The theory of constraints in manufacturing

The Theory of Constraints in Manufacturing

R. Y. Qassim

Department of Mechanical Engineering. School of Engineering and

Program of Mechanical Engineering. COPPE

Federal University of Rio de Janeiro. RJ. Brazil

In this paper, a systematic and quantitative view is presented for the application of the theory of constraints in manufacturing. This is done employing the operational research technique of mathematical programming. The potential of the theory of constraints in automated manufacturing is demonstrated.

Keywords: Theory of Constraints, bottlenecks, manufacturing management

Introduction

Manufacturing has undergone a number of changes in the last few years, in view of the economic environment in which companies are operating and of the introduction of advanced manufacturing technology (AMT). In the new economic environment, the tendency is for more products per company with decreasing product lifecycles. This has resulted in the necessity of integrating design, production, maintenance, quality control and post-sales services. This integration has been achieved through the introduction of AMT such as computer-aided design (CAD), computerised numerical control (CNC) and predictive maintenance. The increasing competitiveness in industry to capture larger market shares has led to the introduction of modern manufacturing management strategies such as just-in-time, material resources planning, and the theory of constraints. The last of those is the subject of this paper.

The objective of this paper is two–fold:

• Provide a systematic and quantitative view of the theory of constraints.

• Describe the popential of the theory of constraints in automated manufacturing.

Theory of Constraints

The theory of constraints, hereafter abreviated as TOC, is a production and operations management strategy centered on the concept of capacity-constrained resources (CCR), more commonly called bottlenecks. TOC starts from the assumption of the existence of one or more CCR in any system. This assumption tends to lose its validity in systems with balanced loads. Any resource, be it manpower, machines, procedures, raw materials, or markets, is said to be a CCR if it limits the performance metric of the system due to the fact its capacity is less than the demand placed upon it. The performance may be profit, production volume, or any other suitable criterion. A simple example of a CCR is the slowest operation in a continuos simple flow line of production of discrete parts. In more complex production system configurations such as a job shop with multiple routeing of parts, it may not be obvoius which is the CCR in the system.

The implementation of TOC in practice is achieved in a sequence of logical steps; e.g., Chase and Aquilano (1995). These steps are:

1. CCR identification

2. CCR management

3. Performance improvement

These steps will be decribed in some detail in the following sections.

CCR Identification

As stated in the last section, a CCR is defined as a resource which prevents the system from achieving a higher level of preformance. It is then necessary to define precisely a metric for performance. In TOC, as it is applied in manufacturing environments, performance is considered to be profit which is defined as

PROFIT = THROUGPUT – OPERATING EXPENSES ,

or

PR = TH – OE.

In turn, thoughput is defined as the sale prices of finished products, and operating expenses are defined as the costs of raw materials employed in obtaining these finished products. Clearly, performance is associated with a given time horizon; e.g., week, month, year.

Assuming that a manufacturing facility such as a workshop can make a number of several finished products, each with its unit sale price, unit raw material costs, and market demand, then maximun profit is obtained by making the most profitable mix of finished products subject to multi-resource capacities available. A CCR is defined as the resource which has the highest ratio of utilization to availabllity. There exists a number of methods to identify a CCR. Fredendall and Lee (1997) provide a heuristic procedure for determining a CCR. The most general and efficient method is based on linear programming (Leubbe and Finch, 1992; Plenert, 1993; Mabin and Gibson, 1998).

Now we develop a linear programming (LP) model for CCR identification in manufaturing systems by introducing necessary notation.

i – index denoting part; i = 1, ,I

j – index denoting resource; j = 1, ,J

A_ij – uniti processing time of part i in resource j

B_i – unit profit of part i

C_j – capacity of resource j in planning horizon

D_i – market demand for part i in planning horizon

R_i – amount produced of part i in planning horizon

S_j – idle time of resource j in planning horizon

The LP model for CCR identification may be set down as:

Maximise

subject to

The objective function (1) represents the total profit obtained over the planning horizon. The constraints (2) ensure the capacity limit for each resource j. The constraints (3) ensure that production of each part does not exceed demand. Finally, the constraits (4) and (5) guarantee the nonnegativity of the decision variables R_i and S_j of the LP model, whose input parameters consist of the set A_ij, B_i, and C_i. The resource with the highest ratio of utilisaton,

to availability, C_j, is the same as the resource with the minimun idle time, S_j. Consequently, any resource with S_j = 0 is a CCR. It can be seen that the LP model seeks to identify the optimun part mix and the CCR (s) in the manufacturring system. The LP model provides the master production schedule (MPS) which maximises throughput.

CCR Management

Once identified, a CCR must be used effectively in such a way so as to obtain the desired performance of the manufacturing system. This is known as CCR management, which is implemented by a scheduling technique refered to as drum-buffer-rope (DBR). It is well established that scheduling in job shops is extremely complex due to its combinatorial nature. DBR reduces this complexity by focusing attention on CCRs as opposed to all resources. In order to achieve the highest performance possible, CCRs must be scheduled with a view to avoiding unnecessary idle time which implies lost throughput.

The drum-buffer-rope constraint scheduling problem (DBRCSP) has been derved from a modification of the general resource constrained project schedule problem (GRCPSD) by Simon et al (1996). As with other project scheduling approaches, GRCPSD suffers from two inherent limitations:

1. Each task/activity happens only once.

2. Precedence among tasks/activities is specified explicitly than through the nature and amounts of material movement between tasks/activities.

These two limitations are removed by employing the state task network (STN) representation, originally developed by Kondili et al (1993). In the next section, we present the STN problem for DBR scheduling.

STN – DBR Scheduling

Before developing the model, we introduce necesssary notation

i = index denoting task; i=1, ,I C_s = maximum storage capacity dedicated to state S j = index denoting production unit; j=1, J I_j = Set of tasks which can be performed by production unit j t = index denoting time; t=1, H = maximum capacity of production unit j when used for performing task i s = index denoting material state; s=1, S = minimum capacity of production unit j when used for performing task i S_i = set of states which has task i as input W_ijt = binary decision variable = 1 if production unit j starts processing task j at the start of period t ; = 0 otherwise = set of states which has task i as output B_ijt = amount of material which starts undergoing task i in production unit j at the start of period t r_is = proportion of input of task i from state S Î S_i S_st = amount of material stored in state S at the start of period t = Proportion of output of task i to state S Î S_i M = sufficiently large number P_is = processing time for output of task i to F_sjt = amount of material of state s being held in production j during the tome interval t P_i = completion time of task i, b = index denoting buffer unit K_i = set of production units capable of performing task c = index denoting CCR unit T_s = set of tasks which has input from state S F_s = set of states whose members are finished products = set of tasks which has output to state S R_st = quantity of finished products in state S scheduled for delivery at time t

The STN model for DBR scheduling may then be set down as:

Minimise

subject to:

In the objective function (7), we seek to minimise the maximum difference between delivered quantities and stored finished products.

The constraints (8) ensure that at any given time, a production unit which is not a CCR can only start at most one task. For a CCR, the constraints (9) guarantee that it is never idle, since by definition a CCR is the resource that sets an upper limit to manufacturing system performance. The constraints (10) serve to ensure that tasks are performed non-preemptively.

The constraints (11) ensure that the quantity of material undergoing a task in a production unit is bounded by the minimum and maximum capacities of that unit. The constraints (12) guarantee that the quantity of material stored in a state does not exceed the maximum storage capacity for that state. The constraints (13) and (14) constitute material balances for production units and buffers, respectively. The binarity and nonnegativity of the appropriate decision variables are ensured by the contraints (15)-(17).

In the STN model, we have assumed that the only unit possessing an input buffer is a CCR. Neither setup non maintenance tasks are taken into consideration. At the same time, we note that all the aforementioned aspects can be easily incorporated into a more general STN model. We purposely restricted our scope so as to focus on the attainment of scheduled deliverices of finished products.

Performance Improvement

The identification and management of CCRs serve to achieve maximun performance for given capacity and demand levels. In order to go beyond this level of performance, measures have to be taken and the corresponding investments have to be made to increase capacity and/or demand.

The operational performance criteria are (Rahan, 1998):

1- Throughput (TH) defined as the rate at which the manufacturing system generates revenue.

2- Inventory (IN) defined as the investment made to generate revenue.

3- Operating expense (OE) defined aas the cost of transforming inventory into throughput.

In the context of TOC, system performance is improved by increasing inventory in order to increase throughput and/or decrease operating expense. It is clear that the performance measurements of TOC are very different from traditional management accounting approaches.

Apart from external constraints, such as market demand, internal CCRs by definition limit the performance of manufacturing system. Consequently, CCRs must be the focus of all improvement efforts. For a CCR machine, its capacity may be increased by such measures as setup reduction, breakdown reduction, and processing speed enhancement. All such measures naturally involve investments. In order to evaluate investments in system performance improvement, we consider setup reduction to illustrate the invesment evaluation procedure. We employ the notation already persented above in the section on CCR identification, and we introduce extra notation.

b – index denoting CCR machine

E – investment in setup reduction

- maximum available time of CCR machine that can be achieved by setup reduction

Normally, E is a function of the amount of setup reduction. Trevino et al (1993) have shown that E is given by:

The TOC model for setup reduction investment may then be stated as:

Maximise

subject to:

In the objective function (19), we seek to maximise return (throughput) on investment (inventory). The constraints (20) ensure a capacity limit for each non-CCR machine, and the constraint (21) plays the same role for the CCR machine, presumably already identified by the LP model shown above. The constraints (22) and (23) set lower and upper bounds, respectively, for the available time of the CCR machine.

The constraints (24) and (25) ensure nonnegativity of part production volumes and non-CCR machine idle times respectively. Finally, the constraint (28) ensures that the investment in setup reduction is considered only when the CCR machine remains as such.

CCR in Automated Manufacturing

In a manufacturing system which is completely or partially automated, TOC is particularlçy useful, as we attempt to show in this section. Automated systems may take variious forms such as computerised numerical control (CNC) machines, robots, automatic guide vehichle (AGV) transport machanisms, and flexible manufacturing systems (FMS). Such systems may exist as standalone or as part of an overall conventional fabrication system involving extrusion, forming, milling, and cutting machines.

In view of the fact that substantial investments are involved in the acquisition, installation, and operation of such automated systems, it is necessary that CNC, AGV, robotic cells and FMS be utilised to the maximum level. This in turn implies that such systems are highly likely to be unbalanced having one more CCR. Despite this fact, the author is unaware of any work applying TOC specifically to automated manufacturing systems. In what follows, we present two problems where TOC may be useful.

1. In a manufacturing system consisting of an FMS and conventional cutting machines, it is necessary to divide the workload between these two subsystems. In the FMS, we need to identify the CCR in a multi-part multi-route setting. We start by introducing necesssary notation.

i – index denoting part; i=1, , I

j – index denoting FMS machine, j = 1, , J

k – index denoting operation

A_ijk – unit processing for operation k of part i in machine j

B_i – unit profit of part i

C_j – capacity in time units of resource j in planning horizon

D_i – market demand for part i in planning horizon

R_i – amount produced of part i in planning horizon

S_j – idle time of machine j in planning horizon

T_ijk – amount of part i having its k th operation on machine j

K_i – number of operations on part i

The part mix-route mix problem may be stated as:

Maximise

subject to

The objective function (27) represents the net profit to be maximised. The constraints (128) reflect the capacity limitation of each machine. The constraints (29) ensure that production does not exceed demand for each part. The constraints (30) and (31), guarantee the nonnegativity of the decision variables.

2. Because of the capability of CNC machining centres to continuously perform a variety of operations on a part by automatically changing the cutting tools, such machining centres constitute CCRs in the manufacturing system in which they are inserted. Consequently, their effective scheduling is critical to achieving maximum system perormance. Consider a CNC machining centre where it is required to determine the part mix to be machined and the corresponding tool set with a view to maximising net profit. Now, we introduce necessary notation.

p – index denoting part ; pÎ P

q – index denoting period; q Î Q

t – index denoting tool; t Î T

C_pt – unit cost of machining part p using tool t

D_tq – deterioration condition of tool t in period q

E_tq – unit cost of replacement of tool t in period q

F_tq – unit cost of reset of tool t in period q

G_pq – unit cost of producing part externally in period q

R_pq – unit revenue from part p in period q

S_tq – unit setup cost of tool t in period q

U_tq – binary decision variable = 1 if tool t is replaced in period q, = 0 otherwise

V_tq – binary decision variable = 1 if tool t is reset in period q, = 0 otherwise

W_tq – binary decision variable = 1 if tool t is neither replaced non reset in period q, = 0 otherwise

X_pt – binary decision variable = 1 if part p is machined by tool t, = 0 otherwise

Y_t – binary decision variable = 1 if tool t is loaded, = 0 otherwise

The part selection and allocation model may be stated as:

Maximise

subject to

The objective function (32) represents the net profit we wish to maximise. The constraints (33) ensure that each part is machined by exactly one tool or is not machined at all. Tha constraints (34) ensure that all needed tools are loaded on the carousel. The constraints (35) reflect the tool carousel capacity limitation. Tha constraints (36) reflect the deterioration dynamics of the tools. The constraints (37)-(41) guarantee the decision basics as to tool replacement, resetting, or otherwise. Finally, the constraints (42) guarantee the binary of the decision variables.

Conclusions and Sugestions

For unbalanced manufacturing systems, TOC constitutes a useful strategy for maximising and improving system performance. We have shown that the operational research technique of mathematical programming provides a systematic basis for the implementation of TOC in practice. The author is involved in two projects at present on the application of mixed-integer linear programming and queueing networks to TOC implementation in industrial fabrication/assembly systems and computadorised numerical control/flexible manufacturing systems (Azevedo Filho, 2000; Qassim, 2000). A substantial scope exists for developing mathematical models of TOC in automated manufacturing systems and their validation in industrial practice.

Acknowlegment

The author takes this opportunity to thanks the Brasilian Council of Research and Development (CNPq) for a scholarship to support this work.

Manuscript received: July 2000. Technical Editor: Átila P. Silva Freire.

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Publication Dates

• Publication in this collection
  14 Sept 2001
• Date of issue
  2000

History

• Received
  July 2000