Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-09T01:30:57.527Z Has data issue: false hasContentIssue false
  • English
  • Français

LOG-CONCAVITY OF COMPOUND DISTRIBUTIONS WITH APPLICATIONS IN OPERATIONAL AND ACTUARIAL MODELS

Part of: Distribution theory Distribution theory - Probability

Published online by Cambridge University Press:  23 August 2019

F. G. Badía ,
C. Sangüesa Open the ORCID record for C. Sangüesa [Opens in a new window]  and
A. Federgruen
F. G. Badía
Affiliation:
Department of Statistical Methods and IUMA, University of Zaragoza, Zaragoza50018, Spain E-mail: [email protected]; [email protected]
C. Sangüesa
Affiliation:
Department of Statistical Methods and IUMA, University of Zaragoza, Zaragoza50018, Spain E-mail: [email protected]; [email protected]
A. Federgruen
Affiliation:
Graduate School of Business, Columbia University, New York10027, United States E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We establish that a random sum of independent and identically distributed (i.i.d.) random quantities has a log-concave cumulative distribution function (cdf) if (i) the random number of terms in the sum has a log-concave probability mass function (pmf) and (ii) the distribution of the i.i.d. terms has a non-increasing density function (when continuous) or a non-increasing pmf (when discrete). We illustrate the usefulness of this result using a standard actuarial risk model and a replacement model.

We apply this fundamental result to establish that a compound renewal process observed during a random time interval has a log-concave cdf if the observation time interval and the inter-renewal time distribution have log-concave densities, while the compounding distribution has a decreasing density or pmf. We use this second result to establish the optimality of a so-called (s,S) policy for various inventory models with a stock-out cost coefficient of dimension [$/unit], significantly generalizing the conditions for the demand and leadtime processes, in conjunction with the cost structure in these models. We also identify the implications of our results for various algorithmic approaches to compute optimal policy parameters.

Keywords

actuarial modelcompound distributioninventory modellog-concavityreplacement model

MSC classification

Primary: 62E10: Characterization and structure theory
Secondary: 60E15: Inequalities; stochastic orderings
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 35 , Issue 2 , April 2021 , pp. 210 - 235
Copyright
Copyright © Cambridge University Press 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Albrecher, H., Cheung, E.C.K., & Thonhauser, S. (2011). Randomized observation periods for the compound Poisson risk model dividends. Astin Bulletin 41: 645–672.Google Scholar
2.Albrecher, H., Cheung, E.C.K., & Thonhauser, S. (2013). Randomized observation periods for the compound Poisson risk model: the discounted penalty function. Scandinavial Actuarial Journal 6: 424–452.CrossRefGoogle Scholar
3.An, M.Y. (1998). Log-concavity versus log-convexity: A complete characterization. Journal of Economic Theory 80: 350–396.CrossRefGoogle Scholar
4.Archibald, B. & Silver, E. (1978). (s,S) policies under continuous review and discrete compound Poisson demands. Management Science 24: 899–908.CrossRefGoogle Scholar
5.Arrow, K., Karlin, S., & Scarf, H. (1958). Studies in the mathematical theory of inventory and production. California: Stanford University Press.Google Scholar
6.Asmussen, S. & Albrecher, H. (2010). Ruin probabilities. Hackensack, NJ: World Scientific Publishing.CrossRefGoogle Scholar
7.Badía, F.G. & Sangüesa, C. (2015). Inventory models with nonlinear shortage costs and stochastic lead times; Applications of shape properties of randomly stopped counting processes. Naval Research Logistics 62: 345–356.CrossRefGoogle Scholar
8.Bagnoli, M. & Bergstrom, T. (2005). Log-concave probability and its applications. Economic Theory 26: 445–469.CrossRefGoogle Scholar
9.Beckmann, M. (1961). An inventory model for arbitrary interval and quantity distribution of demand. Management Science 8: 35–57.CrossRefGoogle Scholar
10.Bertsimas, D. & Paschalidis, I.Ch. (2001). Probabilistic service level guarantees in a Make-to-Stock manufacturing system. Operations Research 49: 119–133.CrossRefGoogle Scholar
11.Cai, J. & Willmot, G.E. (2005). Monotonicity and aging properties of random sums. Statistics and Probability Letters 73: 381–392.CrossRefGoogle Scholar
12.Chain Store Age (2002). Inventory mangement 2002: data, detail and discipline: Supermarkets. December I.P. 17A.Google Scholar
13.Chen, F. (2000). Optimal policies for multi-echelon inventory problems with batch ordering. Operations Research 48: 376–389.CrossRefGoogle Scholar
14.Chen, F. & Zheng, Y.S. (1993). Inventory models with general backorder costs. European Journal of Operations Research 65: 175–186.CrossRefGoogle Scholar
15.Chen, F.Y. & Krass, Y.D. (2001). Inventory models with minimal service level constraints. European Journal of Operations Research 134: 120–140.CrossRefGoogle Scholar
16.Chen, X. & Simchi-Levi, D. (2006). Coordinating inventory control and pricing strategies: the continuous review model. Operations Research Letters 34: 323–332.CrossRefGoogle Scholar
17.Choi, S., Ruszczyński, A., & Zhao, Y. (2011). A multiproduct risk-averse newswendor with law-invariant coherent measures of risk. Operations Research 59: 346–364.CrossRefGoogle Scholar
18.Cohen, M.A., Deshpande, V., & Wang, Y. (2003). Managing supply chains with differentiated service requirements in Practise of Supply Chain Management: Where Theory and Applications Converge, Harrison, T., Lee, H., & Neale, J. (eds.), Chapter 11. Boston: Kluwer, pp. 175–195.Google Scholar
19.Esary, J.D., Marshall, A.W. & Proschan, F. (1973). Shock models and wear processes. Annals of Probabability 1: 627–649.Google Scholar
20.Federgruen, A., Liu, Z., & Li, L. Synthesis and Generalization of Structural Results in Inventory Management: A Generalized Convexity Property to appear in Mathematics of Operations Research. https://doi.org/10.1287/moor.2019.1001CrossRefGoogle Scholar
21.Federgruen, A. & Schechner, Z. (1983). Cost formulas for continuous review inventory models with fixed delivery lags. Operations Research 31: 957–965.CrossRefGoogle Scholar
22.Federgruen, A. & Wang, M. (2015). Inventory models with shelf age and delay-dependent inventory costs. Operations Research 63: 155–186.CrossRefGoogle Scholar
23.Federgruen, A. & Zheng, Y. (1992). An efficient algorithm for computing an optimal (r,Q) policy. Operations Research 40: 808–813.CrossRefGoogle Scholar
24.Feinberg, E.A. & Liang, Y. (2017a). Structure of optimal policies to periodic review inventory models with convex costs and backorders for all values of discount factors. Annals of Operations Research https://doi.org/10.1007/s10479-017-2548-6.CrossRefGoogle Scholar
25.Feinberg, E.A. & Liang, Y. (2017b). On the optimality equation for average cost Markov decision processes and its validity for inventory control. Annals of Operations Research, https://doi.org/10.1007/s10479-017-2561-9.CrossRefGoogle Scholar
26.Gallego, G. (1998). New bounds and heuristics for (Q, r) policies. Management Science 63: 701–715.Google Scholar
27.Gallego, G. (2001). Minimax analysis for finite-horizon inventory models. IIE Transactions 33: 861–874.CrossRefGoogle Scholar
28.Gallego, G. & Moon, I. (1993). The distribution free newsboy problem: review and extensions. Journal of the Operations Research Society 44: 825–834.CrossRefGoogle Scholar
29.Gallego, G. & Moon, I. (1994). Distribution free procedures for some inventory models. Journal of the Operations Research Society 45: 651–658.Google Scholar
30.Hadley, G. & Whitin, T. (1963). Analysis of inventory systems. Englewood Cliffs: Prentice-Hall.Google Scholar
31.Huh, W.T., Jankiraman, G., Muharremoglu, A., & Anshul, S. (2011). Inventory systems with a generalized cost model. Operations Research 4: 1040–1047.CrossRefGoogle Scholar
32.Klincewicz, J.G. & Whitt, W. (1984). On approximations for queues, II: shape constraints. AT & T Bell Laboratories Technical Journal 63: 139–161.CrossRefGoogle Scholar
33.Lee, H.L., Billington, C., & Carter, B. (1993). Hewlett-Packard gains control of inventory for service through design for localization. Interfaces 23: 1–11.CrossRefGoogle Scholar
34.Levi, R., Perakis, G., & Uichanco, J. (2015). The data-driven newsvendor problem: New bounds and insights. Operations Research 63: 1294–1306.CrossRefGoogle Scholar
35.Levi, R., Roundy, R., & Shmoys, D.B. (2015). Provably near-optimal sampling based policies for stochastic inventory control models. Mathematics of Operations Research 32: 821–839.CrossRefGoogle Scholar
36.Muckstadt, J.A. (2000). Analysis and algorithms for service part supply chains. New York: Springer.Google Scholar
37.Ninh, A. & Prekopa, A. (2013). Log-concavity of compound distributions with applications in stochastic optimization. Discrete Applied Mathematics 161: 3017–3027.CrossRefGoogle Scholar
38.Özer, O. & Xiong, H. (2008). Stock position and performance etimation for distribution systems with service constraints. IIE Transactions 40: 1141–1157.CrossRefGoogle Scholar
39.Perera, S., Janakiraman, G., & Shun-Chen, N. (2018). Optimality of (s,S) Inventory Policies under Renewal Demand and General Cost Structure. Production and Operations Management 27: 368–383.CrossRefGoogle Scholar
40.Peterson, R.E. & Silver, E.A. (2007). Decision systems for inventory control and production management. New York: Wiley.Google Scholar
41.Porteus, E.L. (1971). On the optimality of generalized (s,S) policies. Management Science 17: 411–426.CrossRefGoogle Scholar
42.Porteus, E.L. (2002). Foundations of stochastic inventory theory. Stanford, CA: Stanford University Press.Google Scholar
43.Rosling, K. (2002). Inventory cost rate functions with nonlinear shortage costs. Operations Research 50: 797–809. (Appendix with the proofs available at http://www.informs.org/content/download/16442/184983/file/Rosling.pdf).CrossRefGoogle Scholar
44.Ross, S.M. (1983). Stochastic processes. New York: Wiley.Google Scholar
45.Ross, S.M., Shanthikumar, J.G., & Zhu, Z. (2005). On increasing-failure-rate random variables. Journal of Applied Probability 42: 797–809.CrossRefGoogle Scholar
46.Sahin, I. (1979). On the stationary analysis of continuous review (s,S) inventory systems with constant lead times. Operations Research 27: 717–729.CrossRefGoogle Scholar
47.Sahin, I. (1983). On the continuous-review (s,S) inventory model under compound renewal demand and random lead times. Journal of Applied Probability 1: 213–219.CrossRefGoogle Scholar
48.Samworth, R.J. (2018). Recent Progress in Log-Concave Density Estimation. Statistical Science 4: 493–509.Google Scholar
49.Scarf, H. (1958). A min-max solution of an inventory problem. In Arrow, K.A., (ed.), Mathematical methods in the social sciences. Standford: Stanford University Press, pp. 201–209.Google Scholar
50.Scarf, H. (1960). The optimality of (s,S) policies in the dynamic inventory problem. In Arrow, K. A., Karlin, S. & Suppes, P. (eds.), Studies in the mathematical theory of inventory and production. Standford: Stanford University Press.Google Scholar
51.See, C.T. & Sim, M. (2010). Robust approximation to multiperiod inventory management. Operations Research 58: 583–594.CrossRefGoogle Scholar
52.Sengupta, D. & Nanda, A.K. (1999). Log-concave and concave distributions in reliability. Naval Research and Logistics 46: 419–433.3.0.CO;2-B>CrossRefGoogle Scholar
53.Shaked, M. & Shanthikumar, J.G. (2007). Stochastic orders. New York: Springer.CrossRefGoogle Scholar
54.Shanthikumar, J.G. (1988). DFR property of first-passage times and its preservation under geometric compounding. Annals of Probability 16: 397–406.CrossRefGoogle Scholar
55.Shaoxiang, C. (2004). The infinite horizon periodic review problem with setup costs and capacity constraints: a partial characterization of the optimal policy. Operations Research 52: 409–421.CrossRefGoogle Scholar
56.Shaoxiang, C. & Lambrecht, M. (1996). X−y band and modified (s,S) policy. Operations Research 44: 1013–1019.CrossRefGoogle Scholar
57.Shapiro, A.D., Dentcheva, D., & Ruszczyński, A. (2009). Lectures on Stochastic Programming: Modeling and Theory. Philadelphia: SIAM.CrossRefGoogle Scholar
58.Shi, J., Katehakis, M., & Melamed, B. (2013). Martingale methods for pricing inventory penalties under continuous replenishment and compound renewal demands. Annals of Operations Research 208: 593–612.CrossRefGoogle Scholar
59.Shi, J., Katehakis, M., Melamed, B., & Xia, Y. (2014). Production-inventory systems with lost sales and compound Poisson demands. Operations Research 62: 1048–1063.CrossRefGoogle Scholar
60.Taylor, H.M. (1975). Optimal replacement under additive damage and other failure models. Naval Research and Logistics 22: 1–18.CrossRefGoogle Scholar
61.Tijms, H. (1972). Analysis of (s,S) inventory models. Amsterdam: Mathematical Centre Tracts, Mathematisch Centrum.Google Scholar
62.Tijms, H. (1986). Stochastic Modelling and Analysis: A Computational Approach. New York: Wiley.Google Scholar
63.Veinott, A. Jr (1965). The optimal inventory policy for batch ordering. Operations Research 13: 424–432.CrossRefGoogle Scholar
64.Veinott, A. Jr (1966a). On the optimality of (s,S) inventory policies: new conditions and a new proof. SIAM Journal of Applied Mathematics 14: 1067–1083.CrossRefGoogle Scholar
65.Veinott, A. Jr (1966b). The statuts of mathematical inventory theory. Management Science 12: 745–777.CrossRefGoogle Scholar
66.Vigil, R. (2017). Suppliers, Don't Get Burned by Tight Delivery Windows & Steep Penalties at Big-Box Retailers (Available at https://www.loaddelivered.com/blog/suppliers-dont-get-burned-by-tight-delivery-windows-steep-penalties-at-big-box-retailers).Google Scholar
67.Wang, Y. & Yeh, Y.N. (2006). Log-concavity and LC-positivity. Journal of Combinatorial Theory. Series A 114: 195–210.CrossRefGoogle Scholar
68.Wei, L., Jasin, S., & Xin, L. (2017). Asymptotic optimality of Order-Up-to-Control for stochastic inventory systems with sequential probabilistic service level constraints University of Michigan, working paper.CrossRefGoogle Scholar
69.Yu, Y. (2009). On the entropy of compound distributions on non-negative integers. IEEE Transactions on Information Theory 55: 3645–3650.CrossRefGoogle Scholar
70.Yu, Y. (2011). Concave renewal functions do not imply DFR interrenewal times. Journal of Applied Probability 2: 583–588.CrossRefGoogle Scholar
71.Zheng, Y. (1991). A simple proof of the optimality of (s,S) policies in infinite-horizon inventory systems. Journal of Applied Probability 28: 802–806.CrossRefGoogle Scholar
72.Zheng, Y. & Federgruen, A. (1991). Finding optimal (s,S) policies is about as simple as evaluating a single policy. Operations Research 39: 654–665.CrossRefGoogle Scholar
73.Zipkin, P. (1980). Simple ranking methods for allocation of one resource. Management Science 26: 34–43.CrossRefGoogle Scholar
74.Zipkin, P. (1986). Stochastic leadtimes in continuous-time inventory models. Naval Research and Logistics 33: 763–774.CrossRefGoogle Scholar
75.Zuckerman, D. (1977). Replacement models under additive damage. Naval Research and Logistics 24: 549–558.CrossRefGoogle Scholar