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Nahmias Chapter 5: Harvey's Specialty Shop (Q, r) policy with a stock-out cost

ToolInventory Policy
SourceNahmias, S. (2009). Production and Operations Analysis, 6th ed. McGraw-Hill/Irwin, New York. Chapter 5, Harvey's Specialty Shop and Mac's newsstand examples.
LevelGraduate

Problem

Harvey's Specialty Shop sells 200 jars of a mustard a year; an order costs $50, a jar costs $10 and is held at 20 % a year, a stock-out costs $25 per jar, and the lead-time demand is normal with mean 100 and standard deviation 25. Find the order quantity and reorder point that minimize the expected annual cost, and the safety stock, costs and fill rate of that policy.

Inputs to Enter

  • Model (Q, r) continuous review (loads the example), period year: demand 200, order cost 50, unit cost 10, holding rate 20 %, shortage cost 25
  • Lead-time demand entered directly: mean 100, SD 25; service basis cost
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Answers

  • Iteration: Q0 = EOQ = 100, 1 − F(r0) = 100 × 2/(25 × 200) = 0.04, r0 = 100 + 1.75 × 25 = 144; converges to Q = 111, r = 143 (book, two-decimal z table)
  • Policy (111, 143): safety stock 43 jars, z = 1.72, holding h(Q/2 + r − μ) = $197, setup Kλ/Q = $90.09, n(r) = 25 G(1.72) = 0.435, stock-out cost $19.6, 0.78 jars short a year, fill rate 0.996
  • Type 1 service 95 %: r = 100 + 1.645 × 25 = 141; fill rate 98 % at Q = 100: G(z) = 0.08, z = 1.02, r = 126
  • Mac's newsstand (newsvendor, c 0.25, p 0.75, demand N(11.73, 4.74)): critical ratio 0.67, z = 0.43, Q* = 13.77 → 14 copies

Notes

Choose the (Q, r) model; the select fills the published inputs. The continuous solution of the tool lies within one unit of the book's rounded hand iteration.

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