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3n values average yield rates batch Cn(In,Qn conditions for existence conditions on costs constant demand cost for periods cost per unit d-IN demands and costs deterministic demands discount factor discounted cash flow discounted expected cost dmax dmin dynamic programming ensure equation expected good output f pf(p)dp F(pN finite and infinite Gerchak global optimum h E(P hE(P heuristic hN)F(pN holding the expected INFINITE HORIZON POLICIES infinite horizon problem inventory holding cost last period lot sizes milder conditions multiplicative policy non-negative opportunity cost OPTIMAL FINITE optimal input quantity optimal policy optimal solutions optimal value order condition POLICIES FOR SINGLE-STAGE policy is multiplicative pQn-i problem with constant procedure production and inspection profitable to produce RANDOM YIELDS right hand side satisfy demand setup costs shortage cost SINGLE-STAGE PRODUCTION SYSTEMS subsequent period taking the limit time-varying demands value of Qn variable cost variable production cost Variable Yield Yano yield rate distribution