Multi level Programming in Multi Site Production Systems Analysis

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Vol-7, Special Issue3-April, 2016, pp1341-1356 http://www.bipublication.com

Research Article

Multi level Programming in Multi Site Production

Systems Analysis

N. Bakhshizadeh1, M. Memarpour2

and A. Makui3

1

Phd student in industrial engineering, Dept Of industrial engineering & Mechanics, Qazvin Islamic Azad University, Qazvin-Iran.

2

Phdstudent in industrial engineering,

Islamic Azad University Science and Research Branch, Tehran - Iran 3

Associate Professor, Dept of industrial engineering, Science and industry University - Iran

*Corresponding Author Email: bakhshizadeh.nastaran @gmail.com

ABSTRACT

Considering the significance of the Supply Chain Management Systems and the due planning related to it, cause that researcher has gathered data on various modeling types about Multi Site Production planning Systems that are actually Supply Chains located in different places geographically. It is worth noting that the presented models are of very short history and they are just under the focus of study after 80's. In the present article, the writer has surveyed the Multi Level Programming Model and after studying the nature and application of such mathematical planning and its due relevance to the subject under the study, the Multi Site Production Systems were selected. It is completely obvious that the active elements in a Supply Chain may not be in total agreement and cooperation with each other and also they may not be in complete opposition with each other, therefore the modeling of the above mentioned systems does not seem to be proper through a usual one-zero or linear planning. The significant points in Multi Level Programming Model are: First, it is hierarchical and therefore it is more efficient in supply chains in which some active elements are more influential and powerful than the others. Second, in this system the decision makers in the upper levels determine their goals and decisions and ask the lower levels of the organization to optimize the goals separately. The lower-level decision makers would present their decisions to the upper-levels and the decisions would be reformed or modified by them, considering the general interests of the organization. The process continues to an optimal or a more satisfactory solution.Thus, the researcher has presented his analytical model (which is itself a consideration of just two steps of this chain: production and supply) based on a two-level planning model and has eventually tested his model in a car company.

Keywords:supply chain management, multi site production systems, multi level programming institutional

1. INTRODUCTION

Different techniques emerged from the various managerial philosophies and further also newer viewpoints were grown from such techniques, of which supply chain management may be mentioned. Supply chain management is a managerial viewpoint suggested for the promotion and growth of the organizations. Such viewpoint, appeared from the basis of the previous methods and thoughts, seems to be an integrated viewpoint focusing on both qualitative and quantitative discussions. In

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customers expectation caused by the promoted knowledge and information of the same, the buyers are those who determined the products features. Eventually, in order to achieve the aforementioned capabilities which are principally considered as the pre-requisites to enter into the global markets, certain discussions such as development to the qualitative performance in the field of quality and/or supply chain management in the companies’ major managerial systems occurred. Additionally, certain pressures were imposed to the companies fro, different aspects which resulted in inability of the company in fulfillment of all the tasks. Therefore, certain activities such as supply/ demand planning, procurement of materials, production and planning the products or services, goods storage, controlling inventory, quality control, goods and services distribution, etc, which all used to be completed on the company level have not been transferred to the supply chain level. In fact, in supply chain, the operations level has significantly increased and became more complex and in supply chain management we learn that instead of merely watching ourselves, we consider and supervise the entire process.(Eftekhari ,2006). Naturally, the supply chain planning issues include the stages of modeling the multi decisions related to each other hierarchically. Whereas the different activities of a chain are often organized by its separated elements, each having separate and sometimes conflicting purposes, the performance and control of the entire network shall be based on some aspects. Many of the planning models are based on this assumption that all the supply chain networks activities are directed without considering the different aspects by a general organizer. In this regards, reviewing the works of McDonald (1997) and Tsiakis (2001) shall be useful(McDonald & Karimi,1997)( Tsiakis et al.,2001). The focus of operations planning and control for manufacturing firms has expanded successively over the last 50฀years. New principles, techniques, and systems have emerged that have allowed for new approaches. The perspective for planning and control has expanded from internal production operations to supply chain operations

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product. Multi site planning may model different issues with different target functions. For instance, problems with the target function of total fixed and variable costs or the problem with the target function of timing for those systems having irrelevant parallel mechanism. In this regard, several approaches had been posed which will be studied here. The middle-term planning algorithms for the multi site production networks are classified into two groups:

1.1. Middle-term planning algorithms with focused/ non-focused approach

1.2. Middle-term planning algorithms based on definite/ indefinite demands (random)

2.literature :

Although the primary studies made on two-level planning date back to the 70’s, but until early 1980 the application of such math programs was not highly considered in the modeling of the hierarchical decision making processes and engineering designing problems. One of the earliest studies in regarding this issue was made(1985)by Kolstad. Originally, two-level planning problems are difficult and it is not surprising that often algorithmic searches have focused on the simplest two-level states until now. For instance, linearity, quadratic and/or concavity of the target functions and/or other similar limitations are simplifying assumptions. Specially, the samples which have more been studied during a long period of time are relevant to two-level planning issues, in which all the functions are linear. Therefore, this subset is the subject of many of the studies presented up to now. Through time, more complex two-level programs were studied and even those including discrete variables were considered. From amongst the studies made in this regard, the works of Savard and Gauvin (1994) may be mentioned. Therefore, more generalized studies were also presented. For instance, Vincent and Calami (1994) defined the two-level planning non linear problems and math programs with limitations in equal forms .The nature of two-level planning combinations have been studied in colson et al.(2005) presented a method based on the reliable region for the non linear

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integer linear programming (MILP) model for general two-phase aggregate production planning systems. Due to NP-hard class of APP, he implement a genetic algorithm and tabu search for solving this problem. The computational results show that these proposed algorithms obtain good-quality solutions for APP and could be efficient for large scale problems .Fahimnia et al. (2012) noted that production plan concerns the allocation of resources of the company to meet the demand forecast over a certain planning horizon and a distribution plan involves the management of warehouse storage assignments, transport routings and inventory management issues. A production–distribution plan integrates the decisions in production, transport and warehousing as well as inventory management. The overall performance of a supply-chain is influenced significantly by the decisions taken in its production–distribution plan and hence one key issue in the performance evaluation of a supply chain is the modeling and optimization of the production–distribution plan considering its actual complexity. Based on the integration of Aggregate Production Plan and Distribution Plan, this article develops a mixed integer non-linear formulation for a two-echelon supply network (i.e. a production-distribution network) considering the real-world variables and constraints. Genetic Algorithm (GA), known as a robust technique for solving complex problems, is employed for the optimization of the developed mathematical model due to its ability to effectively deal with a large number of parameters. To demonstrate the applicability of the methodology, a real-life case study will be finally studied incorporating the production of different types of products in several manufacturing plants and the distribution of finished products from plants to a number of end-users via multiple direct/indirect transport routes.After that they examined how a complex supply chain yields cost reduction benefits through the global integration of production and distribution decisions. The research is motivated by a complex real world supply chain planning problem facing a large automotive company. A mixed-integer nonlinear production-distribution

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demonstrates the feasibility and efficiency of the proposed method .Sakawa et al. (2012) in their paper considered Stackelberg solutions for decision making problems in hierarchical organizations under fuzzy random environments. Taking into account vagueness of judgments of decision makers, fuzzy goals are introduced into the formulated fuzzy random two-level linear programming problems. On the basis of the possibility and necessity measures that each objective function fulfills the corresponding fuzzy goal, together with the introduction of probability maximization criterion in stochastic programming, we propose new two-level fuzzy random decision making models which maximize the probabilities that the degrees of possibility and necessity are greater than or equal to certain values. Through the proposed models, it is shown that the original two-level linear programming problems with fuzzy random variables can be transformed into deterministic two-level linear fractional programming problems. For the transformed problems, extended concepts of Stackelberg solutions are defined and computational methods are also presented. A numerical example is provided to illustrate the proposed methods. Non cooperative fuzzy random two-level linear programming problems are considered .Fuzzy goals are introduced into the formulated two-level programming problems .New models are proposed through possibilities and stochastic programming .Extended concepts of Stackelberg solutions under fuzziness and randomness are defined. Computational methods for the defined Stackelberg solutions are presented. They considered interactive decision making methods for random fuzzy two-level linear programming problems. Assuming that the decision makers concern about the probabilities that their own objective function values are smaller than or equal to certain target values, fuzzy goals of the decision makers for the probabilities are introduced. Then, the possibility-based probability model to maximize the degrees of possibility with respect to the attained probability is considered. Interactive fuzzy nonlinear programming to obtain a satisfactory solution for the decision maker at

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programming problems. A numerical example for a three-level 0–1 programming problem is provided to illustrate the proposed method(Sakawa & Matsu,2014).Biswas and Bose (2013) developed a fuzzy goal programming methodology for solving quadratic multi objective multilevel programming problems in a hierarchical decision making environment. In the proposed procedure, the tolerance membership functions in fuzzy set theory are defined first for measuring the degree of satisfactions of the objectives of each decision maker at each level. Then a nonlinear fuzzy goal programming model is formulated to achieve highest degree of each of the defined membership goals at each level to the extent possible on the basis of priorities of importance of optimizing the objectives. Afterwards the nonlinear fuzzy goals are converted into linear forms by applying a linear approximation technique. The model is then solved for measuring the degree of satisfaction of the objectives of decision makers at each level by arriving at a compromise decision regarding the optimality of the sets of decision variables controlled individually by each of them. To illustrate the proposed methodology a numerical example is solved and compared the solution with existing approaches .Li and Liu (2013) noted that Supply chain management is important for companies and organizations to improve their business and enhance competitiveness in the global marketplace. The bullwhip effect problem of supply chain systems with vendor order placement lead time delays in an uncertain environment is addressed in this paper. Among the numerous causes of bullwhip effect, he focus on uncertainties with respect to demand, production process, supply chain structure, inventory policy implementation and especially vendor order placement lead time delays. Minimizing the negative effect of these uncertainties in inducing bullwhip effect creates a need for developing dynamical inventory policy that increases responsiveness to demand and decreases volatility in inventory replenishment. First, a dynamic model of supply chain with above uncertainties is developed. Then, a novel uncertainty-dependent robust

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uncertainties considering demand, supply, and process sides .The potential risks are managed by the fuzzy logic. Developing new measures weighted by a multi-attribute decision-making technique .Benders decomposition method is applied as the solution approach.

3.Integrating and integrated planning model : Integrating has a deep relationship with today’s world. Collapsing the economic and political borders of the recent decade imposes a new responsibility on the integration process. The companies have to establish complete integrity and such activity for them is like an important business opportunity and a certain part of the strategic operations. For the time being, such activity for the companies are like general practice, by which they may pass through the national borders and may use the new opportunities of deals, decreasing production costs, collecting capital, etc. This issue has deeply affected the companies architecture: on products, technologies, production equipment and organization which are increasingly expanding. Such changes have consequences and also it shall be noted that the mew viewpoints need complete taking benefit from the company potentials. Therefore, we plan for the productions sites global networks. Multi site production network discloses suitable managerial problems especially in the middle-term domain (formulating the middle-middle-term planning), when the decision making factors shall be set (e.g. assigning product/ industrial unit, inventory policies, main time scheduling, etc.)Integration and simulation the flow of information and materials in the production site dependent on a supply chain provide important requirements for the modern industrial organizations. In order to assure optimized interaction between customer satisfaction and production costs, automotive industrial groups make structural changes in their production systems. In an analysis made on the level of vulnerability of the supply chain of a car manufacturing system in the average level it was known as that the turbulences occurred in the subcontractors’ level flow in the upstream and downstream of the supply chain. Eventually, in order to prevent and reduce such turbulence in

the supply chain, integration and participation in all the production stages as well as in the subcontractors level have been recommended. The integration approach for the Multi site system has been taken as the basis for this study and the model given in this regard is a work from Kanyalkar and Adil (2005). Modeling method has also been designed considering the multi site system integration. Therefore, here we study this part: Integrated planning or multi site systems has three dimensions: supply (purchase), production and distribution which are in fact the steps of materials reshaping (purchase, production, distribution), planning levels (general and side plans) and planning cycles (plans determined in the previous and present cycles). In multi site system configuration, the suppliers positions (quays and industrial units) with the goods receiving centers (selling outlets and industrial units) have either been determined in advance or are defined dynamically. The plant considered for each of the steps of supplying/ production/ distribution requires arranging and conforming them to each other. In order to do so, two approaches may be used:

a. Hierarchical approach: in which all the steps are considered hierarchically. Here, the decisions are transferred from a step to another and in each step simple planning is made which results in optimization of such subset (each of the subsets separately).

b. Simulation approach: in this method all the step are considered simultaneously which, however, results in planning complexity while its advantage is that it generates a general optimized plan.

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set) identifies the entire monthly production throughout the country to face all the demands of sales outlets considering the total set limitations. The decision making partial plans identify weekly production of industrial unit, inventories, quantities sent from each industrial unit to each sales outlet, raw materials contents (which are either considered as source or ordered to the suppliers/ offloading sites for each industrial unit). Usually this state is available in the detailed industries. The partial and total plans are set across the reshaping steps. 4. two-level planning problems :

Planning for non-focused planning for long term has been considered as the most important problems of decision making. Several solutions and approaches based on the concept of systemic analysis the large domains lack the needed ability to model certain samples from the independent subsystems which often exist in practice (Roghanian ,2006). There are a certain group of the mathematical planning problems which optimize a variety of targets in a hierarchical structure. In such type of problems there is several decision-making levels, “each controlling a certain part of the decision making variables existing in the decision making environment”. In such type of problems each level has its own special target function and each target function in each hierarchical levels has its own limitations, while there may also be common limitations for the entire problem. The following problem is of two-level type. It should be mentioned that in the two-level problems the high level decision maker, either an individual or an organization, is called pioneer, while the lower level problem decision maker is called follower.

) , ( min

,

y x F Y X

X (1)

st : G(x,y)0

) , ( min f x y

y

st :g(x,y)0

The problems variables are categorized into 2 groups of higher level (xRn1) and lower level (yRn2) which may either be discrete or continuous. Similarly, the functions

(F :Rn1Rn2 R ) and ( f :_Rn1_Rn2 _R )

denote the higher and lower levels target functions, respectively which may be linear, non linear, fraction, etc. Meanwhile, (G :Rn1Rn2Rm) and (g :_Rn1_Rn2_Rm) vector functions denote the higher and lower levels target functions, respectively which may also be linear, non linear, fraction. Multilevel optimization problems are certain mathematical plans while a subset of their variables shall be optimized solution of another plans parameterized by their remained variables. When the existing plan is a mathematical planning in its second level then the problem shall be a two-level planning problems. The triple-level planning optimization problems are referred to those planning problems in which the plan existing in its second level is a two-level planning problem. Through expanding such idea the multilevel problems may be defend in a more general state manner. Multilevel planning models details control on the decision variables along certain regular levels across the stepped planning structure. Multilevel planning is often proper tools to model certain samples of the decision making process which are not participating and cooperating with each other but are conflicting. For instance, a player optimizes as subset of decision variables, while during the same it simultaneously considers the independent reaction of each of the other players with respect to its action. Such type of planning, though there are not many well known techniques on the same, and has occasionally been used, enjoys several abilities and potential. Historically speaking, multilevel optimization has a close relationship with the Steckelberg (1952) economic problems in the games theory. Therefore, an economic planning process is studied which includes the conflicting organization: pioneer and follower. Within the special framework of the Steckelberg games, it is assumes that the pioneer predicts the follower elements (Thoai et al.,2002). Furthermore, this issue provides the former with the possibility of choosing the best strategy, in other words, the pioneer chooses the X strategy in the (X Rn₎

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strategy (Yi(x)Rm1) set as per each (xX ).

The Yi(x) sets are assumed to be closed and concave. In case the two-level planning is considered we have:

) , ( min

,y F x y

x (2)

subject to g(x,y)0

Where y in terms of any value of x; the solution of another second level (lower level) problem is as per the following:

)

,

(

min

f

x

y

y (3)

subject to

h

(

x

,

y

)



0

The relevant simplified model may be given as per the following:

) , ( min

,y F x y

x (4)

subject to 0

) , (x y 

g ,h(x,y)0

Of which the optimized answer is the two-level planning problem optimized value lower level (minimum target function) (Roghanian ,2006). 5. two-level planning model :

As mentioned also earlier, presenting a suitable model for a supply chain in way to be able to cover all its aspects (including: supply, production and distribution) is a difficult task. In this regard, the supply and production steps have been considered in this model. It should be mentioned that the problem has been modeled by using two-level planning approach, while two plans have been identified in the following hierarchical levels, each control a set of decision variables and also a set of conflicting targets in a totally independent manner. The first level of the planning is related to the producer company and the second level is related to the supplier. The target function of the producer includes costs of fluctuations from the production and keeping inventory and also deficiency in the contingency reserve. The producer limitations include the limitation related to the planned and real production volumes; limitation related to the relation between produced, demanded and available volumes; limitation related to the merged production capacity; warehouse capacity limitation and the limitation relevant t the contingency inventory policy making .The target function of suppliers includes minimizing the

transportation costs and the costs of keeping inventory with the suppliers.

The limitations of suppliers include limitation related to the balance level of inventory and receive volume of goods from suppliers; limitation related to the relation between final product and produced parts considering the parts consumption coefficient; limitation related to the maximum sent volume of a type of goods by the supplier; limitation related to the keeping capacity and finally limitation related to the percentage of supplied demand from each supplier. The case study has been made on an automotive company and the presented model has been studied in terms of the information obtained from such company (concerning the production and suppliers). Now, first of all the designed model is posed in two-level manner in two steps of production and supply, and then the case study is made and updated as per the model and presented as per the practical data:





HT IM

CS C

PV PV

CP Z

jt j

jt jt

t t man

 

 

 

 



 

min

s.t: Wjt XMjtPVjtPVjt0

R T t j  

, (1)

IM D XM

IMj,t1 jt jt jt (2)

E XM j j

jt 

  ₍₃₎

WM V

IM j j

t j  

 , ₍₄₎

D CS

CS

IMjt jt jtjt j,t (5)



]

) [

min

,

, sup

RC IR

RT RP

Z

dl l dlt

d

dl l

t dl t d

 



 



 



s.t:

IR RR

RP

IRdl,t1 dl,t dl,tRLTd dl,tRLTd

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IP XM

RR

IP dlt jt jl lt d

t

l,1 ,    , (7)

IR

RRdl,t dl,t (8)

RS

IPl,t l (9)

) ( _, ₁

, lj

j jt dl

t

dt RCS XM

RP     (10

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The parameters and variables of decision have been defined as per the following:

j type of produced product of the factory t time period (month)

CPt cost of changing the plan from t-1

period to t period ($/ unit of product)

PVjt deviation from the plan from t-1 period

to t period for j product

WMjt deviation from planned final product

inventory volume at the end of t period

CSjt deviation from contingency inventory

volume for j product in t period

Cj cost of having contingency inventory

less than determined volume

IMjt j product inventory in producer

location in the end of t period

HTjt cost of keeping one unit of j product

for a month

Wjt j product monthly production timetable

in t period

XMjt j product monthly production volume

in t period T planning horizon R re-planning period term

IMjt j product inventory in t period

Djt j product demand in t period

j j product conversion coefficient to the

capacity merged unit E factory merged capacity

v

j

required volume to store a unit of j product

WM total storage capacity of producer



jt a certain percentage of the demand for j

product in t+1 period which shall be kept as contingency reserve in t period

RTdl cost of transporting one unit of l

product from d supplier

IRdl,t inventory of d supplier goods in the

end of t period

RCdl cost of keeping each unit of l product

supplied from d supplier

RPdl,t planned goods receiving from d

supplier in the end of t period

RRdl,t volume of goods receiving from d

supplier in the end of t period

RLTd required time to transport goods from d

supplier to the producer

IPd,t inventory of d supplier in the producer

premises in the end of t period

dj volume of d supplier goods applied to

produce one unit of k product

RSd goods keeping capacity of d supplier in

producer premises

RCSd percentage of d supplier goods from

future period demand

In the presented model Zmanis the target

function of the producer which is composed of 3 parts. First part includes the costs of deviation from the production plan. Second part expresses the cost of failure to supply contingency inventory as defined. Third part includes the inventory keeping costs.

Eq. (1) is related to the balance of planned and real productions while the deviation variables in the same generate such balance.

Eq. (2) is related to the balance of inventory and produced volume and the demand of a period with the remained inventory for the next period. Eq. (3) is related to the observation of merged production capacity.

Eq. (4) is related to the warehouse capacity. Eq. (5) is related to a certain percentage of product demand which shall be kept as contingency reserve.

Eq. (6) is related to the balance of inventory and the received goods volume from the suppliers. Eq. (7) establishes a proper relation between the final product and produced parts by the suppliers considering the parts consumption factor. E.q (8) indicates that the maximum sent volume of a type of goods by the supplier cannot exceed its inventory.

Eq. (9) is related to the keeping capacity, and E.q (10) is related to the percentage of demand supplied from each supplier.

5.2. Case Study :

Considering the variety of Iran Kaveh Saipa Company range of product, in order to fulfill such case study the author has focused on the three-axel tanker product, which is composed of the following subsets:

 Chassis

 Underpinning

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Each of them includes sub parts and suppliers as per the following:

 Chassis

Beam core: two suppliers

Power and wind post: one supplier

 Underpinning One supplier

 Tanker

Manhole cover: one supplier 4’flange: two suppliers 5.3. Model Solving Guide :

 B-1 model shall mean the producer (Iran Kaveh Saipa Co) model, while B-2 model shall mean the supplier model.

 In B-2 model the limitation on (IP40) were removed (due to being excessive)

 In B-2 model, the limitation on (RRIR) were removed (due to being excessive)

 (_RCS_dl) was considered to be zero

 Supplying period RR has been delegated to one period before the same.

 (



_lj) was taken as 1.



_D

₃_t20,

_D

₂_t20,

_D

₁_t20, demand in all the period for j1,2,3 is assumed to be 20

 j=subset chosen from the abovementioned tanker product: chassis, underpinning and tanker

(j= 1,2,3) so that: j=1 chassis, j=2 underpinning, j=3 tanker

 l- chosen part from each subset

(l= 1,2,3,4,5) so that l=1 beam core; l=2 power and wind post, l=3 entire underpinning, l=4 manhole cover, l=5 4” flange

 d= supplier of the relevant part (d= 1,2,3,4,56,7,8)

5.4. Model solving method:

First of all the B-1 model is solved and then the optimized values of the decision making variables are fond and such values are placed in B-2 model and the latter model was solved. The results of B-2 were considered as totally along with the opinions of the suppliers. therefore, the same results were considered as the optimized solution.

5.4.1. Solving the case study model by Lingo software:

Table 2: units of goods that recived from supplier at the end of period(_RRd,t)

Period (monthly) supplier part from each subset product 12 11 10 9 8 7 6 5 4 3 2 1 7 15 3 20 0 18 15 7 15 3 20 0 d=1 L=1

J=1 d=2 2 23 0 10 7 16 18 2 23 0 10 7

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14 25 3 43 2 36 31 14 25 3 43 2 d=8 L=5

Table 3:values of RPdt, IRdt, RRdt, IPdt, XMjt and IMjt related to beam core (2ndpart) of chassis(1st subset) by two suppliers

supplier Period ( monthly )

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7 15 3 20 0 18 15 7 15 3 20 0

RR

14 0 0 0 0 0 0 0 0 0 0 0 0

IP

6 14 0 0 0 0 0 0 0 0 0 0 0

IP

7 0 25 3 43 2 36 31 14 25 3 43 2

XM

3 5 25 20 37 14 32 16 5 11 6 23 0 18

IM

3

Table 7: values of RPdt, IRdt, RRdt, IPdt, XMjt and IMjt related to beam core (2ndpart) of tanker (3st subset) by one supplier

RP

0 2 16 41 44 87 89 125 156 170 195 198

IR

7 10 0 23 2 18 16 7 10 0 23 0

RR

14 0 0 0 0 0 0 0 0 0 0 0 0

IP

8 0 25 3 43 2 36 31 14 25 3 43 2

XM

3 5 25 20 37 14 32 16 5 11 6 23 0 18

IM

3 6. CONCLUSION AND SUGGESTIONS:

Multi site production systems planning in the form of mathematical common and typical planning cannot meet all the possible conditions and states. As mentioned in the earlier parts, in order to enable a multi site system to have a proper performance considering the different involved objectives in the steps of supply, production and distribution, the relevant elements shall correctly been recognized and also the effect of each of the same shall be studied. Sometimes the chain members are neither in complete cooperation and participation nor in the absolute conflicting state. Therefore, it should be noted that sometimes there is an “in-between” state established between them. Thus, we should take benefit from a model which may have a better result. Although not highly considered by the scholars, and the commencement of its activities dates back to the 80’s, but two (multi) level planning and may be effective in this field (multi site production system), because, as mentioned in the earlier chapters, in this type of planning the higher level decisions makers determine their targets and decisions. Then they ask the lower organizational level to optimize the targets separately. The lower levels decision makers present their decisions to the higher levels, by whom such decisions are modified

and improved, for which the general interest of the organization is also considered. This process continues until a satisfactory solution is obtained. This process of decision making is quite practical for non-focused systems such as agriculture, governmental and sovereign policies, economic, financial, military, transportation systems and network designing and is also quite special for the analysis of conflict state. In this article the two-level model has been presented through considering the production and supplying steps from the supply chain in multi site production system. Several decision making variables are available in the supply chain which play a major role in the same and shall be considered in modeling and planning of such systems, as ignoring any of the same results in unrealistic nature of the relevant model. Therefore, in this regard the author hereby presents his suggestions for further studies of the scholars in the field of multi site production systems modeling as per the following:

 Considering all the three steps of supplying, production and distribution in the chain and applying triple-level mathematical modeling

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 Considering higher number of parts and products and also suppliers related to the same in modeling

 Considering higher number of decision making variables affecting such modeling

 Extending and applying such type of models in different systems planning

We hope to see deeper consideration concerning the nature of multi site production systems in the future and the advantages of using the same for those companies tend to have extensive commercial activities and also have several advancements in the field of planning multi site production systems and the managers shall consider the importance of this issue that today, due to the technological developments, increased customers awareness and knowledge, market economic policies, variable and dynamic conditions of the commercial environment, they shall continuously make extraordinary changes in their structure and system to survive in the field of competition. Also, it should be noted that mathematics and its affect on all the stages and conditions of survival are more than obvious for all. Therefore, by taking benefit from such technique and knowledge we shall direct our systems towards a direction where there is not even a smallest problem and defect on them.

7. REFERENCES:

1. Archimede , B.,Charbonnaud, P.Mercier , N.(2003).Robustnwss evaluation of Multi site distributed schedule with peturbed virtual jobshops .Production Planning and Control,14,55-67.

2. Arntzen, B. C., Brown, G. G., Harrison, T. P., & Trafton, L. L. (1995). Global supply chain management at Digital Equipment Corporation. Interfaces, 25(1), 69-93.

3. Baky, I. A. (2013). Interactive TOPSIS algorithms for solving multi-level non-linear multi-objective decision-making problems. Applied Mathematical Modelling. 4. Bard, J. F. (1988). Convex two-level

optimization. Mathematical

Programming,40(1-3), 15-27. ons on, 30(10), 986-993.

5. Bitran, G. R., Haas, E. A., & Hax, A. C. (1982). Hierarchical production planning: A two-stage system. Operations Research, 30(2), 232-251.

6. Blackburn, J. D., & Millen, R. A. (1982). The impact of a rolling schedule in a multi-level MRP system. Journal of Operations Management, 2(2), 125-135.

7. Biswas, A., & Bose, K. (2013, July). A fuzzy goal programming technique for quadratic multiobjective multilevel programming. In Fuzzy Systems (FUZZ), 2013 IEEE International Conference on (pp. 1-8). IEEE. 8. Caridi, M., & Sianesi, A. (2000). Multi-agent

systems in production planning and control: An application to the scheduling of mixed-model assembly lines.International Journal of Production Economics, 68(1), 29-42.

9. Chandra, P., & Fisher, M. L. (1994). Coordination of production and distribution planning. European Journal of Operational Research, 72(3), 503-517.

10.Colson, B., Marcotte, P., & Savard, G. (2005). A trust-region method for nonlinear bilevel programming: algorithm and computational experience.Computational Optimization and Applications, 30(3), 211-227.

11.Dhaenens-Flipo, C. (2000). Spatial decomposition for a multi-facility production and distribution problem. International Journal of Production Economics, 64(1).

12.Eftekhari M.(2006). An Introduction on Supply Chain Management.Master’s Degree Thesis in Industrial Engineering. science and industry University Tehran- Iran.

13.Escudero, L. F., & Kamesam, P. V. (1995). On solving stochastic production planning problems via scenario modelling. Top, 3(1), 69-95.

14.Fahimnia, B., Luong, L., & Marian, R. (2012). Genetic algorithm optimisation of an integrated aggregate production–distribution plan in supply chains.International Journal of Production Research, 50(1), 81-96.

15.Fahimnia, B., Farahani, R. Z., & Sarkis, J. (2013). Integrated aggregate supply chain planning using memetic algorithm–A performance analysis case study.International Journal of Production Research, 51(18), 5354-5373.

16.Ferber, J. (1999). Multi-agent systems: an introduction to distributed artificial intelligence (Vol. 1). Reading: Addison-Wesley.

17.Fumero, F., & Vercellis, C. (1999). Synchronized development of production, inventory, and distribution schedules. Transportation science, 33(3), 330-340.

(15)

Computer-Integrated Manufacturing, 20(3), 191-198.

19.Gnoni, M. G., Iavagnilio, R., Mossa, G., Mummolo, G., & Di Leva, A. (2003). Production planning of a multi-site manufacturing system by hybrid modelling: A case study from the automotive industry. International Journal of production economics, 85(2), 251-262.

20.Guinet, A. (2001). Multi-site planning: A transshipment problem. International Journal of production economics, 74(1), 21-32.

21.Horst, R., Thoai, N. V., & Tuy, H. (1989). On an outer approximation concept in global optimization. Optimization, 20(3), 255-264. 22.Ivanov, D., Sokolov, B., & Pavlov, A. (2013).

Dual problem formulation and its application to optimal redesign of an integrated production– distribution network with structure dynamics and ripple effect considerations. International Journal of Production Research, 51(18), 5386-5403

23.Jackson, J. R., & Grossmann, I. E. (2003). Temporal decomposition scheme for nonlinear multisite production planning and distribution models. Industrial & Engineering Chemistry Research, 42(13), 3045-3055.

24.Jeroslow, R. G. (1985). The polynomial hierarchy and a simple model for competitive analysis. Mathematical programming, 32(2), 146-164.

25.Jolayemi, J. K., & Olorunniwo, F. O. (2004). A deterministic model for planning production quantities in a multi-plant, multi-warehouse

environment with extensible

capacities. International Journal of Production Economics, 87(2), 99-113.

26.Johansen, S. G. (1999). Lot sizing for varying degrees of demand uncertainty.International journal of production economics, 59(1), 405-414.

27.Kanyalkar, A. P., & Adil*, G. K. (2005). An integrated aggregate and detailed planning in a multi-site production environment using linear programming.International Journal of Production Research, 43(20), 4431-4454. 28.Kolstad, C. D. (1985). A review of the

literature on bi-level mathematical programming. Los Alamos, New Mexico: Los Alamos National Laboratory.

29.Kratica, J., Dugošija, D., & Savić, A. (2013). A new mixed integer linear programming model for the multi level uncapacitated facility location problem.Applied Mathematical Modelling.

30.Leung, S. C., Wu, Y., & Lai, K. K. (2003). Multi-site aggregate production planning with multiple objectives: a goal programming

approach. Production Planning & Control, 14(5), 425-436.

31.Li, C., & Liu, S. (2013). A robust optimization approach to reduce the bullwhip effect of supply chains with vendor order placement lead time delays in an uncertain environment. Applied Mathematical Modelling, 37(3), 707-718.

32.Li Zhu, D., Xu, Q., & Lin, Z. (2004). A homotopy method for solving bilevel programming problem. Nonlinear Analysis: Theory, Methods & Applications,57(7), 917-928.

33.Matsumoto, S., Kato, K., & Katagiri, H. (2012, November). Interactive decision making for random fuzzy two-level programming problems through probability-possibility maximization. In Soft Computing and Intelligent Systems (SCIS) and 13th International Symposium on Advanced Intelligent Systems (ISIS), 2012 Joint 6th International Conference on (pp. 1610-1614). IEEE.

34.Mazzola, J. B., & Neebe, A. W. (1999). Lagrangian-relaxation-based solution procedures for a multiproduct capacitated facility location problem with choice of facility type. European Journal of Operational Research, 115(2), 285-299.

35.McDonald, C. M., & Karimi, I. A. (1997). Planning and scheduling of parallel semicontinuous processes. 1. Production planning. Industrial & Engineering Chemistry Research, 36(7), 2691-2700.

36.Metters, R. (1998). General rules for production planning with seasonal demand.International journal of production research, 36(5), 1387-1399.

37.Mirzapour Al-E-Hashem, S. M. J., Malekly, H., & Aryanezhad, M. B. (2011). A objective robust optimization model for multi-product multi-site aggregate production planning in a supply chain under uncertainty. International Journal of Production Economics, 134(1), 28-42.

38.Ng, N. K., & Jiao, J. (2004). A domain-based reference model for the conceptualization of factory loading allocation problems in

multi-site manufacturing supply

chains. Technovation, 24(8), 631-642.

39.Olhager, J. (2013). Evolution of operations planning and control: from production to supply chains. International Journal of Production Research, 51(23-24), 6836-6843. 40.Ozdamar, L., & Yazgac, T. (1999). A

(16)

Journal of Production Research,37(16), 3759-3772.

41.Petkov, S. B., & Maranas, C. D. (1997). Multiperiod planning and scheduling of multiproduct batch plants under demand uncertainty. Industrial & engineering chemistry research, 36(11), 4864-4881.

42.Pontrandolfo, P. (1999). Global manufacturing: a review and a framework for planning in a global corporation. International Journal of Production Research,37(1), 1-19.

43.Ramezanian, R., Rahmani, D., & Barzinpour, F. (2012). An aggregate production planning model for two phase production systems: Solving with genetic algorithm and tabu search. Expert Systems with Applications, 39(1), 1256-1263.

44.Ristroph, J. H. (1990). Simulation of ordering rules for single level material requirements planning with a rolling horizon and no forecast error. Computers & Industrial Engineering, 19(1), 155-159.

45.Roghanian E.(2006).Applying Multipurpose Multilevel Planning Under Uncertainty Conditions In Multi-Rank Supplying Chains Planning.PhD thesis subject suggestion. science and industry University Tehran- Iran. 46.Roux, W., Dauzère-Pérès, S., & Lasserre, J. B.

(1999). Planning and scheduling in a multi-site environment. Production planning & control, 10(1), 19-28.

47.Saati, S., & Memariani, A. (2004). Bilevel Programming and Recent Approaches. Journal of applied mathematics, Islamic Azad University, Lahijan, Iran, 22-35.

48.Sakawa, M., & Matsui, T. (2013). Interactive random fuzzy two-level programming through possibility-based probability model. Information Sciences,239, 191-200. 49.Sakawa, M., & Matsui, T. (2013). Interactive

fuzzy random cooperative two-level linear programming through level sets based probability maximization. Expert Systems with Applications, 40(4), 1400-1406.

50.Sakawa, M., & Matsui, T. (2014). Interactive fuzzy stochastic multi-level 0–1 programming using tabu search and probability maximization. Expert Systems with Applications, 41(6), 2957-2963.

51.Sakawa, M., & Matsui, T. (2012). Interactive fuzzy programming for random fuzzy two-level programming problems through possibility-based fractile model.Expert Systems with Applications, 39(16), 12599-12604.

52.Sakawa, M., Katagiri, H., & Matsui, T. (2012). Stackelberg solutions for fuzzy random two-level linear programming through probability

maximization with possibility. Fuzzy Sets and Systems, 188(1), 45-57.

53.Sauer, J., & Bruns, R. (1995, July). Intelligent multi-site coordination and scheduling. In Proceeding of the international conference on improving manufacturing performance in a distributed enterprise: Advanced systems and tools, Edinburgh (pp. 207-214).

54.Savard, G., & Gauvin, J. (1994). The steepest descent direction for the nonlinear bilevel programming problem. Operations Research Letters, 15(5), 265-272.

55.Shen, W., & Norrie, D. H. (1999). Agent-based systems for intelligent manufacturing: a state-of-the-art survey. Knowledge and information systems,1(2), 129-156.

56.Simpson, N. C. (2001). Questioning the relative virtues of dynamic lot sizing rules. Computers & Operations Research, 28(9), 899-914.

57.Stadtler, H. (2000). Improved rolling schedules for the dynamic single-level lot-sizing problem. Management Science, 46(2), 318-326. 58.Tabrizi, B. H., & Razmi, J. (2013). Introducing a mixed-integer non-linear fuzzy model for risk management in designing supply chain networks. Journal of Manufacturing Systems, 32(2), 295-307.

59.Timpe, C. H., & Kallrath, J. (2000). Optimal planning in large multi-site production networks. European Journal of Operational Research, 126(2), 422-435.

60.Tönshoff, H. K., Seilonen, I., Teunis, G., & Leitão, P. (2000). A mediator-based approach for decentralised production planning, scheduling and monitoring.

61.Torabi, S. A., & Moghaddam, M. (2012). Multi-site integrated production-distribution planning with trans-shipment: a fuzzy goal programming approach.International Journal of Production Research, 50(6), 1726-1748. 62.Tsiakis, P., Shah, N., & Pantelides, C. C.

(2001). Design of multi-echelon supply chain networks under demand uncertainty. Industrial & Engineering Chemistry Research, 40(16), 3585-3604.

63.Vercellis, C. (1999). Multi-plant production planning in capacitated self-configuring two-stage serial systems. European Journal of Operational Research, 119(2), 451-460. 64.Vicente, L. N., & Calamai, P. H. (1995).

Geometry and local optimality conditions for bilevel programs with quadratic strictly convex lower levels. InMinimax and Applications (pp. 141-151). Springer US.

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