The newsvendor problem with normal, worst-case and binomial distribution of demand: Managerial implications with examples

Tomasz Brzęczek; Dušan Hrabec

doi:10.3934/jimo.2024029

Article Contents

2024, Volume 20, Issue 12: 3628-3646. Doi: 10.3934/jimo.2024029

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The newsvendor problem with normal, worst-case and binomial distribution of demand: Managerial implications with examples

Tomasz Brzęczek^1, , and
Dušan Hrabec^2,

1.
Poznan University of Technology, Poland
2.
Tomas Bata University, Czech Republic

^*Corresponding author: Tomasz Brzęczek
^*Corresponding author: Tomasz Brzęczek

Received: September 2023

Revised: February 2024

Early access: March 2024

Published: December 2024

Abstract / Introduction Full Text(HTML) Figure(8) / Table(7) Related Papers Cited by

Abstract

The paper examines the newsvendor problem with demand distributions commonly used in the literature. Optimal order convergence is checked numerically. An important contribution is that the expected profits differ considerably in nominal and relative terms when the profit-loss ratio is low valued while the demand variability is at least moderate. Missed expected profit reaches even 10% of total order cost for a mark-up that equals to the worst case break even one. The optimal order quantities are compared to counterparts derived from sales data. The main managerial implication indicates that the normal distribution solution outperforms distribution-free solutions in predicting the maximal expected profit under empirical demand. Therefore, the MaxiMin solution is recommended to be used in practice to simplify the maximal expected profit calculus and for break-even mark-up evaluation. However, it should not be used to solve the optimal order quantity because the worst-case distribution asymmetry is determined by the profit-to-loss ratio and can be contradictory to the asymmetry of sales data. Moreover, the normal distribution optimum bounded with 0.8 service level commitment shows negative expected profit under mark-up from the range of 5-20% while demand variability is low or moderate proportionally to mark-up.

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Mathematics Subject Classification: Primary: 90-02, 90-08; Secondary: 90-11.

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Figure 1. The optimal order quantities $ Q^{N}, Q^{B}, Q^{S} $ for demand $ D(\mu = 10,\sigma = 2) $

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Figure 2. The optimal order quantities $ Q^{N} $, $ Q^{B} $ and $ Q^{S} $ for $ D(\mu = 10,\sigma = 5) $

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Figure 3. Maximum expected profit from various distributions of demand $ D(\mu = 10,\sigma = 2) $

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Figure 4. Maximum expected profit from various distributions of demand $ D(\mu = 10,\sigma = 5) $

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Figure 5. Histogram of 30 weekly sales $ D(\mu = 29.8, \sigma = 15.35) $ versus the normal distribution function

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Figure 6. The optimal order quantities for various distributions of demand $ D(\mu = 29.8, \sigma = 15.35) $

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Figure 7. The maximal expected profit for theoretical and empirical demand distributions with $ (\mu = 29.80; \sigma = 5.35) $

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Figure 8. Expected profits for the optimal and non-optimal order quantities under the worst-case and the normal distributions of $D\sim N\left( \mu = 10,\sigma = \left\{ 2,\ \ 5 \right\} \right)$ for break-even mark-up values

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Table 1. The standard deviation $ S $ of profit payoffs $ Z $ around the maximal expected profit $ E(Z) = \Pi $ for discount $ d = 1 $ for demand $ D(\mu = 10,\sigma = 5) $

		$ m $
		0.05	0.1	0.2	0.4	0.6	0.8	1	1.2	1.6	1.75	2	2.4
$ S(P) $	Normal	2.9	4.7	8.1	13.3	18.4	21.7	28.9	34.2	44.4	48.0	54.2	64.0
	Worst-case	3.5	4.2	6.1	9.4	13.0	17.4	22.6	28.5	41.9	47.2	56.5	72.1
	Binomial	1.1	2.2	3.9	6.8	7.8	11.9	13.3	14.6	21.6	22.9	25.0	28.3

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Table 2. Break-even mark-up $\underline{m}$ values for analyzed demand settings

Demand	Worst-case		Normal		Binomial		Empirical
$ (\mu,\sigma) $	$d' = 0.5$	$d = 1$	$d' = 0.5$	$d = 1$	$d' = 0.5$	$d = 1$	$d = 1$
(10, 2)	0.020	0.040	0	0.001	0	0.0001	NA
(10, 5)	0.125	0.250	0.042	0.085	0	0	NA
(29.80, 15.35)	0.135	0.270	0.030	0.060	0	0	0
NA - not applicable

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Table 3. Expected profit for the optimal order quantity from various distributions $ (\mu = 10,\sigma = 2),\ d = 1 $

	$ {\underline{m}}^{N} = 0.001 $		$ {\underline{m}}^{S} = 0.04 $			$ m = 0.6 $			$ m = 1.2 $
	$ Q^{N} $	$ Q^{B} $	$ Q^{S} $	$ Q^{N} $	$ Q^{B} $	$ Q^{B} $	$ Q^{N} $	$ Q^{S} $	$ Q^{B} $	$ Q^{S} $	$ Q^{N} $
	3.80	5.00	5.20	6.46	7.00	9.00	9.36	9.48	10	10.18	10.23
$\Pi^{N}(Q)$	0.123	0.10	1.02	1.15	1.11	23.48	23.56	23.40	50.50	50.55	50.56
$\Pi^{S}(Q)$	-0.260	-0.260	0	0	0	22.25	22.25	22.25	49.04	49.04	49.04
$\Pi^{B}(Q)$	0.018	0.022	1.03	1.22	1.28	24.58	24.37	24.33	51.82	51.46	51.37

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Table 4. Expected profit for the optimal order quantity from various distributions $ (\mu = 10,\sigma = 5),\ d = 1 $

Optimal order quantity	$ {\underline{m}}^{N} = 0.085 $		$ {\underline{m}}^{S} = 0.25 $			$ m = 0.6 $			$ m = 1.2 $
	$ Q^N $	$ Q^B $	$ Q^N $	$ Q^S $	$ Q^B $	$ Q^N $	$ Q^S $	$ Q^B $	$ Q^B $	$ Q^S $	$ Q^N $
	2.92	7.00	5.79	6.25	8.00	8.41	8.71	9.00	10	10.18	10.57
$\Pi^{N}(Q)$	0	-2.77	3.44	3.40	2.98	14.40	14.35	14.28	37.44	37.57	37.58
$\Pi^{S}(Q)$	-2.35	-3.04	0	0	0	10.64	10.64	10.64	32.61	32.61	32.61
$\Pi^{B}(Q)$	1.24	2.51	7.09	7.46	8.64	22.67	22.96	23.25	50.30	50.10	50.04

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Table 5. The optimal order and maximum expected profit for service level constraint (18) when discount $ d = 1 $ and demand volatility $ \sigma = 2 $

		$ m $
		0.05	0.1	0.2	0.4	0.6	0.8	1	1.2	1.6	1.75	2	2.4
$ Q^{'} $	Normal	10.6										10.8	11.0
	Worst-case	10.4	10.6	10.9	11.3	11.5	11.8	12	12.2	12.5	12.6	12.8	13.1
	Binomial	10.0						10	10	10	10	10	10
$ \Pi^{'} $	Normal	-3.3	1.0	3.7	13.2	22.6	32	41.4	50.9	69.7	76.8	88.6	107.8
	Worst-case	0.3	1.8	5.5	13.7	22.3	31.1	40	49.0	67.4	74.3	85.9	104.5
	Binomial	-1.5	0.9	5.5	14.7	24	33.2	42.4	51.7	70.2	77.1	85.9	107.2

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Table 6. The optimal order and maximum expected profit for service level constraint (18) when discount $ d = 0.5 $ and demand volatility $ \sigma = 5 $ constraint (18)

		$ m $
		0.05	0.1	0.2	0.4	0.6	0.8	1	1.2	1.6	1.75	2	2.4
$ Q^{'} $	Normal	12.9								13.4	13.6	13.9	14.5
$ Q^{'} $	Worst-case	11.6	12.2	13.2	14.5	15.5	16.3	17.1	17.7	18.9	19.4	20	21
$ \Pi^{'} $	Normal	-7.4	-5.1	-0.6	8.5	17.5	26.6	35.6	44.7	62.8	69.8	81.4	100.1
$ \Pi^{'} $	Worst-case	-1.5	-0.6	2.1	8.8	16.3	24.2	32.3	40.6	57.6	64.1	75	92.6

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Table 7. The optimal order and maximum expected profit for service level constraint (18) when discount $ d = 1 $ and demand volatility $ \sigma = 5 $

		$ m $
		0.05	0.1	0.2	0.4	0.6	0.8	1	1.2	1.6	1.75	2	2.4
$ Q^{'} $	Normal	12.9
	Worst-case	11.1	11.6	12.2	13.2	13.9	14.5	15	15.5	16.3	16.6	17.1	17.7
	Binomial	11	12
$ \Pi^{'} $	Normal	-17	-14.8	-10.2	-1.2	7.9	16.9	26	35	53	59.9	71.2	89.3
	Worst-case	-3.1	-2.9	-1.2	4.2	10.6	17.6	25	32.6	48.4	54.4	64.6	81.2
	Binomial	-8.8	-6.4	-1.5	8.2	18	27.7	37.5	47.2	66.7	74	86.2	105.7

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