application of triplet notation and dynamic ...

International Journal of Business Intelligence Research, 1(2), 9-20, April-June 2010 9

application of triplet notation and dynamic programming to single-line, Multi-product dairy production scheduling Virginia Miori, St. Joseph’s University, USA Brian W. Segulin, RoviSys Co., USA

aBstraCt The application of optimal methods for production scheduling in the dairy industry has been limited. Within supply chain terminology, dairy production was generally considered a push process but with advancements in automation, the industry is slowly transforming to a pull process. In this paper, the authors present triplet notation applied to the production scheduling of a single production line used for milk, juice, and carnival drinks. Once production and cleaning cycles are characterized as triplets, the problem is formulated. Lagrange relaxation is applied and the final solution is generated using dynamic programming. Keywords:

Dynamic Programming, Production Scheduling, Pull Process, Push Process, Triplet

introduCtion The application of optimal methods for production scheduling in the dairy industry has been limited. The predominate need that has been addressed is the need to forecast supply. As the dairy industry implements advanced production equipment technologies and demand forecasting becomes stronger, dairies may begin to push back on suppliers and enable efficient production schedules. This paper addresses this emerging area of application.

DOI: 10.4018/jbir.2010040102

Raw milk is initially processed into two variations, skim, and whole. One percent and two percent milk are the result of blending appropriate ratios of skim and whole milk. Therefore, a single filling line may simultaneously produce all four varieties of milk (Mans, 2007). Other product families such as orange juice, carnival drinks, and buttermilk are also processed using the same filling line although separate storage tanks are used for each product family. Product is piped into one of two bowls for filling. Between batches of different product families, the equipment must be cleaned. Less extended cleaning can also be required between products within the same product family.

Copyright © 2010, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.

10 International Journal of Business Intelligence Research, 1(2), 9-20, April-June 2010

A primary goal of this scheduling approach is to facilitate the transition of the dairy operation from a push system to a pull system. Order due dates will act as constraints in the problem formulation and ultimately guide the order of production. Customers have standing due dates each week for their orders and these dates must be met. Additionally the scheduling approach will reduce inventory held. In the past excess inventory was carried when it was uncertain as to whether an order could be slotted into the production schedule. A very limited amount of research has been completed in this area. We begin with presentation of literature both directly and indirectly addressing the dairy industry. Chemical industry scheduling models and other type of food processing scheduling models offer insight into appealing approaches to the dairy problem. We then formulate the scheduling problem using triplet notation where the first leg of the triplet represents a production run and the second leg of the triplet represents a cleaning cycle. Constraints are relaxed using Lagrange relaxation and the problem is them solved using dynamic programming.

literature Truckload vehicle scheduling provided the first stepping off point for this research. The nature of productive vehicle movements followed by unproductive movements presented by Miori (2006, 2008) paralleled the nature of productive scheduling as composed of productive activity followed by unproductive activity such as cleaning and packaging transitions. Batch production scheduling within other industries has been studied at length. The most immediate carry-over in the literature occurs in the chemical industry. Brucker and Hurink (2000) applied a two-phase tabu search to the problem of scheduling batch production to a particular facility. The batches were scheduled in order to meet order deadlines. Production and cleaning times were examined in the tabu search approach as well as in a general job-shop scheduling ap-

proach. Wang and Guignard (2002) created a MILP formulation for continuous-time batch processing in the chemical industry. Burkard and Hatzl (2006) applied a heuristic minimizing makespan to batch scheduling problems in the chemical industry. The heuristic was an iterative construction algorithm with recommended diversification and intensification strategies to obtain good suboptimal solutions. Tang and Huang (2007) applied a neighborhood search within a two-stage heuristic to rolling batch scheduling for seamless steel tube production. Cheng and Kovalyov (2001) solved the scheduling of multiple batches on a single machine much like the scheduling that must be performed in this dairy example. The primary objective was to minimize cost while also minimizing maximum lateness, the number of late jobs and the weighted completion time. The authors offered a classification of computational complexities and present efficient dynamic programming algorithms for the problem. Li and Yuan (2006) too discussed scheduling on single machines with the three hierarchical criteria of minimizing makespan, minimizing machine occupation time, and minimizing stock-out cost. Dynamic programming was also used to solve the problem. Yuan, Liu, Ng, and Cheng (2006) applied dynamic programming to single machine batch scheduling problems. Their objective was to minimize makespan when faced with product-family setup times and order release dates. Continuous and discontinuous material flow scheduling in process industries was presented by Neumann, Schwindt, and Trautmann (2005). The interpretation of the problem as both continuous and discontinuous carries over nicely into the dairy problem. Though individual orders are being processed, a single batch may actually provide the needed product for multiple orders. Therefore the production remains continuous but with established discontinuities. The basic scheduling problem is solved using a branch and bound technique. Batch scheduling with identical process times was discussed by Quadt and Kuhn (2007). The production lines in use were flexible flow line much like those we will examine in this



paper. Setup or cleaning is incurred between jobs of different product families. The authors minimize setup time and the mean flow time using two nested genetic algorithms. Production control in the dairy industry was the subject of work by Nakhla (1995). He addressed the overall impact of efficient production scheduling within a dairy operation and examined the appropriate balance between optimization models and work rule approaches. The concept of optimization was discussed with an objective of minimizing cost and traditional constraints including resource availability, demand, and order delivery dates. Basic rules of thumb that describe the carry over from an optimal solution to operations adjustments were presented. Early research was published by Schuermann and Kannan (1978) on a production forecasting and planning system. After forecasting supply for 1-gallon packages of milk, the forecast acted as input to a production scheduling model. The production scheduling was performed using an LP that minimized total cost, and provided a starting solution for the scheduler to customize. Supply forecasting also figured prominently in Benseman (1986). Production planning was performed for the New Zealand Co-operative Dairy Company based on quotas determined by dairy farmers. An LP that maximized profit was generated. It used a yield analysis to determine final production quantities and produced an acceptable starting production schedule. Mellalieu and Hall (1983) followed up on this work and presented a network-planning model that used the production schedule as input and made higher-level decisions for the co-operative. Killen and Keane (1978) took supply forecasting to the point of predicting the distribution of cow calving dates. They used an LP model to establish seasonal patterns in supply that improved their ability to schedule production. The most significant overlap between our research and published literature existed in a case study performed by Classen and van Beek (1993). They developed a MILP that resolved bottlenecks in the packaging facility in order to produce an improved master schedule for a cheese packaging operation. Changeovers and

order due dates were considered while examining clusters of jobs. The MILP minimized mean flow time.

dairy operation Milk, orange juice, carnival drinks, and buttermilk are all produced by the dairy and all held in their own storage tanks. Product is transferred from the storage tanks into two filling bowls, from which product is dispensed into the appropriate size bottle to fulfill the current order. For all products other than milk, unique product types are transferred into each filling bowl. Though the scheduling model handles all products and all transitions between products, for the purpose of illustration, in this paper we focus primarily on milk. Three possible configurations can be used for the filling bowls. The first two are simpler in nature. Both bowls contain skim milk or both bowls contain whole milk. These configurations are employed when long runs of a single type of milk, skim or whole, are scheduled. The homogeneous use of the bowls provides greater speed and efficiency in production. The third configuration is heterogeneous: one bowl contains whole milk and one bowl contains skim milk. Every variety of milk can be produced in the heterogeneous configuration by adjusting the ratios of whole and skim milk dispensed into the milk bottles. The non-milk products are made off line and stored, before being placed in a single filling bowl. Note that variations between product types result in the need to clean the filling bowls, supply lines, and valves between production runs. Regardless of the product, the procedure remains the same. Bottles are labeled and filled; each label not only states the contents of the bottle, but the specific customer name. All products can be generalized into product families, the necessary cleaning cycles are determined based on both product family and purity of product. In addition, transitions must be scheduled when packaging changes are required. The size of the final bottle can change (ex: gallon to half gallon) and the final configuration of bottles



can change (ex: single bottle to two-pack to four-pack of bottles).

product families Though orders are placed for individual products, production times and cleaning times are determined by product family. Table 1 provides a sample of product families and individual products. This is certainly not an exhaustive list, but it is useful in illustrating production and cleaning sequences.

Cleaning Matrices A full cleaning cycle (FC) is required between consecutive production runs of any two product families. Cleaning may also be required between production runs of individual products within a product family. For example, a rinse (R) would be required after high pulp OJ if no pulp OJ were the next product to be produced. A rinse (R) would not be required however, if the production were to proceed in the opposite order. In any sequential production runs between which cleaning is not required, a cleaning type of no-clean (NC) is designated with a completion time of zero (0) minutes. The required cleaning cycles are presented in Table 2.

packaging transitions Non-milk products are produced and sold in single gallon quantities. The milk products however, may be packaged in three different ways: single gallon, two-pack of gallons, or four-pack of gallons with each pallets holding only one package type. We must not only account for cleaning between products, but we must also account for any package transitions. The production sequence is altered when shifting between single and multi-packs, multi-packs and singles and within multi-packs. A bundler is used to combine single gallons into multi-packs. Single gallons are diverted from the bundler while gallons to be combined into multi-packs are sent to the bundler. When transitioning from a single gallon to the multi-gallon packaging (two-pack or four-pack) and the products are no longer diverted away from the bundler, the bundler then shifts from an idle state to an active state. The shift also requires confirmation from the operator that the bundler is configured for the proper bundle type. When transitioning from a multi-pack to single pack, the bottles must be stopped at the diverter while the bundler is flushed of all bundles destined for the previous pallet. Transitions between multi-pack products require a manually triggered change in the bun-

Table 1. Product families Product Family

Product Subtypes

Milk (M)

Whole Milk 2% Milk 1% Milk Skim Milk

Orange Juice (OJ)

No Pulp OJ Medium Pulp OJ High Pulp OJ Calcium Fortified OJ

Carnival Drink (CD)

Orange Drink Fruit Punch Grape Drink

Buttermilk (B)

Whole Buttermilk Low fat Buttermilk



Table 2. Required cleaning cycles Milk W

Milk

2

OJ

1

S

W

NC

R

R

R

2

NC

NC

R

R

1

NC

NC

NC

R

S

NC

NC

NC

NC

NP OJ

NP

MP HP

FC

C

MP

CD HP

C

FC

NC

NC

NC

F

R

NC

NC

R

R

NC

R

R

R

R

NC

FC

FC

G B

L W

FC

G

L

W

FC

FC

FC

FC

NC

R

R

R

NC

R

R

R

NC

FC

dler configuration, and confirmation from the operator that the transition has occurred. This includes transitions from two-pack to four-pack as well as four-pack to two-pack. The time consumed by any of these package transitions is stochastic and ties directly to the current state of production. For purposes of this paper, we will however assume deterministic time and leave the characterization of the stochastic time to subsequent research. The package transition matrix follows. NC designates no change in the line configuration and consumes zero (0) transition time. AB is the activation of the bundler, DB is diversion from the bundler, and RB is the reconfiguration of the bundler.

F

NC

R

O CD

O

B

FC

FC

NC

NC

R

NC

sChedulinG Model Before discussing the solution method, we first review the use of triplets in production schedule. We then move in to the use of these triplets in the solution formulation and ultimate sample problem solution.

triplets The formulation of the scheduling model presented in this paper utilizes a triplet concept (Miori, 2006). The use of triplets has been established in transportation modeling where the first leg of the triplet designates a loaded movement and the second leg designates an empty movement. A scheduling triplet is composed of

Table 3. Packaging transitions Single

Two-Pack

Four-Pack

Single

NC

AB

AB

Two-Pack

DB

NC

RB

Four-Pack

DB

RB

NC



a productive activity (first node), an unproductive activity (second node) and followed by a subsequent productive activity (third node). For dairy production, this means milk production, followed by transition between the first node production run and the third node production run. The third node acts only as a placeholder. The unproductive activity, or transition, must logically follow from the productive activity in that triplet. The transition will leave the equipment in one of several states. The subsequent triplet to be scheduled must require the equipment to begin in this state. As triplets are chained together to create a schedule, the third node of a triplet must always match the state of the first node in the subsequent triplet. This creates a logical connection between the production activities. Production and transition times are determined for each node of the triplet. The triplets are generated from the orders for products so that any triplet corresponds to a single order. Multiple triplets are created for every order to allow more sequencing options to be considered, though once the order has been produced; all associated triplets are eliminated from consideration. The triplet time is the sum of the individual node times. The third node is always considered to have a time of zero (0). The production time reflects a base production rate of 5 minutes per pallet. The transition time includes the accumulation of cleaning and packaging transitions necessary between the first and third nodes. Recall that the cleaning transition times are deterministic and for purposes of this paper, the stochastic packaging transitions are treated as deterministic. Costs are accumulated in the same manner as times. The schedule is made up of a series of triplets, the duration of which is a single day in the production operation. Multiple schedules are created from a single repository of demand for multiple production lines. Solution Formulation Binary (0,1) decision variables, represent the use of a triplet (value of 1) or the lack of use (value of 0). These are designated pc as x tuv and the cost of a triplet is designated as

cpc , where t = production start time, u = production end time (coincident with transition start time), v = transition end time, p(o) = product demanded in order o, and c = transition type. Demand and order delivery dates are known in advance and completed orders must be staged for transport by their delivery date. The scheduling horizon is 6 days long, with two shifts per day. At the end of the second shift each day, the filling line is emptied and cleaned so that no production runs carry over to the following day. The production-scheduling objective(1) is to minimize production cost that implicitly minimizes machine downtime: Min

å

p (o ),ctuv

p (o ),c cp(o ),c x tuv .

(1)

The initial constraint ensures that all demand scheduled for immediate receipt by customers be served (2). If production time is available, the next day’s demand may begin processing before close of the current day’s production. Demand is denoted as y p(o ),d (d = day order due).

∑y ctuv

p (o )d

p (o ),c x tuv ≥1

∀y p(o ),d ≥ 1

(2) As in the traveling salesman problem, conservation of flow (3) must also be achieved.

∑

p (o ),ctuxy

p (o ),c p (o ),c (x tuv −x vxy )=0

∀v

(3)

Schedule feasibility must be maintained. We accomplish this through constraint (4), which maintains successive process times and constraint (5), which forces on-time order fulfillment. t ≤u ≤v


p (o ),c ∀x tuv

(4)


p (o )c ∀x tuv

u ≤ Dp(o )

(5) Finally, the decision variables are all binary (6). (6)

p (o ),c x tuv Î (0, 1)

solution Method Dynamic programming is ultimately used to solve this problem, but before doing so we apply a penalty method with Lagrange relaxation.

penalty Method Two constraints are removed from the formulation, the service of demand constraint (2) and the on-time production constraint (5). These constraints tie directly to our first priority of overall customer service. By relaxing these constraints and applying penalties, we are able to perform extended what-if analysis. The new scheduling model formulation follows with penalty parameters λ1 and λ2. The conservation of flow constraint and schedule feasibility constraint remain in the model. Min

∑

p (o ),ctuv

p (o ),c p (o ),c cp(o ),c x tuv − l1 (∑ y p(o )d x tuv − 1) + l2 (u − Dp(o ) ) ctuv

(7) Subject to:

∑

p (o ),ctuxy

p (o ),c p (o ),c (x tuv −x vxy )=0

∀v

(8) t ≤u ≤v

p (o ),c ∀x tuv

(9) p (o ),c x tuv Î (0, 1)

(10)

dynamic programming We now apply dynamic programming to this formulation. Note that the dynamic programming recursion specifically addresses the revised objective function, but the remaining constraints must also be addressed. The recursion is written to progress forward in time when building the schedule, therefore it will ensure the proper progression through time and satisfy constraint (9). In addition, the iteration procedure through the possible triplets and the end condition for the recursion fulfill the conservation of flow constraint (8). All production schedules start at time 0 with the equipment in idle mode. The development of a production schedule requires the ending time in each triplet to be coincident with the starting time of the next triplet. The schedule ends at the completion of a day’s production, again a time based measurement and all the equipment ends in the idle state. The recursion requires the following notation: A(k+1,k) = Available triplets on day k+1, depends on triplet chosen on day k S(k) = The triplet chosen on day k C s (k ), j = Cost of adding triplet j after the triplet chosen on day k f(i,k) = the cost to go if we choose triplet i on day k f(1,0) = 0 U(I,k) = The number of unserved orders if we choose triplet i on day k a(s(k )) = The initial time in the triplet chosen on day k (t,,) b(s(k )) = The intermediate time in the triplet chosen on day k (,u,) d(s(k )) = The finall time in the triplet chosen on day k (,, v) Each day we select the triplet z, which minimizes the cost of the triplet selected at the current time and the cost to go from the current time forward. We also select the Lagrange multipliers λ1, and λ2. The full recursion takes the form:



Figure 1. Schedule decision tree

f (s(k ), k ) =

min

z ∈A(k +1,k ),l1 ,l2

p (o ),c {C s((kk)),z − l1 (∑ y p(o )d x tuv − 1) + ctuv

l2 (u − Dp(o ) ) + f (z, k + 1)}

(11) The dynamic program is solved for each day of the schedule using custom C++ code. This code manages the demand and, allowing new orders to be considered and completed orders to be classified as such. It creates the ability to maintain a rolling scheduling horizon which can then yield a week’s schedule (6 working days and two shifts per day) to be produced.

saMple solution We present a short manual example to illustrate the triplet schedule building process. Figure 1 provides the full decision tree for a problem with three triplets: a, b and c. These same triplets recur at each branch of the decision tree and each level (L0, L1, L2, L3, and L4) represents a scheduling decision. The Procedure begins at the source or start node of the network at L0. The first triplet in L1 to be considered is triplet a. The schedules which emanate from triplet a at L1 are schedules 1 – 9 in Table 4. Those from triplet b at L1 are schedules 10-18

and the schedules emanating from triplet c at L1 are numbered 19-27. The production schedule is initialized at time zero (0) in an idle mode. No matter the length of the shifts each day, we reset the entire schedule every morning. The production schedule must also end in an idle mode. Recall that the end time for each scheduled triplet is coincident with the starting time of the subsequent triplet. We must also meet the stated constraints that ensure the proper progression of the schedule through time. We must satisfy all time windows as well as satisfying conservation of flow. The orders for the sample problem are contained in Figure 2. These orders are used to demonstrate the solution method and extend for one shift. Eight large volume orders must be scheduled and these eight orders accumulate to 96 pallets. The orders have two different production due dates that are derived from the order delivery due dates. The vehicle time in route and work time required to load and unload vehicles are accumulated and then backed off from the order delivery due dates to yield the production due date. The standard approach to scheduling production at the dairy had been based solely on delivery timing requirements. No production



Table 4. Potential schedules designated as sequence of triplets Schedule #

L0

L1

L2

L3

L4

1

a

a

a

a

a

2

a

a

a

b

a

3

a

a

a

c

a

4

a

a

b

a

a

5

a

a

b

b

a

6

a

a

b

c

a

7

a

a

c

a

a

8

a

a

c

b

a

9

a

a

c

c

a

10

a

b

a

a

a

11

a

b

a

b

a

12

a

b

a

c

a

13

a

b

b

a

a

14

a

b

b

b

a

15

a

b

b

c

a

16

a

b

c

a

a

17

a

b

c

b

a

18

a

b

c

c

a

19

a

c

a

a

a

20

a

c

a

b

a

21

a

c

a

c

a

22

a

c

b

a

a

23

a

c

b

b

a

24

a

c

b

c

a

25

a

c

c

a

a

26

a

c

c

b

a

27

a

c

c

c

a

efficiencies were attempted because greater emphasis was placed on achieving 100% customer service satisfaction. A typical schedule would begin with a load diagram. The product required for each vehicle would be produced in succession, eliminating any production efficiencies. Often unnecessary transitions would occur due to product type and package type. Four types of milk are produced. As we would expect, these are skim, 1%, 2% and

whole milk. The milk production is modeled as the first node of the triplet. The second node of the triplet models the required transitions, which include (bowl) cleaning and packaging transitions. The third node represents subsequent milk production that again may be any of the four milk types. Figure 3 shows the triplets under consideration in this example. Note that this is significant reduction from the complete set of triplets. By imposing the link between



production runs and transitions, we have been able to minimize the number of triplets under consideration. This provides significant benefits in processing time, especially as the scheduling horizon and the number of products increase. After running the data through the dynamic programming model, a production sequence is created. The sequence for this set of orders is presented in Figure 4. Completion of the schedule allows for the creation of a load diagram. This is a visual representation of the physical placement of the orders once they are loaded into the waiting trailers. They are loaded in the opposite of the delivery sequence. Figure 5 shows the load diagram. We noted that this schedule considers only milk products; currently juices and carnival drinks are restricted to production on the weekends. As the dairy operation matures, more complexity will be added to the schedule due to increased packaging variations and additional products. The production scheduling model supports all of these increased complexities. Production scheduling becomes even more important. The use of the scheduling model allows the dairy to build these products into their regular production schedule, affording them to opportunity to make the most efficient schedule for all products.

ConClusion The triplet notation and dynamic programming provide an effective way to automate production scheduling in the dairy industry. This method is made possible by advancements in the equipment and processing of dairy products. We are able to leverage off these developments and take dairy from an entirely push process to a pull process yielding reductions in cost, consistency in scheduling and greater plant efficiency.

extensions The ultimate goal of the dairy operation is find a balance between optimal routes and optimal

production scheduling. To accomplish this, we can add an out of-route constraint to the production scheduling model. This constraint may be relaxed in the Lagrange relaxation and assigned an appropriate penalty parameter before using the dynamic programming engine to solve the problem. It allows the model to provide a benchmark optimal schedule and then produce a scenario of the schedule that reflects the route requirements. The dairy scheduler then has the opportunity to determine whether this cost increase is adequately offset by the decrease in routing cost. This step is necessary to coordinate the use of two separate scheduling systems (one for production and one for routing). In this way, the dairy may leave the routing system as is and use the flexibility in the production scheduling system to achieve the best achievable performance. Further planned extensions of this work include applying the dynamic programming engine to pipeline scheduling, primary steel scheduling and fenestration scheduling.

referenCes Benseman, B. R. (1986). Production Planning in the New Zealand Dairy Industry. The Journal of the Operational Research Society, 37(8), 747–754. Brucker, P., & Johan, H. (2000). Solving a chemical batch scheduling problem by local search. Annals of Operations Research, 96, 17–38. doi:10.1023/A:1018959704264 Burkard, R. E., & Hatzl, J. (2006). A complex time based construction heuristic for batch scheduling problems in the chemical industry. European Journal of Operational Research, 174, 1162–1183. doi:10.1016/j.ejor.2005.03.011 Cheng, T. C. E., & Kovalyov, M. Y. (2001). Single machine batch scheduling with sequential job processing. IIE Transactions, 33, 413–420. doi:10.1080/07408170108936839 Classen, G. D. H., & van Beek, P. (1993). Planning and scheduling packaging lines in food industry. European Journal of Operational Research, 70(2), 150–158. doi:10.1016/0377-2217(93)90034-K



Killen, L., & Keane, M. (1978). A Linear Programming Model of Seasonality in Milk Production. The Journal of the Operational Research Society, 29(7), 625–631. Mans, J. (2007). Computer-controlled filler runs all fat levels of milk. Packaging Digest, 44(2), 22–26. Mellalieu, P. J., & Hall, K. R. (1983). An Interactive Planning Model for he New Zealand Dairy Industry. The Journal of the Operational Research Society, 34(6), 521–532. Miori, V. M. (2006). A Novel Approach to the Continuous Flow Truckload Routing Problem. Applications of Management Science . In Productivity, Finance, and Operations, 12. New York, NY: Elsevier JAI. Miori, V. M. (2006). A Dynamic Programming and Econometric Modeling Theoretic Approach to the Truckload Routing Problem Using a Triplet Formulation. Unpublished doctoral dissertation, Drexel University, Philadelphia, PA. Miori, V. M. (2008). A Dynamic Programming Approach to the Stochastic Truckload Routing Problem. Manufacturing, Distribution and Transportation in the Supply Chain: Modeling, Optimization and Applications. London: CRC Press. Nakhla, M. (1995). Production control in the food processing industry, The need for flexibility in operations scheduling. International Journal of Operations & Production Management, 15(8), 73–88. doi:10.1108/01443579510094107

Neumann, K., Schwindt, C., & Trautmann, N. (2005). Scheduling of continuous and discontinuous material flows with intermediate storage restrictions. European Journal of Operational Research, 165, 495–509. doi:10.1016/j.ejor.2004.04.018 Quadt, D., & Kuhn, H. (2007). Batch scheduling of jobs with identical process times on flexible flow lines. International Journal of Production Economics, 105, 385–401. doi:10.1016/j.ijpe.2004.04.013 Schuerman, A. C., & Kannan, N. P. (1978). A production Forecasting and Planning System for Dairy Processing. Computers & Industrial Engineering, 2(3), 153–158. doi:10.1016/0360-8352(78)90025-6 Tang, L., & Huang, L. (2007). Optimal and nearoptimal algorithms to rolling batch scheduling for seamless steel tube production. International Journal of Production Economics, 105, 357–371. doi:10.1016/j.ijpe.2004.04.011 Wang, S., & Guignard, M. (2002). Redefining Event Variables for Efficient Modeling of Continuous-Time Batch Processing. Annals of Operations Research, 116(1), 113–126. doi:10.1023/A:1021372029962 Wenhua, J., & Yuan, J. (2006). Single machine parallel batch scheduling problem with release dates and three hierarchical criteria to minimize makespan, machine occupation time and stocking cost. International Journal of Production Economics, 102, 143–148. doi:10.1016/j.ijpe.2005.02.003 Yuan, J. J., Liu, C. T. N., & Cheng, T. C. E. (2006). Single machine batch scheduling problem with family setup times and release dates to minimize makespan. Journal of Scheduling, 9, 499–513. doi:10.1007/ s10951-006-8776-2

Virginia Miori is an Assistant Professor at St. Joseph’s University and was a 2006 doctoral graduate of the LeBow College of Business at Drexel University, specializing in optimization of Supply Chains. She also holds an MS in Operations Research from Case Western Reserve University and an MS in Transportation from the University of Pennsylvania. Dr. Miori has ten years of teaching experience and has accumulated over fifteen years of experience in developing and implementing operations research models applied to problems in the chemical industry, manufacturing, logistics, transportation and supply chain management. She has published a number of articles and has received an outstanding dissertation award from Drexel University and an outstanding research award from St. Joseph’s University.



Brian W. Segulin is a Senior Software Developer at the Rovisys Company. He has over 20 years of experience designing and developing software solutions for integration, scheduling and adaptive modeling. He has done work in the process industries including metals, glass, oil and gas, paper, and food and beverage. He specializes in integrating legacy systems with state of the art control solutions exposing process data for use in evaluating process performance.


application of triplet notation and dynamic ...

application of triplet notation and dynamic ...

Suggest Documents

Dynamic Network Notation

Racah notation and Paschen notation

Triplet-triplet energy-transfer coupling: Theory and

Application of Dynamic Programming

triplet states and triplet excitons in chemically mixed crystals of

Impact of triplet absorption and triplet-singlet annihilation on the

Triplet-triplet absorption spectra of phthalocyanine and ... - Springer Link

Algebra (notation and equations) Algebra (notation and equations)

Application-dependent dynamic monitoring of distributed and ...

notation and performance

Application of Known Triplet Phases in the ... - IUCr Journals

Sequences and Summation Notation

AN APPLICATION OF DYNAMIC PROGRAMMING

Asymptotic Notation Θ-notation Ornek O-notation Ω-notation Θ, Ω, O ...

S-notation: A complete musical notation system for scratching and ...

Mathematical Notation Numbers Notation Meaning Example N ...

The Dynamic Network Notation: Harnessing Network Effects in PaaS ...

Dreamweaver - Dynamic Application Development

Sigma notation

Notation Systems

A Laser Photolysis Study of Triplet-Triplet Annihilation Reactions of

Detecting triplet locking by triplet synchronization indices

63 triplet

dependence of the efficiency of triplet-triplet energy ... - Springer Link