relation between production capacity and variety of products on a

International Journal of InnovativeComputing, Information and Control ICIC International c⃝2011 ISSN 1349-4198Volume 7, Number 5(B), May 2011 pp. 2821–2835

RELATION BETWEEN PRODUCTION CAPACITY AND VARIETYOF PRODUCTS ON A SCHEDULING WITH PERISHABLE ITEMSFOR MINIMIZING THE SUM OF EARLINESS AND TARDINESS

Shimpei Matsumoto1, Tomoko Kashima2 and Hiroaki Ishii3

1Department of Information Systems and ManagementFaculty of Applied Information Science

Hiroshima Institute of Technology2-1-1 Miyake, Saeki-ku, Hiroshima 731-5193, Japan

[email protected]

2Department of Information and Systems EngineeringFaculty of Engineering

Kinki University1 Takaya Umenobe, Higashi-Hiroshima, Hiroshima 739-2116, Japan

[email protected]

3Department of Mathematical SciencesSchool of Science and Technology

Kwansei Gakuin University2-1 Gakuen, Sanda, Hyogo 669-1337, Japan

[email protected]

Received February 2010; revised July 2010

Abstract. This paper addresses a practical scheduling problem with perishable items asan example of food-processing industry, which has not been studied so far, and presentsa scheduling method and some experimental results. This paper focuses on the problemof boiling process in a jam plant. The problem in this paper includes the constraint ofperishability, which is a constraint when products are processed too much earlier thantheir due date, the values are going to reduce by reason of inferior quality, on the otherhand, when the production is not in time for due date, the shortage of supply is goingto happen. In the boiling process, setup time to change the kinds of products can beassumed as production bottleneck, so, it might be important to control setup operationsfor obtaining an effective management. To find a useful sequence to boil ingredients, anew management method is considered to minimize the deterioration in quality caused bythe sum of earliness and tardiness. As basic study, this paper shows a general productionline producing relatively small kinds of products in comparison with high-mix low-volumeproduction line. In experiments, this paper mainly considers the possibility of variety ofproducts corresponding to the production capacity.Keywords: Food-processing, Perishable, Scheduling, High-mix low-volume, Minimizingthe sum of earliness and tardiness

1. Introduction. Over the last several years, practical scheduling problems have beenstudied by many researchers (Ma et al., 2010, Smith et al., 2009) including food-processingindustry which is motivating our study. For example, Zhang and Wu presented a decom-position based algorithm for large-scale job shop scheduling problems, and numerical com-putations were conducted for real-life production environment of a speed-reducer factory(Zhang and Wu, 2009). Tseng et al. proposed a dynamic scheduling heuristic that utilizedtask deadline and task assignment time to enhance the completeness of the experimentand to more closely approach real Grid computing (Tseng et al., 2009).

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2822 S. MATSUMOTO, T. KASHIMA AND H. ISHII

Among them, the food-processing industry presents unique technical challenges to au-tomation (Jakeman, 1994). The qualities of grocery items and raw materials in food-processing factory will drastically decline with passage of time. The characteristic iscalled as “Perishability”, and a factor, passage of time, is considered to be extremelyimportant for perishable items. In general, the production scale of food-processing in-dustry is relatively small in comparison with any other industries such as automotive,consumer electronics and mining and manufacturing (Hui et al., 2007), so, the food-processing industry has traditionally been less automated than other sectors. In spite ofsuch environment, today’s food-processing industry has to carry out high-mix low-volumeproduction to meet diversified customer’s demands. Moreover, domestic food-processingfactories should compete with overseas companies with low-end products in price compe-tition due to the globalization of food distribution. Food-processing industry has facedmany challenges, such as ingredients, packaging, transportation, energy and labor, but atthe same time, customers have required higher service and customization for food prod-ucts (Arbib et al., 1999). Therefore, at present, it is thought to need to introduce aneffective management system with science-based approach even though the production infood-processing is complex and difficult (Sweet, 2006).In the food-processing, current automation and control have been primarily imple-

mented at unit process or single machine level. Processing of raw materials has beencontinuous or batch process (Sung et al., 2003, Ha et al., 2000), such as ingredient mix-ing, forming, frying and baking, with single loop or multivariable controls. The objectiveis usually given to maintain the consistent sensory and nutritional quality (Harikrishnan1et al., 2005), subject to variations in raw material properties which are often greater andmore difficult to measure than in other manufacturing sectors. Minimizing waste includ-ing machine idle time between the processes is in general more important than laborcosts, and its advanced control is dependent on innovation in sensors which are capableof measuring properties associated with product attributes (Soman et al., 2007).We have developed a scheduling system by the cooperative research with our university

and a food-processing factory manufacturing fruits-processing products as a case study(Yagi et al., 2005), and have considered many approaches to improve: total optimiza-tion across the supply chain, collaborative action between processes and scheduling ina certain process (thawing of raw materials, boiling process, inspection process, fillingprocess). This paper concentrates on the processing line of boiling process on the basisof investigation results, which the boiling process is assumed as the most critical processthan any others. In addition, the working site has expected a computer aided system tocalculate a useful schedule of operations within practical time while keeping various con-straints. This paper considers to minimize the sum of earliness and tardiness dependingon the perishability, and in this problem, the setup operation is assumed as peculiarlyimportant. To obtain exact numerical value for setup operation, we examine product’scharacteristics such as color, smell and taste (Leung et al., 2004). To solve the problem,this paper applies a method based on Lagrangian relaxation (Luh et al., 1990, Nishi etal., 2010). The problem relaxing machine capacity constraints can be decomposed intoindividual job-level sub-problems which can be solved by dynamic programming.

2. Surroundings. The food-processing factory has supplied fruits-processing products.The production line consists of 2 boiling cauldrons, inspection operation, one coolingmachine, one filling machine and one packaging machine. This factory has 4 key phases.

(1) Unfreezing process: This process unfreezes frozen fruits not to waste the raw mate-rials on the basis of the production information with changes of demands, processingcapacity of the production lines and due dates of products.

A SCHEDULING WITH PERISHABLE ITEMS 2823

(2) Blending and boiling process: This process blends some flavors for unfrozen fruits.The unfrozen fruits are perishable. If too much fruits are boiled, it exceeds thecapacities of cauldrons, inspectors and filling machine. At the end or beginning ofeach working day, switching operation is usually executed to change another kind ofingredient.

(3) Inspecting process: In this process, specialized workers inspect the ingredientsblended to maintain the quality of products. The number of workers give a directeffect for production cost, so, it is necessary to manage the number of workers. Mostof inspectors work 4 hours in a day. Fruits-processing production is fully influencedby seasonal demands, so, managing temporary workers is important to follow theincrease of demands. It is also important to control the number of temporary workersas much as possible because the wage of the temporary workers is more expensivethan regular workers.

(4) Filling and packing process: This process fills contents into several types of bottles.When the types of products and the size of bottles are switched, it needs setupoperations to clean the machine and to replace the labels. The filling machine has tobe stopped when the flow of process is disrupted at the inspecting phase.

This paper focuses on a scheduling problem at the blending and boiling process on thebasis of our analysis for all processes. The problem is under the assumption that theproducts with same type of raw materials, such as fruits, sugar and pectin, are consideredas same job, which are putted into and boiled by cauldron together, even though thebottle or label of products are different.

At first, this paper analyzed pretreatment processes before the boiling process. In thepretreatment process, the maintenance of raw materials and frozen fruits is considered ascritical factor. Operating raw fruits is not so easy because of the high risk for spoilingand dispersion of quality. Due to the reason, all fruits are frozen by advanced quick frozentechnology even though they are in season. Fruits are thawed for 2 – 4 days to keep theirtaste, color and flavor. If the thawing amount is insufficient, it cannot meet productionrequirement in time because the quick thawing might lose fruits’ flavor and their specifiedquality. On the contrary, if the thawing amount is overfull, the surplus fruits are justwaste because it is impossible to suspend the operation and to re-freeze the raw materialsafter thawing. To operate proper thawing operation, the production amount (the boilingschedule) has to be scheduled before one week at least. By deciding the boiling schedule,not only an efficient thawing amount but also the exact time to select size, shape and colorof raw materials, and the time to allocate resources (staff, machines) can be obtained.

Next, this paper also analyzed processes after the boiling process. In the inspectionprocess just behind the boiling process, the most of workers are part-time. To give theeffective boiling schedule, the available staff assignment can be obtained. Similarly, themanagement behind the inspection process can be also carried out by using available topromise (ATP) and delivery car securing.

3. Problem Finding.

3.1. STEP1: Analysis of production planning table. In the processing line, pro-duction planning tables are used to decide the combination and schedule of products. Thispaper viewed many production planning tables to uncover the bottleneck that should beimmediately improved. The tables show “commodity names”, “commodity codes”, “pro-duction amounts”, “raw materials”, “the number of containers”, “processing time” and“expected date to process”. There are 3 types of production planning tables, long-term,middle-term and short-term. The details are as follows.


• Long-Term Production Planning Table: This table has information for next 3months – half a year. The schedule is decided based on the performance of previousyear, market trend, weather and marketing forecast information in cooperation withemployees at the shop-floor and sales workers. This table is frequently updated atany time depending on the changes in social environment.

• Middle-Term Production Planning Table: This table has information for a lastmonth. The production schedule is decided by using the information and the salesforecast information with inventory amount confirmed at the passage of 20 days on apast month. This table mainly shows the production sequence, the production dateand the production amount.

• Short-Term Production Planning Table: This table has information of a lastweek. This table is based on the middle-term production planning table, and it isusually re-scheduled by interrupt jobs. This table mainly shows the raw materialsto use, and staff allocation.

As the results of analysis, the middle-term production is decided as the schedulingproblem in this paper.

3.2. STEP2: Summary of analysis. This paper obtained various types of informationby the interview with actual workers, and analyzed the manufacturing information. Thispaper used the fishbone diagram for analysis (Mouritsen et al., 2005, Li et al., 2005), whichclassifies the imaginable factors for the attributes “Why is the food-processing difficult?(For the understanding of issues)” and “How should we do to improve the manufacturingefficiency (For the understanding of solution)”. Some important factors were found bythe analysis. Some issues and solutions are described as follows (each issue is linked witheach solution by its number).

Issues

(1) The manager has decided an executable schedule and inventory amount only by hisempirical rules.

(2) At the selection process of raw materials (size, shape and color), there is some surplusworkforce.

(3) Processing fresh fruits is technically difficult because it is necessary to consider the rotrisk of raw materials. Fruits are preserved by freezing, and defrosted in the thawingroom on the thawing process. The thawing operation requires for 3 – 4 days, so, whenthe amount of thawing is not enough, the production might not be in time, otherwisewhen the amount of thawing is too enough, it might make a lot of waste.

(4) About the staff assignment in the inspection process, there is surplus workforce, butat the same time, the production target is not sometimes satisfied due to the shortageof workforce.

(5) The quality of products is gradually and rapidly deteriorated with stocking time. Thedeterioration is often occurred due to the pre-production.

(6) When the factory accepts orders, delivery due date and delivery cars are arranged byconsidering the production capacity.

Solutions

(1) It is necessary to develop the production scheduling system not depending on theexpert’s empirical knowledge and intuition.

(2) A proper number of part-time staffs might be obtained by deciding the manufacturingsequence at the boiling process.

(3) The efficient amounts of raw materials can be decided by fixing the production amountbefore one or more week.


(4) An efficient amount of workforce can also be decided by fixing the production schedulebefore one or more week.

(5) From the viewpoint of the freshness management, it might be necessary to producethe production sequence close to the delivery date.

(6) The actual production has occurred some opportunity loss. The opportunity lossmight be reduced as much as possible by replying the delivery due date based on therigorous production schedule. A production sequence might also be easily obtainedwith short computing time by giving capability that can freely update the priority ofproducts with same delivery due date.

4. Modeling. This section shows various factors concerning the production schedule.

• It needs two or more weeks from order receipt to shelves at stores. In general, theproducts are delivered within one month from the shipping order.

• A package used in warehouse consists of 24 bottles. The maximum productioncapacity of one day is about 6000 packages. 6000×24 = 144000 bottles are producedat full capacity.

• The weight of a bottle is about 150(g) on average, so the total production amountof one day is 144000(bottles)× 150(g) ≈ 21.5(t).

• The capacity of container to process the fruits is 500(kg) – 650(kg), which is differentaccording to the kinds of fruits. The amount of boiled fruits has to be changed tokeep the filling speed constantly due to the capacity constraint at the inspectionprocess.

• By assuming the average capacity of container as 550(kg), the production amountcan be written as 21.5(t)÷ 500(kg) = 43(containers).

• The filling process pulls the boiled partly-finished products into bottles. The filltime is about 10 – 15 minutes per one container.

• By using the average fill time as 11 minutes, the operation hour of one day is ex-pressed as 43(Containers)× 11(min) = 7.9(hrs) ≈ 8.0(hrs).

As for this study, managing setup operations with machine idle time to change thekinds of fruits is the most important factor. The setup time is calculated as follows.

At first, based on the production circumstance described above, the processing timeto boil fruits is given as 1p for the scheduling problem, where 1p is a batch, and 1pcorresponds to 11(min). By using parameter p, the maximum production time of one day(production capacity of one day) is expressed as pmax, and the problem in this paper isgiven pmax = 43p. The average time to change the kinds of fruits is 3(hrs), and the averagetime to change the types of bottles is 1.33(hrs), which are found by our investigation.Therefore, the setup time to change the kinds of fruits is also given as 16p, and the setuptime to change the types of bottles is also given as 5p.

This paper considers not only the setup time but also the washing time, the clearanceoperation of fruits seeds, the influence of peculiar soil bacterium of fruits and the differenceof color or taste of products (strong or weak) as a setup operation. First of all, productsvariety is given as n = 24, and products are defined as Ji∈N . In parallel with Ji, the kindsof fruits are given as K(Ji). This factory processes 13 kinds of K(Ji), so, it is describedas K(Ji) = {1, 2, · · · , 13}, where each number corresponds to each fruit.

The combinations between each product including the types of bottles, the deliverydue date and labor cost are expressed numerically on the basis of parameters describedabove. To calculate each factor, at first this paper arranges the fruits in the ascendingorder according to the strength of their color and taste. By providing the setup timeCK(Ji)K(Jj) from the fruit K(Ji) to K(Jj), the total labor time to setup is given by

CK(Ji)K(Jj) = EaK(Ji)K(Jj) + EbK(Ji)K(Jj) + EcK(Ji)K(Jj) + 16p; i, j ∈ N, i ̸= j, (1)


where 16p is the cost to change the kind of products, Eaij is a distance between fruits,Ebij is the labor time to change the type of bottles and Ecij is a prohibition rule.The distances between each fruit are given as Eaij to consider the labor time linked to

the setup operation. The distance is given by coordinates shown as follows.

EaK(Ji)K(Jj) =

If K(Ji) > K(Jj) CdK(Ji) − CdK(Jj),

If K(Ji) < K(Jj) CbK(Jj) − CbK(Ji),

Otherwise 0,

(2)

Cdk = {0.0, 0.8, 1.0, 1.2, 1.5, 1.6, 1.7, 2.0, 2.2, 2.5, 2.6, 2.8, 3.0},Cbk = {0.0, 0.6, 0.7, 0.8, 1.0, 1.1, 1.2, 1.3, 1.4, 1.6, 1.7, 1.8, 2.0}.

Equation (2) expresses the labor time to change the product with strong flavor to theproduct with weak flavor, and Equation (3) also expresses the labor time to change theproduct with to weak flavor to the product with strong flavor. Naturally, the labor time tochange the same kind of products meets Equation (4). By adjusting coefficient constantsbased on the standard value p, the values of Cdk and Cbk are determined numerically.The labor time to change the types of bottles is given by

EbK(Ji)K(Jj) = 5p× σB(Ji)K(Jj), (3)

B(Jk) = {0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0},where B(Jk) expresses the types of bottles, σij is a 0-1 decision variable, which is givenas σij = 0 if i = j, and is given as σij = 1 if i ̸= j.The prohibition rule Ecij given in Equation (1) is a constraint to keep the quality of

products; Ecij is to keep the fruits’ color, taste and flavor. In particular, the purposegiving the prohibition rule is to protect the problem of food allergy. This paper gives oneprohibition rule for the production line as

EcK(Ji)K(Jj) = IbK(Ji) × IaK(Jj) ×Xp, (4)

IbK(Jk) = {0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},IaK(Jk) = {0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1},

where Ibk is the product k processed in arbitrary time, Ial is the product l processed justafter the product k and X is an extremely large value.Each product is given the priority depending on the characteristic and sales perfor-

mance, and the priority is applied for the problem described later to evaluate the sum ofthe earliness and tardiness and the total inventory cost. To evaluate the sum of earlinessand tardiness, this paper uses the average inventory amount obtained by the inventorydetention period, the inventory level (upper and lower), the safety amount of inventory,the amount of shipping requirement and the compatibility between fruits such as penaltymainly due to the allergy. This paper defines earliness penalty, which means that whenthe process of product is completed earlier than its delivery due, the quality of productsmight be deteriorated. As the results, the wholesale price will be decreased even thoughit meets delivery requirement and the shipment is enough possible. The earliness is justlinked with the inventory cost, so, the problem assumes the earliness and tardiness as theinventory cost depending on the priority of products. This paper also gives the tardinesspenalty, which is the due date delay, and the factory of this paper does not allow the duedate delay. So, the tardiness can be considered as a constraint in the scheduling problem.


To give the priority of products, this paper divides all products roughly into threepatterns, the main products, the high-mix low-volume products and the seasonal products,depending on the characteristics of each product.

• Main products: These types of products are processed by large quantities, andthe partly-finished products are different. Most of the products have same content,but the types of bottles are different. Sales are predictable by the marketing result.The material cost is cheap due to the mass purchase and mass production, so, theinventory cost is the lowest than the any other types of products.

• High-mix low-volume products: The costs of production and raw materials arehigh due to the small-lot production and low-volume purchase, so, the inventory costis relatively high. At the same time, the products can be given high sales price, so,the profit meets high. The sales are comparatively predictable.

• Seasonal products: These types of products are premiere products, and they aremainly produced as the purpose of advertisement. They are produced as trial basisat every season, so, the customer trends are usually unpredictable because there isno sales information. The products are extremely low-volume purchase, so, the costsof production and raw materials are extremely high. Additionally the products aremore perishable than other products. As the result, the inventory cost of these typesof products meets extremely high, and the profit is also extremely high.

5. Formulation and Solution. The boiling process is modeled to reduce the deteri-oration rate of ingredients (fruits) caused by perishability, which is one of the criticalconstraint of food products, with the objective to minimize the sum of earliness and tar-diness. In the problem, fruits are considered as jobs, the boiling process is consideredas a machine. The perishable materials cannot be allowed by mass production with alarge amount of inventory. In actual, when the order with a large production amount isrecieved, one kind of job is divided into several times to process the varous kinds of jobs ina production period. To obtain an effective production schedule, the various kinds of jobshave to be processed while reducing the sum of earliness and tardiness with decreasingthe influence of setup operation. This paper assumes the most important factor in theboiling process as the setup operation to improve the manufacturing efficiency.

Preconditions

• The problem is formulated by a single stage with infinite production capacity and asingle machine, and there is no time constraint.

• The number of jobs, the production amount, the setup time and the job’s processingtime shown in the production schedule table are already known in advance, so, theyare given as constants.

• There is no idle time to operate setup, so, the setup operation is processed just aftereach job’s operation.

• The setup cost is not considered, and also neither the cost nor the time to prepareeach job’s operation are considered.

• Two or more jobs are not processed at the same time, and each operation is processedonly once.

• Intermittence, interruption and replacement of each job’s operation are prohibited.• Each operation is definitely completed within operation hours for each day.

Sets

• T – Set of Time; T = {0, 1, 2, · · · , tmax|tmax = 20× 43}.• N – Set of jobs; N = {1, 2, · · · , n}.

Decision Variable


• Ci – Completion time of job i.

Parameters

• tmax – Maximum production time.• Oi – Operation time of job i (including preparation time to setup job i).• Sij – Setup time between job i and job j.• di – Due date of job i.• ji – Priority of job i.• δit – 0-1 variable. If the operation of job i or setup operation St

ij is operated at timet, δit = 1 is given, otherwise δit = 0.

Problem

Minimizen∑

i=0

ji(di − Ci)2, (5)

Subject ton∑

i=1

δit ≤ 1, (6)

Ci ≤ di, (7)

Ci −Oi ≥ 0, (8)

i ∈ N, t ∈ T.

Equation (5) denotes the minimization of the sum of earliness and tardiness. Equation(6) expresses the constraint not to process two or more jobs simultaneously. Equation (7)is the constraint to prohibit the due date delay and Equation (8) is the constraint thatthe operation of initial job must be started after the beginning time of production.This paper applies a solution method based on Lagrangian relaxation. Relaxation

problem with a nonnegative parameter Λt (Luh et al., 1990) is described as

Minimizen∑

i=1

ji(di − Ci)2 +

tmax∑t=1

λt

(n∑

i=1

δit − 1

), (9)

Subject to (7), (8).

Equation (9) is transformed as

n∑i=1

ji(di − Ci)2 +

tmax∑t=1

λt

(n∑

i=1

δit − 1

)=

n∑i=1

(ji(di − Ci)

2 +tmax∑t=1

λtδit

)−

tmax∑t=1

λt. (10)

The first term of Equation (10) is the summation for i, the second term is fixed numberbecause λt is a constant. Constraint equations of the relaxation problem is based on t,the problem for each job can be written as

Minimize ji(di − Ci)2 +

tmax∑t=1

λtδit, (11)

Subject to (7), (8).

6. Experimental Results. This paper defined following criterions to evaluate the sched-uling problem.

• Upper Bound: The sum of earliness and tardiness obtained by the solution method.• Lower Bound: The sum of earliness and tardiness that naturally arises due to theoverlapping of due dates. The difference between the upper and the lower boundsare described as “Gap”. The minimum value of the gap is zero, so, it can be said tobe an optimal result when the value of gap is zero.


In numerical experiments, the correlations and relations of some types of productioninformation and the number of containers were mainly considered. This paper gave 300types of samples (a sample is used for the same meaning as an input data) for simulations,which were obtained from the historical information used by actual production, and thesesamples consisted of various types of production situations. The degree of optimizationmight be different depending on the types of samples because the number of jobs withearliness and tardiness will eventually increase when many jobs might have the same duedate. This problem is often found in the case with the large number of containers. On theother hand, the production with the small number of containers and jobs can easily reducethe sum of earliness and tardiness because the possibility of the confliction of due datemight be low. Therefore, it is possible to flexibly schedule the production in responseto the production capacity of the factory. So, the distribution of samples was firstlyexamined to understand the characteristics of samples based on the relation between thenumber of samples and containers, and the lower bounds are shown in Figures 1(a) and1(b). From the tendency as shown in Figure 1(a), this paper examined the result as shownin Figure 1(b) because the existence of correlation between the number of containers andthe lower bounds might be an efficient information for making a decision. From Figure1(b), it can be thought that the number of containers did not depend on the lower bound.Figure 1(b) also shows that most of containers fell within the range of 600 – 800 for eachlower bound. Therefore, this paper understood that the lower bounds and the numberof containers were uncorrelated. Additionally, this paper gave Figure 2 which shows thefrequency distribution of containers. Figure 2 shows that about 80% of samples werewithin 650 – 800 containers.

The Number of Samples

Lower Bound Lower Bound

The Number of Container

500

550

600

650

700

750

800

850

900

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16500

550

600

650

700

750

800

850

900

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0

5

10

15

20

25

-1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0

5

10

15

20

25

-1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

(a) (b)Lower Bound Lower Bound

The Number of Samples The Number of Containers

Figure 1. Distribution of input data


The Total of Containers in Input Data

0

5

10

15

20

25

30

35

40

575 590 605 620 635 650 665 680 695 710 725 740 755 770 785 800 815 830 845

Figure 2. Variance of the number of containers


Figure 3 shows the distribution of jobs with earliness and tardiness obtained by thesolution method for all samples. The lower bounds of samples were almost evenly dis-tributed in the range of 0 – 14, but Figure 3 shows the tendency, which the number ofsamples with 12 – 13 jobs with earliness and tardiness got high frequency. The result givesthe possibility to improve the production efficiency, because the samples with 12 and 13jobs with earliness and tardiness might have uncovered important factors. On the otherhand, this paper found that there is no correlation between the number of containers andthe lower bounds.

0

10

20

30

40

50

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19


The Number of Jobs with Earliness or Tardiness

Figure 3. The number of jobs with earliness and tardiness depending onthe number of samples

Figure 4 shows the relation between the upper bounds (results obtained by the solutionmethod) and the lower bounds obtained by 300 simulations. In Figure 4, it sometimeshappened that two or more results overlapped at one point. Figure 4 also shows the ap-proximated line y = 0.8x+4.1 of the experimental results, and y = x is ideal value becausethe upper bounds are coincided with the lower bounds. We can see the performance ofsolution method by the degree of coincidence between the upper and lower bounds. Thegap between the upper and lower bounds got smaller according to the increase of lowerbound. As the gap became large with decreasing the lower bound, this tendency might bedue to the influence of the prohibition rule and each job’s priority (The higher priority wasgiven depending on the magnitude of benefit). Based on the result, this paper considersthat it is necessary to give the multi-objective formulation with not only the reductionof the number of jobs with earliness and tardiness, but also the maximization of totalbenefits.

0123456789

10111213141516171819

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Lower Bound

Upper Bound

Figure 4. Averages of upper and lower bounds


This paper examined “operating rate” and “real operating rate” defined as follows.

• Operating rate: The manufacturing time includes real manufacturing time, setuptime and idle time. The operating rate consists of the total of manufacturing andsetup time, and it is the percentage of operating time to process products not in-cluding idle time in the total capacity of this factory. This is also described asthe production line usage rate. The factory has 20 manufacturing days described as20(slots), and each slot consists of 43p discrete manufacturing containers above men-tioned, so, the total manufacturing capacity of this factory is 20(slots)×43p = 860p.

• Real operating rate: This is the percentage of net operating time in the totaloperating time. The operating rate includes the setup time to operate preparatoryworks such as the setup or maintenance operation, so, this is removed the preparatoryworks from the total operating time.

At first, this paper examined the operating rate and the real operating rate obtainedby the solution method. Figure 5 shows the production line usage rate and the numberof containers for the real operating rate. As shown the left graph in Figure 5, the realoperating rate rose with increasing the operating rate. From Figure 5, two situationswere considered as a factor raising the operating rate. The first was that the sample hasa lot of containers and the second was that the production schedule spends a lot of timeto operate setup. To find out a primary factor from the two causes, the real operatingrate was examined. Under the samples used by experiments, the operating rate waspositively correlated with the real operating rate. From the right graph in Figure 5, thereal operating rate also rose (linearly-increased) with increasing the total of containers,so the solution method is said to be effective for samples with a lot of containers.

Production Line Usage Rate The Number of Containers

Real Operating Rate Real Operating Rate

84.0

86.0

88.0

90.0

92.0

94.0

96.0

98.0

65.0 75.0 85.0 95.084.0

86.0

88.0

90.0

92.0

94.0

96.0

98.0

550 600 650 700 750 800

Figure 5. Production line usage rate and the number of containers

Usually, increasing the kinds of jobs enhances the possibility of occurrence of the setupoperations. The real operating rate will drop when incompatible combinations of jobsmake a lot of setup operations even though there is small number of jobs. Therefore, toexamine which the factor gives the most significant impact for the real operating rate,this paper examined the relation between the number of containers and the number ofjobs with earliness and tardiness as shown in Figure 6. Figure 6 mainly shows the sumof the earliness and tardiness with the variance of containers. The numbers described inthe circle under the graph in Figure 6 mean the frequencies of samples, and each radiusgets bigger depending on the frequency of samples. Only the samples of this paper cannotconclude that the cause of setup operations was due to the number of containers, however,it seems that incompatible jobs reduced the real operating rate. The sizes of circles werenearly depending on the number of samples for each container, so, the dispersion of


samples for the sum of earliness and tardiness might be influenced by the dispersion ofcontainers given as input data. Usually we can find the trend that increasing containersmakes many jobs with earliness or tardiness, but Figure 6 did not show the trend. Thecorrelation between the number of containers and the sum of earliness and tardiness was0.08. Given the lower bounds as shown in Figure 1, it was found that the solution methodcan reduce the earliness and tardiness as much as possible.

Distribution of Containers in Input Data

Sum of Earliness and Tardiness

5 6 8 9 10 11 12 13 15 17

590 620 650 680 710 740 770 800 830

0

3

6

9

12

Figure 6. Sum of the earliness and tardiness with the variance of containers

Figure 7 shows the percentages of setup time in the operating rate for the numbers ofcontainers. The percentage of setup time decreased with increasing the containers. Fromthis result, we can find that the reduction of setup time might directly contribute to theimprovement of real operating rate. When there were a lot of containers in a sample, notonly the current objective but also the minimization of setup time were simultaneouslyachieved by giving the sum of earliness and tardiness as objective function. On the otherhand, when there were a small number of containers, the total of setup time was not fullyminimized. To prepare some kinds of formulations depending on the scale of problems isthought to be important.

10

12

14

16

0

2

4

6

8

550 600 650 700 750 800 850

The Number of Containers

The Percentage of total setup time with

Respect to All of Production Line Usage

Figure 7. The percentage of total setup time

Figure 8 shows the change of average values of the gaps with respect to the lower bounds,and the numbers of jobs with earliness and tardiness are also described here for reference.In Figure 8, the label of vertical axis on the left side denotes the gaps between the upperand lower bounds, and the label of vertical axis on the right side denotes the numbers ofjobs with earliness and tardiness, where the upper bound is the value of objective function.


At first, the change of the numbers of jobs with earliness and tardiness were examined.The numbers of jobs with earliness and tardiness increased approximately linearly withincreasing the lower bounds. On the other hand, the gaps between the upper and lowerbounds became smaller with increasing the lower bounds. Therefore, the solution methodmight be more effective for samples with large lower bound, because the smaller the gapwas, the better the results were obtained. With this results, the main cause increasingthe jobs with earliness and tardiness is considered as the overlapping of due date. Thesolution method was relatively-ineffective for samples with small lower bound, so, theperformance of solution method is thought to be directly influenced by the setup time.As the real operating rate of the small number of containers was low due to the increaseof setup time, the number of containers might have a strong correlation with the lowerbounds. So, this study considers that this issue might be solved by giving proper duedates based on setup operations of all jobs. This study will address the problem aboutdue dates scheduling as a new challenge of this production site, and based on the availabledue dates obtained by the new scheduling model, the solution method might obtain moreefficient results.

Lower Bound

The Number of Jobs with

Earliness and Tardiness

Gaps between the Upper

and Lower Bounds

0

1

2

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4

5

6

1 3 5 7 9 11 13 150

2

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8

10

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Figure 8. Gaps between upper and lower bounds

Comparison of Average

• Real Operating Rate: 91.2%• Production Line Usage Rate: 86.1%• Sum of Earliness and Tardiness: 4.7

This paper summarized the performance of solution described above. The more con-tainers the sample had, the more the percentage of the total setup time in the operatingrate reduced; the loss of freshness due to setup operation will be avoided. The solutionmethod is effective for scheduling with a lot of containers or a lot of due date overlap, so,it is expected to apply the solution method for multi-production line.

7. Conclusions. This paper examined a scheduling problem in a food processing factorywith an important constraint, perishability and aimed to manage the deterioration rate ofproducts. Especially, this paper concentrated on the boiling process, and by modeling itsprocessing flow, the relation between the production capacity and the variety of productswas examined for reducing the sum of earliness and tardiness. At first, all processes inthe factory were observed, and a problem at the boiling process was considered based onthe analysis results. Next, this paper developed a computer aided support system with


a heuristic solution method using Lagrangian relaxation, which can show the processingsituation visually and schedule immediately. In the actual production field, there hadbeen some interrupt jobs, and the managers had been required to reschedule frequently.From now only experts had been able to make useful production planning, but the sup-port system can obtain available schedule constantly in a short time without experts’techniques. The system also gave the capability to reflect the modification of scheduleinstantly in response to the production situation.In the experiments, the solution method showed the trend that the real operating rate

was positively correlated with the number of containers. Therefore, the solution methodmight be available for the case with many containers, and it also might make the bestpossible use of the factory’s capacity. This paper aimed to minimize the sum of earlinessand tardiness, but the real operating rates were 90% or above for both the operating rateand the number of containers. The objective of this study satisfied the reduction of notonly the sum of earliness and tardiness but also the setup time concurrently. The morethe samples had many containers, the more the solution method got effective results. Thispaper gave higher priority for minimizing the sum of earliness and tardiness than reducingthe setup costs, so, as a future work, we will examine a different strategy depending onthe scale of input data. As a point we should improve rapidly, this paper will consider thebalance between minimizing the sum of earliness and tardiness and maximizing the realoperating rate. The introduction of factors such as the percentage of setup time in totalmanufacturing time, and the priority or penalty to minimize the deterioration of qualitywill be considered as an idea to expand the problem.In addition, we will examine following subjects. At present, all variables and coefficients

in the scheduling problem were previously given only by our observation and analysis forthe actual field. The predetermined problem and solution have usually been difficult todeal with accidental events such as machine breakdown and drastic changes of demands(priority of products). The major cause of these difficulties is thought that some coefficientconstants in the scheduling problem were fully based on the evaluation standards given byour subjective view. We consider that the proper factors might be only known by expertswho make production schedule, and their knowledge at actual manufacturing field canunify many factors into one evaluation measure. Experts can make adequate decision-making by their specialized technique, and they can deal with unexpected events as quitesimple trouble by finding general rules from accidental events. The present difficulty willbe solved by implementing interactive feature which can store the expert’s know-how, andthen based on the expert’s decision, some weights of various factors will be determined.As the result, a system with experts’ technical know-how will be constructed. We alsoaim to success the experts’ know-how. By expressing the experts’ techniques definitely,ineffectual procedures might be discovered, and then, a further improvement might beexpected. Similarly, the expansion of the scheduling problem for high-mix low-volumeproduction line, the shortening of makes span, and the reduction of setup cost will beconsidered as future works.

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relation between production capacity and variety of products on a

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