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INTERDISCIPLINARY JOURNAL OF CONTEMPORARY RESEARCH IN BUSINESS
COPY RIGHT © 2012 Institute of Interdisciplinary Business Research
845
SEPTEMBER 2012
VOL 4, NO 5
OPTIMIZATION OF PRODUCTION PLANNING USING MATHEMATICAL MODEL
(CASE STUDY :BEHNOUSH IRAN COMPANY
Yaser Ghorbanzad Master of Industrial Management(OR), Department of Management and Economy, Science and Research Branch, Islamic Azad
University, Tehran, Iran
Abbas Toloie Eshlaghy Industrial Management Dept., Islamic Azad University, Science and Research Branch, Tehran, Iran
Mohammadali Afshar Kazemi Industrial Management Dept., Islamic Azad University, Science and Research Branch, Tehran, Iran
Abstract
The present survey has been conducted to optimize production planning in production systems
based on product using a mathematical model in a beverage manufacturing company for
modeling by means of ideal planning techniques.
Objective of this survey is to study the current status of manner of production and planning in the
factory under study and the existing problems in production process. Therefore dependent
variables in production planning have been defined. Accordingly effective production ideals on
production planning are identified given to managers' viewpoints and mathematical model of
planning is designed in the above factory using ideal planning technique. Given to the capability
of Lingo software in solving mathematical problems this software was used to solve the designed
mathematical model.
Keywords: Production planning, ideal planning, mathematical modeling, Lingo software
1- Introduction
Today modern challenges are dominant in industrial organizations and companies parallel to
previous issues. Exhaustibility and constraint of raw materials, human resources, production
capacity, place, time and capital are among the previous issues. Nowadays manufacturing
companies are faced with important apprehensions which have increased necessity of using
scientific methods in encountering with the related issues following commercial phenomena like
competition in the market.
Dynamism of the commercial and industrial environment and multiplicity of effective factors on
performance of organizations encounter organizational decision-makers with various purposes
that gratifying satisfaction levels of them becomes important. "Operation research" is one of the
scientific tools that has high capability in gratifying the above needs as a scientific branch.
Techniques of this branch of management science have a significant impact on playing decision-
making role given to the high capability in formulating organizational issues and considering
limitations and needs of managers of the organization. Also, it is a powerful tool in
measurement, directing and controlling of organizational indigenous and exogenous factors for
management (Bransson, 1993).
It has been tried in this survey to formulate and model this activity scientifically using the
knowledge of operation research and studying planning production process of the above factory.
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2- Research literature
The vast majority of the works reviewed opt for the linear programming-based modeling
approach, particularly mixed integer linear programming models Conversely, nonlinear
programming is only used in two references (Benjamin, 1989; Lababidi et al., 2004). Six
references refer to the multi-objective programming-based modeling approach, of which three
use either multi-objective linear or integer linear multi-objective programming, while the other
three opt for nonlinear modeling. The inclusion of uncertainty in the various models is achieved
by fuzzy programming with stochastic programming. Both kinds of mathematical programming
appear as either a complementary modeling approach or the main approach as in Sakawa et al.
(2001), Demirli and Yimer (2006) and Aliev et al. (2007) for fuzzy programming, or as in Sabri
and Beamon (2000) and Goetschalckx et al. (2002) for stochastic programming. Likewise,
heuristic solution algorithms and metaheuristics are used as complementary techniques to solve
mathematical programming models, mainly integer linear programming. gramming and, to a
lesser extent, nonlinear, multi objective or fuzzy programming. The use of simulation tools to
complement the mathematical models is considered in the hybrid modeling approach referred to
in four references (Lee and Kim, 2000, 2002; Lee et al., 2002; Lim et al., 2006). Next, the details
of each modeling approach used by the different works reviewed are provided. Martin et al.
(1993) presented a linear programming model for planning production, distribution and
inventory operations in the glass sector industry. Chen and Wang (1997) proposed a linear
programming model to solve integrated supply, production and distribution planning in a supply
chain of the steel sector. Ryu et al. (2004) suggested a bi-level modeling approach comprising
two linear programming models, one for production planning and one for distribution planning.
These models subsequently consider demand uncertainty, resources and capacities when they are
reformulated by multi-parametric linear programming. Kanyalkar and Adil (2005) proposed a
linear programming model for aggregated and detailed production and dynamic distribution
planning in a multiproduct and multiplant supply chain. Oh and Karimi (2006) put forward a
linear programming model that integrates production and distribution planning for a
multinational firm in the chemical sector in a multi-plant, multi-period and multi-product
environment. This model also works with tax and financial data, such as taxes related with the
firm’s business activity or amortiza tions. Jung et al. (2008) compared linear programming
models for centralized and decentralized production and transport planning environments.
2-1 Advantages of production planning
Applying appropriate method of production planning will create advantages as below for the
organization. Enhancement of productivity and better using of possibilities such as raw materials,
machineries, human force and etc
Reduction of inventory level of raw materials and parts, incomplete products and
final product in the warehouse that is leaded to decrease costs.
Increasing of human force efficiency due to coordination and decreasing of
pressures due to lack of prediction in activities
On-time accomplishment of commitments to customers and enhancing their
satisfaction Reduction of stops and increasing efficiency of machineries
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VOL 4, NO 5
Realization of long-term purposes of production in the form of production planning
Reduction of losses arising from missed sales
Reduction of dependence of production system on the individual and prohibition of
doing works by taste
Creating balance among productive stations and workshops and optimal application
of production capacities (Korajewski & Ritzman, 2001).
2-2 Techniques of production planning
Usually techniques of production planning could be divided into two major classes in terms of
finding the best possible response or optimal solution:
A) Techniques that give optimal response
B) Techniques that don't necessarily give optimal response
C) Linear programming and its applications
D) Many of the management decisions intend to make using resources of the organization
effective. These resources typically include machineries, labor force, money, space of the
warehouse and raw materials which are used to produce goods and services. Linear
programming is a mathematical technique with extensive application that has been
designed to help managers in programming and decision-making about resource
allocation.
E) In quantitative analysis field modeling and solving a problem mathematically is called
programming. Also, computer programming has played a significant role in advancement
and using linear programming (Mehregan, 1996).
3- Designing mathematical model of production planning in production systems based on
the product
3-1 Main hypotheses to design the model
The following hypotheses are regarded as fundamental assumptions in the proposed model given
to conditions of the above company:
1- There are four production lines in Iran Behnoush Company that each one can produce a
family of products. It is noteworthy that production or not production of each product
doesn't affect production or not production of other products.
2- Demand level of each product that is determined by the business unit and by obtaining
orders of customers is specified for the programming unit.
3- Price of each product is determined given to the contract between the company and
customers.
4- Length of the period for production planning of the company is monthly due to high
variety of products, variability of customers' demand and cold or hot temperature (by
assuming that coldness and hotness of air temperature during months of a year is
specified).
5- Number of workers (available human force) will be stable during the programming
period. It is notable that number of available human force will be reviewed at the
beginning of each month and given to demand prediction.
6- Given to day and night production of the company there will be no problem regarding
preparing incomplete products. In other words, prerequisites of ordered products will
always be provided.
7- Production capacity of each product is specified given to the allocated human force to it.
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8- Wastage is not involved in designing the mathematical model because of its low level
that is created during production process.
9- Cost of every product is determined due to the used materials in it, side costs such as
electricity, wind, personnel expenses of non-productive forces at the regular time of
production and overtime.
3-2 Major elements of the proprietary model
First, variables, specifications and the applied parameters in the survey are represented by
studying production process. Then systemic limitations which affect production capacity and
modeling manner of the issue in the production process will be identified. Where a part of
process or operational station has no impact on production capacity or the model and doesn't
create any limitation for the model practically it is omitted from the model.
3-2-1 Characteristics (specifications) of production
Given to variety of the effective factors on production level of each product and the allocated
force to each production line it is first essential to be familiar with characteristics
(specifications).
Insert Table 1
4-2-2 Decision variables
In systemic viewpoint the major portion of outputs of the mathematical model is its decision
variables. Decision variables of production planning mathematical model in this survey are
defined based on the following characteristics:
Insert Table 2
4-2-3 Parameters of the mathematical model (fixed amounts of model)
Each mathematical model needs specified amounts that have a direct impact on final results of its
solution as the model input. Technical coefficients of limitations, amounts at the right side of
ideals and coefficients of the variables used in the target function are elements of inputs in the
mathematical model. Fixed amounts that must be determined from documents and analysis of the
collected data before solving the model include the below amounts:
Insert Table 3
4-3 Systemic limitations
As we know systemic limitations constitute one part of ideal programming model. These
limitations are restraints which limit having access to purposes. Among such limitations that are
considered in this survey we can refer to limitation of the needed raw materials to produce
nonalcoholic beer and limitation of production capacity of production lines. Parameters related to
these limitations were obtained through studying internal information of the company and the
transmitted information from various units such as industrial accounting, warehouses, production
saloons, marketing, sales and production planning.
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VOL 4, NO 5
4-3-1 Limitation of providing the required raw materials
Preparing of raw materials is one of the most important difficulties that manufacturing
companies and factories are faced. This issue becomes more complicated when such materials
are of imported type. Given that Iran Behnoush Company imports a considerable volume of its
raw materials limitation of raw materials must be regarded among essential limitations in
designing the mathematical model for production planning of this company.
Management of the company determines amount of the consumption raw materials from type m
to be purchased at the beginning of each year given that volume of consumption raw materials in
the previous years is specified. Due to the point that percentage amount of combination of raw
materials in one liter of nonalcoholic beer and annual purchasing amount of materials (Rm) are
specified limitation of raw materials is defined as below:
≤
m=1, 2, …, 5 , i= 1, 2,…, 4 , j=1, 2, …, 5 , t= 1, 2, …, 12
Where:
i= characteristic of nonalcoholic beer based on packaging type of the produced nonalcoholic beer
(1- bottle, 2-can, 3-rotational, 4-pet)
j= characteristic of nonalcoholic beer based on capacity (1- 280 cc, 2- 330 cc, 3- 500 cc, 4- 1000
cc, 5- 1500 cc)
t= characteristic of months of production planning (1- April, 2- May, 3- June, …, 12- March)
m= characteristic of the applied raw materials in producing nonalcoholic beer (1- malt, 2- hop, 3-
HFCS, 4- citric acid, 5- ascorbic (vitamin c))
= amount of raw materials of type m that is used to produce nonalcoholic beer with capacity
j
= production amount of nonalcoholic beer with packaging i and liter j in time period t
= amount of raw materials of type m that is available in one year.
4-3-2 Production capacity limitation of production lines
Iran Behnoush Company has four production lines that are named based on packaging type of
nonalcoholic beer of these lines including: 1-bottle, 2- can, 3- rotational, 4- pet. Every production
line has limited productive capacity. As we know each line has a nominal production capacity
and a real production capacity that real production capacity is considered in this model so that
the model becomes closer to reality.
Meanwhile, it is noted that amount of real production capacity of production lines don't have
considerable difference in monthly time periods with each other.
Thus, limitation of real production capacity of production lines in one month is as below:
≤
i=1, 2,…,4 , j=1, 2, …, 5, t=1, 2, …, 12
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Where
i= characteristic of nonalcoholic beer based on packaging type of the produced nonalcoholic beer
(1-bottle, 2- can, 3- rotational, 4- pet)
j= characteristic of nonalcoholic beer based on capacity (1- 280 cc, 2- 330 cc, 3- 500 cc, 4- 1000
cc, 5- 1500 cc)
t= characteristic of months of production planning (1- April, 2- May, 3- June, …, 12- March)
= production amount of nonalcoholic beer with packaging i and liter j in time period t
Ci= amount of real production capacity of production line i during the year
It must be noted that real production capacity of production lines during each time period
(monthly) doesn't have perceptible difference with each other. Therefore, real production
capacity of each production line is considered similar during various time periods. Real
production capacity of each production line is as below.
4-4-4 Managerial purposes
As we know ideal is a certain position (or quantity) in time and place that the decision maker
intends to access it. Amounts related to these ideals are obtained through counseling with experts
and the clear-sighted and about some issues like market share it is exploited from comprehensive
programs of the company. Ideal level of managerial purposes in the present survey has been
designed in the model given to studying of compiled programs and the represented policies in
programs as well as interviewing with managers of Iran Behnoush Company.
4-4-4-1 The ideal to increase return on sales
One of the most important purposes that management of Iran Behnoush Company considers in
determining the programs is to reach maximum return on sales during the planning period.
Increasing of selling amount of products will be leaded to more profit and increased liquidity
potential as well as maintenance of the market share in management view. Since all products will
have profit for the company through their selling ideal of maximum return on sales is the product
of sum of productions in profit of each unit of products and is stated as below:
+ 1+- 1
-) =
i=1, 2,…,4 , j=1, 2, …, 5, t=1, 2, …, 12
Where
i= characteristic of nonalcoholic beer based on packaging type of the produced nonalcoholic beer
(1-bottle, 2- can, 3- rotational, 4- pet)
j= characteristic of nonalcoholic beer based on capacity (1- 280 cc, 2- 330 cc, 3- 500 cc, 4- 1000
cc, 5- 1500 cc)
t= characteristic of months of production planning (1- April, 2- May, 3- June, …, 12- March)
= profit of each unit of nonalcoholic beer with packaging i and liter j
= production amount of nonalcoholic beer with packaging i and liter j in time period t
1+= positive deviation from ideal of maximum return on sales
1-= negative deviation from ideal of maximum return on sales
= maximum expected return on sales
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4-4-4-2 Ideal of decreasing the loss due to missed sale of products
Given to the policy of Iran Behnoush Company's management the loss arising from missed sale
must be decreased as much as possible to maintain the market share and obtain more customer
satisfaction. Therefore, management of the company intends to minimize the difference between
production level of products of the company and demand level of the market. Lower production
level with regard to demand level of customers is leaded to lose a portion of market demand and
this is resulted in the loss of missed sale. Ideal of decreasing the loss due to missed sale of
products is the product of profit of each nonalcoholic beer unit by amount of unprovided demand
of each unit of products. It is stated as the following:
+ ( 2+- 2
-) =
i=1, 2,…,4 , j=1, 2, …, 5, t=1, 2, …, 12
Where
i= characteristic of nonalcoholic beer based on packaging type of the produced nonalcoholic
beer (1-bottle, 2- can, 3- rotational, 4- pet)
j= characteristic of nonalcoholic beer based on capacity (1- 280 cc, 2- 330 cc, 3- 500 cc, 4- 1000
cc, 5- 1500 cc)
t= characteristic of months of production planning (1- April, 2- May, 3- June, …, 12- March)
= profit of each unit of nonalcoholic beer with packaging i and liter j
= amount of unprovided demand (missed sale) of nonalcoholic beer with packaging i and liter
j in time period t
2+= positive deviation from ideal of amount of unprovided demand (missed sale) of products
2-= negative deviation from ideal of amount of unprovided demand (missed sale) of products
G2= maximum amount of the intended cost due to missed sale of products
4-4-4-3 Ideal of decreasing storage time of products
When amount of products is more than demand level of customers the company is forced to store
them. Therefore, this brings about imposing of additional storage cost which is an unfavorable
cost for the company. Management of the company is determined to decrease storage of products
as much as possible:
+ ( 3+- 3
- ) =
i=1, 2,…,4 , j=1, 2, …, 5, t=1, 2, …, 12
Where
i= characteristic of nonalcoholic beer based on packaging type of the produced nonalcoholic
beer (1-bottle, 2- can, 3- rotational, 4- pet)
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j= characteristic of nonalcoholic beer based on capacity (1- 280 cc, 2- 330 cc, 3- 500 cc, 4- 1000
cc, 5- 1500 cc)
t= characteristic of months of production planning (1- April, 2- May, 3- June, …, 12- March)
= storage cost of each unit of nonalcoholic beer with packaging i and liter j
= number of nonalcoholic beer with packaging i and liter j stored in time period t.
3+= positive deviation from ideal of decreasing storage time of products
3-= negative deviation from ideal of decreasing storage time of products
G3= maximum amount of the intended cost due to storage of products
5-4-3 Objective function
Now objective function of the mathematical model is designed based on principles of ideal
planning technique by considering the above political limitations and managerial ideals.
We know that objective function in ideal planning is the strength of this technique because of the
possibility to lay several ideals beside each other in objective function of this technique.
Structure of the objective function is according to minimization of inappropriate deviations of
ideals.
Given to the intended ideals of Iran Behnoush Company's management unfavorable ideal
deviations are determined as below:
Insert Table 4 Cardinal method is used to design objective function as importance of minimization of each
unfavorable deviation is identical for company managers. The considerable point of this
technique regarding the necessity that elements of objective function should have an identical
scale (unfavorable deviations) is based on a software that it is not needed to use this issue in the
present survey because all ideals are of one type (money type).
Therefore, objective function of the mathematical model under study is proposed as below:
Minimize Z= 1- + 2
++ 3
+
Subject To:
+ 1-- 1
+) =
+ ( 2-- 2
+) =
+ ( 3-- 3
+ ) =
≤
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≤
i= 1, 2, …, 8
j=1, 2, …, 4
t=1, 2, …, 12
m=1, 2, …, 5
All Variable ≥0
5- Conclusion
Finally a model was represented for production planning in Iran Behnoush Company after
conducting various studies, studying the existing conditions and limitations, interviewing with
managers and experts and defining the variables. The represented model could be effective on
income increase, decreasing overtime cost and cost of lost sales in addition to conducting weekly
production planning. Obtained results and claims of the company's managers and the proposed
mathematical model could be effective on saving time besides providing the management's
needs.
Given that several purposes were considered by management of Iran Behnoush Company such as
increased income, decreased overtime and decreased cost of loss in production (lost sales) it
could be argued that sum of these different and almost opposite purposes is only possible
through ideal planning. This is confirmed by observing the results and verifying their accuracy
by users and managers of this company.
Confirmation of the general director of Iran Behnoush Company is provided below about
usefulness of the proposed model given to obtained results regarding the model implementation.
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78-120.
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Production Planning and Control 13, 35–46.
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Annexure
Table 1- Specifications of decision variables of mathematical model
Table 2- Decision variables of the mathematical model
Table 3- Parameters of the mathematical model
Variable Description
qijt Production amount of nonalcoholic beer with packaging i and liter j in period t
d1+ Positive deviation from ideal of maximum return on sales
d1- Negative deviation from ideal of maximum return on sales
d2+ Positive deviation from ideal of products' missed sale (unprovided demand)
d2- Negative deviation from ideal of products' missed sale (unprovided demand)
d3+ Positive deviation from ideal of decreasing the amount of time to store productions
d3- Negative deviation from ideal of decreasing the amount of time to store productions
Variable Description
Sijt Amount of unprovided demand (missed sale) of nonalcoholic beer by packaging i and
liter j in period t
Nijt Number of nonalcoholic beer with packaging i and liter j stored in period t
Bij Profit of each unit of nonalcoholic beer with packaging i and liter j
Fij Warehouse cost of each unit of nonalcoholic beer with packaging i and liter j
Emj Amount of raw materials of kind m that are used to produce nonalcoholic beer with
capacity j
Rm Amount of raw materials of kind m that are available in one year
Ci Amount of capacity of real production of production line i during the year
G1 Maximum level of the expected return on sales
G2 Maximum level of the intended cost due to missed sale of products
G3 Maximum level of the intended cost due to the stored products
Specification Description Range
i Characteristic of nonalcoholic beer based on kind of the product's packaging (1-
bottle, 2- can, 3- rotational, 4- pet)
i=1, 2, 3, 4
j Characteristic of nonalcoholic beer based on capacity (1- 280 cc, 2- 330 cc, 3-
500 cc, 4- 1000 cc, 5- 1500 cc)
j=1,2,…,5
t Characteristic of months of production planning (1- April, 2- May, 3- June, …,
12- March)
t= 1, 2, …, 12
m Characteristic of the usable raw material in producing nonalcoholic beer (1-
malt, 2- hop, 3- HFCS, 4- citric acid, 5- ascorbic (vitamin C)
m= 1, 2, .., 5
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Table 4- Information related to unfavorable deviations in objective function Variable of
unfavorable
deviation
unfavorable deviation Ideal
1- Negative deviation Ideal of increasing return on sales
2+ Positive deviation Ideal of decreasing amount of loss due to missed sale of products
3+ Positive deviation Ideal of decreasing storage time of products