A goal programming approach to multi-period production line scheduling

A GOAL PR~~MMING APPROACH TO MULTI-PE~OD PRODUCTION LINE SCHEDULING SANG M. LEE* Department of Management, ColIege of Business Administration, The University of Nebraska, Lincoln, NB 68588 EDW~

R. CuyToNt

and BERNARD W. TAYLOR, IIISQ

Department of Business Administration, College of Business, Virginia Polytechnic Institute and State University, Blacksb~g, VA 24061 Seeps and Pmpare-Numerous studies have been published concerning the problem of scheduling multiple products on several production lines. However, these sdutioo approaches are often unrealistic and limited in their applicabiliiy. This paper describes a realistic akenmtive goal programming approach for the production schedulii problem with muItipIe objectives. SpeeiticaIly a model is developed which considers the achievement of production demand, cost, overtime requirements and capacity as objectives in the production problem. The goal programming model is developed and demonstrated via an in-depth case example. The model results assign the amount of several product types to be produced on multiple production lines over a finite planning horizon. IO addition, sensitivity analysis is discussed. Abstract-This paper considers the problem of schedtding multiple products on several dilTerentproduction lines (i.e. the n-job, m-machine scheduli problem) wheo multiple objectives exist. Spectically a goal programming model is developed and demonstrated via a case example. The production setting modeled in this paper includes three separate production lines that merge into a singk inspection facility. Prior to the description and demoos~tion of the goal prolix pmdel, a brief overview of the muIti-pr~uct scheduling problem and the various solution approaches to the problem is presented. INTRODUCTION

have been various approaches suggested in the literature directed at the production scheduling problem. The n-job (product), siugie machine (production facility) flowshop is perhaps the simplest form of the production scheduling problem. Solution approaches include models which rn~~e pr~uction costs by sched~g via optimal economic tot sizes[4,15]; integer programming to minimize job lateness[lb]; linear programmiug[l8]; dynamic programming models to minimize cost under conditions of stochastic demand[6]; and, several heuristic approaches IS]. The more complex version of the scheduling problem is the expanded n-product, mmachine production setting. This general problem with its numerous variations has been

There

*Sang M. Lee is Professor and Chairman of the Department of Management at the University of Nebraska at Lincoln. He received the Ph.D. in hfanagemeot Science from the University of Georgia, and an M.B.A. from Miami University of Ohio. He is currently Series Editor of the Modera Decision Atudysis Series, PetroceIli/Cbarter Publishers, Inc. His books published include Goal Programmingfor &&ion Analysis (1972), Zntmductionto Axision Sciince (1975). tia L. J. Moore, and Linear Optimization for Management (1976).In addition, he has published over sixty articles in such journ& as DC&ion Sciences, Managemenl Science, Journal of ZGance, AZZETransactions, Sloan Management Reyiew and n~rous others. He is a memba of TIMS, ORTA and AIDS; currently serving as Secretary of AIDS, and has previously served as Vice President and President of National AIDS and Southeast AIDS respectively. tEdward R. Clayton is Professor and Coordinator of Management Science at Veia Polytechnic Institute and State University, Blacksburg, Virginia. He received a B.S. from the University of Florida and a Ph.D. from Clemson University in Engineering Management. He is a co-author of GERT Modding and Simubion: Fun&ment& and ~piicatioas, abuse by Pe~~~/C~r iii 1976.He is a charter member of AIDS and a member of TIMS aud is currently chairman of S.E. TIMS. His research interests are in network analysis, simuIatioo, and appticatioas of mathematical programming. He has published numerous articles in journals including Management Science, Journal of Mar&.etingand others. He serves as a consultant to the U.S. Army Concepts Analysis Agency on math programming and simulation. SBernard W. Taylor, III, is Associate Professor of ~meot Science at Virginia Polytechnic Institute and State &iversity. He received the Ph.D. and M.B.A. from the University of Georgia and a B.I.E. from the Georgia Institute of Technology. Dr. Taylor has published articles in such journals as Management Science, Lkcision Sciences, ADZE Transacfions, Computers and Operations Reseatvh, Omega, Production and ZrtventoryManagement, Operational Research Quarluly (forthcoming) and numerous others. He is a member of AIDS, TIMS and the Academy of Management. BPlease address ail correspondence to: Dr. Bernard W. Taylor, Department of Business A~rnis~~~, College of Business, Vii Polytechnic Institute and State University, Blaeksburg, VA 24061. 205

S. M. LEE,E. R. CLAYTON and B. W. TAYUIR,III

206

analyzed by several researchers. A large segment of these solution proposals have employed linear programming models including those by Eilon[2] who developed a large-scale model for a chemical production process and then compared this model with an alternative multiproduct batch-scheduling approach; and by Gorenstein [3] who described linear programming models which determine optimal economic lot sizes. Pritsker et al.[17] offer a &l integer linear programming model in which several alternative objective functions are considered. In addition to these generalized linear programming approaches, Sadlier [ 191 proposes the transportation method of linear programming to assign jobs to machines while Kornbluth and LePage[8] describe a separable programming algorithmic approach. While linear programming models seem to comprise the majority of solution approaches, there have been several alternatives including a dynamic programming model employing the gradient solution technique by Lee and Shaikh[lO]; a column generation technique for direct solution[9]; heuristic approaches [5]; and production disaggregation[20]. As one can observe, a number of different solution approaches to the production scheduling problem have been proposed. The number of differing approaches is in part due to the unique variations which can result from a particular application to a specific production environment and the type of solution results which are desired from a model. However, there has also been a substantial amount of criticism questioning the validity and applicability of many of the solution approaches, resulting in a continuous search for alternative approaches [ 1,7]. Since there are a large assortment of relevant variables related to the scheduling problem, it is extremely diicult to develop a comprehensive model; thus, true optimality of solution results is often questionable. An additional drawback of many of the solution approaches is their inability to entertain multiple production objectives. For this reason, goal programming surfaces as a logical solution alternative. Also, since *goal programming attempts to achieve a “satisfactory” rather than optimal solution in the face of conflicting goals, it appears to be a more realistic approach. There have been a limited number of applications of goal programming to the production scheduling problem, including a model described by JZskeltiinen[7]. This model establishes goals for attaining production, employment, and inventory levels. However, this goal programming application only determines the number of units to schedule per period and does not actually schedule products or production facilities. As a result, the goal programming model to be described in this paper is directed at the more prevalent product to machine scheduling problem. THE PRODUCTION

SCHEDULING

PROBLEM

The production scheduling problem treated in this paper consists of five different product models which are assembled on one of three production lines. AU models on the three lines are channeled into an inspection facility consisting of dual inspection stations. Figure 1 depicts the process of production, inspection and final product completion. The planning horizon is six days. At the beginning of the six-day planning cycle, a demand forecast is generated by marketing for each of the five models. While final delivery of the units MODEL

I PRODUCTION PROCESS \\

\

BEGIN

END

MODEL 2

INSPECTION FACILITY

MODEL 3

MODEL 5

Fii. 1. The production process.

A goal programming approach to production line scheduling

207

produced is not required until the end of the week (i.e. the sixday period) there is a strong desire to avoid in-process inventories. Thus, units produced during a single day must also be inspected and placed in final inventory. Production times and costs are assumed to be linear and normal production capacity is limited to the line hours available. Given these conditions the problem becomes one of how best to schedule production on the three assembly lines over the six-day planning horizon. Because of the presence of multiple objectives within the pr~uction process (as opposed to the limited objective of cost rnin~i~tion), goal programming is employed as the solution technique. However, because it is a well-known technique, goal programmingwill not be discussed in this paper. For a comprehensive review of goal programming see Lee[ll-141. FORMULATION

OF THE MODEL

The goal pro~ming model designed for the production scheduling problem consists of several goal constraints relating to production demand, costs, utilization, overtime and inspection. Decision vatiables

The decision variables for the model define the number of units to be produced of each model on each line per day: Xi* = number of product of model i produced in day i on line k,

i = 1,s models

i= 1,6days k = I,3 assembly lines, Modei demand

At the beginning of each sixday period production requ~ements for each of the five models are determined. Each of the models can be produced on any one of the three assembly lines. The goal for model demand is developed as: 3

6

where Mi = production requirement for model i, d.-*’ = deviational variables from 1 to n, where n = 5, the number of goal equations for model demand. This formulation results in 5 goal equations each corresponding to the demand for one of the five models. Since customer service is more important to management than excess production, the min~~tion of d,,- is more spout than d.’ (excess pr~uc~on). In, fact meeting the demand goals is the highest priority of the firm. The five different model demands for this example are shown in Table 1. Productioncapacity

The daily production capacity for each assembly line is the same, eight regular hours of time available per day. An exception is the sixth day (i.e. today) which has only four av~able hours per line. However, the required production times differ for each model on each of the Table 1. Demand requirements for model i wi)

Model 1

2

3

4

-5

Demand 3200 2600 1500 2700 3600

S. M. LEE,E. R. CLAYTON and B. W. TAYLOR, III

208

three assembly lines. The goal constraints for daily production capacity are formulated as follows:

where, Pti = the production requirement per model i on line k, n = 6+23, Di = daily capacity (hours), Dj = 8.0 hours for j = 1,5 and Dj = 4.0 for j = 6. This formulation results in 18 goal equations. Each equation represents the production on one assembly line of,all five models for one day. Thus, for each working day three goals exist (one for each assembly line). The different production times required for the different models and assembly lines for this example are shown in Table 2. Table 2. Production requirements(hr) for model i on line j

Line Day j

I 2 3

1

Model 3

2

j=1,5 j=6 Capacity 4

5

0.0075 0.0060 0.0090 0.0100 0.0080 0.0060 0.0070 O.OO!XI0.0085 O.OO!N 0.0070 0.0065 0.0105 O.OO!N 0.0085

IlOWS

8 8 8

4 4 4

The different processing times are a result of several factors. Fist, and most obvious, each model requires a different amount of time to produce. The different line times are a result of two factors: (1) varying worker proficiencies between lines; and (2) varying equipment age and quality. The minimization of all d,- representing underutilization of daily production capacity is the second highest priority goal of management. To do otherwise could result in layoffs for certain lines on particular days. Management desires to avoid the occurrence of layoffs for the obvious reasons of labor morale and good union relations. Overtime While overtime is available on all three lines on a daily basis, management desires to limit it

to four hours per line per day. This results in 15 goals formulated as follows: d6’ + d% = 4.0, d,+ + d; = 4.0, d& + d, = 4.0.

Note that in the above set of overtime equations there are no equations reflecting overtime for d&, d& and dj. These three deviational variables represent overtime on the three assembly lines during day six. Because of management policy overtime (above the four normal production hours) is not permitted on day six. Inspection

After production on the three lines each product is channeled into an inspection facility (Fig. 1). The inspection facility consists of two inspection stations in parallel. Each product model requires a different time for inspection, however, the inspection times do not vary according to the assembly line the model was produced on. This results in six goal equations formulated as: 3

5

P=l

t=l

IJijk

+

dm- = 16

where, Ii = the inspection required for model i, n = 39+45. This formulation reflects the fact that 16 hours (i.e. 8 hours for each of the two stations) are available daily for inspection. The inspection times, Ii, for each model are given in Table 3.

A goal programming approach to production line scheduling Table 3. Inspection times (4)

Inspection time (hr)

1

2

Model 3

4

5

0.005

0.007

0.006

0.004

0.005

Note that d,,+representing overtime, is deleted in the above formulation.,This is a result of management policy since it is believed that the quality of inspection decreases rapidly after an eight-hour day. However, if one of the assembly lines works overtime then the eight-hour inspection day must be staggered (even more than normal) so that all units can be inspected to avoid in process inventories. Since the daily inspection capacity is strictly limited, units produced tend to be uniform over the six-day period resulting in uniformly utilized lines. However, if enough inspection time is not available then underutilization can occur. Thus, the model formation can also determine the required number of inspection stations for management. Production cost Each model costs a different amount to produce. Costs also vary between assembly lines (as a result of the variations in labor and equipment between lines mentioned earlier). The production budget for the six-day planning period is based on the amount of production required. The product cost goals are as formulated as follows:

where C&= the per unit cost of model I on line k, n = 45, B = the six-day production budget (e.g. $12,ooO). This formulation results in one goal equation reflecting the entire period production cost. The model per unit production costs for each assembly are given in Table 4. Table 4. Production costs, C,, for model i on line k Line

I

2

Model 3

4

5

1 2 3

$1.10 1.25 1.15

$0.85 0.80 0.95

$0.95 0.90 0.92

$1.00 1.03 1.07

$0.80 0.85 0.75

The objective function In a goal programming model the objective function represents the minimization of deviation from the specified goal levels according to preemptive priorities. The priority ranking for this model example are as follows (in order of decreasing priority level): PI: the achievement of demand, P1: the minimization of production underutilization, P3: the production budget, P4: not exceeding demand, Ps: the minimization of overtime. These priorities result in the following objective function formulation: minimize 2 = {P,(d,-+d5-),

Pz(d6-+d&), P3d4;, P4(dl++d6+), Ps(da’+dto)). MODEL RESULTS

The example presented in this paper resulted in a model with 90 decision variables and 45 constraint equations. The problem was solved using the LEESGP computer program[2]. The model required 63 simplex iterations to produce the solution, however, at a minimal amount of time and cost. The model results are presented in Table 5.

210

S. bi. LEE,E. R. CLAYTON and B. W. TAYLOR, III Table 5. Model results MY

Day Line 1

1

1

2

Model 3 4

1067

2 3

476

530 762 667

177 2

: 3

4

: 3 1 2 3

5

: 3

6

: 3

49

256

689

319 889

1022 1039 889

889

1067 945 889

228 889

E 889

444 444

400 444 444

1067

800 700

400

Total

1067 totI6 762

889 133 671

line production

844 1143 889

1143

889

3

5

3200 2600 1500 2700 5467

14,967

Priority one was achieved reflecting the fact that demand was met. Priority two was also achieved indicating that production capacity was fully utilized. In addition priority five was met indicating that there was no overtime (i.e. overtime was minimized). Priorities three and four were not achieved. The priority three goal, represented the production budget and was exceeded by $1,934. The priority four goal for demand showed that demand was exceeded by 1867 units. It can be seen from Table 5 that the entire amount of-the excess production occurred for model five (noting that model five demand was 3600 units). The excess production occurred primarily becauSe of the priority one requirement minimizing production capacity underutilization. When this situation occurs, the excess production should simply be stored and carried over to the next planning period and subtracted from demand.

The primary uses of the model described in this paper is as a plain and scheduling tool. It fuElIs this use by generating the daily schedule for the five models on the three assembly lines as shown in Table 5. However, by changing the parameters in the model and observing the effects, the production manager can test alternative scheduling strategies and production configurations. For example, in this model a test was performed where the inspection facility was reduced to one station (i.e. 8 hr). The result was a drastic decrease in production on all three lines resulting in numerous layoffs (since a unit produced must be inspected and inspection is a strict constraint). Other tests which can be made include varying the production budget, changing the priority structure, and changing the production time and cost requirements (as a result of equipment and labor changes on the different lines). In addition, lines can be shut down (i.e. eliminate the goal constraints for certain lines) on different days. As one example, the priority structure for this model was changed to reverse the priority one and two goals. As a result, the top priority became the avoidance of pr~uction undeN~i~~on. The model results from this test showed that this goal was achieved but it was done by overproducing certain models and not producing anything for other models-obviously an undesirable result. An additional area of testing (not included in this model) relates to the possibility of decreasing worker proficiency as the six-day period passes. This occurrence can be reflected by changing the per unit production time parameters for different days.

A goal programming approach to production line scheduling

211

CONCLUSIONS

The model described and demonstrated in this paper is for a general production operation with multiple product models, multiple assembly lines and a multi-period planning cycle. The existence of multiple objectives makes goal programming an appropriate solution approach. Actual implementation of the model is dependent on the particular firm and their own unique production characteristics. It may be necessary to alter the complexity of the model to make it applicable to a specific pr~u~tion setting. If it is necessary to change the model to reflect the existence of in-process inventories or multiple production stages the model parameters will expand greatly. Data collection for a model of this type should be of little ditIiculty since most of the parameters are those normally measured in production operations.

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