Bipolar Interval-valued neutrosophic set is another generalization of fuzzy set, neutrosophic set, bipolar fuzzy set and bipolar neutrosophic set and thus when applied to the optimization problem handles uncertain data more efficiently and flexibly. Current work is an effort to design a flexible optimization model in the backdrop of interval-valued bipolar neutrosophic sets. Bipolar interval-valued neutrosophic membership grades are picked so that they indicate the restriction of the plausible infringement of the inequalities given in the problem. To prove the adequacy and effectiveness of the method a unified system of sustainable medical healthcare supply chain model with an uncertain figure of product complaints is used. Time, quality and cost are considered as satisfaction level to choose best supplier for medicine procurement. The proposed model ensures 99% satisfaction for cost reduction, 63% satisfaction for the quality of product and 64% satisfaction for total time taken in medicine supply chain.

Fuzzy set theory offers not only an influential, significant and powerful role in depiction of imprecision and uncertainties, but also caters the subjectivity and vagueness of natural languages. To handle with uncertainty and imprecision, notion of fuzzy sets is formerly proposed by Zadeh [

In real life scenario, wherever fuzzy sets and generalizations are taken into account, one merely has to rely on experts’ opinion to determine membership, non-membership and indeterminacy grades. For this purpose, real numbers between 0 and 1 are used to accommodate and quantify more elements while assigning them membership grades based on quantitative and deterministic attributes. This approach may fulfill the purpose theoretically but in real life it would be an unreliable judgment. There is also a common observation that a rigid numerical value cannot be considered realistic and error free if assigned to mere an experts opinion. Therefore Wang et al. [

Bosc et al. [

Optimization strategies and theory plays a significant part in the assortment of domains to manage various real-world or genuine issues. Various methods have been acquainted to deal with problems in wide fields including engineering and supply issues. Over the most recent couple of years optimization approaches readily be given more attention. Progression in computing skills and increased dependency on real-like problems based on optimization are behind this change and many amazing proposition have been presented by numerous analysts and researchers. These issues give rise to the current expansions in the field of optimization methods, theory and applications. Some more recent applications of optimization problems in engineering, medical diagnosis, synthetic biology and transport infrastructure are given in [

Usually in optimization problems, it has been seen that a little change in the given conditions or constraints may provide better solutions to the given problem. A few times it isn’t helpful to consider exact requisites as a large number of these are acquired by assessment, or by researchers’ observations. Due to uncertainty involved in subjectivity of human languages and also in natural matters we cannot exactly predict or define the results of these matters. Such types of real-life problems always demand special attentions for solution [

The resources required to provide products or services to a target customer are referred to as the supply chain. Strategic sourcing in healthcare is often a highly complicated and fragmented operation. This process and the allocation of resources become more complex if the data available contains fuzziness. Recent enriched technological and innovative, organizational and financial progressions in health-care organizations have given expanded admittance of dealing with patients [

Current work aims to construct and crack a medicine supply-chain model in interval-valued bipolar neutrosophic environment. This supply chain model is designed for a healthcare system that is integrated as-well as uncertain in-terms of product complaints. In its modest form a supply chain is the activities needed by the association to provide services or goods to the buyers. The conventional medical health-care facilities are restricted to drug organizations, patients and hospitals. Prescribed study comprises of a unified medi-care framework which likewise incorporates the role of public authority and healthcare department. The current work is divided into six sections. The first section provides a quick overview of fuzzy sets and extensions, focusing on bipolar interval-valued neutrosophic sets. Section two describes the whole medical supply chain model in detail. In addition, objective functions and model constraints are briefly examined. Section three discusses the proposed approach. Section four contains a numerical example based on the model developed in section two. A solution based on the prescribed approach is given in section five. For comparison in this section, satisfaction levels are considered. Lastly, section six comprises the conclusion. Development of proposed technique to determine the solution of this multi-objective and multi-period optimization problem in interval-valued bipolar neutrosophic backdrop is the motivation and objective of current study.

For the proposed study a group of contractors: a health division and a network of clinics and hospitals have been decided. The chain of clinics and hospitals need to choose the most appropriate dealer for the medicines based on the business trio that are cost, time and quality.

Indicators

Parameters

Decision variable

The production limit of each dealer and price of specific medicine remains the same for the whole year.

In each period quality assurance cost remains the same.

The number of complaints regarding quality of product received by the manufacturer is uncertain.

The quality of units of medicine delivered over in the last year is known.

Labor cost remains the same throughout the year.

Carbon emission cost for handling and energy cost and carbon emission tax for transportation of medicines are known and fixed.

It is an integrated medi-care design in which chain of health centers pays the transportation expenditures.

The brief account of every objective function and constraints used to model the problem mathematically are as follow:

There are numerous factors effecting cost of medicine supply chain. At various level of this network different kinds of charges and dues are involved. Despite that, this particular problem is bound to the following costs.

The first term in the above equation depicts the cost of production, labour, energy and carbon emission handling. Labour cost, carbon emission handling cost and energy cost will remain the same throughout the specific period. The second term comprises of transportation cost and carbon emission tax for transportation and the cost of quality inspection done by health care department

Three types of times are involved in medicine supply network. First is the manufacturing time, second is the quality inspection time by the health care authorities and lastly the transportation time.

Here, the first term of the objective function describe the time taken by the suppliers or dealers in the manufacturing and second term is the sum of transportation time and time taken by health care department for quality inspection.

There are also many parameters to measure the quality level in the medicine supply chain; however the prescribed model rely on the complaints launched by the customers to maintain the quality level of suppliers. Mathematical modeling of quality function is described below:

Since number of complaints by the consumers is taken as quality assurance parameter; therefore, quality function is also minimized in the model. These complaints are highly uncertain because the suppliers do not know exactly about the number of complaints received regarding quality of certain products. Thus, fuzzy theory is used to handle the uncertainty associated with the model.

The process to determine the deviation in the fuzzy variable associated with quality function and comprises of three steps. In the first step, any of the fuzzy membership grades like trapezoidal or triangular can be taken. Afterwards, in the second step, fuzzification technique of conversion of a crisp function into a fuzzy function will be applied and at the last step defuzzification process will be followed. Again for defuzzification any of the already developed techniques like centroid method, first or last maxims, weighted average method, signed-distance method, center of largest area, and many more. Therefore, the equivalent defuzzified quality complaint function using signed-distance method is given below as

All the above stated objectives are subjected to the constraints given below:

This constraints limit the purchase by putting sum of quantity of medicine provided by dealers equals to the demand generated by health care units.

Quality constraints guarantees the standard acceptance quality limit.

Capacity constraint shows that the manufacturing capacity of dealers for any specific medicine should be greater than or equal to the quantity of medicine provided by dealer.

The value of binary variable is 1 when the quantity is released by the dealer and otherwise 0.

Multi-objective optimization problems occurred while dealing with optimization of non-commensurable, conflicting and multiple objective functions subject to certain conditions and circumstances. A general representation of a multi objective optimization problem with

where

Problem

Find

where

To build up bipolar interval-valued membership functions of different target capacities we could initially acquire the table of positive solutions. Lower and upper bound of each membership grade is acquired by using this positive solution table.

The confluence

Solution Methodology: In this model, bipolar interval-valued neutrosophic optimization technique is employed to solve multi-objective and multi-period problem. Following steps are followed to solve prescribed model.

Step 1: Table of positive solutions is attained by solving each objective function from set of

Step 2: Lower and upper bound of every objective function is attained by using below given relations

Step 3: Possible membership grades for bipolar interval valued neutrosophic linear programming problem is constructed as below.

Step 4: Transformed bipolar interval-valued neutrosophic optimization problem is given by

such that

also,

where,

This part portray the use and efficacy of the prescribed model and approach. A real-life integrated medical supply chain application is used for further illustration and validation of the model as well as of solution approach.

Numerical Example: To tackle the prescribed model, we assume an example which is just hypothetical. Three medical healthcare units are situated in the central city, which are responsible for providing medi-care facilities to the nearby regions and surrounding areas (see

Hospital administration buys those medicines approved by the healthcare department, which means that the hospitals require verification and approval of the department to purchase medicines. Expected demand by the hospitals for each medicine for the coming three months is provided in

Time period | Hospitals | Medicines | |||
---|---|---|---|---|---|

18146 | 15648 | 15467 | 12786 | ||

17423 | 12796 | 13454 | 13899 | ||

12378 | 17584 | 13477 | 16581 | ||

18569 | 17852 | 18526 | 16545 | ||

15488 | 18524 | 13692 | 12332 | ||

14588 | 17650 | 17414 | 14520 | ||

16540 | 13214 | 16589 | 12542 | ||

15465 | 12111 | 19545 | 18541 | ||

15647 | 16587 | 12548 | 16298 |

Medicine | Dealer |
|||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Capacity (Kgs) | Time (hrs/batch) | Batch size) | ||||||||||

1 | 2 | 3 | 4 | 1 | 2 | 3 | 4 | 1 | 2 | 3 | 4 | |

13255 | 23852 | 13217 | 11748 | 8 | 6 | 7 | 8 | 1124 | 960 | 1277 | 783 | |

17852 | 22155 | 16257 | 14855 | 9 | 7 | 10 | 8 | 859 | 1844 | 652 | 1233 | |

15425 | 12352 | 18542 | 23659 | 9 | 4 | 8 | 7 | 759 | 1247 | 1127 | 1025 | |

18564 | 12511 | 15212 | 12144 | 8 | 7 | 9 | 7 | 751 | 1050 | 1241 | 659 |

Medicine | Dealer |
|||||||
---|---|---|---|---|---|---|---|---|

Price | Units | |||||||

1 | 2 | 3 | 4 | 1 | 2 | 3 | 4 | |

5 | 4 | 3 | 4 | 125632 | 384521 | 525415 | 175421 | |

4 | 6 | 3 | 4 | 221217 | 136516 | 185421 | 186544 | |

6 | 4 | 5 | 3 | 194529 | 146520 | 201452 | 134527 | |

4 | 4 | 3 | 2 | 356426 | 125243 | 375423 | 215246 |

Hospital | Dealer |
|||
---|---|---|---|---|

82 | 78 | 51 | 79 | |

74 | 64 | 65 | 97 | |

65 | 45 | 74 | 52 |

Medicine | Dealer |
|||||||
---|---|---|---|---|---|---|---|---|

Complaints | QA level | |||||||

1 | 2 | 3 | 4 | 1 | 2 | 3 | 4 | |

124 | 230 | 312 | 393 | 3 | 3.5 | 1.2 | 2.3 | |

246 | 360 | 386 | 161 | 2.5 | 1.8 | 3.9 | 4.1 | |

119 | 131 | 185 | 361 | 2.5 | 3.2 | 1.6 | 4.4 | |

371 | 346 | 257 | 121 | 1.6 | 3.4 | 4 | 2.8 |

Standard acceptance level for this particular model is

Bipolar interval-valued neutrosophic optimization technique is applied to solve given numerical example. Step by step solution procedures are discussed in Section 3. Payoff table is constructed as per the procedure discussed in step 1.

The proposed technique includes the conversion of multi-objective problems to single objectives. Step 2 is applied to calculate the lower and upper bounds of each objective from the payoff matrix. Step 3 is followed to construct whole sets of membership grades. And lastly, the optimal value of each objective function is attained by following the procedure explained in step 4. The numerical findings were made on a personal computer with 4 GB RAM and a 2.50 GHz processor using MATLAB (R2017a).

Objective function | Satisfaction level w.r.t proposed approach | Satisfaction level w.r.t fuzzy optimization | Objective function value |
---|---|---|---|

Cost | 99% | 85% | 5427 |

Quality | 64% | 58% | 3797 |

Time | 64% | 62% | 62789.92 |

Medicine | Dealer | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | ||

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||

1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||

0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | ||

1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | ||

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||

0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | ||

0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | ||

1 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | ||

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||

1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Medicine | Dealer | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

4929 | 4206 | 0 | 5352 | 2271 | 1371 | 3323 | 2248 | 2430 | ||

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||

13217 | 13217 | 12378 | 13217 | 13217 | 13217 | 13217 | 13217 | 13217 | ||

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||

0 | 0 | 0 | 0 | 0 | 0 | 0 | 12111 | 0 | ||

15648 | 12796 | 17584 | 17852 | 18524 | 17650 | 13214 | 0 | 16587 | ||

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |||

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||

0 | 1102 | 1125 | 0 | 1340 | 0 | 0 | 0 | 196 | ||

0 | 12352 | 12352 | 0 | 12352 | 12352 | 0 | 12352 | 12352 | ||

15467 | 0 | 0 | 18526 | 0 | 5062 | 16589 | 7193 | 0 | ||

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||

12786 | 13899 | 16581 | 16545 | 12332 | 14520 | 12542 | 18541 | 16298 | ||

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Selection of the most appropriate supplier in the procurement process of a healthcare system is always very challenging. Current work is an effort to model an integrated multi-objective and multi-period medicine supply chain problem to evaluate suppliers on the account of less number of quality complaints against per million units’ release of medicine, production, transportation and external audit cost and time taken for manufacturing and logistics. Current model is also taking care of the environmental aspect and effect of structured network. To minimize the effects of greenhouse gases a carbon emission cost and carbon emission tax for transportation of medicines are also infused in the prescribed model. To ensure the maintenance of quality this model also incorporates QAL constraint and significant input of health care department. On time delivery is also assured in the given model by minimizing manufacturing time, delivery time and quality inspection time. For illustration purpose a numerical model of three health care units, four dealers, and four medicines for three months of planning time period is considered. Bipolar interval-valued neutrosophic optimization is employed to attain optimality of the proposed model and the results achieved are quite satisfactory. Proposed approach not only convert this complicated multi-objective model into single objective but also provided 99% satisfaction for cost reduction, 63% satisfaction for the quality of product and 64% satisfaction for total time taken. To purchase different medicines procurement department can use the data of decision variables to select best dealer for each hospital. Future line of business may involve the purchase of life saving medical equipment from national and international markets, making it a global supply chain model. Since the environmental impact is also taken into account in the proposed model so in future it can be modified and upgarded for the managemet of any complex sustainable supply chain network.