A Fuzzy Dynamic Programming for the Optimal Allocation of Health Centers in some Villages around Baghdad

Abstract

The Planning and Resource Development Department of the Iraqi Ministry of Health is very interested in improving medical care, health education, and village training programs. Accordingly, and through the available capabilities of the ministry, itdesires to allocate seven health centers to four villages in Baghdad, Iraq therefore the ministry needs to determine the number of health centers allocated to each of these villages which achieves the greatest degree of the overall effectiveness of the seven health centers in a fuzzy environment. The objective of this study is to use a fuzzy dynamic programming(DP) method to determine the optimal allocation of these centers, which allows the greatest overall effectiveness of these health centers to be achieved, which is the expected increase in the average life years in the village population in a fuzzy environment. The results of this studyareproved after a real-life problem was solved that the proposed method is an effective mathematical model for making a series of related decisions, and it provides us with a systematic procedure to determine the optimal combination of decisions.

Constraints Optimal Classical Continuous Control Vector Problem for Quaternary Nonlinear Hyperbolic System

“…where 𝑦 ⃗ = (𝑦 1 , 𝑦 2 , 𝑦 3 , 𝑦 4 ) ∈ 𝑯 𝟏 (𝛀) = (𝐻 1 (Ω)) 4 is the quaternary solution vectors (QSVs), corresponding to the quaternary classical continuous control vector (QCCCV) 𝑢 ⃗⃗ = (𝑢 1 , 𝑢 2 , 𝑢 3 , 𝑢 4 ) ∈ 𝑳 𝟐 (𝐐) = (𝐿 2 (𝑄)) 4 and (𝑓 1 , 𝑓 2 , 𝑓 3 , 𝑓 4 ) ∈ 𝑳 𝟐 (𝐐) is a vector of a given function on (𝑄 × ℝ × 𝑈 1 ) × (𝑄 × ℝ × 𝑈 2 ) × (𝑄 × ℝ × 𝑈 3 ) × (𝑄 × ℝ × 𝑈 4 ), with 𝑈 𝑖 ⊂ ℝ, ∀𝑖 = 1,2,3,4 .…”

“…Let𝑉 ⃗⃗ = {𝑣 ⃗ = (𝑣 1 , 𝑣 2 , 𝑣 3 , 𝑣 4 ) ∈ 𝑯 𝟏 (𝛀), 𝑣 1 = 𝑣 2 = 𝑣 3 = 𝑣 4 = 0 𝑜𝑛 𝜕Ω},V ⃗ ⃗⃗ = 𝑯 𝟎 𝟏 (𝛀) = (𝐻 0 1 (Ω)) 4 , 𝑳 𝟐 (𝑰, 𝑽) = (𝐿 2 (𝐼, 𝑉)) 4 and 𝑉 = 𝐻 0 1 (Ω), the inner product (IP) and the norm(Nr) in 𝑳 𝟐 , and 𝑳 𝟐 (𝑰, 𝑽 * ) is the dual of 𝑳 𝟐 (𝑰, 𝑽).…”

“…on 𝐼, (39) ( 𝑍 3 (𝑇) , 𝑣 3 ) = (𝑍 3𝑡 (𝑇) , 𝑣 3 ) = 0 , (40) (𝑍 4𝑡𝑡 , 𝑣 4 ) + (∇𝑍 4 , ∇𝑣 4 ) + (𝑍 1 , 𝑣 4 ) − (𝑍 2 , 𝑣 4 ) + (𝑍 3 , 𝑣 4 ) + (𝑍 4 , 𝑣 4 ) = (𝑍 4 𝑓 4𝑦 4 , 𝑣 4 ) + (𝑔 4𝑦 4 , 𝑣 4 ) , ∀𝑣 4 ∈ 𝑉 a.e. on, (41) (𝑍 4 (𝑥, 𝑇) , 𝑣 4 ) = (𝑍 4𝑡 (𝑇) , 𝑣 4 ) = 0, (42) The WF ((35-(42)) has a unique solution 𝑍 ⃗ = (𝑍 1 , 𝑍 2 , 𝑍 3 , 𝑍 4 ) ∈ (𝐿 2 (𝑄)) 4 (this it can proved so as the proof of existence a unique QSVs for the WF (( 11)-( 15)). Now, replacing 𝑣 𝑖 = 𝛿𝑦 𝑖𝜀 in (35), (37), ( 39) and (41), for 𝑖 = 1,2,3,4 resp.…”

“…Optimal control problems (OCPs) are important in a wide range of practical applications, including robotics robotics [1], economics [2], weather conditions [3], community health [4], and a variety of other scientific fields. Nonlinear ODEs [5]or nonlinear PDEs (NLPDEs) [6] usually dominate OCPs.…”

“…From Theorem 1, corresponding to the Seq.QCV {𝑢 ⃗⃗ 𝑘 } the WF of the QSEs has "a unique" solution {𝑦 ⃗ 𝑘 = 𝑦 ⃗ 𝑢 𝑘 } and ∥ 𝑦 ⃗ 𝑘 ∥ 𝑳 𝟐 (𝑰,𝑽) , ∥ 𝑦 ⃗ 𝑘𝑡 ∥ 𝑳 𝟐(𝑸) are bounded, then by Alaoglu's theorem (ATH), there exists a Subsequence of {𝑦 ⃗ 𝑘 } and {𝑦 ⃗ 𝑘𝑡 }, say again {𝑦 ⃗ 𝑘 } and {𝑦 ⃗ 𝑘𝑡 }, s.t. 𝑦 ⃗ 𝑘 → 𝑦 ⃗ WK in 𝑳 𝟐 (𝑰, 𝑽), 𝑦 ⃗ 𝑘𝑡 → 𝑦 ⃗ 𝑡 WK in (𝐿 2 (𝑄))4 . Now for each 𝑘. and by applying the ACTH[14] , there is a Subsequence of{𝑦 ⃗ 𝑘 } say a gain {𝑦 ⃗ 𝑘 } s.t.…”