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## A New Approach to Economic Production Quantity Problems with Fuzzy Parameters and Inventory Constraint

### Citations

6240 |
Fuzzy sets
- Zadeh
- 1965
(Show Context)
Citation Context ...for a special case of inventory constraints. Often is desirable to try to solve the EOQ-models with their extensions analytically through the solution of the derivatives (as also done originally by Harris, [8]). There are also other optimization approaches used in the EOQ literature. If the uncertainties in the EOQ-models can be modeled stochastically (as done in [9]), the track of probabilistic models should be conducted, but this is not always possible in the process industry. For the uncertainties relevant to this paper it is better to use fuzzy numbers instead of probabilistic approaches ([17,18]). In the line of research of fuzzy EOQ-models, there are contributions for instance like Chang [6], who worked out fuzzy modifications of the model of [13], which took the defective rate of the goods into account. Ouyang and Wu and [11] Ouyang and Yao [12] solved an EOQ-model with the lead times as decision variables as well as the order quantities. Taleizadeh and Nematollahi [15] presented again an EOQmodel with a final time horizon, with perishable items, backordering and delayed payments. Sanni and Chukwu [14] did a EOQ-model with deteriorating items, ramp-type demand as well as shortages.... |

892 |
Outline of a New Approach to the Analysis of Complex Systems and Decision Processes
- Zadeh
- 1973
(Show Context)
Citation Context ...for a special case of inventory constraints. Often is desirable to try to solve the EOQ-models with their extensions analytically through the solution of the derivatives (as also done originally by Harris, [8]). There are also other optimization approaches used in the EOQ literature. If the uncertainties in the EOQ-models can be modeled stochastically (as done in [9]), the track of probabilistic models should be conducted, but this is not always possible in the process industry. For the uncertainties relevant to this paper it is better to use fuzzy numbers instead of probabilistic approaches ([17,18]). In the line of research of fuzzy EOQ-models, there are contributions for instance like Chang [6], who worked out fuzzy modifications of the model of [13], which took the defective rate of the goods into account. Ouyang and Wu and [11] Ouyang and Yao [12] solved an EOQ-model with the lead times as decision variables as well as the order quantities. Taleizadeh and Nematollahi [15] presented again an EOQmodel with a final time horizon, with perishable items, backordering and delayed payments. Sanni and Chukwu [14] did a EOQ-model with deteriorating items, ramp-type demand as well as shortages.... |

532 |
Fuzzy sets as a basis for a theory of possibility,”
- Zadeh
- 1978
(Show Context)
Citation Context ... 0) = 1 2 ∫ 1 0 (Bl(α) + Bu(α))dα 2.2 Chance Constrained Programming Chance constrained programming, originally introduced in probabilitic environment by Charnes and Cooper [7], is a widely-used method to handle uncertain parameters in optimization problems. The original approach was later modified to incorporate fuzzy parameters and possibility and necessity measures [10]. According to this approach, it is not necessary to use any defuzzification method, the extent to which the constraints of the models are satisfied in terms of possibility or necessity are calculated. Possibility measure [19] is a maxitive normalized monotone measure, i.e. Pos ( ⋃ i Bi ) = sup i Pos(Bi). where {Bi} is any family of sets in the universe of discourse. The dual measure of possibility, termed as necessity, is defined as: Nec(B) = 1− Pos(BC). EPQ with Fuzzy Parameters and Inventory Constraint 409 We can consider fuzzy numbers as possibility distributions on the real line using the formula Pos(C ⊂ R) = sup x∈C μB(x), where μB(x) is the membership function of the fuzzy number B. In this paper we will calculate the possibility of the fulfilment of constraint with the left-hand side being a fuzzy expressio... |

106 |
Chance-constrained programming.
- Charnes, Cooper
- 1959
(Show Context)
Citation Context ...zy set. Definition 2 Let B a fuzzy set on R and 0 ≤ α ≤ 1. The α-cut of B is the set of all the points x such that μB(x) ≥ α, i.e. B(α) = {x|μB(x) ≥ α} . Let Ω be the family of all fuzzy sets B defined on R for which the α-cut B(α) = [Bl(α), Bu(α)] exists for every 0 ≤ α ≤ 1, and both Bl(α) and Bu(α) are continuous functions on α ∈ [0, 1]. Definition 3 For B ∈ Ω define the signed distance of B to 0 as d(B, 0) = 1 2 ∫ 1 0 (Bl(α) + Bu(α))dα 2.2 Chance Constrained Programming Chance constrained programming, originally introduced in probabilitic environment by Charnes and Cooper [7], is a widely-used method to handle uncertain parameters in optimization problems. The original approach was later modified to incorporate fuzzy parameters and possibility and necessity measures [10]. According to this approach, it is not necessary to use any defuzzification method, the extent to which the constraints of the models are satisfied in terms of possibility or necessity are calculated. Possibility measure [19] is a maxitive normalized monotone measure, i.e. Pos ( ⋃ i Bi ) = sup i Pos(Bi). where {Bi} is any family of sets in the universe of discourse. The dual measure of possibility... |

46 |
Economic production quantity model for items with imperfect quality.
- Salameh, Jaber
- 2000
(Show Context)
Citation Context ...erivatives (as also done originally by Harris, [8]). There are also other optimization approaches used in the EOQ literature. If the uncertainties in the EOQ-models can be modeled stochastically (as done in [9]), the track of probabilistic models should be conducted, but this is not always possible in the process industry. For the uncertainties relevant to this paper it is better to use fuzzy numbers instead of probabilistic approaches ([17,18]). In the line of research of fuzzy EOQ-models, there are contributions for instance like Chang [6], who worked out fuzzy modifications of the model of [13], which took the defective rate of the goods into account. Ouyang and Wu and [11] Ouyang and Yao [12] solved an EOQ-model with the lead times as decision variables as well as the order quantities. Taleizadeh and Nematollahi [15] presented again an EOQmodel with a final time horizon, with perishable items, backordering and delayed payments. Sanni and Chukwu [14] did a EOQ-model with deteriorating items, ramp-type demand as well as shortages. This paper has a track of research development behind. Already Bjork and Carlsson [3] solved analytically an EOQ problem with backorders, with a signed di... |

40 |
Ranking fuzzy numbers based on decomposition principle and signed distance. Fuzzy Sets and Systems,
- Yao, Wu
- 2000
(Show Context)
Citation Context ...t behind. Already Bjork and Carlsson [3] solved analytically an EOQ problem with backorders, with a signed distance defuzzification method. Bjork [1] solved again a problem with a finite production rate and fuzzy cycle time, which was extended in [2] to a more general fuzzy case. The approach used in this paper is novel since there are no papers (to our knowledge) that focus on the realistic modeling of inventory constraints. Our solution methodology is to one part similar also to Bjork and Carlsson [4] and Bjork [1], , where the fuzzy model is defuzzified using the signed distance method [16], however, the solution is here not found through the derivatives, but numerically, since our fuzzy problem is more difficult to solve. This paper extends the results in the recent publication by Bjork [2] with the limited storage capacity restriction with a more complex, but much more realistic inventory space constraint model. In addition, we consider not only the crisp case, but also the case of chance constrained formulation (in the fuzzy sense) of the storage limitations. The rest of the paper is structured as follows. First we will explain some preliminaries, then we will present the cr... |

25 |
How many parts to make at once.
- Harris
- 1913
(Show Context)
Citation Context ...cases: in the first case the optimization problem will be defuzzified with the signed distance measure and in the second case, the storage constraint needs to be fulfilled, only to a certain degree of possibility. Both cases are solved and illustrated with an example. Keywords: Economic Production Quantity, Triangular fuzzy numbers, Inventory constraint, Signed distance, Chance constrained optimization. 1 Introduction Even with more than 100 years of EOQ (Economic Order Quantity) development, current stream of new findings and results do not tend to decrease. Even if the first model by Harris [8] was very simple, it has been very popular in industry and also an inspiration to many researchers. In this basic model the order size needed to be determined given holding costs, order setup costs and annual demand. This model has been altered in many ways to capture more complex and realistic situations in the industry. For instance, the EPQ (Economic Production Quantity) solved a problem where the product is produced to stock, also multi item, storage capacity limitation and so on is further extensions of the basic model. These additions may be very crucial, even to the extent of only havin... |

25 |
Chance constrained programming with fuzzy parameters,
- Liu, Iwamura
- 1998
(Show Context)
Citation Context ... defined on R for which the α-cut B(α) = [Bl(α), Bu(α)] exists for every 0 ≤ α ≤ 1, and both Bl(α) and Bu(α) are continuous functions on α ∈ [0, 1]. Definition 3 For B ∈ Ω define the signed distance of B to 0 as d(B, 0) = 1 2 ∫ 1 0 (Bl(α) + Bu(α))dα 2.2 Chance Constrained Programming Chance constrained programming, originally introduced in probabilitic environment by Charnes and Cooper [7], is a widely-used method to handle uncertain parameters in optimization problems. The original approach was later modified to incorporate fuzzy parameters and possibility and necessity measures [10]. According to this approach, it is not necessary to use any defuzzification method, the extent to which the constraints of the models are satisfied in terms of possibility or necessity are calculated. Possibility measure [19] is a maxitive normalized monotone measure, i.e. Pos ( ⋃ i Bi ) = sup i Pos(Bi). where {Bi} is any family of sets in the universe of discourse. The dual measure of possibility, termed as necessity, is defined as: Nec(B) = 1− Pos(BC). EPQ with Fuzzy Parameters and Inventory Constraint 409 We can consider fuzzy numbers as possibility distributions on the real line using the... |

11 |
An application of fuzzy sets theory to the EOQ model with imperfect quality items.
- Chang
- 2004
(Show Context)
Citation Context ...ir extensions analytically through the solution of the derivatives (as also done originally by Harris, [8]). There are also other optimization approaches used in the EOQ literature. If the uncertainties in the EOQ-models can be modeled stochastically (as done in [9]), the track of probabilistic models should be conducted, but this is not always possible in the process industry. For the uncertainties relevant to this paper it is better to use fuzzy numbers instead of probabilistic approaches ([17,18]). In the line of research of fuzzy EOQ-models, there are contributions for instance like Chang [6], who worked out fuzzy modifications of the model of [13], which took the defective rate of the goods into account. Ouyang and Wu and [11] Ouyang and Yao [12] solved an EOQ-model with the lead times as decision variables as well as the order quantities. Taleizadeh and Nematollahi [15] presented again an EOQmodel with a final time horizon, with perishable items, backordering and delayed payments. Sanni and Chukwu [14] did a EOQ-model with deteriorating items, ramp-type demand as well as shortages. This paper has a track of research development behind. Already Bjork and Carlsson [3] solved anal... |

9 |
J.-S.: A minimax distribution free procedure for mixed inventory model involving variable lead time with fuzzy demand.
- Ouyang, Yao
- 2002
(Show Context)
Citation Context ...ation approaches used in the EOQ literature. If the uncertainties in the EOQ-models can be modeled stochastically (as done in [9]), the track of probabilistic models should be conducted, but this is not always possible in the process industry. For the uncertainties relevant to this paper it is better to use fuzzy numbers instead of probabilistic approaches ([17,18]). In the line of research of fuzzy EOQ-models, there are contributions for instance like Chang [6], who worked out fuzzy modifications of the model of [13], which took the defective rate of the goods into account. Ouyang and Wu and [11] Ouyang and Yao [12] solved an EOQ-model with the lead times as decision variables as well as the order quantities. Taleizadeh and Nematollahi [15] presented again an EOQmodel with a final time horizon, with perishable items, backordering and delayed payments. Sanni and Chukwu [14] did a EOQ-model with deteriorating items, ramp-type demand as well as shortages. This paper has a track of research development behind. Already Bjork and Carlsson [3] solved analytically an EOQ problem with backorders, with a signed distance defuzzification method. Bjork [1] solved again a problem with a finite pr... |

8 |
The EOQ model under stochastic lead time.
- Liberatore
- 1979
(Show Context)
Citation Context ...only a few things may be uncertain. These fuzzy uncertainties may come from the fact that the demand may be uncertain, but still reliable data is not found to make justified probabilistic statements. This case is tackled in our paper for a special case of inventory constraints. Often is desirable to try to solve the EOQ-models with their extensions analytically through the solution of the derivatives (as also done originally by Harris, [8]). There are also other optimization approaches used in the EOQ literature. If the uncertainties in the EOQ-models can be modeled stochastically (as done in [9]), the track of probabilistic models should be conducted, but this is not always possible in the process industry. For the uncertainties relevant to this paper it is better to use fuzzy numbers instead of probabilistic approaches ([17,18]). In the line of research of fuzzy EOQ-models, there are contributions for instance like Chang [6], who worked out fuzzy modifications of the model of [13], which took the defective rate of the goods into account. Ouyang and Wu and [11] Ouyang and Yao [12] solved an EOQ-model with the lead times as decision variables as well as the order quantities. Taleizade... |

5 | Soft computing and the Bullwhip effect.
- Carlsson, Fuller
- 1999
(Show Context)
Citation Context ...age capacity limitation and so on is further extensions of the basic model. These additions may be very crucial, even to the extent of only having storage capacity for one weeks production (this was the case in a Nordic plywood production facility that we have collaborated with). It is obvious that we will produce to stock in the process industry environment. In these settings we need the EOQ-models with some proper extensions. The uncertainties in the process industry can sometimes be measured probabilistically, but sometimes data is not enough and therefore fuzzy measures may be needed, c.f [3,5]. There have also been a lot of research contributions in this line of research. For instance [3] solved an EOQ model with backorders and infinite replenishment lead time A. Laurent et al. (Eds.): IPMU 2014, Part I, CCIS 442, pp. 406–415, 2014. c© Springer International Publishing Switzerland 2014 EPQ with Fuzzy Parameters and Inventory Constraint 407 with fuzzy lead times. However, sometimes the environment may be more stable, and only a few things may be uncertain. These fuzzy uncertainties may come from the fact that the demand may be uncertain, but still reliable data is not found to make ... |

2 |
A Multi-item Fuzzy Economic Production Quantity Problem with a Finite Production Rate.
- Bjork
- 2012
(Show Context)
Citation Context ...es as decision variables as well as the order quantities. Taleizadeh and Nematollahi [15] presented again an EOQmodel with a final time horizon, with perishable items, backordering and delayed payments. Sanni and Chukwu [14] did a EOQ-model with deteriorating items, ramp-type demand as well as shortages. This paper has a track of research development behind. Already Bjork and Carlsson [3] solved analytically an EOQ problem with backorders, with a signed distance defuzzification method. Bjork [1] solved again a problem with a finite production rate and fuzzy cycle time, which was extended in [2] to a more general fuzzy case. The approach used in this paper is novel since there are no papers (to our knowledge) that focus on the realistic modeling of inventory constraints. Our solution methodology is to one part similar also to Bjork and Carlsson [4] and Bjork [1], , where the fuzzy model is defuzzified using the signed distance method [16], however, the solution is here not found through the derivatives, but numerically, since our fuzzy problem is more difficult to solve. This paper extends the results in the recent publication by Bjork [2] with the limited storage capacity restric... |

2 |
The Outcome of Imprecise Lead Times on the Distributors. In:
- Bjork, Carlsson
- 2005
(Show Context)
Citation Context ...age capacity limitation and so on is further extensions of the basic model. These additions may be very crucial, even to the extent of only having storage capacity for one weeks production (this was the case in a Nordic plywood production facility that we have collaborated with). It is obvious that we will produce to stock in the process industry environment. In these settings we need the EOQ-models with some proper extensions. The uncertainties in the process industry can sometimes be measured probabilistically, but sometimes data is not enough and therefore fuzzy measures may be needed, c.f [3,5]. There have also been a lot of research contributions in this line of research. For instance [3] solved an EOQ model with backorders and infinite replenishment lead time A. Laurent et al. (Eds.): IPMU 2014, Part I, CCIS 442, pp. 406–415, 2014. c© Springer International Publishing Switzerland 2014 EPQ with Fuzzy Parameters and Inventory Constraint 407 with fuzzy lead times. However, sometimes the environment may be more stable, and only a few things may be uncertain. These fuzzy uncertainties may come from the fact that the demand may be uncertain, but still reliable data is not found to make ... |

2 |
The Effect of Flexible Lead Times on a Paper Producer.
- Bjork, Carlsson
- 2007
(Show Context)
Citation Context ... items, ramp-type demand as well as shortages. This paper has a track of research development behind. Already Bjork and Carlsson [3] solved analytically an EOQ problem with backorders, with a signed distance defuzzification method. Bjork [1] solved again a problem with a finite production rate and fuzzy cycle time, which was extended in [2] to a more general fuzzy case. The approach used in this paper is novel since there are no papers (to our knowledge) that focus on the realistic modeling of inventory constraints. Our solution methodology is to one part similar also to Bjork and Carlsson [4] and Bjork [1], , where the fuzzy model is defuzzified using the signed distance method [16], however, the solution is here not found through the derivatives, but numerically, since our fuzzy problem is more difficult to solve. This paper extends the results in the recent publication by Bjork [2] with the limited storage capacity restriction with a more complex, but much more realistic inventory space constraint model. In addition, we consider not only the crisp case, but also the case of chance constrained formulation (in the fuzzy sense) of the storage limitations. The rest of the paper is... |

2 |
An inventory control problem for deteriorating items with back-ordering and financial considerations.
- Taleizadeh, Nematollahi
- 2014
(Show Context)
Citation Context ...obabilistic models should be conducted, but this is not always possible in the process industry. For the uncertainties relevant to this paper it is better to use fuzzy numbers instead of probabilistic approaches ([17,18]). In the line of research of fuzzy EOQ-models, there are contributions for instance like Chang [6], who worked out fuzzy modifications of the model of [13], which took the defective rate of the goods into account. Ouyang and Wu and [11] Ouyang and Yao [12] solved an EOQ-model with the lead times as decision variables as well as the order quantities. Taleizadeh and Nematollahi [15] presented again an EOQmodel with a final time horizon, with perishable items, backordering and delayed payments. Sanni and Chukwu [14] did a EOQ-model with deteriorating items, ramp-type demand as well as shortages. This paper has a track of research development behind. Already Bjork and Carlsson [3] solved analytically an EOQ problem with backorders, with a signed distance defuzzification method. Bjork [1] solved again a problem with a finite production rate and fuzzy cycle time, which was extended in [2] to a more general fuzzy case. The approach used in this paper is novel since there ar... |

1 | The economic production quantity problem with a finite production rate and fuzzy cycle time. In:
- Bjork
- 2008
(Show Context)
Citation Context ... goods into account. Ouyang and Wu and [11] Ouyang and Yao [12] solved an EOQ-model with the lead times as decision variables as well as the order quantities. Taleizadeh and Nematollahi [15] presented again an EOQmodel with a final time horizon, with perishable items, backordering and delayed payments. Sanni and Chukwu [14] did a EOQ-model with deteriorating items, ramp-type demand as well as shortages. This paper has a track of research development behind. Already Bjork and Carlsson [3] solved analytically an EOQ problem with backorders, with a signed distance defuzzification method. Bjork [1] solved again a problem with a finite production rate and fuzzy cycle time, which was extended in [2] to a more general fuzzy case. The approach used in this paper is novel since there are no papers (to our knowledge) that focus on the realistic modeling of inventory constraints. Our solution methodology is to one part similar also to Bjork and Carlsson [4] and Bjork [1], , where the fuzzy model is defuzzified using the signed distance method [16], however, the solution is here not found through the derivatives, but numerically, since our fuzzy problem is more difficult to solve. This paper ... |

1 |
W.I.E.: An Economic order quantity model for Items with Three-parameter Weibull distribution Deterioration, Ramp-type Demand and Shortages.
- Sanni, Chukwu
- 2013
(Show Context)
Citation Context ... paper it is better to use fuzzy numbers instead of probabilistic approaches ([17,18]). In the line of research of fuzzy EOQ-models, there are contributions for instance like Chang [6], who worked out fuzzy modifications of the model of [13], which took the defective rate of the goods into account. Ouyang and Wu and [11] Ouyang and Yao [12] solved an EOQ-model with the lead times as decision variables as well as the order quantities. Taleizadeh and Nematollahi [15] presented again an EOQmodel with a final time horizon, with perishable items, backordering and delayed payments. Sanni and Chukwu [14] did a EOQ-model with deteriorating items, ramp-type demand as well as shortages. This paper has a track of research development behind. Already Bjork and Carlsson [3] solved analytically an EOQ problem with backorders, with a signed distance defuzzification method. Bjork [1] solved again a problem with a finite production rate and fuzzy cycle time, which was extended in [2] to a more general fuzzy case. The approach used in this paper is novel since there are no papers (to our knowledge) that focus on the realistic modeling of inventory constraints. Our solution methodology is to one part s... |