J4 J2 J2 J2 J2 J2 J2 J2 J2 J2 J1 J1 J1 J4

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1869 0 0 1665 1391 0 0 1138 0 0 8186 0 0 0 0 0 0 0 0 0 0 2022 0 0 4035 14175 0

1314 0 0 0 0 0 0 0 0 2650 0 0 0 0 0 0 0 0 0 0 2673 0 178 0 32783 0 0

0 0 0 0 3683 0 0 0 0 0 0 0 0 0 0 0 1343 0 0 0 0 0 0 0 32783 0 0

2650 6245 2299 2650 1555 0 1221 0 0 0 0 0 0 0 3726 0 0 0 0 2553 1997 0 0 0 0 26438 0

11689 6432 0 6034 4488 0 0 0 0 1673 0 0 0 1376 0 2786 7114 10652 1351 0 0 0 1050 10500 0 17438 2783

0

34482

39597

37808

51333

85365

F07 F08 F09 F10 F11 F12 F13 F14 F15 F16 F17 F18 F19 F20 F21 F22 F23


711

product family, during a regular production day (weekdays) and 2500 €/h in overtime periods (Saturday and Sunday). Moreover, the cost for the production of any fermentation recipe is approximately 500 € in a regular production day and 750 € in overtime periods. The fixed utilization and cleaning packaging unit , for any packaging line is 1000 € in a regular unit cost, Costjn production day and 2000 € in overtime periods.

product family

n0

n1

n2

n3

n4

n5


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4895 0 0

0 0 0 729 0 0 1138 0 2225 0 0 0 0 0 0 0 4264 240 0 0 4895 19170 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1997 240 178 0 36818 19170 0

0 0 0 0 0 0 0 0 0 0 0 0 1343 0 0 0 1997 240 178 0 0 19170 0

0 2712 1166 0 0 0 0 1673 0 1376 1026 0 1343 0 0 0 1997 0 178 0 0 36363 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

daily inventory

4895

32660

58403

22928

47834

0

Table 17. Case Study I: Inventory Profile for the FM Case (kg) production periods (days) product family

n0

n1

n2

n3

n4

n5


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 729 0 0 1138 0 2225 0 0 0 0 0 0 0 0 240 0 0 4035 14175 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 240 178 0 36818 14175 0

0 0 0 0 0 0 0 0 0 0 0 0 1343 0 0 0 0 240 178 0 0 14175 0

0 2712 1166 0 0 0 0 1673 0 1376 1026 0 1343 0 0 0 1997 0 178 0 0 36363 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

daily inventory

0

22542

51411

15936

47834

0

Table 18. Case Study I: Inventory Profile for the FM&M Case (kg) production periods (days) product family

n0

n1

n2

n3

n4

n5


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 729 0 0 1138 0 2225 0 0 0 0 0 0 0 0 240 0 0 4035 14175 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 240 178 0 36818 14175 0

0 0 0 0 0 0 0 0 0 0 0 0 1343 0 0 0 0 240 178 0 0 14175 0

0 2712 1166 0 0 0 0 0 0 0 1026 0 1343 0 0 0 1997 0 178 0 0 36363 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

daily inventory

0

22542

51411

15936

44785

0

712


Figure 7. Case I: Breakdown of total costs.

Figure 8. Case I: Gantt charts for all cases.

Remarks. A closer look at the yogurt production line reveals the following points about the current line configuration:

(i) Fruit-mixers are common resources that limit the total plant production capacity. For instance, packaging unit J3 and


713

Table 19. Case Study II: Product Demand (Cups) Per Production Day product

n1

P01 P02 P03 P04 P05 P06 P07 P08 P09 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 P20 P21 P22 P23 P24 P25 P26 P27 P28 P29 P30 P31 P32 P33 P34 P35 P36 P37 P38 P39 P40 P41 P42 P43 P44 P45 P46 P47

n2

n3

n4

n5

n6

2327 3680 3680 18800

3124 4216

5152 19500 6624 1104 2900 1472 736 415

212 1472 7100 6000

736 876 354 102 1104

3680 458 11000

9500 33112

2208 568 1248 2715 4148 1074 2900 4000 2600 1220 1232 1232 6624 8832 9245 10300 12900 5120 2673 972 780 1782 2400

model

n1

P48 P49 P50 P51 P52 P53 P54 P55 P56 P57 P58 P59 P60 P61 P62 P63 P64 P65 P66 P67 P68 P69 P70 P71 P72 P73 P74 P75 P76 P77 P78 P79 P80 P81 P82 P83 P84 P85 P86 P87 P88 P89 P90 P91 P92 P93

562 4000

n2

4700 2600

n3

n4

6000 785 6800

6100

n5

n6

108 108

3000 465

564 1188 1008 1188 1188 1008 108 108 1914 1600 972 2600 1620 1900 1944 2200 1012 13365 4581 568 1000 900 120 180 545 594 480 920 650 625 716 215 10000 300 342

120 4480

4213 1200 2150

135

Table 20. Case Study II: Computational Resultsa

NFM FM FM&M

product

objective continuous binary function equations variables variables 307535 295722 274219

3607 3194 3558

1358 1232 1386

1050 875 973

nodes

CPU time (s)

109054 1595 977

59.57 3.72 3.92

a Solved in GAMS 22.8, CPLEX 11, on a Dell Inspiron 1520, 2.0 GHz, 2 GB of RAM.

packaging unit J4 cannot package flavored yogurt simultaneously (see Figure 3); the same statement holds for packaging unit J1 and packaging unit J2. Therefore, a relatively low-cost fruit-mixer investment seems to be an alternative to increase the yogurt production line capacity. (ii) According to Table 3, each product p and/or product family f can be produced only on one packaging unit. Thus, the production process seems to be lacking in flexibility. Discussions with the plant manager revealed that it is possible to install a low-cost manifold investment in packaging unit J4

in order to package more product families. With the current operating policies, packaging unit J4 can process only product family F23. Table 6 lists the packaging rates for all product families after this investment. Note that packaging unit J4 could process five product families (F6, F8, F9, F10, and F23), instead of just one product family (F23). Table 7 gives the product family relative sequence in a production day, and Tables 8 and 9 give the additional changeover times and costs, respectively, for this case. For the sake of clarity of presentation, the current plant configuration will be referred as NFM, the fruit-mixer retrofit design option as FM, and the joint fruit-mixer and manifolds investment as FM&M. All case studies were resolved for all plant configurations using CPLEX 11.0 solver through the GAMS 22.814 interface. Case Study I. Product demands for this case study are given in Table 10. There is no minimum safety stock for any period. Computational statistics for all plant configuration models are given in Table 11. The optimal solution was achieved in all

Table 21. Case Study II: Economic Comparative Results (€) model

total cost

inventory cost

operating cost

recipe cost

unit cost

changeover cost

NFM FM FM&M

307535 295722 274219

134064 122951 107053

102321 102321 103017

16750 17250 16750

17000 17000 20000

37400 36200 27400

714


Figure 9. Case II: Cost (€) comparative analysis. Table 22. Case Study II: Production Schedule for the NFM Case (kg) production periods (days) product family F01 F02 F03 F04 F05 F06 F07 F08 F09 F10 F11 F12 F13 F14 F15 F16 F17 F18 F19 F20 F21 F22 F23

pkg unit

n0


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4082 0 4366 0 3974 0 0 0 1200 0 0 0 691 0 0 8870 1462 2200 0 4383 638 0 13163 0 0 1506 0 0 0 6432 0 2089 1200 3614 0 0 0 0 0 3540 972 0 0 0 0 4317 0 0 4562 0 6785 0 4700 0 6800 6208 1300 0 1787 0 0 0 0 1872 0 10890 0 1094 2767 0 2145 0 1568 0 0 900 0 0 0 4225 14400 0 0 15549 11120 0 26438 7220 1806 0 0 0

0

56442 28959 61482 55949 77483 26438


Figure 10. Case II: Inventory cost (€) per production period comparison.

cases at a very low computational cost. The larger-size NFM model gave the optimal solution in 22 CPU s, whereas the FM and FM&M cases were each solved in less than 0.5 CPU s. The solution of the FM model represents a 7.57% improvement over the NFM model. On the other hand, the total cost of the FM&M case is 10.79% lower than that of the NFM model. Table 12 shows the detailed cost breakdown. A visual representation is illustrated in Figure 5. It should be noted that the FM and FM&M configurations lead to lower total inventory costs than the NFM case (12.17% and 16.42%, respectively).

n1

n2

n3

n4

n5

n6

13048 0 14791 0 2086 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3453 0 0 0 0 0 0 0 0 0 1200 0 0 0 13034 0 0 0 0 0 29871 0 0 26438 0 0

Figure 6 illustrates the daily inventory costs for all plant configurations. Tables 13-15 provide the produced quantities for every product family in each production day for the NFM case, the FM case, and the FM&M case, respectively. It can be seen that the dairy plant starts its operation on Sunday (n0) in the FM case, in order to achieve full demand satisfaction. This overtime explains the higher operating costs of this configuration. The other two configurations are capable of satisfying the demand profile without operating the plant on Sunday. Furthermore, Tables 16-18 give the exact inventory levels for every product family in each production day for the NFM, FM, and FM&M cases, respectively. A total cost breakdown can be found in Figure 7. It is important to note the 3% reduction of the inventory cost contribution in the total cost regarding the proposed FM and FM&M configurations over the current plant configuration. Finally, the Gantt charts for all cases are illustrated in Figure 8.

Ind. Eng. Chem. Res., Vol. 49, No. 2, 2010 Table 23. Case Study II: Production Schedule for the FM Case (kg)

Table 25. Case Study II: Inventory Profile for the NFM Case (kg)

production periods (days) product family F01 F02 F03 F04 F05 F06 F07 F08 F09 F10 F11 F12 F13 F14 F15 F16 F17 F18 F19 F20 F21 F22 F23

pkg unit

n0

n1


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4082 3974 1200 691 1462 1200 0 0 1200 0 972 0 4562 4700 1300 0 0 0 2468 0 14400 9460 1806


0

n2

n3

n4

n5

production periods (days) n6

0 4366 0 13048 0 0 0 0 14791 0 0 0 0 2086 0 0 0 8870 0 0 2200 0 4383 0 0 0 12600 0 0 0 1506 0 0 0 0 6432 0 2089 0 0 3614 0 0 0 0 0 0 3540 0 0 0 0 0 3453 0 4317 0 0 0 0 0 6785 0 0 0 0 6800 6208 0 0 0 1787 0 0 0 0 0 0 3072 0 7875 0 4109 0 0 2436 2145 3750 9615 0 0 0 0 0 0 0 0 4225 0 0 0 0 12638 32783 0 0 21348 13969 0 26438 0 0 0 0 0

53477 28380 55831 63780 78848 26438

Table 24. Case Study II: Production Schedule for the FM&M Case (kg) production periods (days) product family

pkg unit

n0

n1

F01 F02 F03 F04 F05 F06

J3 J3 J3 J3 J3 J3 J4 J3 J3 J4 J3 J4 J3 J4 J2 J2 J2 J2 J2 J2 J2 J2 J2 J1 J1 J1 J4

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4082 3974 1200 691 1462 0 726 474 150 0 0 1109 0 0 972 0 4562 4700 1300 0 524 0 2468 0 14400 9460 1806

0

54061 24731 41085 70904 89534 26438

F07 F08 F09 F10 F11 F12 F13 F14 F15 F16 F17 F18 F19 F20 F21 F22 F23


n2

n3

n4

n5

715

n6

0 4366 0 13048 0 0 0 0 14791 0 0 0 0 2086 0 0 0 8870 0 0 2200 0 4383 0 0 0 0 0 0 0 0 0 4476 8598 0 0 0 1032 0 0 0 0 0 0 0 6282 0 0 2089 0 0 0 3705 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3540 0 0 0 0 0 3453 0 4317 0 0 0 0 0 6785 0 0 0 0 6800 6208 0 0 0 1787 0 0 0 0 0 0 3072 0 7351 0 4109 0 0 4581 0 3750 9615 0 0 0 0 0 0 0 0 4225 0 0 0 0 12638 32783 0 0 21348 13969 0 26438 0 0 0 0 0

Case Study II. Product demand quantities and due dates for this case study can be found in Table 19. There is no minimum safety stock for any period. The computational statistics for all cases are given in Table 20. All cases were solved to global optimality at a low computational cost. The larger-size NFM model reached the optimal solution in about 1 CPU min, whereas the FM and FM&M cases each solved this industrial scheduling problem in less than 4 CPU s.

product family

n0

n1

n2

n3

n4

n5


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 317 0 0 196 0 0 1200 0 972 0 0 0 0 0 0 2767 0 0 0 1660 852

0 0 317 0 0 196 1032 0 4814 0 0 135 0 0 0 0 6849 2767 0 0 0 1660 852

440 0 317 0 0 13245 1032 0 3705 0 0 0 0 0 54 0 3015 331 0 0 0 28098 852

440 0 317 883 2208 13245 0 2089 3705 0 0 0 0 108 54 1872 0 331 0 0 15549 35318 852

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 30063 0

daily inventory

0

7964

18622

51088

76970

30063

Table 26. Case Study II: Inventory Profile for the FM Case (kg) production periods (days) product family

n0

n1

n2

n3

n4

n5


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 317 0 0 758 0 0 1200 0 972 0 0 0 0 0 0 0 900 0 0 0 852

0 0 317 0 0 758 1032 0 4814 0 0 135 0 0 0 0 3834 2436 900 0 0 0 852

440 0 317 0 0 13245 1032 0 3705 0 0 0 0 0 54 0 0 0 900 0 0 21348 852

440 0 317 883 2208 13245 0 2089 3705 0 0 0 0 108 54 0 0 3750 0 0 12638 35318 852

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 30063 0

daily inventory

0

4999

15078

41893

75605

30063

Table 27. Case Study II: Inventory Profile for the FM&M Case (kg) production periods (days) product family

n0

n1

n2

n3

n4

n5


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 317 0 0 284 474 150 1109 0 972 0 0 0 0 0 524 0 900 0 0 0 852

0 0 317 0 0 284 0 0 1109 0 0 135 0 0 0 0 3834 4581 900 0 0 0 852

440 0 317 0 0 171 0 0 0 0 0 0 0 0 54 0 0 0 900 0 0 21348 852

440 0 317 883 2208 4647 0 0 3705 0 0 0 0 108 54 0 0 3750 0 0 12638 35318 852

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 30063 0

daily inventory

0

5582

12012

24081

64919

30063

716


Figure 11. Case II: Breakdown of total costs.

Figure 12. Case II: Gantt charts for all cases.

The solution of the FM configuration represents a 3.84% improvement over the NFM case. On the other hand, the FM&M model leads to a 10.83% better solution than the NFM model.

Table 21 presents a detailed breakdown of the objective function, along with a visual representation in Figure 9. Once again, note that FM and FM&M appear to provide lower total inventory


costs than the NFM case. The inventory costs of the FM and FM&M configurations are 8.29% and 20.15% lower, respectively, than those of the NFM case. Figure 10 illustrates the daily inventory costs for all plant configurations. It is important to note that the FM&M configuration leads to significantly lower inventory costs on almost all production days. The quantities of each product family during each production day for the NFM, FM, and FM&M cases are listed in Tables 22-24, respectively. More specifically, Tables 25-27 give the exact inventory levels for every product family in each production day for the NFM, FM, and FM&M cases, respectively. The total cost breakdown is illustrated in Figure 11. Once again, both the FM and FM&M configurations lead to a 5% decrease in the inventory cost contribution in the total cost compared to the current plant configuration. Finally, the Gantt charts for all cases are shown in Figure 12. 5. Conclusions In this work, a mathematical framework has been proposed for the simultaneous lot-sizing and production scheduling of yogurt production lines. This model aims at being the core element of a computer-aided advanced scheduling and planning system in order to facilitate decision-making in dairy plant industrial environments. At this point, it is emphasized that it can be difficult to directly quantify the benefits of computeraided scheduling, because the precomputer situation is not usually known in detail, so there is no sufficient basis for comparison. However, this single fact is an excellent argument in favor of computer-aided scheduling as discussed by Jakeman: 15 “If you do not know how well you are doing, how can you improve your performance?” A salient feature of the dairy industry is that the customers usually confirm (i.e., change) their order quantities just prior to dispatch. For some products, this could be handled by carrying stocks, but for yogurt (or other fresh food products), the quantities must be changed on the fly.15 Yogurt is a perishable product, and strategies of building up inventories are inappropriate because they compromise its quality, selling price, and freshness. Thus, a production planning and rescheduling approach to address these problems in this type of industry seems to be a promising and challenging research direction. Acknowledgment Financial support from the Spanish Ministry of Education (FPU grant) and Project DPI2006-05673 is gratefully acknowledged. The authors also thank Mr. Nikolas Polydorides, production manager at KRI-KRI S.A., for the provision of data and the fruitful comments and suggestions. Nomenclature Indices f, f ′ ) product families j, j′ ) equipment units n, n′ ) scheduling time periods p ) yogurt products r ) fermentation recipe types Sets Ffruit ) fruit-flavored stirred product families f FJf ) packaging units j ∈ Jpack able to package product family f FJJfj ) defines the connection among the available fruit-mixer units j ∈ Jmix with the packaging lines j ∈ Jpack when packaging family f ∈ Ffruit

717

FPf ) yogurt products that belong to the same product family f Jmix ) fruit-mixer equipment units Jpack ) packaging lines JFj ) product families f that can be packaged in packaging unit j ∈ Jpack RFr ) product families f that have the same fermentation recipe type origin r Parameters Costffchange ′jn ) sequence-dependent changeover cost between product family f and product family f ′ in packaging unit j ∈ Jpack at period n (€); accounts for cleaning and sterilizing operations hold ) inventory cost for product family f at period n (€/kg) Costfn oper ) variable operating cost for packaging product family f Costfjn in packaging unit j ∈ Jpack at period n (€/h); includes labor and utilities costs rec ) cost for producing fermentation recipe r at period n (€) Costrn unit Costjn ) fixed cost for utilizing packaging unit j ∈ Jpack at period n (€) demfn ) demand for product family f at period n (kg) cup ) demand for product p at period n (cups) dempn horjn ) production time horizon for packaging unit j (h) QFfjmax ) maximum production amount of product family f in packaging unit j ∈ Jpack (kg) QFfjmin ) minimum production amount of product family f in packaging unit j ∈ Jpack (kg) QRrmax ) maximum production capacity of fermentation recipe r (kg) QRmin ) minimum produced quantity of fermentation recipe r (kg); r accounts for capacity restrictions of pasteurization and fermentation tanks ratefj ) packaging rate for product family f at packaging unit j ∈ Jpack (kg/h) sdff ′j ) sequence-dependent setup time between product family f and product family f ′ in equipment unit j (h); accounts for cleaning and sterilizing operations setupjn ) daily opening setup time for every unit j at period n (h); accounts for pasteurization and homogenization stages shutdownjn ) daily shutdown time for every unit j at period n (h); accounts for cleaning of yogurt production line for hygienic and quality reasons Stplant ) total plant storage capacity (kg) Stfmax ) maximum storage capacity for product family f (kg) safety Stfn ) safety stock for product family f at period n (kg) tferm ) minimum fermentation time for preparing fermentation recipe f r for producing stirred yogurt product family f (h); reflects the minimum cooling time for set yogurt products before the packaging stage max tpack ) maximum run time for product family f on unit j at fjn period n (h) min tpack ) minimum run time for product family f on unit j at fjn period n (h) weightpcup ) weight of product p per cup (kg) Continuous Variables Cfjn ) completion time for product family f in unit j at period n (h) pack Qfjn ) packaged amount of product family f in unit j ∈ Jpack at period n (kg) Stfn ) inventory of product family f at period n (kg) Tmix fjn ) processing time for product family f in fruit-mixer equipment j ∈ Jmix at period n (h) pack Tfjn ) packaging time for product family f in packaging unit j ∈ Jpack at period n (h)

718


Binary Variables Xff ′jn ) if product family f is processed exactly before f ′, when both are assigned to the same unit j at the same period n, then Xff ′jn ) 1; otherwise, Xff ′ jn ) 0 Yfjn ) if product family f is allocated to unit j at period n, then Yfjn ) 1; otherwise, Yfjn ) 0 YJjn ) if packaging unit j ∈ Jpack is used at period n, then YJjn ) 1; otherwise, YJjn ) 0 YRrn ) if fermentation recipe r is produced at period n, then YRrn ) 1; otherwise, YRrn ) 0

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ReceiVed for reView June 23, 2009 Accepted November 12, 2009 IE901013K

Optimal Production Scheduling and Lot-Sizing in ... - ACS Publications

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