Optimization of Product Mix Planning in High-Mix-Low-Volume
Industries Using Genetic Algorithms
SIEW CHIN NEOH1, NORHASHIMAH MORAD2, CHEE PENG LIM1, and
ZALINA ABDUL AZIZ1
1
School of Electrical and Electronic Engineering, University Science Malaysia
Engineering Campus, 14300 Nibong Tebal, Penang, MALAYSIA
2
School of Industrial Technology, University of Science Malaysia
11800, Minden, Penang, MALAYSIA
Abstract: - This paper describes a product mix optimization model using Genetic Algorithms (GAs) for HighMix-Low-Volume (HMLV) industries. The main objectives of the proposed GA-based planner are to reduce
the high conversion time due to frequent product changeovers and to allow a balanced workload throughout a
particular planning horizon. In addition, the product mix model is developed to fulfill the customer demands
as well as to prevent over-loaded capacity. The proposed approach is applied to data obtained from a
semiconductor manufacturing plant and the results indicate that the optimized product mix planner performs
better than those obtained from conventional planning methods.
Key-Words: - Product Mix, Genetic Algorithms, High Mix Low Volume, Optimization
1 Introduction
Product mix analysis is described as the process of
determining the percentage of each product in
different production and business activities [1]. This
process is complex as it involves various inputs such
as resource utilization, resource limitation, product
variety, product characteristics, and market demand.
Thus, product mix optimization plays an important
role in capacity planning especially in High-MixLow-Volume (HMLV) manufacturing environments
which need to produce more product variety with
lower volume in the dynamic business climate.
In product mix planning, the theory of
constraints (TOC) and linear programming (LP) are
two widely used approaches. Luebbe and Finch
compared TOC and LP and clarify the differences
between TOC philosophy and LP technique [2].
TOC heuristic is easily understandable and simpler
to use than LP [2], but LP is a better planning tool
than TOC when multiple constrained resources exist
[3-4]. In [3], it is argued that traditional accounting
methodologies cannot determine optimal product
mix and that the dropping marginally profitable
products based on the accounting rules will give less
than optimal profitability. Beside TOC and LP,
Genetic Algorithms (GAs) have been used in
product mix planning problems. GA-based TOC has
been proposed in [5] as it can easily find very high
quality solutions for both small and large problems
at reasonably small computation time. GA has also
been applied to the optimization of product mix and
material match in [6]. In this paper, a GA-based
product mix optimization model is proposed to
minimize conversion time, to fulfill predicted
demand, to prevent overloaded capacity, and to
achieve a balanced workload throughout a planning
horizon.
Since HMLV requires enhanced organization
and process discipline to embrace a high degree of
multitasking [7], it has always been a competitive
manufacturing strategy for HMLV industries to
obtain systematic product mix planning.
The
proposed approach is applied to a semiconductor
manufacturing plant to solve practical, real-world
product
mix
problems
encountered
by
manufacturing firms. The problem of HMLV
manufacturing systems that requires high conversion
time has been addressed. Time means money, and
the reduction in changeover and conversion time can
allow the production line to maximize its resource
utilization for more value-added processes.
2 Problem Formulation
Multi-product production in the HMLV industries
involves a large number of parameters and
constraints. In Fig. 1, there are n types of products
to be planned, Pi (i  1,2 ,...,n) represents the ith
type of product in the product mix whereas
Di (i  1,2 ,...,n) represents the ith product’s demand
that must be fulfilled within the planning horizon.
The planning time index is given as
Wk (k  1,2 ,...,t) ,in which it determines the interval
within the product mix planning horizon (e.g.
weekly for a month or monthly for a year).
In the proposed model, the product mix problem
is represented by randomly generating product
occurrence, Oik, and product volume, Vik, where i
represents the product type and k represents the
planning index. Product occurrence is used to
indicate whether or not a product has been selected
for production for that particular week or month
(Equation 1).
1 if product i will be produce in week k
Oik  
0 if otherwise
(1)
After obtaining the product occurrence, product
volume for each particular time index, Vik will then
be generated randomly by fulfilling the product
demand, Di and production loading rules.
On the production floor, minimum production
volume per product per week is a constraint.
Mi (i  1,2 ,...,n) is used to represent the minimum
production volume per product i per week. When
the demand of product i, Di, is nonzero, the sum of
product occurrence throughout the planning horizon
has to be more than 1 but less than the total product
demand, Di, divided by the minimum production
volume per product i per week, Mi (Equation 3).
Note that Mi ≠0. On the other hand, if Di is zero, the
sum of product occurrence should be zero (Equation
4). In other words, there is no product needed to be
produced if there is no demand for it.
t
1   Oik 
k 1
t
O
ik
Di
,
Mi
 0 ,
k 1
Product
type
(Pi)
P1
P2
P3
:
:
Pn
Product
type
(Pi)
P1
P2
P3
:
:
Pn
W1
O11
O21
O31
:
:
On1
W1
V11
V21
V31
:
:
Vn1
Week
(Wk)
W2
O12
O22
O32
:
:
On2
W3
O13
O23
O33
:
:
On3
Week
(Wk)
W2
V12
V22
V32
:
:
Vn2
W3
V13
V23
V33
:
:
Vn3
……
……
……
……
Wt
O1t
O2t
O3t
:
:
Ont
Predicted
Demand
(Di)
D1
D2
D3
:
:
Dn
Wt
V1t
V2t
V3t
:
:
Vnt
Predicted
Demand
(Di)
D1
D2
D3
:
:
Dn
2.1 Demand constraint
The planning of product mix requires each product
to fulfill the demand constraint of the whole
planning horizon. Therefore, the sum of the
generated product volume throughout the planning
horizon has to be equal to the product demand, i.e.,
t
ik
 Di , for i  1,2,3,..., n
k 1
2.2 Production loading constraint
(3)
if Di  0
(4)
2.3 Capacity Constraint
Capacity constraints refer to the resource ability and
availability to process a particular set of product
mix. Equation (5) shows that the available capacity
of week kth, ACk is equal to the difference between
total capacity of week kth, TCk and the down time
capacity of that week, DCk which includes all
capacity used for scheduled down time (e.g.
preventive maintenance and engineering experiment
) and buffer time.
ACk  TCk  DCk , for k  1,2,3,..., t
(2)
(5)
The capacity usage based on the generated
product mix in week kth, UCk (k  1,2 ,...,t) has to
be always less than or equal to the available capacity
in the production floor in week kth, ACk (k  1,2 ,...,t)
as shown in Equation (6).
UCk  ACk
Fig.1: Product mix structure in the proposed model
V
if Di  0
(6)
3 GA-based Product Mix Planning
GAs are stochastic global search and optimization
methods that mimic the metaphor of natural
biological evolution [8]. Throughout the genetic
evolution, better solution to the problems is
obtained. Its effectiveness in industrial applications
and ease of use has been demonstrated by a number
of researchers [9].
Product
type (Pi)
P1
P2
P3
:
:
Pn
W1
…..
…..
…..
:
:
…..
Week
W2
…..
…..
…..
:
:
…..
(Wk)
W3
…..
…..
…..
:
:
…..
W4
…..
…..
…..
:
:
…..
Product
type (Pi)
P1
P2
P3
:
:
Pn
W1
…..
…..
…..
:
:
…..
Week
W2
…..
…..
…..
:
:
…..
(Wk)
W3
…..
…..
…..
:
:
…..
W4
…..
…..
…..
:
:
…..
Fig.2: Randomly Selected Crossover
3.1 Chromosome Representation
In GA, the representation scheme determines how
the problem is structured. In the proposed model,
the product mix for a particular planning horizon is
represented by a 2D matrix, and two types of
encoding are used: 1) binary encoding and 2) real
number encoding. Binary encoding is used to
indicate the product occurrence Oik whereas real
number encoding is used to indicate the weekly
product volume, Vik that should be tested. The
structure of the chromosome representation is shown
in Fig.1.
3.2 Initialization and Selection Function
The model will randomly generate solutions for the
entire population. Stochastic Universal Sampling
which is a single phase sampling algorithm with
minimum spread and zero bias is used as the
selection function to keep the expected number of
copies of each chromosome into the next generation.
3.3 Genetic Operators
3.3.1 Crossover
In the proposed model, Randomly Selected
Crossover is used. The principle of Randomly
Selected Crossover is depicted in Fig.2. The
crossover probability is given as Pxovr, in which
each individual has Pxovr chances to be selected for
a crossover. For each crossover, number of rows
(number of product) will randomly be selected from
a 2D matrix (an individual) to be inter-exchanged
with another individual to produce offspring (new
individuals).
3.3.2 Mutation
Two Mutation techniques are suggested in the
model: Weekly Mutation (Fig.3) and Randomly
Selected Mutation (Fig.4). Weekly Mutation is to
swap the volume for randomly selected product in a
particular week to another week whereas Randomly
Selected Mutation is to regenerate random product
volume for a randomly selected product. The
probability for an individual to be selected to
undergo mutation is Pmu for Weekly Mutation and
Pm for Randomly Selected Mutation.
Product
type (Pi)
P1
P2
P3
:
:
Pn
W1
…..
…..
…..
:
:
…..
Week
W2
…..
…..
…..
:
:
…..
(Wk)
W3
…..
…..
…..
:
:
…..
W4
…..
…..
…..
:
:
…..
Fig.3: Weekly Mutation
Product
type (Pi)
P1
P2
P3
:
:
Pn
W1
…..
…..
…..
:
:
…..
Week
W2
…..
…..
…..
:
:
…..
(Wk)
W3
…..
…..
…..
:
:
…..
W4
…..
…..
…..
:
:
…..
Fig.4: Randomly Selected Mutation
3.4 Evaluation Function
In this paper, an aggregating approach is applied to
minimize conversion capacity and frequent
changeover, to prevent overloaded capacity, and to
achieve a balanced workload. Equation (7) shows
the formulation of the objective value.
(7)
k 1
CVCk (k  1,2 ,...,t) refers to conversion capacity
for week kth. It includes the hardware and software
changing time, temperature waiting time, and lot set
up time. The minimization of conversion capacity is
needed to reduce the impact of rapid changeovers
brought by HMLV systems.
OVPk (k  1,2 ,...,t) is a penalty function given
to product mixes that overload the weekly resource
capacity for week kth. It indicates that the production
floor is no longer capable of fulfilling the weekly
demand based on the available resource capacity. In
other words, OVPk is applied when UCk is more than
ACk.
UBPk (k  1,2 ,...,t) is a penalty function given
to unbalanced workload throughout the planning
horizon. Fig.5 shows an example of unbalanced
workload in two weeks capacity consumption. The
product mix distribution is not systematically
planned: machine utilization may be too high in
week 1 but too low in week 2.
Thus, the
introduction of UBPk is used to address unbalanced
workload problems in product mix planning. GAP
in Fig.5 refers to unused capacity or difference
between available capacity and used capacity.
Week 1
Week 2
20%
4 Results and Discussion
As the proposed objective function is a minimization
function, the lower the objective value, the better the
product mix is. Fig.6 shows that the objective value
decreases (better product mix is obtained) as the
number of generation increases.
Fig.6: Best Objective Value for each generation
Weekly Capacity Comsumption
(Original Product Mix)
Capacity (%)
t
ObjectiveValue   (CVCk OVPk  UBPk )
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
21.17
24.63
38.07
52.58
Available Weekly
Capacity Time
Frame
66.33
63.46
46.45
31.35
11.90
16.07
1
2
15.48
12.50
3
4
Week
90%
DT
Available
Capacity
GAP
GAP
Used Capacity
Fig.7(a): Weekly capacity usage based on original
product mix
Weekly Capacity Usage
(Proposed Model)
Fig.5: Unbalanced workload in capacity usage.
3.5 Termination and Parameters Selection
The termination criterion is met once the same
solution has been continuously obtained by the GA
for more than 40 reproduction trials. The population
size used in the case study is 70 and the maximum
generation used is 100. The probabilities for genetic
operators are given as: Pxovr = 0.7, Pm = 0.7 and
Pmu = 0.3.
Capacity (%)
Used Capacity
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
31.88
56.22
11.90
1
26.55
28.09
28.97
57.38
56.44
58.53
16.07
15.48
12.50
3
4
2
Down Time
Week
Gap
Used Capacity
Fig.7(b): Weekly capacity usage
proposed product mix
based
on
Using data from a semiconductor plant, weekly
capacity consumptions based on the proposed GA
product mix and the original product mix are
compared. As shown in Fig.7(a) and 7(b), the
proposed model produced a more balanced workload
throughout the four weeks planning horizon as
compared to the original product mix using
predicted demand method. In Fig.7(a), the weekly
workload is not distributed equally. More balanced
workload can be observed in Fig.7(b) where the
GAP is more or less the same for each week.
The performance of the original and GA-based
product mix are showed in Table 1. In comparison,
the overall objective value has been improved by
65.93%. GA-based product mix shows better
performance in conversion capacity (CVC) and
unbalanced workload penalty (UBP) in which CVC
has been reduced by 28.19% and UBP has been
reduced to zero. Both original and GA-based
product mix has zero value for OVP because there
are no overloaded capacity and the production is still
able to support the demand. The results show that
the proposed model has managed to reduce
conversion time caused by high product mix and, at
the same time, is able to achieve a more balanced
workload in the weekly product mix distribution.
5
Summary
In this paper, chromosome representation using both
binary and real-value encodings is suggested to
represent the HMLV product mix structure. Based
on the analysis and comparison of GA-based product
mix planning and the original product mix planning,
the GA approach shows its effectiveness in HMLV
product mix optimization. The proposed GA-based
model outperforms the original one in the reduction
of conversion time and achievement of a balanced
workload. Instead of having time lost due to
frequent changeover, more time can be utilized for
more value-added jobs. Besides, the model has also
managed to fulfill each product demand throughout
the planning horizon and, at the same time, prevent
overloaded capacity. As a result, the GA-based
model has proved its efficacy to be deployed as a
product mix planning tool for HMLV manufacturing
systems.
Table 1: Comparisons of GA based product mix and original product mix using the proposed objective
function evaluation
Evaluation Function
Original Product
Mix
GA based Product
Mix
Improvement
(%)
Objective Value
380.56
129.66
65.93
OVP (overloaded penalty)
0
0
0
CVC (conversion capacity)
180.56
129.66
28.19
UBP (unbalanced workload penalty)
200
0
100
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