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A min-plus algebra is a linear algebra over the semiring R_ε', equipped with the operations “"⊕'=min" ” and “⊗=+”. In min-plus algebra, there is the concept of characteristic polynomial obtained from permanent of matrix. Min-plus algebra can be extended to an interval min-plus algebra, which is a set 〖 I(R)〗_ε' equipped with the operations ¯("⊕'" ) and ¯("⊗" ). Matrix over interval min-plus algebra has some special forms, one of which is a triangular matrix. The concept of characteristic polynomial and eigenproblem can be applied to a triangular matrix. There is a more concise formula for solving the eigenproblem of triangular matrix because this matrix is a special form of matrix. This research will discuss the characteristic polynomial and solving eigenproblem of triangular matrix over interval min-plus algebra using its characteristic polynomials. The research method used is a literature study. From the research results, the permanent formula and characteristic polynomial formula of the triangular matrix are obtained. It is also obtained that the smallest corner of the characteristic polynomial is the principal eigenvalue and the vector eigen corresponding to the principal eigenvalue can be obtained through the matrix A_λ. For readers who are interested in this topic, can research about characteristic polynomial and eigenproblem of matrices with other special forms over min-plus interval algebra.

Health insurance company have to determine claim reserve that’s suitable with the existing condition. There is three party that’s involved in the health insurance management, namely the policy holder, Admedika as the third party administration, and also the insurance company itself as the (insurer). When the policy holder obtained treatments whose financing is done through a health insurance, then the health insurance company have the obligation to finish the financial matters. Delays in payments from insurance companies to health facilities are caused, among others, by the administrative process. Thus, every claim submitted by the insured party to the insurance company will be settled in stages to the health facilities. The data presented from these conditions form a triangle matrix (run-off triangle) which then becomes the basis for estimating the amount of IBNR claims reserves. The Bornhuetter-Ferguson (BF) method involves the amount of premium that has become income for the company and calculates the Ultimate Claim value in estimating the amount of claim reserves. This method is the result of the development of the previous method, Chain-Ladder (CL), which only relies on historical data on claim payments. Premium calculations need to be involved in health insurance, because the insurance period is short, which is only one year. Insurance companies haven't had time to turn around the money to invest, so payment of claims will depend more on the premium that becomes income for the company (earned premium). The estimated claim reserve value is more suitable and robust than the CL method. Estimated claim reserves that occur in the 2nd event period amount to IDR50,658,714 with an estimated interval for the 2nd event period between IDR10,215,477 and IDR91,101,950