Assumption 6,1. Việc kiểm kê, chi phí tồn đọng là dễ chịu, có nghĩa là, nếu cn U ncn U n, sau đó cn U ncn U n và ngược lại, n, n, trong đóCác biến đồng nhà nước, n (t), đáp ứng phương trình vi phân sau đây với điều kiện chu kỳ tương ứng | 356 6 SUSTAINABLE COLLABORATION IN SUPPLY CHAINS c- and cn are the unit costs of storage inventory and backlog of producttype n respectively. We assume relatively large backlog costs are assigned to products that cause large inventory costs and vice versa as formalized below. Assumption . The inventory and backlog costs are agreeable that is if c -Un c -Un then c Un c ríUrí and vice versa for n n e Y where Y 1l n . Without losing generality we also assume that if c-Un c- Un then n n and if n n then c nUn c -Urí n n e Y where Y 1lN . Analysis of the problem n Applying the maximum principle to problem - the Hamiltonian is formulated as follows H -Â kX- t - c X t - Â t un t - dn . The co-state variables n t satisfy the following differential equations with the corresponding periodicity boundary condition c if Xn t 0 n t - c if Xn t 0 ts n tf a a e -c- c- if Xn t 0 To determine the optimal production rate un t when A t 0 we consider the following four possible regimes which are defined according to Un t . Full Production regime FP This regime appears if there is an n such that Un n t 0 andU n t Uri ri t Vrl n n n eY. In this regime according to we should have un t Un and urt t 0 Vn n n rleY. to maximize the Hamiltonian. No Production regime NP If U n t 0 VneY. In this regime we should have un t 0 VneY to maximize the Hamiltonian. Singular Production regime S-SP This regime appears if there is S cY the rank of S the rank of S is defined as the number of units in S and denoted R S is greater than 1 and U n t Uri riO 0 Vn n eS andU n t Um m t VneS m S. SUPPLY CHAINS WITH LIMITED RESOURCES 357 In this regime there is a set of products S the active set for which the Hamiltonian gradients U n t 0 are equal to each other and are greater than all the other gradients at an interval of time. Singular Production regime Z-SP This regime appears if there is a Z such that U n t Un n t 0 n n eZ andU n t Um m t n e Z m Z . In this regime there is a
