Download presentation

Presentation is loading. Please wait.

Published byJack Ludwick Modified over 2 years ago

1
1 Outline single-period stochastic demand without fixed ordering cost base-stock policy minimal expected cost maximal expected profit (s, S) policy: minimal expected cost

2
2 Single-Period Problem

3
3 Common Features stochastic demands how many items to order too many: costly and with many leftovers obsolescence too few: opportunity loss real-life problems multi-product multi-period or continuous-time starting with a single-product, single-period problem

4
4 Useful Policies in Inventory Control any simple way to control inventory? yes, some simple rule-type policies base-stock policy base-stock level S an order of S-p units for an inventory position p units on review (s, S) policy reorder point s an order of S-p units only for an inventory position p s on review special case: (S-1, S) policy for continuous review under unit demands (R, Q) policy R in (R, Q) s in (s, S) all replenishment orders being of Q units (R, nQ) policy a variant of (R, Q) to replenish multiple batches of Q units to make inventory position above R

5
5 How Good Are the Policies? how good are the policies? would they be optimal in some sense? what are the conditions to make them optimal? would the conditions for single- and multi-period problems be different? would the conditions change with the setup cost?

6
6 Answers basically yes to the questions single-period problems base-stock policy being optimal among all policies: zero setup cost under convexity (and certain other assumptions) (s, S) policy being optimal among all policies: positive setup cost under K-convexity (and certain other assumptions) multi-period problems similar conditions as the single-period problems

7
7 Single-Period Problem Optimality of the Base-Stock Policy Minimizing Expected Cost

8
8 Ideas of the Optimality of the Based-Stock Policy for Single-Period Problems let H(q) be the expected cost of the period if the period starts with q items (ignoring the variable unit cost for the time being) let I j be the initial inventory on hand how to order? I1I1 I3I3 q H(q)H(q) I2I2

9
9 Ideas of the Optimality of the Based-Stock Policy for Single-Period Problems let H(q) be the expected cost of the period if the period starts with q items (ignoring the variable unit cost for the time being) let I j be the initial inventory on hand how to order? I1I1 I3I3 I1I1 I3I3 q H(q)H(q) q H(q)H(q) I2I2 I2I2 I1I1 I2I2 I4I4 q H(q)H(q) I3I3

10
10 Ideas of the Optimality of the Based-Stock Policy for Single-Period Problems unique minimum point q * of H(q) the base- stock policy being optimal order nothing if initial inventory I 0 q * ; else order q * I 0 (ignoring the variable unit cost for the time being) I1I1 I3I3 I1I1 I3I3 q H(q)H(q) q H(q)H(q) I2I2 I2I2 I1I1 I2I2 I4I4 q H(q)H(q) I3I3

11
11 Newsvendor Models: Optimality of the Base-Stock Policy a single-period stochastic inventory problem order lead time = 0 cost items c = the cost to buy each unit h = the holding cost/unit left over at the end of the period = the shortage cost/unit of unsatisfied demand D = the demand of the period, a continuous r.v. ~ F v(q) = the expected cost of the period for ordering q items objective: minimizing the total cost of the period

12
12 Newsvendor Models: Optimality of the Base-Stock Policy cost: variable ordering cost, holding cost, and shortage cost ordering q units variable ordering cost = cq how about holding cost and shortage cost?

13
13 Newsvendor Models: Optimality of the Base-Stock Policy first assume that no inventory on hand inventory on hand after ordering = q expected holding cost when D is a constant d holding cost = 0, if d q holding cost = h(q d), if d < q holding cost = hmax(0, q d) = h(q d) + in general when D is random, expected holding cost = hE[q-D] +

14
14 Newsvendor Models: Optimality of the Base-Stock Policy expected shortage cost when D is a constant d shortage cost = 0, if q d shortage cost = (d q), if q < d shortage cost = max(0, d q) = (d q) + in general when D is random, expected holding cost = E[D-q] +

15
15 Newsvendor Models: Optimality of the Base-Stock Policy the problem: min v(q) = cq + hE[q-D] + + E[D-q] + how to order? what is the shape of v(q)? q v(q)v(q) complicated optimal ordering policy if the shape of v(q) is complicated

16
16 Example 6.1.1. let us play around with a concrete example standard Newsvendor problem c = $1/unit h = $3/unit = $2/unit D ~ uniform[0, 100] q * = ?

17
17 Example 6.1.1. min v(q) = cq + hE[q-D] + + E[D-q] + = q + 3E[q-D] + + 2E[D-q] + D ~ uniform[0, 100]

18
18 Newsvendor Models: Optimality of the Base-Stock Policy for I 0 q* the expected total cost to order y = v(I 0 +y) c (I 0 +y)+ cy = v(I 0 +y) c I 0 the expected total cost not to order = v(I 0 ) c I 0 optimal not to order because v(q) is increasing for q q* for I 0 < q* the expected total cost of ordering q* I 0 = v(q*) cq*+c(q* I 0 ) = v(q*) cI 0 the expected total cost ordering y = v(I 0 +y) c(I 0 +y)+cy = v(I 0 +y) cI 0 optimal to order q* I 0 because v(I 0 +y) v(q*) for y q* I 0 ordering policy: base-stock type for I 0 < q *, cost of the buying option = v(q * ) cI 0 ; cost of the non- buying option = v(I 0 ) cI 0

19
19 Newsvendor Models: Optimality of the Base-Stock Policy min v(q) = cq + hE[q-D] + + E[D-q] + would v(q) be always a convex function? Yes optimal policy: always a base-stock policy [q-d]+[q-d]+ d [d-q]+[d-q]+ d

20
20 Newsvendor Models: Optimality of the Base-Stock Policy can we differentiate v(q)? v(q) = cq + hE[q-D] + + E[D-q] + first-order condition: c + hF(q) - F c (q) = 0

21
21 Example 6.1.1. q * = F 1 (( c )/( +h )) c = $1/unit, h = $3/unit, = $2/unit q * = F 1 (0.2) F = uniform[0, 100] q * = 20 another way to show the result

22
22 Newsvendor Models: Optimality of the Base-Stock Policy a similar problem as before, except unit selling price = p objective: maximizing the total profit of the period skipping shortage and holding costs for simplicity r(q) = the expected profit if ordering q pieces max r(q) = pE[min(D, q)] – cq r(q) concave

23
23 Newsvendor Models: Optimality of the Base-Stock Policy r(q) = pE[min(D, q)] – cq min (D, q) = D + q – max(D, q) = D – [D–q] + let = E(D) r(q) = p pE(D q) + cq r(q) concave first-order condition gives Example 6.1.2.

24
24 Example 6.1.2. revenue: p = $5/unit cost terms unit purchasing cost, c = $1 unit inv. holding cost, h = $3/unit (inv. holding cost in example) unit shortage cost, = $2/unit (shortage in example) the demand of the period, D ~ uniform[0, 100]

25
25 Example 6.1.2. expected total profit r(q) = pE[min(D, q)] – cq – hE(q–D) + – E(D–q) + min (D, q) = D + q – max(D, q) = D – [D–q] + = E(D) r(q) = p[ – E(D–q) + ] – cq – hE(q–D) + – E(D–q) + r(q) = p – cq – (p+ )E(D–q) + – hE(q–D) + concave function r (q) = – c + (p+ )F c (q) – hF(q) q * = F c (0.6) = 60

26
26 Example 6.1.2. deriving from sketch r(q) = p – cq – (p+ )E(D–q) + – hE(q–D) + D ~ unif[0, 100] E(D–q) + = E(q–D) + =

27
27 Single-Period Problem Optimality of the (s, S) Policy Minimizing Expected Cost

28
28 Fixed-Cost Models: Optimality of the (s, S) Policy a similar problem, with fixed cost K per order the sum of the variable cost in ordering, the expected inventory holding cost, and the expected shortage cost: v(q) = cq + hE[q D] + + E[D q] + S * : the value of q that minimizes v(q) s * < S * such that v(q * ) = K+v(S * )

29
29 Fixed-Cost Models: Optimality of the (s, S) Policy for I 0 > S * : no point to order ordering y: v(I 0 +y)+K+cy not ordering: v(I 0 ) for s I 0 S * : do not order not ordering: v(I 0 ) cI 0 ordering y: v(I 0 +y) c(I 0 +y)+K+cy = v(I 0 +y)+K cI 0 for I 0 s: order up to S * not ordering: v(I 0 ) cI 0 ordering y: v(I 0 +y) c(I 0 +y)+K+cy = v(I 0 +y)+K cI 0 optimal policy: (s, S) policy q v(q)v(q) Figure 1. The Definition of s * and S * S*S* v(S*)v(S*) K v(s*)v(s*) s*s* can break I 0 S* into two cases, I 0 < s and s I 0 S *

30
30 Example 6.2.1. cost terms fixed ordering cost, K = $5 per order unit purchasing cost, c = $1 unit inventory holding cost, h = $3/unit unit shortage cost, = $2/unit the demand of the period, D ~ uniform[0, 100] the expected total cost for ordering q items

31
31 Example 6.2.1. v'(S * ) = 0 S* = 20 v(S * ) = 90 s * = the value of q < S * such that v(s * ) = v(S * ) s* = 5.8579 optimal policy if on hand stock x < 5.8579, order S-x otherwise do nothing

Similar presentations

Presentation is loading. Please wait....

OK

Introduction to Basic Inventory Control Theory

Introduction to Basic Inventory Control Theory

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on online education system Ppt on review writing resources Download ppt on fundamental rights and duties of filipinos Download ppt on mutual fund Ppt on digital library system Ppt on hard gelatin capsule processing Ppt on water quality standards Ppt on pi in maths the ratio Ppt on bluetooth based smart sensor networks Ppt on insurance policy management system