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A New Approach for Master Production Schedule under Theory of Constraints
Davood Golmohammadi a and S. Afshin Mansouri b
a College of Management, University of Massachusetts Boston, USA.
[email protected] Brunel Business School, Brunel University West London, UK.
POMS 22nd Annual Conference
Reno, Nevada, U.S.A.
April 29 to May 2, 2011
Abstract: Product mix optimization is one of the main challenges in a production system.
Several algorithms have been developed based on the Theory of constraints (TOC) approach
to determine an optimized master production schedule (MPS). Most of these algorithms are
evaluated based on simple examples and they may not be very efficient in dealing with real-
world operations. We investigate the inefficiency of recent algorithms, and demonstrate some
of the fundamental factors that have not been considered in any of the current algorithms.
Finally, we propose a new algorithm under TOC approach to create an efficient MPS.
Key Words: Theory of constraints, Master Production Schedule (MPS)
1. Introduction:
The Theory of constraints (TOC) is a successful operations philosophy, focused on the
concept of marginal attention to the constraints that inhibit the performance of the entire
system (Linhares, 2009). A constraint prevents a system from achieving its goal and may
include a machine whose capacity limits the throughput of the entire production process,
a highly specialized operator or scarce tool. The TOC attempts to identify constraints in
the system, and to exploit and elevate them to improve the overall output of the system
(Fawcett 1991). Principles for achieving a continuous improvement process are as
follows:
Identify the system’s constraint(s).
Decide how to exploit the system’s constraint(s).
Subordinate everything else to the above decision.
Elevate the system’s constraint(s).
If in any of the previous steps a constraint is broken, then go back to the first step.
In Step 1, the constraint or bottleneck is identified. In Step 2, an MPS is developed to
maximize the throughput. Step 3 develops a detailed schedule for production, ensuring that
the constraint resource can fulfill the MPS schedule established in Step 2. Step 4 encourages
continuous improvement to fully utilize the existing capacity and to increase the capacity at
the constraint. Step 5 continually reevaluates the entire system to see if there is a new
constraint after making the improvements identified in Step 4. If the constraint has changed,
then the heuristic starts over again with Step 1 (Fredendall and Lea, 1997).
TOC emphasizes that all activities of a system should be harnessed by the constraint, and
the constraint’s capacity should be fully utilized. How to optimize the resource utilization to
maximize the enterprise’s throughput within the determined product types is a dilemma for
operations managers. It involves the determination of the type and quantity of products which
is known as the product mix optimization problem. Product mix optimization is one of the
main challenges in a production system with a direct impact on the manufacturing
enterprise’s performance, such as profit, work-in-process (WIP), and customer service. The
product mix heuristic generates a master production schedule (MPS) which is capable of
maximizing the firm’s net profitability.
Several algorithms have been developed based on TOC to determine an optimized MPS.
Some of the algorithms (Goldratt,1990; Patterson,1992; Luebbe and Finch,1992; Lea and
Plenert, 1993; Lee and Plenert, 1996) are only proven to find the optimal solution of a single
constraint example. Also, in some problems, the results are inefficient and produce non-
optimal or infeasible solutions when certain new product alternatives are available. Posnack
(1994) and Maday (1994) argued that TOC approach should be properly used, and for non-
integer solutions, partial products should be allowed to be manufactured in the next planning
horizon. The other issue arises when there are multiple constraints; TOC approach might
generate an optimal solution (Luebbe and Finch 1992; Lee and Plenert,1993), or an infeasible
solution (Plenert, 1993; Fredendall and Lea,1997; Hsu and Chung,1998; Balakrishnan and
Chen, 2000). Other scholars developed improved algorithms (Fredendall and Lea, 1997; Hsu
and Chung, 1998; Aryanezhad and Komijan, 2004 and Komijan et al., 2009). However some
of the main drawbacks still exist in their method. Mainly, most of these algorithms are
validated based on simple examples and may not be very efficient when dealing with large-
scale or real-world operations in a job-shop system. Operation scheduling in a job-shop
system is generally complicated. Linhares (2009) criticized the current algorithms and
claimed that an efficient and optimum heuristic is simply impossible.
In this study, we investigate inefficiency of the recent algorithms, and demonstrate some
of the fundamental factors that have not been considered in them. In the next step, we
develop an algorithm under TOC approach to create MPS. In this unique method,
identification of constraints is explicitly different from the current procedure within all
algorithms. We demonstrate that constraints are not just those resources that do not have
adequate capacity to meet the demand.
In order to conduct a comprehensive study, we implement the proposed model in a
production line inspired from a real case that has a great deal of complexity and similarity to
most job-shop systems in the real world. Setup times and different processing times are
incurred in this system. More details about the case and its exclusive features will be
illustrated later.
2. Research methodology
The MPS planning is developed based on the first two principles of the TOC. In situations
where there are complexities, it may be difficult to create an efficient MPS. In the following
conditions, complexity may occur (Fox et al., 1986):
One constraint feeds another constraint.
There are a number of setups in the constraints and many parts use them.
There is a significant difference among the production lead times (from the constraint
to the end of the line) for products.
The different parts of one product need to use the constraint.
In our case study, we faced not only the above complexity conditions, but also the following
situations:
The difference between available capacity and required capacity for some of the
resources (machines) was not significant (this may create constraint shiftiness).
The sequence of operations showed that most of the machines (constraint and non-
constraint) were used by all products.
Processing times and setup times were different for the same operation of different
products.
We consider complex situations in operations such as different parts of one product need to
use the constraint, or a number of setups need to be done in the constraints for many
products. Without careful consideration to these situations which are common in real-world
operations, the accuracy of MPS is questionable. This is one of the crucial drawbacks of the
current methods in the literature that we address in our method. We define the key factors
which play a vital role to create an accurate MPS, and take them into consideration for the
model development. A function is defined and scores are assigned to all resources based on a
decision making model. Simply, these scores expose the potential of resources to act as
bottlenecks. The highest score shows the primary bottleneck. This is a unique manner in
creating MPS. Then, via several steps, the best combination of products is finalized as MPS.
2.1 MPS Design
We propose our unique algorithm in this section. We emphasize that all resources may play
the role of constraints; even those that might show excess capacity; i.e. the difference
between the available capacity and required capacity to meet the demand is positive. There
are n available resources, ri (i=1 to n) is a resource belonging to the resource constraints set
with a degree of caution from [0 to ∞]. The higher degree means that the resource can have
more effect on the operations as a bottleneck. The highest degree is the primary bottleneck. In
this algorithm, we take into account the number of setups and the sequence of operations as
well as the capacity requirement to determine bottlenecks. The method’s steps are illustrated
in two phases as follows:
Phase 1: Identify the system’s constraint(s):
In this phase, via several steps, the bottlenecks and their priority are determined.
These steps are as follows:
a) Calculate the following ratio for all resources
Avaiable Capacity−Required CapacityAvaiable Capacity * (-100)
b) Count the number of products that use ri.
c) Count the number of times a product uses ri to produce one unit of a final product.
Follow this step for all products
d) Add all above scores
e) Normalize them based on the highest score achieved in part d
f) Add the results of parts e and a
g) Divide the results of part f into the absolute value of the summation of the results
in part f. The normalized scores which are greater than 0 are the membership
value of ri for (i=1 to n). The highest membership is the primary bottleneck.
h) Sort the machines in descending order of the ratio obtained in part g.
Phase 2: MPS Design
In this phase, MPS is developed based on the bottlenecks and improved via
several steps.
i) Consider the first machine as the main bottleneck (B); develop a feasible MPS
through the following steps which have partly been inspired by the algorithm of
Fredendall and Lea (1997):
i.1) Sequence products in non-decreasing order of Ri ratio which is defined as
Ri=CM i / ti , B. In the event of a tie, give priority to the product with the higher
CMi.
i.2) Develop the initial feasible MPS by allocating market demand of products
(Pi) in the above order. Make sure the MPS is feasible at all stages, i.e. there is
always enough capacity on all critical machines identified in step (a).
i.3) Calculate total throughput (TP) of the initial MPS: TP=∑i=1
n
P i×CM i.
i.4) Verify that the total throughput can be improved by trading-off
production quantities (Pi) between higher and lower ranked products in the
sequence.
i.4.1) Set i=1; examine potential trade-offs between Pi and Pj; j=i+1,…,n.
Accept those trade-offs that improve throughput.
i.4.2) Set i=i+1.
i.4.3) If i<n-1, go to step i-4-1 otherwise, stop.
i.5) Consider the resultant MPS as a final solution.
We will explain the proposed algorithm by an example in the next section.
3. Case Study
We applied MPS scheduling logic to an automotive part manufacturer that operates based
on a job-shop system. We selected the operations of five products, A, B, C, D and E which
have 2000, 5000, 2000, 3000 and 3000 units in demand, respectively, for one month as the
period of production. Product B consists of two types of raw materials, B1 and B2. Figure 1
shows the different operations routings of the products. The processing times and capacities
are shown in Table 1.
Figure 1. Operations Routings of the Products
Table 1. The Processing Times and Capacities
3.1 Proposed Algorithm Implementation
Phase 1:
The results of the first phase of the proposed algorithm are shown in Table 2. This algorithm
considers the first machine in the priority list as the primary bottleneck.
Table 2. Identifying the Bottlenecks- Results of Phase 1 Implementation
Phase 2:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161 0.9 1.5 1 1.5 0 1 0.5 0 0 0 0 0 0 0 0 2,000 1,500
0.5 0 1 1 2.3 1 0.5 0 0.5 0.5 3 0 0 0 0 0 5,000 2,2000.5 0 0 0.5 0 2.6 2 0 0.5 0 0 4.75 9 2 1 0.6 2,000 3,0001 2 0 0.5 2 0 0.75 2 0 0 0 0 1 0 0 0 3,000 1,700
0.5 0 0 2 1 1 1 0 0 0 0 0 0 0 0 0 3,000 2,8008000 17000 8640 15700 24520 13500 10500 16920 7200 3360 20000 11520 21200 4800 2400 1440
10000 7800 8000 15500 23500 13200 13750 7000 3500 2500 15000 9500 21000 4000 2000 1200-2,000 9,200 640 200 1,020 300 -3,250 9,920 3,700 860 5,000 2,020 200 800 400 240
E
Machine (Resource) Number Market Demand
Contribution Margin (CMi)
Capacity Difference (dj)Required Capacity
Available Capacity (CPj)
DCBA
Product
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 165 2 2 5 4 3 5 2 2 1 1 1 2 1 1 1
A 2 2 2 2 3 1 1B 1 1 3 4 2 1 1 1 1C 1 1 3 1 1 3 2 1 1 2D 1 1 2 2 1 1 1E 1 3 2 1 1
11 5 5 16 15 9 10 4 4 2 1 5 5 2 2 368.75 31.25 31.25 100 93.75 56.25 62.5 25 25 12.5 6.25 31.25 31.25 12.5 12.5 18.75
25.000 -54.118 -7.407 -1.274 -4.160 -2.222 30.952 -58.629 -51.389 -25.595 -25.000 -17.535 -0.943 -16.667 -16.667 -16.66793.750 -22.868 23.843 98.726 89.590 54.028 93.452 -33.629 -26.389 -13.095 -18.750 13.715 30.307 -4.167 -4.167 2.0830.249 -0.061 0.063 0.262 0.238 0.144 0.248 -0.089 -0.070 -0.035 -0.050 0.036 0.081 -0.011 -0.011 0.006
Number of Times used in the
Operations Sequence of Product
Number of Products
Sub Total
b: %Capacity Shortagec: a+b
Normalized c
a: Normalized Sub
Machine (Resource) Number
The MPS is created and improved in several steps. Table 3 shows a group of critical
machines or bottlenecks based on the priority resulting from Phase 1. The Priority Index for
products based on machine 4 is shown in Table 4.
Table 3. Considered Resources as Possible Bottlenecks
Table 4. Priority Index for Products based on Machine 4
Product Contribution Margin (CMi)
Priority index (PRij)M4
A 1,500 1500B 2,200 2200C 3,000 6000D 1,700 3400E 2,800 1400
Total 14500.00
The next step is to create the initial MPS. The results are shown in Table 5.
Table 5. Initial MPS
4 1 7 5 6 131 1 1 1.5 0 0 2,000 1,5001 0.5 0.5 2.3 1 0 5,000 2,200
0.5 0.5 2 0 2.6 9 2,000 3,0000.5 1 0.75 2 0 1 3,000 1,7002 0.5 1 1 1 0 3,000 2,800
15,700 8,000 10,500 24,520 13,500 21,20015,500 10,000 13,750 23,500 13,200 21,000
200 -2,000 -3,250 1,020 300 200
B
Product First group of critical machines according to Normalized c Market Demand
Contribution Margin (CMi)
A
CDE
Available Capacity (CPj)Required Capacity
Capacity Difference (dj)
Trade-off analysis is performed in the next step after the initial MPS development to
determine whether throughput can be improved. The results of rounds 1 and 2 of the trade-off
analysis are shown in Tables 6 and 7.
Table 6. Trade-off Analysis- Round 1
Table 7. Trade-off Analysis- Round 2
Order on Machine: M4 M4 M1 M7 M5 M6 M13 M4 M1 M7 M5 M6 M13C 0.5 0.5 2 0 2.6 9 2,000 2,000 14,700 7,000 6,500 24,520 8,300 3,200 3,000 6,000,000 D 0.5 1 0.75 2 0 1 3,000 3,000 13,200 4,000 4,250 18,520 8,300 200 1,700 5,100,000 B 1 0.5 0.5 2.3 1 0 5,000 5,000 8,200 1,500 1,750 7,020 3,300 200 2,200 11,000,000 A 1 1 1 1.5 0 0 2,000 1,500 6,700 0 250 4,770 3,300 200 1,500 2,250,000 E 2 0.5 1 1 1 0 3,000 0 6,700 0 250 4,770 3,300 200 2,800 -
Available Capacity (CPj) 15,700 8,000 10,500 24,520 13,500 21,200 Total 18,350,000
tij Market Demand
Quantity Remaining capacity Contribution Margin (CMi)
Throughput
Order on Machine:
M4 M4 M1 M7 M5 M6 M13 M4 M1 M7 M5 M6 M13C 0.5 0.5 2 0 2.6 9 2,000 2,000 14,700 7,000 6,500 24,520 8,300 3,200 3,000 6,000,000 D 0.5 1 0.75 2 0 1 3,000 2,500 13,450 4,500 4,625 19,520 8,300 700 1,700 4,250,000 B 1 0.5 0.5 2.3 1 0 5,000 5,000 8,450 2,000 2,125 8,020 3,300 700 2,200 11,000,000 A 1 1 1 1.5 0 0 2,000 1,500 6,950 500 625 5,770 3,300 700 1,500 2,250,000 E 2 0.5 1 1 1 0 3,000 625 5,700 188 0 5,145 2,675 700 2,800 1,750,000
Available Capacity
(CPj) 15,700 8,000 10,500 24,520 13,500 21,200 Total 19,250,000
ThroughputtijMarket
DemandQuantity Remaining capacity
Contribution Margin (CMi)
Order on Machine:
M4 M4 M1 M7 M5 M6 M13 M4 M1 M7 M5 M6 M13C 0.5 0.5 2 0 2.6 9 2,000 2,000 14,700 7,000 6,500 24,520 8,300 3,200 3,000 6,000,000 D 0.5 1 0.75 2 0 1 3,000 2,500 13,450 4,500 4,625 19,520 8,300 700 1,700 4,250,000 B 1 0.5 0.5 2.3 1 0 5,000 5,000 8,450 2,000 2,125 8,020 3,300 700 2,200 11,000,000 A 1 1 1 1.5 0 0 2,000 0 8,450 2,000 2,125 8,020 3,300 700 1,500 - E 2 0.5 1 1 1 0 3,000 2,125 4,200 938 0 5,895 1,175 700 2,800 5,950,000
Available Capacity (CPj) 15,700 8,000 10,500 24,520 13,500 21,200 Total 21,200,000
tijMarket
DemandQuantity Remaining capacity
Contribution Margin (CMi)
Throughput
Table 7 shows the best MPS resulting from the proposed algorithm. The next step is to
evaluate the proposed model and its capability.
4. Comparison and Validation
Ultimately, to prove the capability of the proposed algorithm, the MPS for the above complex
job-shop system, inspired from a case in auto industry, is generated using the proposed
algorithm as well as a benchmark algorithm. We considered the well-known algorithm of
Fredendall and Lea (1997) in the literature as a benchmark. After following the MPS
development based on their algorithm, a new MPS is developed and the results are shown in
Table 8.
Table 8. MPS Development based on Fredendall and Lea Algorithm (1997)
The results are then evaluated and compared by a simulation technique. A discrete event
simulation model of the proposed model was developed for the purpose of experimentation.
Generally, a discrete event simulation model of a system is constructed by defining the events
that can occur and then modelling the logic associated with each event to capture the
changing status in the system. We used Arena software version 12 to model and analyze the
dynamic of the system. To find an optimal or a very satisfactory solution based on the best set
of input variables, we used OptQuest optimization software. OptQuest overcomes the above
limitation by automatically searching for optimal solutions within Arena simulation models.
1 7 Used Left Used LeftB 0.5 0.5 5,000 2,200 4400.00 5,000 2,500 8,000 2,500 5,500 11,000,000 0.64E 0.5 1 3,000 2,800 2800.00 3,000 3,000 5,000 1,500 4,000 8,400,000 0.81D 1 0.75 3,000 1,700 2266.67 3,000 2,250 2,750 3,000 1,000 5,100,000 0.66C 0.5 2 2,000 3,000 1500.00 1,375 2,750 0 688 313 4,125,000 1.00A 1 1 2,000 1,500 1500.00 0 0 0 0 313 0 0.00
Available Capacity (CPj) 8,000 10,500 Total throughput: 28,625,000Required Capacity 10,000 13,750
Capacity Difference (dj) -2,000 -3,250
Product Machine Market Demand
Contribution Margin (CMi)
Ri=CMi/(ti,M7) MPS Macine 7 Machine 1 Througput Ri+1(tleft,j+ti,BNj)/Cmi
Without an appropriate tool, finding an optimal solution for a simulation model generally
requires that you search in a heuristic or ad hoc fashion. This usually involves running a
simulation for an initial set of decision variables, analyzing the results, changing one or more
variables, re-running the simulation, and repeating this process until a satisfactory solution is
obtained. This process can be very tedious and time-consuming, even for small problems, and
it is often not clear how to adjust the variables from one simulation to the next.
All MPS were implemented in the production line, and the results of simulation are shown in
Table 9.
Table 9. Simulation Results and Comparisons
MPS Simulation Results Total Throughputs
ProductsFredendall
and Lea (1997)
Proposed Algorithm
Fredendall and Lea (1997)
Proposed Algorithm
Fredendall and Lea (1997)
Proposed Algorithm
C 1375 2,000 850 847 $8,961,400 $9,751,800D 3000 2,500 1400 1452B 5000 5,000 1160 1392A 0 0 0 0E 3000 2,125 500 600
The results show that the developed algorithm could accurately identify the main bottlenecks
and create an efficient MPS. The role of defined factors in modelling was salient. The
proposed algorithm could provide more throughputs than the Fredendall and Lea (1997)
algorithm. The output and profit of this MPS is very satisfactory. However, the performance
of the current algorithm shows more deviation from the expected output of the production
line.
5. Conclusion
Product mix optimization is one of the main challenges in production management. We
defined the key factors which play a vital role in creating an accurate MPS for real-world
operations. We demonstrated that the common constraints consideration in a production
system based on TOC may not be appropriate for a job-shop system in the real-world. We
have included the sequence of operations, number of setups, and number of products using
resources for MPS development. Moreover, we considered how products utilize resources in
the proposed algorithm. This is one of main contributions of this research, to consider key
factors for MPS development. We developed a new and unique approach to identify
constraints and create an efficient MPS. Furthermore, the research addresses the complexity
of creating an efficient MPS, and creates new avenues for future research. We will validate
our proposed model with more examples in the near future.
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