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Thus when the market needs change, the production cost and the subcontracting cost change but the fixed amount of earnings mentioned in the contract stays the same. The system’s optimal production plan is the same when the difference between the production cost and the Patent Translation Provider subcontracting cost stays constant as well as the difference between the costs of local and global optimization is constant (property Using this property we are not obliged to change the production plan when the production cost changes. In addition, in some cases, we could be able to avoid one of two analyses.Proof: If for factory then it is enough to demonstrate that the optimal value of the objective function as well as the optimal production plan are the same when the production cost and the subcontracting cost are cp and when the production cost and the subcontracting cost we take the following objective function:which does not influence the optimization). This results the same minimum value and exactly the same production plan due to the same group of constraints When the centralized optimization gives an optimal solution for to subcontract the extra demand regardless of plan, the decentralized optimization gives exactly the same solution (property Proof: In this case obtains the demand curve which is exactly the same to the curve of the final product. In the case of decentralized optimization (which gives the optimal solution for in the worst scenario we will get a production plan which follow the demand or a mix plan (subcontracting and inventory). The satisfaction of the first curve (centralized optimization) is more expensive for than the satisfaction of the second (decentralized optimization) because the supplementary (to the production capacity) demand is greater. For this reason the production cost in decentralized optimization is greater than or equal to the production cost of the centralized optimization and using property we prove that centralized and decentralized optimal production cost for should be the same Finally, we have demonstrated that when at the decentralized optimization, the extra demand for is satisfied from inventory then the centralized optimization has the same optimal plan Proof: In this case of decentralized optimization, has the best possible curve of demand because satisfy the extra demand without subcontracting. In centralized optimization in the best scenario we take the same optimal solution for mix policy. If we take the case of mix policy then the centralized optimal solution of will be greater than or equal to the decentralized optimal solution and using property we prove that centralized and decentralized optimal production cost for should be the same.