Product damage and free sampling: a newsvendor model with passive and proactive self-consumption

Abstract

A retailer sells a single product for a single period. During transportation and storage, some of these products are consumed by the retailer either (1) due to unavoidable damages (passive self-consumption), or (2) distributed for free to the customers (proactive self-consumption). This creates a mismatch between the amount purchased by the retailer and the amount available for sale. We study passive self-consumption with (i) fixed and (ii) proportional consumption, and proactive self-consumption with (iii) additive and (iv) multiplicative demand. Under proactive self-consumption, the retailer holds more inventory and receives a higher profit; the reverse is true under passive self-consumption. Yet, (i), (iii) and (iv) result in a higher order quantity and same fill rate compared to no self-consumption, (ii) may result in a higher or lower order quantity with a lower fill rate. When both types of self-consumption coexist, the optimal policy can be complicated. We characterize the optimal policy and show through numerical studies that the optimal policy can take at most three formats: sell to the market with positive proactive self-consumption, sell to the market with zero proactive self-consumption and do not sell to the market. Interestingly, the optimal order quantity is not smooth in the fraction of the proportional self-consumption. Further we find that when the market adoption rate is uncertain, the optimal strategy preserves a similar structure. The retailer benefits from expediting if the difference between the high and the low market adoption rates is high and the probability of a high market adoption rate is low.

Keywords:

• newsvendor model
• self-consumption
• optimal order quantity
• additive and multiplication demand

Notes

1 If E[D²]<∞, then can take the value of infinity, otherwise

2 We similarly define ( ) and () for the latter part of the paper.

3 The total number of adoptions as a function of distributed quantity has been studied in Iyengar and Berger (2014), they find that the number of influence doses within a given period has a non-linear and sub-additive impact on adoption: More doses are associated with increased adoption, but with sharply diminishing marginal impact. In our paper, we choose to use exponential functions to approximate the increasing and sub-additive properties found in Iyengar and Berger (2014).

4 To verify that the threshold is increasing in p, we take derivative of the threshold over p. After some simplification, the expression of the derivative is This hence proves the increasing property of the threshold. The threshold is decreasing in c can be shown similarly.

5 To show that Q* is unimodal in α, first note that β=1/(1−α) is strictly decreasing in α. Thus, it is sufficient to show Q* is unimodal in β. Taking derivative of the optimal order quantity over β and multiplying by we have the modified derivative equals Note that the modified derivative is in the form of where Thus, the statement holds because of the IFR property and the fact that 1/x is strictly decreasing in x.

6 In general, the optimal order quantity is neither increasing nor decreasing in the consumption percentage α because (1)/(1−α) is increasing in α but is decreasing in α.

7 Either the market size N or the adoption rate λ is small, or the profit margin p−c is small

8 It is easy to show that q* is unimodal in λ, and solving for the first order condition we can compute the critical points of λ.

9 See Shaked and Shanthikumar (2006) for definition of stochastic convex orders.

10 For well-known research work on demand information updating, see for example Li and Zhang (2013), Gurnani and Tang (1999), Yan et al (2003), etc.

11 For multiplicative demand, the analysis and insights are similar, so we omit them in this paper.

12 Note that if the retailer consumes zero, then the demand becomes ε, which is independent of the market adoption rate, and hence the retailer’s problem reduces to the classical newsvendor problem, and the retailer will not expedite.

13 Because π(Q, q) is continuous and concave in Q, the optimal order quantity satisfies the first order condition. Thus, if either the solution to the first order condition under case (2) or (3) is feasible, then that solution is optimal. Otherwise, the boundary of case (2) and (3) gives the optimal order quantity.

14 Indeed, the quantity that is available to the market should exceed the optimal order quantity with no self-consumption.

15 Note that c−ρ_Hc_e>ρ_Lp, will never happen because c_e>c and p>c.

16 This is because (i) the left hand side converges to zero if q→0 or q→∞, (ii) the left hand side is always non-negative, and (iii) there is only one q such that the derivative of the left hand side equals zero.

17 If Equation (4) never holds, then we pick any q₁>q₂>0.

18 Note that in both cases, Q⩾q is a feasible solution.