Unformatted text preview: 2
when the inventory reaches a zero level?
This is either at the end of period 2, period 3, or period 4:
If inventory reaches a zero level at the end of period 2, then we just pay a fixed cost for period 2 plus the remaining cost C3 , which would
give 2 + C3 = 2 + 2.4 = 4.4
If inventory reaches a zero level at the end of period 3, then we need to buy enough inventory at period 2 to cover periods 2 and 3
That’s r2 + r3 = 5 units; we pay a onetime fixed cost, plus a holding cost for the 3 remaining units at the end of period 2, plus the
remaining cost C4 , which is 2 + 3 × 0.2 + C4 = 4.6
If inventory reaches a zero level at the end of period 4, then we need to buy enough inventory at period 2 to cover periods 2, 3, and 4
That’s r2 + r3 + r4 = 7 units; we pay a onetime fixed cost, plus a holding cost for the 5 remaining units at the end of period 2, plus a
holding cost for the 2 remaining units at the end of period 3, which is 2 + 5 × 0.2 + 2 × 0.2 = 3.4 Thus, we have C2 = 3.4 J. G. Carlsson, U of MN ISyE Lecture 10: Inventory Models November 7, 2013 53 / 55 Periodic review dynamic program
To find C1 we need to consider four cases: when is the first time after period 1
when the inventory reaches a zero level?
This is either at the end of period 1, period 2, period 3, or period 4:
If inventory reaches a zero level at the end of period 1, then we just pay a fixed cost for period 1 plus the remaining cost C2 , which would
give 2 + C2 = 2 + 3.4 = 5.4
If inventory reaches a zero level at the end of period 2, then we need to buy enough inventory at period 1 to cover periods 1 and 2
That’s r1 + r2 = 5 units; we pay a onetime fixed cost, plus a holding cost for the 2 remaining units at the end of period 1, plus the
remaining cost C3 , which is 2 + 2 × 0.2 + C3 = 2.4 + 2.4 = 4.8
If inventory reaches a zero level at the end of period 3, then we need to buy enough inventory at period 1 to cover periods 1, 2, and 3
That’s r1 + r2 + r3 = 8 units; we pay a onetime fixed cost, plus a holding cost for the 5 remaining units at the end of period 1, plus a
holding cost for the 3 remaining units at the end of period 2, plus the remaining cost C4 , which is 2 + 5 × 0.2 + 3 × 0.2 + C4 = 5.6
If inventory reaches a zero level at the end of period 4, then we need to buy enough inventory at period 1 to cover periods 1, 2, 3, and 4
That’s r1 + r2 + r3 + r4 = 10 units; we pay a onetime fixed cost, plus a holding cost for the 7 remaining units at the end of period 1, plus
a holding cost for the 5 remaining units at the end of period 2, plus a holding cost for the 2 remaining units at the end of period 3, which is
2 + 7 × 0.2 + 5 × 0.2 + 2 × 0.2 = 4.8 Thus, we have C1 = 4.8, which we can get by either buying 10 units at the start,
or by buying 5 units at the start and 5 units in period 3
The general formula:
Ci = min j=i,i+1,...,n J. G. Carlsson, U of MN ISyE {Cj+1 + K + h[ri+1 + 2ri+2 + 3ri+3 + · · · + (j − i)rj ]} Lecture 10: Inventory Models November 7, 2013 54 / 55 References I [1] F.S. Hillier and G.J. Lieberman.
Introduction to Operations Research.
Introduction to Operations Research. McGrawHill Higher Education, 2010. J. G. Carlsson, U of MN ISyE Lecture 10: Inventory Models November 7, 2013 55 / 55...
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This document was uploaded on 02/23/2014 for the course MANAGMENT 2201 at University of Michigan.
 Spring '14

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