Common use of Solving method for Optimization Clause in Contracts

Solving method for Optimization. problem (Q) To solve optimization problem (Q), the Lagrange relaxation technique (▇▇▇▇▇▇▇, 2002) is used. The Lagrangian of problem (Q) is given in Formula 5.3. A new optimization problem, (P ) is developed. This optimization problem (P ) generates a lower bound for problem (Q). In this problem the Lagrangian is minimized by changing values for S and T, for given values of λ1 and λ2. λ1 can be seen as penalty costs for backorders and λ2 can be seen as extra costs in case of expediting. In problem (P ) both λs must be strictly greater than zero. We choose not to include the third constraint in the Lagrangian because this is mathematically more convenient. L(S,T, λ1, λ2) = Σi=0 CiSi + λ1 EBO(Si, Ti) − EBOobj! + λ2 ξ(Ti) − ξobj ! (5.3) Σ Σ (P ) minS,T L(S,T,λ1, λ2) subject to Ti ≤ Si where S is the vector of the turn-around stock of all SKUs (S=S1, S2..Sn) and T is the vector of the threshold of all SKUs (T=T1, T2...Tn). In problem (P ) the values for λ1 and λ2 are not optimized. A way to generate the most optimal lower bound for problem (Q) is by making a new optimization problem (LR) (▇▇▇▇▇▇, 1981). (LR) maxλ1,λ2≥0 minS,T L (S,T,λ1, λ2) subject to Ti ≤ Si