The newsvendor problem with unknown distribution Uri Benzion

The newsvendor problem with unknown distribution Uri Benzion, Yuval Cohen, Tal Shavit Tiber Symposium on Psychology and Economics Tilburg August 28 2008

The newsvendor problem The Newsvendor problem was first presented in Whitin (1955). • The purpose of the newsvendor model is to determine optimal order and inventory. • The vendor can order only once each day. • At the end of the day inventory surplus is thrown out.

For example: • Newspapers. • Rolls and doughnuts in a Bakery • Cellular phones.

The problem is: How many units to buy? Optimal order and expected profit are functions of: (1)Prices: the item's purchase and selling prices, and the salvage price. (We assume that the salvage price is zero in our experiment) (2)The demand distribution. * Supply surplus: when order is grater then demand- Create a lost of the unsold unit. * Demand surplus: When demand is larger then the ordercreate a lost of potential gain from sailing extra unit.

The model assumption (1) Stochastic demand. (2) One-time demand The newspaper of today can use to wrap fish tomorrow.

(3) It is commonly assumed that the decision-maker faces known demand distribution. * In real-life the demand distribution is not always known. For example, when entering new market the vendor might face unknown demand distribution.

The literature Most of theoretical analyses of quantity decisions regarding inventory orders deal with known demand distributions. Azoury (1985), Lovejoy (1990) and Lariviere (1999) developed ordering policies for unknown distributions with exact demand observations.

Eppen and Iyer (1997) developed an updated newsboy heuristic for the evaluation of the stochastic demand process, in which demand is generated by one of several possible pure demanded processes. Burnetas and Smith (2000) provided a nonparametric approach to learning unknown demand proved that in the long run the average profit with incomplete information converges to the optimal profit with full information. Heching et al (2002) found that full information policies can increase revenue more than adaptive policies.

Xu and Hopp (2004) also found a large gap between the revenue achieved by the full information policy and the learning policies when demand signals are noisy. Apostolos and Smith (2000) present theoretically the combined problem of pricing and ordering for a perishable product with unknown demand distribution

Experimental studies Schweitzer and Cachon (2000). • Known Uniform distribution for their experiments • 15 ordering decision rounds for each subject. • Followed each round with information on the profit, cost and total demand. Findings - Their results show that subjects systematically deviate from the expected profit maximizing order quantity. - When profit is larger than the cost subjects tend to order less than the optimal order. - When profit is smaller than the cost subjects tend to order more than the optimal order.

Experimental studies Bolton and Katok (2004): Extended Schweitzer and Cachon (2000), by using 100 decision rounds. They found that: • Enhanced experience improves newsvendor performance, although this improvement is, on average, rather slow. • Additional information (such as moving average ) does not improve profitability.

Bostian et al. (2006) evaluated the effect of learning and adaptive behavior on order decisions in the newsvendor problem. They used five-round intervals to obtain a long-run perspective on period-by-period losses and gains. The results indicate that decision-makers deciding at infrequent intervals seem to behave like inexperienced decision-makers. The individual data showed that participants respond to recent gains and losses, but that inertia often sets in.

Benzion et. al. (2007) Used 100 decision rounds in the newsvendor problem with low/high profit and uniform/normal distributions. They found that: • There is a convergence to a stationary order quantity and stationary mean coefficient throughout the experiments. • The subjects converge away from the optimal level of expected profit. • In the first purchase decision rounds, participants tend to be more biased toward the mean demand than in the last rounds. This bias towards the mean explains the deviation of participants from the optimal order.

Hypotheses Hypothesis 1: Participants’ with known demand will have a “better” order quantity and will tend to have an higher profit compared to participants with Unknown demand. Participants’ order quantity is closer to the optimal order calculated by the newsvendor problem model when they know the demand’s distribution and hence their profit is higher when they know the demand’s distribution. * The optimal solution is according to the mathematical model of the newsvendor problem.

Hypothesis 2: Convergence process All participant will converge toward the subjective optimal order, which may be different from the mathematical optimal order. - Individuals converge to a subjective order in the course of 100 rounds. - As a result the absolute change in the order quantity between two consecutive periods is reduced over time.

Hypothesis 3: Feedback effect. Participants order for the next round will be based on the outcome off previous round in an opposite direction. For example, If the previous order were higher than the demand in previous round, participant will reduce their order. In the mathematical solution the demand order is constant and does not change in response to the actual demand. (* A supply surplus creates a real loss since the subject has ordered more than the actual demand in this round. demand surplus creates a potential loss )

The Experiment • The experiment consist of 100 purchase rounds, that follow 10 rounds of practice. • In each round subject select her order quantity, given the prices of buying and selling. • Each round was followed by a presentation of the: Actual demand, the total cost of the order, the total revenue, the demand /supply surplus, the forfeited profits due to inventory shortage, and the profit. • Before the experiment subjects were handed written instructions, including Information on selling price and cost examples. • The experiment included 121 sophomore and junior management students, who had taken a basic course in statistics. • The experiment took place at a computer Laboratory, and lasted approximately one hour.

Prior to the experiment, participants were divided into eight groups in order to examine the combinations of two profit levels, two variance levels (using different distributions) and known and unknown distributions.

Table 1 - Description of the experiment: Conditions & Distribution Group Distribution Known/Unknown Buying price distribution Selling Profit Optima price margin l order 1 Uniform Known 9 12 Low 75 2 Uniform Known 3 12 High 225 3 Normal Known 9 12 Low 116 4 Normal Known 3 12 High 184 5 Uniform Unknown 9 12 Low 75 6 Uniform Unknown 3 12 High 225 7 Normal Unknown 9 12 Low 116 8 Normal Unknown 3 12 High 184

• The groups were assigned the same cost and selling price as in Schweitzer and Cachon (2000). The uniform demand range consisted of 1 -300 products. The normal demand distribution had a mean of 150 ( =150) and a SD of 50 ( =50) ensuring that 99. 7% of the demand distribution was within the range of 1 -300 (as in the uniform distribution). • Before the experiment, participants were handed written instructions, including examples. • For the groups working with a known distribution, the demand distribution was given to participants as follows: (1) for uniform distribution, participants were told that each value from 1 to 300 has the same likelihood of being chosen. (2) For normal distribution, participants were given a table with demand results of 100 simulated days. This represented the normal distribution in a clear and real way.

• The subjects with unknown distributions were told that the demand was taken from a distribution, without any information about this distribution. * To provide concrete incentives, at the end of the experiment, one of the rounds was randomly selected and the participants were paid proportionally to the profit in the selected round (in cash). The average payment was 20 N. I. S, or about $5

Results Table 2–The average order in different scenarios Optimal All rounds order First Last 20 -round block T-test*, p-value block Uniform Low Known 75 141. 7 148. 6 122. 6 t=3. 22, p<0. 01 Unknown 75 142. 7 141. 2 138. 7 t=0. 21, p=0. 41 Uniform Known 225 175. 7 165. 5 189. 5+ t=-3. 96 , p<0. 01 High Unknown 225 164. 3 161. 8 168. 1 t=-0. 77, p=0. 22 Normal Low Known 116 149. 9+ 154. 6+ 139. 8+ t=2. 6, p=0. 01 Unknown 116 128. 3 130. 0 125. 7 t=1. 1, p=0. 13 Known 184 157. 4 147. 6 167. 0 t=-4. 3, p<0. 01 Unknown 184 160. 4 154. 3 162. 8 t=-1. 5, p=0. 07 Normal High + p-value < 0. 05 for the hypothesis that the average order for known distribution equals the average order for the unknown distribution. * Test the hypothesis that the average order in the first 20 rounds block equals the average order in the last 20 rounds block.

* In dealing with the known distributions, subjects changed their average order in the course of the 100 rounds. * When the distribution is unknown there is no significant difference between the first and the last 20 rounds.

Only in uniform distributions with high profits the order of subjects with known distribution is closer to the optimal order compared to subjects with unknown distribution. In all the other cases we do not find a difference between knowing and unknowing the distribution, or that subjects with unknown distribution are closer to the optimal order (for example, normal distribution with low profit) inconsistent with hypothesis (1).

The subjects who knew the distribution used their knowledge to improve their order. However, as mentioned in Benzion et al (2007) subjects are biased to the distribution’s mean when the distribution is known because subjects have tendency to choose the average ("central tendency bias", e. g. ; Hollingworth 1910; Helson 1964; and Crowford et al 2000).

Table 3 - The average profit in different scenarios Uniform Low Uniform High Normal Low Normal High First Last T-test*, p-value 20 -round block Known -224. 8 -124. 06++ t=-2. 03, p=0. 03 Unknown -243. 3 -219. 6 t=-0. 36, p=0. 36 Known 621. 3++ 704. 7 t=-3. 33, p<0. 01 Unknown 587. 3 724. 0 t=-6. 79, p<0. 00 Known -24. 3+ 98. 8++ t=-3. 64, p<0. 01 Unknown 79. 0 136. 2 t=-3. 70, p<0. 00 Known 918. 8 994. 0 t=-5. 39, p<0. 01 Unknown 883. 9 990. 9 t=-2. 88, p<0. 01 + p-value < 0. 05 for the hypothesis that the average order for known distribution equals the average order for the unknown distribution. * Test the hypothesis that the average order in the first 20 rounds block equals the average order in the last 20 rounds block.

Since there is no basic difference between the orders of subjects with known and unknown distribution we do not find a difference in the profit in the two conditions inconsistent with hypothesis (1).

Table 4 - The average absolute changes of order between rounds All rounds Uniform Low Uniform High Normal Low Normal High First Last 20 -round block T-test*, p-value Known 22. 4+ 41. 2 13. 0+ t=-2. 71, p<0. 01 Unknown 37. 8 43. 4 35. 5 t=-1. 82, p=0. 04 Known 29. 8 33. 9 21. 6 t=-3. 37, p<0. 01 Unknown 35. 8 42. 0 30. 1 t=-1. 59, p=0. 07 Known 15. 0 16. 6 12. 1 t=-1. 20, p=0. 12 Unknown 15. 3 19. 1 8. 7 t=-3. 61, p<0. 01 Known 11. 6++ 14. 8++ 9. 1 t=-2. 66, p=0. 01 Unknown 16. 3 28. 8 14. 7 t=-1. 29, p=0. 1 + p-value < 0. 05 for the hypothesis that the average order for known distribution equals the average order for the unknown distribution. * Test the hypothesis that the average order in the first 20 rounds block equals the average order in the last 20 rounds block.

1. in unknown distribution is not significantly different from the absolute change in known distribution (In most of the cases). 2. Absolute change are lower in the last 20 -round block compared to the first 20 -round block. This convergence process is consistent with hypothesis (2).

Regression analysis • The dependent variable is: The change in order quantity from the previous round (ΔQt). • The independent variables are: (1) the supply surplus in the previous round (Order- Demand = St-1). (2) Dummy variable (δ) for the supply surplus coefficient to test if it is different for the unknown distribution (0 - known distribution, 1 - unknown distribution). We multiply supply surplus by the dummy variable in order to find the effect of the unknown distribution on the coefficient.

Table 5 - Explaining the change in order over time Uniform Low Uniform High Normal Low Normal High α β θ δ R-square 4. 42 -0. 23 0. 12 -0. 07 0. 14 (0. 01) (0. 00) 9. 6 -0. 3 0. 066 0. 02 0. 19 (0. 00) (0. 08) (0. 3) (0. 00) 0. 03 -0. 22 0. 07 -0. 08 0. 23 (0. 97) (0. 00) 10. 2 -0. 26 0. 31 -0. 17 0. 19 (0. 00)

(1) coefficient β is negative and significant, which indicate that the change in quantity order is negatively related to the supply surplus. This is consistent with hypothesis (3) (2) The coefficient (δ) of the dummy variable for unknown distribution is negative. * When the demand’s distribution is unknown the effect of previous round is stronger.

Conclusions We claim that recent experience (e. g. , previous round demand) seems to dominate the decisions to such an extent that there is no significant difference in behavior between those who know the demand distribution and those who do not know it. Indeed, knowing the distribution might create behavioral biases such as the "central tendency bias".

since inventory decision are based on previous experience, investing large budget in surveys and statistical analysis in order to forecast or better understand the demand’s distribution is not an optimal strategy. Knowing the demand’s distribution does not improve subjects' decision and if a large amount of money is invested in knowing the distribution the cost is higher than the gain.

Thank you