Kể từ khi các nhà cung cấp dịch vụ không áp đặt bất kỳ chi phí cố định trật tự, ảnh hưởng của hàng tồn kho ban đầu về gia công phần mềm cho hệ thống tập trung thảo luận trong phần trước, MHC. F (x0 + q * ') = mhh Để nghiên cứu tác động của hàng tồn kho ban đầu sản xuất tại nhà máy của nhà sản xuất, | 96 2 SUPPLY CHAIN GAMES MODELING IN A STATIC FRAMEWORK The effect of initial inventory Since the supplier does not impose any fixed-order cost the effect of initial inventories on outsourcing is identical to that for the centralized system as discussed in the previous section Fx q - Fx. m h h To study the effect of initial inventories on production at the manufacturer s plant let x0 S otherwise it is not optimal to produce at all and x x q-d. Then the profit from not ordering anything is rc x0 x 7 . x 7 Đ mDf D dD Đ mxữ f D dD -J h x0 - D f D dD -J h - D - x0 f D dD. 0 x0 0 X0 On the other hand if the manufacturer produces q 0 products the profit is n q x -cmq-C. The optimal solution for this objective function is determined by m h - - c F q x -------- . Denote S q x then the optimal in-house profit for a given x0 is K0 S -Cm S- x0 -C. Note that if x 0 then assuming that in-house production is profitable under conditions of no initial inventory we have n S -cm S- xo -C 0 while ft x 0 since we do not sell anything when xo 0. That is n S -cm S- x0 -C rc x0 or equivalently n S -CmS -C dx -cj. which implies that it is optimal to produce in-house when x0 0. When initial inventories increase x0 0 then the left-hand part of the inequality remains unchanged while the right-hand part increases towards its maximum which is attained at x0 S. Thus when x S C 0 we have n S -cmS - C ft x -cmx which implies that it is optimal not to produce when x0 S. The right-hand side of the inequality represents the traditional newsvendor objective function n x -cmx which monotonically increases when x0 increases towards S. We conclude that there exists x s S such that n S -cmS - C n s -cms. Thus if x0 s then n S -cmS -C n x -cmx and it is profitable to produce so that S q x . On the other hand if x0 s then n S -cmS - C n x -cmx and it is not profitable to produce. Consequently in contrast to the optimal STOCKING COMPETITION WITH RANDOM DEMAND 97 order-up-to policy when no fixed .
