Debottleneck of Multistage Material-Handling ... - ACS Publications

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Debottleneck of Multistage Material-Handling Processes via Simultaneous Hoist Scheduling and Production Line Retrofit Jie Fu, Chuanyu Zhao, Qiang Xu,* and Thomas C. Ho Dan F. Smith Department of Chemical Engineering, Lamar University, Beaumont, Texas 77710, United States ABSTRACT: Multistage material-handling (MSMH) processes are broadly possessed by industries for manufacturing massive amounts of workpieces (jobs), where hoists are usually employed by following certain movement schedules based on the jobprocessing recipes. In this paper, we consider a common situation that an existing MSMH production line has reached its maximum productivity; in order to debottleneck the production and increase its productivity, some units will be retrofitted to increase their job-processing capacity under a fixed budget. Under this situation, which units need to be retrofitted and how many additional capacities will be added to those retrofitted units have to be optimally determined. Correspondingly, since process design has changed, the hoist schedule also needs to be adjusted. Apparently, the best way to deal with such an MSMH debottleneck problem requires the consideration of both the process retrofit and hoist rescheduling at the same time. In this paper, an MILP (mixed integer linear programming) based model is developed to simultaneously identify the best retrofit design and hoist schedule to obtain the maximum productivity under a fixed retrofit budget. On the basis of this development, different retrofit and hoist-scheduling scenarios under different budgets can also be examined, so that the Pareto frontier balancing both retrofit investment and productivity can be identified, which will provide the comprehensive decision support for an MSMH debottleneck problem. The efficacy of the proposed methodology has been demonstrated by case studies.

1. INTRODUCTION Multistage material-handling (MSMH) processes are broadly possessed by industries for manufacturing massive amounts of workpieces (jobs). Surface finishing and printed circuit board fabrication are typical application examples. MSMH processes usually employ controlled hoist(s)/crane(s) to transport jobs along their production lines to accomplish their multistage processing. During the transportation, the hoist follows a given movement schedule based on the processing recipe of every job inline. The job-processing recipe restricts the job-processing sequence and the processing time window at different stages. Sometimes, an MSMH production line can be used to produce multiple types of jobs with different recipes as long as a proper hoist scheduling is provided. Hoist scheduling must be welldeveloped so as to satisfy the manufacturing request of each job, meanwhile to maximize productivity of the production line. Obviously, hoist scheduling is associated with the design of the production line and is one of the most important factors related to productivity. It is reported that as high as 20% reduction in mean job waiting time and 50% improvement in standard deviation of cycle time can be achieved by hoist scheduling.1 Historically, hoist schedules used to be developed based on heuristic experience. The earliest report on computer-aided hoist scheduling was made by Phillips and Unger,2 who investigated the cyclic scheduling problem in a simplified electroplating line, which meant the hoist was scheduled to repeat a fixed movement sequence for processing one type of job. This type of complete movement sequence was referred to as a “cycle”,3 and its corresponding type of scheduling was called “cyclic hoist scheduling” (CHS), which had been proven as an NP-hard problem.4 Since then, a number of other new methods, especially mathematical programming based methods, have been introduced.5−8 Manier and Bloch9 summarized a © 2012 American Chemical Society

classification scheme for hoist scheduling problems. These problems typically require a cyclic schedule involving one or more types of parts, and the most common objective is to minimize cycle time. CHS applications have also been coupled with process design and operation issues to bring multiperspective benefits, such as freshwater/wastewater minimization as well as material and energy savings.10−14 In practice, hoist scheduling may need to handle a situation that new jobs come randomly. This requires hoist scheduling to work in a dynamic and reactive way, i.e., dynamic hoist scheduling (DHS). Yin and Yih15 and Yih16 studied the DHS problem with simplification that the hoist movement schedules for the already inline jobs did not change; only those for the new jobs were added. Lamothe et al.17,18 introduced a heuristic method by resorting to a classical backtrack algorithm. Goujon and Lacomme19 developed a dispatching-rule-based heuristic method, which considered the priority of the most urgent job for processing. Riera and Yorke-Smith20 provided a comprehensive survey on history and classification of hoist scheduling problems, where a hybrid algorithm combined with constraint logic programming and mixed integer programming was presented. Zhou and Li21 then suggested a heuristic method coupling sequence search and linear programming. To increase operability or productivity of a production line, research works on multihoist scheduling have also been studied as well.22−32 In this paper, we consider a common situation that an existing MSMH production line has reached its maximum Special Issue: L. T. Fan Festschrift Received: Revised: Accepted: Published: 123

February 29, 2012 April 6, 2012 April 9, 2012 April 9, 2012 dx.doi.org/10.1021/ie300550a | Ind. Eng. Chem. Res. 2013, 52, 123−133

Industrial & Engineering Chemistry Research

Article

Figure 1. An example of an MSMH process: (a) original production line; (b) new production line after process retrofit.

productivity based on current process design and hoist scheduling; in order to debottleneck the production and increase its productivity, some processing units are planned to be retrofitted to increase their job-processing capacities, such as by adding parallel processing unit(s) or slot(s). However, because of the fixed retrofit budget, the total added capacities are constrained. Under this situation, which units need to be selected for retrofit and how many additional capacities will be added to these retrofitted units have to be optimally determined, so as to debottleneck the production line with a limited resource to maximize the productivity increment. Certainly, once the production line has been retrofitted, the associated hoist schedule should also be adjusted correspondingly. Apparently, the best way to deal with such an MSMH debottleneck problem is to simultaneously consider the process retrofit and hoist rescheduling issues. Unlike previous hoist scheduling studies, this material handling and production line debottleneck problem requires both decision making on process design and operational scheduling areas. Up to now, only few studies have been conducted aiming this area. Ng33 initiated the study on determining the optimal number of duplicated process tanks in a single-hoist circuit board production line. However, such a material-handling system only addressed processing identical jobs, i.e., single-recipe job processing. Then, several works were developed later to discuss the problem of duplicated tanks in production line during manufacturing.34−37 Actually, one production line can be used to produce different types of jobs with different recipes, so that the production line will have inherent flexibility for agile manufacturing. In this paper, an MILP (mixed integer linear programming) model for simultaneous process retrofit and hoist scheduling (SPR-HS) has been developed, which deals with the MSMH debottleneck problem in order to obtain the maximum productivity under a fixed retrofit budget. On the basis of this development, different retrofit and hoist-scheduling scenarios under different budgets can also be examined, so that the Pareto frontier balancing both retrofit investment and productivity can be identified. This study provides a comprehensive decision support to MSMH debottleneck problems from both perspectives of process retrofit and

operational scheduling. The efficacy of the proposed methodology has been demonstrated by case studies.

2. PROBLEM STATEMENT The studied MSMH processes contain a number of processing units with different processing purposes. As shown in Figure 1a, different jobs are processed in the production line. They are picked up from the loading zone initially and, then, dragged into different units for batching processing by following their recipes; finally, the jobs are dropped to the unloading zone to complete their manufacturing. To manufacture a large quantity of jobs in such a cyclic way, the maximum productivity suggests the minimum cycle time of hoist schedule. Apparently, the job processing time in a unit has close relation to hoist cycle time. Thus, those processing units having longer job-processing time are most likely the bottlenecks of the system. A common way to debottleneck the system, so as to increase the productivity of an MSMH production line, is to add extra slots to some units, hence increasing their processing capacities. However, under a fixed budget in terms of a fixed number of extra slots, which units will be selected and how many additional slots will be added need to be optimally determined. For instance, in Figure 1b, units 3 and 7 are selected to be added by one more slot individually. Certainly, the process retrofit will change the number of jobs inline as well as the hoist traveling time. Therefore, CHS has to be simultaneously determined with the process retrofit. To help understand the studied optimization problem, the following terminologies are better to be clarified in advance. Job Processing Recipe. A job processing recipe gives processing requirements for a job, which includes the processing sequence and resident-time window for each processing step. The recipes for different types of jobs are different. Figure 1a gives the different processing sequences according to the recipes for three types of jobs (A, B, and C), respectively. Process Retrofit. In this paper, this means adding additional processing slot(s) to some optimally selected unit(s) in an MSMH production line, so as to boost the overall system productivity. 124

dx.doi.org/10.1021/ie300550a | Ind. Eng. Chem. Res. 2013, 52, 123−133


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Figure 2. Illustrative example for binary variable of zi,n,i′,n′. The term zi,n,i′,n′ is 1 if the hoist releases job n in the unit i and free-moves to the unit i′ to pick up job n′; otherwise, it is 0.

Free Move, Loaded Move, Idle Waiting, and Extra Hoist Trading Time. A hoist free move means the hoist travels without a job; otherwise, if it travels with a job, it is called a loaded move. Due to a scheduling arrangement, a hoist sometimes will simply stay above some unit holding nothing and just waiting for the next loaded move, which is called hoist idle waiting. If a unit is retrofitted by adding additional slots, every time the hoist trades a job (pickup or drop-off a job) with this unit, it takes some extra visiting time than that of the original, which is called extra hoist trading time (EHTT). On the basis of the above descriptions, the studied debottleneck problem for an MSMH process can be summarized as below. (1) Assumptions. A job joins and leaves a production line through a loading zone and an unloading zone, correspondingly, and there are no job capacity limits in the loading/unloading zones (see Figure 1). (2) A unit can have multiple slots and every slot could at most hold one job at a time. (3) The traveling speed of the hoist is fixed, which means the hoist traveling time between a pair of units is totally determined by their locations, and the distance change between two units due to process retrofit is negligible. (4) Whenever a hoist trades a job with a retrofitted unit, the hoist will involve a fixed EHTT. (1) Given Information. An existing MSMH production line with one hoist. (2) Processing recipes of different types of jobs. (3) Number of different types of jobs to be manufactured during one cycle. (4) The hoist traveling time between any pair of units, including the loading and unloading zones. (5) The original job capacities of each processing unit. (6) The number of slots to be added under a fixed retrofit budget. (1) Information to be Determined. Which units need to be retrofitted by adding additional slots and how many additional slots will be added to each retrofitted unit? (2) Detailed schedule of the hoist movement in one cycle after the retrofit. (3) Detailed pickup and dropoff time for each job in the production line. (4) Total cycle time of the retrofitted production line.

and hoist-scheduling plans under different retrofit budgets will be presented. 3.1. SPR-HS Model. In this section, an MILP model simultaneously addressing process retrofit and hoist scheduling is introduced, which includes the objective function, process retrofit constraints, hoist movement constraints, unit processing-capacity constraints, processing time constraints, and variable bounds. For illustration, some logic equations and auxiliary figures are employed to explain the modeling ideas. All the variables and parameters are explained both in the context and the nomenclature. 3.1.1. Objective Function. The goal for the optimal design of process retrofit and hoist scheduling is to maximize the productivity, which is equivalent to minimize the cycle time for processing a production ratio-fixed task, as shown in eq 1.

J = min T

(1)

where T is the cycle time of the hoist schedule. 3.1.2. Process Retrofit Constraints. Suppose an MSMH production line has I chemical units, SI = {1, ..., i, ..., I}. Among these units, unit 1 is the loading zone to accept unprocessed jobs, unit I is the unloading zone to place the completed jobs, and the others i (1 < i < I) are job-processing units for carrying out various manufacturing tasks. Suppose K unit slots, SK = {0, 1, ..., k, ..., K}, rendered by a fixed investment/budget are planned to be added into this production line to increase the productivity. A binary variable xi,k is introduced to determine if a unit needs to be retrofitted for adding extra number of slot(s) (xi,k is 1 if k slots are added to unit i; otherwise, xi,k is 0). Equation 2 shows that unit i can be only added by a fixed number of slots, including 0 slots, which means the unit keeps unchanged. Equation 3 restricts only job-processing units, not the loading/unloading zone, can be retrofitted; and the total number of being added extra slots is K. K

∑ xi ,k = 1,

∀ i ∈ SI ; 1 < i < I

k=0

K

(2)

I−1

∑ ∑ kxi ,k = K k=0 i=2

(3)

3.1.3. Hoist Movement Constraints. Assume the investigated production line can manufacture different types of jobs with different processing recipes, and the job set is defined as SN = {1, ..., n, ..., N}. SIn = {1, ..., in, ..., I} is a directed array of unit indexes, which indicates the unit-processing sequence for manufacturing job n. Note that the processing sequence is fixed by the recipe of job n. Different jobs may undergo different unit-processing sequences according to their recipes; however, their first and last operations will always go through the loading and unloading zones, correspondingly.

3. METHODOLOGY The developed methodology is presented in such a way that the SPR-HS model under a fixed number of being added extra slots (equivalent as a fixed retrofit budget) is introduced first. Then, based on the understanding of this model, the methodology framework for examining the performances of various retrofit 125



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Figure 3. Different scenarios of multijob capacities.

(Mvi,i′) and some possible EHTT due to unit capacity expansion. Note that i + + 1 means the next processing unit from unit i for processing job n, following job n′s recipe. Here, if unit i is selected to expand its capacity by adding some slot(s), then whenever the hoist picks up or drops off a job from/to this unit, an EHTT, DtK, will be spent by the hoist. Similarly, the starting time of a loaded move should be no less than the transitional time from the previous loaded move to the current one as shown in eq 8, where the larger or equal sign indicates the existence of possible waiting time after the hoist arrives at unit i. Equations 7 and 8 include the scenario that the number of total extra slots to be added to the production line equals zero, i.e., K = 0, which means the production line is not retrofitted. As shown in eq 9, when K = 0, DtK = 0, indicating that the EHTT due to unit capacity increment is zero; if K ≥ 1, DtK equals a constant number Dt. Note that the job lifting and releasing time can be customized according to the application. In our methodology, the job lifting and releasing time has been aggregated into free move or loaded move time.

Hoist movement constraints are used to define and restrict variables of the pickup time of job n in unit i (Si,n), drop-off time of job n in unit i (Ei,n), and T. As mentioned before, hoist movements include free moves and loaded moves. Equations 4 and 5 give the integrality constraints for free moves, where a binary variable zi,n,i′,n′ is employed. The variable zi,n,i′,n′ is 1 if the hoist releases job n in unit i and thereafter free moves to unit i′ to pick up job n′; otherwise, it is 0 (as shown in Figure 2). Because only one hoist is employed for a cyclic operation, each hoist free move is actually unique. Equations 4 and 5 respectively describe the unique departure and arrival of a hoist free move. I

N

∑ ∑ zi′ ,n′ ,i ,n = 1,

∀ i ∈ S In ; ∀ n ∈ SN ; i ≠ I

i ′= 1 n ′= 1

(4) I

N

∑ ∑ zi ,n,i′ ,n′ = 1,

∀ i ∈ S In ; ∀ n ∈ SN ; i ≠ 1

i ′= 1 n ′= 1

(5)

A cyclic scheduling problem is a round-table problem. Thus, which unit/job the hoist scheduling starts from does not affect the cycle time and schedule results. S1,1 as shown in eq 6 is designated as the starting time point of the cyclic schedule, which means the loading zone is assigned as the initial starting unit of a cyclic operation. S1,1 = 0

K

Ei ++1, n = Si , n + Mvi , i ++1 +

∑ xi++1,k DtK , k=0

∀ i ∈ S In ; ∀ n ∈ SN ; i ≠ I

(6)

I

Equation 7 describes the relation between the starting and ending time for every loaded move. It suggests the ending time of a loaded move is equal to the starting time of the loaded move, plus the hoist traveling time between unit i and unit i′

Si , n ≥

N

K

∑ ∑ [(Ei′ ,n′ + Mvi′ ,i)zi′ ,n′ , i , n] + ∑ xi ,k DtK , i ′= 1 n ′= 1

∀ i ∈ S In ; ∀ n ∈ SN 126

(7)

k=0

(8)


Industrial & Engineering Chemistry Research ⎧ 0, if K = 0 DtK = ⎨ ⎩ Dt, if K ≥ 1

Article N

(9)

n ′= 1

−ui , n , n ′M < Ei , n ′ − Si , n ≤ M(1 − ui , n , n ′), ∀ i ∈ S In and ∀ i ∈ S In′; ∀ n , n′ ∈ SN ; n ≠ n′

(11) I

Si , n ≥

(10)

∀ i′ ∈ S In′; ∀ i ∈ S In ; ∀ n , n′

∈ SN N

∀ i ∈ S In and ∀ i ∈ S In′; ∀ n , n′ ∈ SN ; n ≠ n′

(18)

k=0

∀ i ∈ S In ; ∀ n ∈ SN

M(wi , n , n ′ − 1) ≤ vi , n , n ′ + ui , n , n ′ + yi , n − 2 < Mwi , n , n ′ ,

(12)


After one cycle operation, the empty hoist will come back to the original position (loading zone) that the cyclic schedule starts from. Thus, eq 13 suggests that the total cycle time should be greater than any ending time of a loaded move plus a free move time to the loading zone. T ≥ Ei , n + Mvi ,1 ,

(17)

−vi , n , n ′M < Ei , n − Ei , n ′ ≤ M(1 − vi , n , n ′),

K

∑ ∑ hi′ ,n′ ,i ,n + ∑ xi ,k DtK , i ′= 1 n ′= 1

(16)

Note that eq 15 is the representation of a logic constraint. To transform eq 15 to the algebraic constraints, linear constraints of eqs 17−19 can be employed.

0 ≤ (Ei ′ , n ′ + Mvi ′ , i) − hi ′ , n ′ , i , n ≤ M(1 − zi ′ , n ′ , i , n),

hi ′ , n ′ , i , n ≤ Mzi ′ , n ′ , i , n ,

k=0


Nonlinear eq 8 can be substituted by its linear equivalent as eqs 10−12, where a positive variable hi′,n′,i,n is introduced to substitute (Ei′,n′ + Mvi′,i)zi′,n′,i,n.

∀ i′ ∈ SIn ′; ∀ i ∈ S In ; ∀ n , n′ ∈ SN

K

∑ wi ,n,n′ + 1 ≤ Cai + ∑ kxi ,k ,

∀ i ∈ S In ; ∀ n ∈ SN

(19)

where another two binary variables, ui,n,n′ and vi,n,n′, are introduced to link Si,n, Ei,n′, and Ei,n. The term ui,n,n′ is 1 if Ei,n′ ≤ Si,n; otherwise, it is 0. Also, vi,n,n′ is 1 if Ei,n ≤ Ei,n′; otherwise, it is 0. 3.1.5. Processing Time Constraints. The processing time for job n in unit i is represented as pi,n. It is the time interval during which a job stays in the unit. Equation 20 gives a general formula for calculating pi,n. Generally, when a job is dropped into unit i, it should be picked up in the next Cai − 1 + ΣKk=0 kxi,k or Cai + ΣKk=0 kxi,k cycles, depending on whether or not the unit is retrofitted, and the time sequence of the job pickup time (Si,n) and drop-off time (Ei,n).

(13)

Equation 14 constrains the logic relation between Si,n and Ei,n based on two scenarios: (i) if Si,n ≤ Ei,n, it suggests the hoist first releases a job in the previous operation cycle and later picks it up in the current operation cycle; then, the binary variable yi,n is 1; (ii) if Si,n > Ei,n, it suggests the hoist first drops the job and later lifts a job in the same operation cycle; then, the binary variable yi,n is 0.

K

−Myi , n < Si , n − Ei , n ≤ M(1 − yi , n ), ∀ i ∈ S In ; ∀ n ∈ SN

pi , n = Si , n − Ei , n + (Ca i − 1 +

k=0

(14)

∀ i ∈ S In ; ∀ n ∈ SN

3.1.4. Unit Processing-Capacity Constraints. Some units of the production line can be designed with multiple jobprocessing capacities. For a unit with a single job capacity; the job stayed in the unit must be moved out before another job can move in. For a multicapacity unit, it has multiple slots so that it can simultaneously process multiple jobs. After the process retrofit, the reconstructed units will have more slot(s), so that their job-processing capacities are increased. Equation 15 shows different scenarios that job n may stay with another job n′ in the same unit i (i.e., two jobs n and n′ are simultaneously processed for certain time in two different slots of unit i). These scenarios are illustrated in Figure 3a−c, and a binary variable wi,n,n′ is defined associated with these scenarios. The term wi,n,n′ is 1 if job n is staying with job n′ in unit i at a time; otherwise, it is 0. In eq 16, ΣNn′=1 wi,n,n′ accounts for the total number of jobs once staying with job n in unit i. Including job n itself, ΣNn′=1 wi,n,n′ + 1 cannot exceed the original capacity of unit i (Cai) plus the number of extra slots added to this unit (ΣKk=0 kxi,k).

(20)

To linearize eq 20, eqs 21−25 are employed. First, a positive variable f i,k is introduced to replace the nonlinear term xi,kT in eq 20. If unit i is not retrofitted with additional k slot(s) (i.e., xi,k = 0), it results in f i,k = 0; otherwise, if xi,k = 1, f i,k = T exists (represented by eqs 21 and 22). Similarly, another positive variable gi,n is introduced to replace the other nonlinear term Tyi,n in eq 20. If Si,n > Ei,n, the binary variable yi,n is 0, resulting in gi,n = 0; otherwise, if Si,n ≤ Ei,n, yi,n is 1, resulting in gi,n = T (represented by eqs 23 and 24). With the auxiliary variables of f i,k and gi,n, eq 20 can be linearly expressed by eq 25. L(1 − xi , k) ≤ T − fi , k ≤ M(1 − xi , k), ∀ i ∈ SI ; ∀ k ∈ SK Lxi , k ≤ fi , k ≤ Mxi , k ,

(21)

∀ i ∈ SI ; ∀ k ∈ SK

(22)

L(1 − yi , n ) ≤ T − gi , n ≤ M(1 − yi , n ),

(Ei , n ≤ Ei , n ′ ≤ Si , n) ∨ (Ei , n ′ ≤ Si , n ≤ Ei , n)

∀ i ∈ S In ; ∀ n ∈ SN

∨ (Si , n ≤ Ei , n ≤ Ei , n ′) ⇔ wi , n , n ′ = 1, ∀ i ∈ S In and ∀ i ∈ SIn ′; ∀ n , n′ ∈ SN ; n ≠ n′

∑ kxi ,k)T + Tyi ,n ,

Lyi , n ≤ gi , n ≤ Myi , n ,

(15) 127

(23)

∀ i ∈ S In ; ∀ n ∈ SN

(24)



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K

pi , n = Si , n − Ei , n + (Ca i − 1)T +

∑ kfi ,k

+ gi , n ,

k=0

∀ i ∈ S In ; ∀ n ∈ SN

(25)

Finally, eq 26 bounds the processing time with the lower and upper limits, RTloi,n and RTup i,n , according to the job processing recipe. RTilo, n ≤ pi , n ≤ RTiup, n ,

∀ i ∈ S In ; ∀ n ∈ SN

(26)

3.1.6. Variable Bounds. All the continuous variables have a lower bound of zero and an upper bound of M. As described, xi,k, yi,n, zi,n,i′,n′, ui,n,n′, vi,n,n′, and wi,n,n′ are defined as binary variables. T , Si , n , Ei , n , pi , n , gi , n , hi , n , i ′ , n ′ , fi , k ≥ 0, ∀ i , i′ ∈ SI ; ∀ n , n′ ∈ SN

(27)

T , Si , n , Ei , n , pi , n , gi , n , hi , n , i ′ , n ′ , fi , k ≤ M , ∀ i , i′ ∈ SI ; ∀ n , n′ ∈ SN

(28) Figure 4. Methodology framework.

xi , k , yi , n , zi , n , i ′ , n ′ , ui , n , n ′ , vi , n , n ′ , wi , n , n ′ ∈ {0, 1}, ∀ i , i′ ∈ SI ; ∀ n , n′ ∈ SN

Table 1. Hoist Traveling Time Mvi′,i (s) for the Case Studies

(29)

unit i

Based on the above equations and explanations, the developed SPR-HS MILP model for general MSMH processes consists of the objective function in eq 1 and constraints described by eqs 2−7, 9−14, 16−19, and 21−29. 3.2. Methodology Framework. To comprehensively evaluate the retrofit design of an MSMH process, multiple design scenarios should be considered. On the basis of the developed SPR-HS model, we have the opportunity to examine and balance the difference of productivity increments under different investment conditions; i.e., under different value of K (total available unit slots), how much the cycle time T can be reduced. Therefore, the methodology framework proposed by this paper to comprehensively evaluate the retrofit design and hoist scheduling of an MSMH process is shown in Figure 4, where the Kup is the maximum extra slots that can be added to the original production line according to the retrofit budget limit. Note that the methodological procedure shown by Figure 4 is equivalent as identifying a Pareto frontier of a duo-objective optimization problem: minimizing capital investment vs maximizing productivity. Therefore, once all of the cycle time T under different K are obtained, the Pareto frontier is also obtained, which can be used for decision support to pick up the most desirable solution for a customized MSMH process.

unit i′

1

2

3

4

5

6

7

8

9

10

1 2 3 4 5 6 7 8 9 10

0 2 4 6 8 10 12 14 16 18

2 0 2 4 6 8 10 12 14 16

4 2 0 2 4 6 8 10 12 14

6 4 2 0 2 4 6 8 10 12

8 6 4 2 0 2 4 6 8 10

10 8 6 4 2 0 2 4 6 8

12 10 8 6 4 2 0 2 4 6

14 12 10 8 6 4 2 0 2 4

16 14 12 10 8 6 4 2 0 2

18 16 14 12 10 8 6 4 2 0

capacity. Meanwhile, three types of jobs, A, B, and C, with different recipes are processed in this production line (see Figure 1a), and each type of job is required to produce one job in every cyclic operation. The processing time windows constrained for each type of job processing in different units according to their recipes are listed in Table 2. Three case studies are conducted for the methodology demonstration and discussion. All the case studies are modeled and solved in GAMS v23.338 with the MILP solver CPLEX.39 Case 1: Debottleneck Result when K = 2. On the basis of the given data, the first case study is conducted when the number of extra slots is specified as two, i.e., K = 2. The developed SPR-HS MILP model has 458 binary variables, 930 continuous variables, and 3605 constraints, and the solving time for this case is about 25 s with an 8-Core Xeon 3.2 GHz Dell server. Figure 5 shows the global optimal result with the developed methodology. The total cycle time is 155.6 s including 54 s of loaded move, 86 s of free move, 14 s of idle waiting time, and 1.6 s of the total EHTT. The optimal retrofit plan is to add one extra slot to each of units 3 and 7. From the simultaneously obtained CHS, the hoist starts from lifting job A from the loading zone; after releasing it to unit 2, the hoist free moves to the loading zone again to lift and transport job C to

4. CASE STUDIES The developed SPR-HS methodology has been used to tackle a debottleneck problem for a multirecipe MSMH process. In this production line (see Figure 1a), 10 units with different functionalities (including the loading and unloading zone) are indexed as 1−10. The loading and unloading zones are designated as unit 1 and unit 10, respectively; and they are allocated separately at the ends of the production line. It takes the hoist 2 s for traveling between two adjacent units (see Table 1). Assume all the units except the loading and unloading zone in the original production line only have single-job processing 128



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Table 2. Processing Time Requirement for the Case Studies unit i

job parameter

n

1

2

3

4

5

6

7

8

9

10

RTloi,n

A B C A B C

− − − − − −

45 75 − 1000 1000 −

75 75 − 1000 1000 −

30 60 − 1000 1200 −

55 45 − 1200 1000 −

75 30 − 1000 1200 −

30 − 55 1000 − 1000

60 − 75 1200 − 1000

45 − 60 1000 − 1000

− − − − − −

(s)

RTup i,n (s)

Figure 5. Optimal debottleneck result when K = 2.

It should be highlighted that during an operational cycle in Figure 5, jobs A, C, and B are picked up from the loading zone at 0, 4, and 69 s, respectively; and jobs C, B, and A are dropped on the unloading zone at 22.2, 87, and 91 s, respectively. For those nonretrofitted units, they are still having single-job capacity; for the double capacity units of 3 and 7, they could hold two jobs at a time. Meanwhile, the optimal hoist schedule contains some idle waiting after a free move. For example, at the time 91 s in Figure 5, the hoist takes 10 s of free move from

unit 7. Following the hoist movements shown in Figure 5, the hoist experiences a series of loaded and free moves according to the scheduling, and finally returns to the loading zone to complete an operation cycle at 155.6 s. Next, the hoist will simply repeat the cyclic schedule and continue the operation. The solution details including every pickup and dropoff time of each loaded move, and job index, as well as the retrofitted units are shown in Figure 5. 129



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Figure 6. Debottleneck result based on adding one slot to each of units 5 and 9.

units 5 and 9 are selected to add one extra slot for each. The developed SPR-HS model can also easily handle this case and the MILP model has 456 binary variables, 930 continuous variables, and 3605 constraints, and the solving time for this case is 2.87 s with an 8-Core Xeon 3.2 GHz Dell server. The

Table 3. Optimal Debottleneck Results under Different Retrofit Budgets scenario no.

no. of extended slots

retrofit location

cycle time (T; s)

solving time (s)

1b 2 3a 4 5 6

0 1 2 3 4 5

unit 7 units 3 and 7 units 3, 7, and 8 units 3, 6, 7, and 8 units 2, 3, 6, 7, and 8

171 166 155.6 146.2 140.4 130.8

0.89 9.54 24.57 118.94 472.38 1392.64

a

Optimal debottleneck result of case study 1. bDebottleneck result when K = 0.

the unloading zone to unit 5 to pick up job A, and then the hoist takes 6.6 s of idle waiting time before picking up A at 107.6 s. Case 2: Debottleneck Result Based on the Fixed Retrofit of Two Units. To highlight the merit of simultaneous optimization of process retrofit and hoist scheduling, the second case study is conducted, in which two units are fixed for capacity expansion and only CHS is optimized. In this case,

Figure 7. Pareto frontier to balance the retrofit investment and the cycle time reduction. 130



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Figure 8. CHS result when K = 0.

to K and T is plotted in Figure 7. It shows that the CHS cycle time is generally decreasing with the increase of the number of K, suggesting that more retrofit investment resulting in larger productivity of the MSMH processes. Thus, the decision maker should balance well between the productivity increment and the retrofit investment. Obviously, Figure 7 is very useful to support such decision makings. It should be noted that when K = 0, there is no unit retrofit conducted. Thus, the SPR-HS model solution is equivalent to the optimal CHS solution for the original MSMH processes. In this case, the involved MILP model has 426 binary variables, 888 continuous variables, and 3504 constraints, and the solving time for this case is 0.9 s with an 8-Core Xeon 3.2 GHz Dell server. Figure 8 shows the global optimal scheduling result. The total cycle time is 171 s including 54 s of loaded move, 98 s of free move, and 19 s of idle waiting time. In the obtained cyclic schedule, the hoist will drag in each job A, B, and C at 0, 79, and 127 s, respectively, from the loading zone to the production line and drag out another three completed jobs of C, B, and A at 18, 97, and 101 s, respectively, to the unloading zone.

cycle time of this CHS solution is 168 s, including 54 s of loaded move, 102 s of free move, and 10.4 s of idle waiting time, as well as 1.6 s of the total EHTT. The detailed information for hoist movements and job processing is shown in Figure 6. It can been seen that the hoist cycle time of case 2 is 12.4 s more than that of case 1, suggesting about 8.0% productivity loss simply because of the inappropriate selection of units for retrofit (i.e., units 5 and 9 are selected but they are not as critical as units 3 and 7 selected by case 1 for this debottleneck problem). Therefore, the selection of the right units for retrofit is very important for productivity debottleneck of an MSMH system. Case 3: Optimal Debottleneck Results when Kup = 5. As disclosed by the methodology framework, multiple optimization cases for simultaneous process retrofit and hoist scheduling under different retrofit budgets will be studied. In this case study, Kup is given by 5. Thus, the SPR-HS model is iteratively solved with respect to K changing from 0 through 5; and under each K, one optimal solution will be obtained. All the optimal solutions under different K are listed in Table 3. On the basis of the obtained solutions, the Pareto frontier with respect 131



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Kup = maximum possible K due to the retrofit budget limit L = a sufficient small number M = a sufficient large number Mvi,i′ = hoist traveling time from unit i to unit i′ lo RTi,n = lower time limit for processing job n in unit i up RTi,n = upper time limit for processing job n in unit i

Note that these completed jobs may be dropped from the previous cycle. Following the indicated movements shown in Figure 8, the hoist experiences a series of free and loaded moves according to the schedule; and finally, it returns to the loading zone to repeat the cycle. Because all the processing units except the loading/unloading zones have single-job processing capacity, the job in those units has to be lifted before another job is released. Meanwhile, the hoist schedule contains large idle waiting after certain free moves. For example in Figure 8, at the time 30 s, the hoist takes 4 s to free move from unit 8 to unit 6 to lift job A. The hoist does not lift A until 49 s, suggesting an idle waiting time of 15 s. Compared with case 1, the hoist cycle time when K = 0 is 15.4 s longer, suggesting about a 9.9% productivity loss. Therefore, simultaneous process retrofit and hoist scheduling is important for productivity debottleneck of an MSMH process. The case studies demonstrate the efficacy of the proposed methodology. Note that the methodology we developed has broad application potentials. In addition to chemical engineering industry, electroplating, coating, and electronic industries, where an MSMH process employs hoists/cranes, may have the potential to use our developed methodology.

Variables

5. CONCLUDING REMARKS The productivity of an MSMH process may reach the inherent bottleneck due to lacking of processing stages/units. Under this situation, certain process units can be retrofitted by additional parallel processing units/slots to boost the productivity, associated with the development of a new operational scheduling strategy. In this paper, an MILP based model is developed to simultaneously identify the best retrofit design and hoist schedule to obtain the maximum productivity under a fixed retrofit budget. On the basis of the development, different retrofit and hoist-scheduling scenarios under different budgets can also be examined, so that the decision support balancing both retrofit investment and productivity can be provided for an MSMH debottleneck problem.

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REFERENCES

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AUTHOR INFORMATION

Corresponding Author

*Phone: 409-880-7818. Fax: 409-880-2197. E-mail: Qiang.xu@ lamar.edu. Notes

The authors declare no competing financial interest.

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Ei,n = ending time point (drop-off time) of the hoist carrying job n into the unit i f i,k = a positive variable to substitute xi,kT gi,n = a positive variable to substitute Tyi,n hi′,n′,i,n = a positive variable to substitute (Ei′,n′ + Mvi′,i)zi′,n′,i,n pi,n = processing time for job n staying in unit i Si,n = starting time point (pickup time) of the hoist carrying job n from the unit i T = total cycle time of the scheduling problem ui,n,n′ = binary variable which is 1 if Ei,n′ ≤ Si,n; otherwise, it is 0 vi,n,n′ = binary variable which is 1 if Ei,n ≤ Ei,n′; otherwise, it is 0 wi,n,n′ = binary variable, which is 1 if job n′ once stays with job n in unit i; otherwise, it is 0 xi,k = binary variable, which is 1 if k slots added to the unit i; otherwise, it is 0 yi,n = binary variable, which is 1 if Si,n ≤ Ei,n; otherwise, it is 0 zi,n,i′,n′ = binary variable, which is 1 if the hoist releases job n in the unit i, and free-moves to the unit i′ to pick up job n′; otherwise, it is 0

ACKNOWLEDGMENTS This work was supported in part by Graduate Student Scholarship from Lamar University. NOMENCLAURE

Sets

SN = {1, ..., n, ..., N} set of processed jobs in a cycle SI = {1, ..., i, ..., I} set of units SIn = {1, ..., in, ..., I} set of unit processing sequence for manufacturing job n SK = {0, 1, ..., k, ..., K} set of extra slots to be added to the units of a production line Parameters

Cai = job processing capacity for unit i DtK = extra hoist trading time (EHTT) K = number of total extra slots that will be added to the original production line 132



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