# American Institute of Mathematical Sciences

April  2017, 13(2): 931-945. doi: 10.3934/jimo.2016054

## New structural properties of inventory models with Polya frequency distributed demand and fixed setup cost

 1 School of Business, East China University of Science and Technology, Shanghai 200237, China 2 The Johns Hopkins Carey Business School, Baltimore, MD 21202, USA

* Corresponding author: Arnab Bisi

Received  December 2014 Revised  June 2016 Published  August 2016

Fund Project: The first author is supported in part by the humanities and social sciences foundation of Chinese Ministry of Education under grant 12YJA630162.

We study a stochastic inventory model with a fixed setup cost and zero order lead time. In a finite-horizon lost sales model, when demand has a Polya frequency distribution (P Fn), we show that there are no more than a pre-determined number of minima of the cost function. Consequently, depending on the relative cost of lost sales and inventory holding cost, there can be as few as one local minimum. These properties have structural implications for the optimal policies and cost functions. A necessary condition for the results to hold for the backordered model has been explained. We further conduct a numerical study to validate our structural results.

Citation: Yanyi Xu, Arnab Bisi, Maqbool Dada. New structural properties of inventory models with Polya frequency distributed demand and fixed setup cost. Journal of Industrial and Management Optimization, 2017, 13 (2) : 931-945. doi: 10.3934/jimo.2016054
##### References:
 [1] S. Ahiska, S. Appaji, R. King and D. Warsing, Markov decision process-based policy characterization approach for a stochastic inventory control problem with unreliable sourcing, International Journal of Production Economics, 114 (2013), 485-496.  doi: 10.1016/j.ijpe.2013.03.021. [2] M. Bijvank, S. Bhulai and T. Huh, Parametric replenishment policies for inventory systems with lost sales and fixed order cost, European Journal of Operational Research, 241 (2015), 381-390.  doi: 10.1016/j.ejor.2014.09.018. [3] S. Bollapragada and T. Morton, Myopic Heuristics for the Random Yield Problem, Operations Research, 47 (1999), 713-722.  doi: 10.1287/opre.47.5.713. [4] X. Chao and P. Zipkin, Optimal policy for a periodic-review inventory system under a supply capacity contract, Operations Research, 56 (2008), 59-68.  doi: 10.1287/opre.1070.0478. [5] S. Chen and J. Xu, Note on the optimality of (s, S) policies for inventory systems with two demand classes, Operations Research Letters, 38 (2010), 450-453.  doi: 10.1016/j.orl.2010.07.005. [6] L. Chen L. Robinson, L. Chen, R. Roundy and R. Zhang, Technical note -New sufficient conditions for (s, S) policies to be optimal in systems with multiple uncertainties, Operations Research, 63 (2015), 186-197.  doi: 10.1287/opre.2014.1335. [7] F. M. Cheng and S. P. Sethi, Optimality of state-dependent (s, S) policies in inventory models with Markov-modulated demand and lost sales, Production and Operations Management, 8 (1999), 183-192. [8] R. Ehrhardt, (s, S) policies for a dynamic inventory model with stochastic lead times, Operations Research, 32 (1984), 121-132.  doi: 10.1287/opre.32.1.121. [9] R. Ehrhardt, Easily computed approximations for (s, S) inventory system operating characteristics, Naval Research Logistics Quarterly, 32 (1985), 347-359.  doi: 10.1002/nav.3800320214. [10] A. Federgruen and P. Zipkin, An efficient algorithm for computing optimal (s, S) policies, Operations Research, 34 (1984), 1268-1285.  doi: 10.1287/opre.32.6.1268. [11] Y. Feng and B. Xiao, A new algorithm for computing optimal (s, S) policies in a stochastic single item/ location inventory system, IIE Transactions, 32 (2000), 1081-1090.  doi: 10.1080/07408170008967463. [12] J. Freeland and E. Porteus, Evaluating the effectiveness of a new method for computing approximately optimal (s, S) inventory policies, Operations Research, 28 (1980), 353-364. [13] E. Huggins and T. Olsen, Inventory control with generalized expediting, Operations Research, 58 (2010), 1414-1426.  doi: 10.1287/opre.1100.0820. [14] D. Iglehart, Optimality of (s, S) policies in the infinite horizon dynamic inventory problems, Management Science, 9 (1963), 259-267.  doi: 10.1287/mnsc.9.2.259. [15] Q. Li and P. Yu, Technical Note -On the quasiconcavity of lost-sales inventory models with fixed costs, Operations Research, 60 (2012), 286-291.  doi: 10.1287/opre.1110.1034. [16] E. Porteus, On the optimality of generalized (s, S) policies, Management Science, 17 (1971), 411-426.  doi: 10.1287/mnsc.17.7.411. [17] E. Porteus, Foundations of Stochastic Inventory Theory, Stanford University Press, Stanford, CA, 2002. [18] H. Scarf, The optimality of (S, s) policies in dynamic inventory problems, Stanford University Press, Stanford, CA, 2002. [19] I. Schoenberg, On Polya frequency functions Ⅰ. The totally positive functions and their Laplace transforms, Journal d'Analyse Mathematique, 1 (1951), 331-374. [20] S. E. Shreve, Abbreviated proof (in the lost sales case) in D. P. Bertsekas, Dynamic Programming and Stochastic Control, Academic Press, New York, 1976. [21] B. Sivazlian, Dimensional and computational analysis in (s, S) inventory problems with gamma distributed demand, Management Science, 17 (1971), B307-B311.  doi: 10.1287/mnsc.17.6.B307. [22] M. Sobel and R. Zhang, Inventory policies for systems with stochastic and deterministic demand, Operations Research, 49 (2001), 157-162.  doi: 10.1287/opre.49.1.157.11197. [23] J. Tijms and H. Groenevelt, Approximations for (s, S) inventory systems with stochastic leadtimes and service level constraint, European Journal of Operational Research, 17 (1984), 175-190.  doi: 10.1016/0377-2217(84)90232-7. [24] A. Veinott Jr., On the optimality of (s, S) inventory policies: New conditions and a new proof, Journal on Applied Mathematics, 14 (1966), 1067-1083.  doi: 10.1137/0114086. [25] A. Veinott Jr. and H. Wagner, Computing optimal (s, S) inventory policies, Management Science, 11 (1965), 525-552. [26] Y. Xu, New bounds of (s, S) policies in periodical review inventory systems, Journal of Shanghai University (English Edition), 14 (2010), 111-115.  doi: 10.1007/s11741-010-0207-2. [27] Y. Xu, A. Bisi and M. Dada, New structural properties of (s, S) policies for inventory models with lost sales, Operations Research Letters, 38 (2010), 441-449.  doi: 10.1016/j.orl.2010.06.003. [28] Y. Zheng and A. Federgruen, Finding optimal (s, S) policies is about as simple as evaluating a single policy, Operations Research, 39 (1991), 654-665.

show all references

##### References:
 [1] S. Ahiska, S. Appaji, R. King and D. Warsing, Markov decision process-based policy characterization approach for a stochastic inventory control problem with unreliable sourcing, International Journal of Production Economics, 114 (2013), 485-496.  doi: 10.1016/j.ijpe.2013.03.021. [2] M. Bijvank, S. Bhulai and T. Huh, Parametric replenishment policies for inventory systems with lost sales and fixed order cost, European Journal of Operational Research, 241 (2015), 381-390.  doi: 10.1016/j.ejor.2014.09.018. [3] S. Bollapragada and T. Morton, Myopic Heuristics for the Random Yield Problem, Operations Research, 47 (1999), 713-722.  doi: 10.1287/opre.47.5.713. [4] X. Chao and P. Zipkin, Optimal policy for a periodic-review inventory system under a supply capacity contract, Operations Research, 56 (2008), 59-68.  doi: 10.1287/opre.1070.0478. [5] S. Chen and J. Xu, Note on the optimality of (s, S) policies for inventory systems with two demand classes, Operations Research Letters, 38 (2010), 450-453.  doi: 10.1016/j.orl.2010.07.005. [6] L. Chen L. Robinson, L. Chen, R. Roundy and R. Zhang, Technical note -New sufficient conditions for (s, S) policies to be optimal in systems with multiple uncertainties, Operations Research, 63 (2015), 186-197.  doi: 10.1287/opre.2014.1335. [7] F. M. Cheng and S. P. Sethi, Optimality of state-dependent (s, S) policies in inventory models with Markov-modulated demand and lost sales, Production and Operations Management, 8 (1999), 183-192. [8] R. Ehrhardt, (s, S) policies for a dynamic inventory model with stochastic lead times, Operations Research, 32 (1984), 121-132.  doi: 10.1287/opre.32.1.121. [9] R. Ehrhardt, Easily computed approximations for (s, S) inventory system operating characteristics, Naval Research Logistics Quarterly, 32 (1985), 347-359.  doi: 10.1002/nav.3800320214. [10] A. Federgruen and P. Zipkin, An efficient algorithm for computing optimal (s, S) policies, Operations Research, 34 (1984), 1268-1285.  doi: 10.1287/opre.32.6.1268. [11] Y. Feng and B. Xiao, A new algorithm for computing optimal (s, S) policies in a stochastic single item/ location inventory system, IIE Transactions, 32 (2000), 1081-1090.  doi: 10.1080/07408170008967463. [12] J. Freeland and E. Porteus, Evaluating the effectiveness of a new method for computing approximately optimal (s, S) inventory policies, Operations Research, 28 (1980), 353-364. [13] E. Huggins and T. Olsen, Inventory control with generalized expediting, Operations Research, 58 (2010), 1414-1426.  doi: 10.1287/opre.1100.0820. [14] D. Iglehart, Optimality of (s, S) policies in the infinite horizon dynamic inventory problems, Management Science, 9 (1963), 259-267.  doi: 10.1287/mnsc.9.2.259. [15] Q. Li and P. Yu, Technical Note -On the quasiconcavity of lost-sales inventory models with fixed costs, Operations Research, 60 (2012), 286-291.  doi: 10.1287/opre.1110.1034. [16] E. Porteus, On the optimality of generalized (s, S) policies, Management Science, 17 (1971), 411-426.  doi: 10.1287/mnsc.17.7.411. [17] E. Porteus, Foundations of Stochastic Inventory Theory, Stanford University Press, Stanford, CA, 2002. [18] H. Scarf, The optimality of (S, s) policies in dynamic inventory problems, Stanford University Press, Stanford, CA, 2002. [19] I. Schoenberg, On Polya frequency functions Ⅰ. The totally positive functions and their Laplace transforms, Journal d'Analyse Mathematique, 1 (1951), 331-374. [20] S. E. Shreve, Abbreviated proof (in the lost sales case) in D. P. Bertsekas, Dynamic Programming and Stochastic Control, Academic Press, New York, 1976. [21] B. Sivazlian, Dimensional and computational analysis in (s, S) inventory problems with gamma distributed demand, Management Science, 17 (1971), B307-B311.  doi: 10.1287/mnsc.17.6.B307. [22] M. Sobel and R. Zhang, Inventory policies for systems with stochastic and deterministic demand, Operations Research, 49 (2001), 157-162.  doi: 10.1287/opre.49.1.157.11197. [23] J. Tijms and H. Groenevelt, Approximations for (s, S) inventory systems with stochastic leadtimes and service level constraint, European Journal of Operational Research, 17 (1984), 175-190.  doi: 10.1016/0377-2217(84)90232-7. [24] A. Veinott Jr., On the optimality of (s, S) inventory policies: New conditions and a new proof, Journal on Applied Mathematics, 14 (1966), 1067-1083.  doi: 10.1137/0114086. [25] A. Veinott Jr. and H. Wagner, Computing optimal (s, S) inventory policies, Management Science, 11 (1965), 525-552. [26] Y. Xu, New bounds of (s, S) policies in periodical review inventory systems, Journal of Shanghai University (English Edition), 14 (2010), 111-115.  doi: 10.1007/s11741-010-0207-2. [27] Y. Xu, A. Bisi and M. Dada, New structural properties of (s, S) policies for inventory models with lost sales, Operations Research Letters, 38 (2010), 441-449.  doi: 10.1016/j.orl.2010.06.003. [28] Y. Zheng and A. Federgruen, Finding optimal (s, S) policies is about as simple as evaluating a single policy, Operations Research, 39 (1991), 654-665.
 Cost and Model Parameters $K$ = fixed setup cost $c$ = unit variable ordering cost $h$ = unit inventory holding cost $l$ = unit lost sales cost ($l > c$) $b$ = unit backorder cost $\alpha$ = discount factor ($0<\alpha\le 1$) $T$ = time horizon
 Cost and Model Parameters $K$ = fixed setup cost $c$ = unit variable ordering cost $h$ = unit inventory holding cost $l$ = unit lost sales cost ($l > c$) $b$ = unit backorder cost $\alpha$ = discount factor ($0<\alpha\le 1$) $T$ = time horizon
 Demand Information $\xi_t$ = the random observation of demand in period $t$, $t = 1, 2,\dots, T$ $f(\cdot)$= the probability density function (PDF) of demand in each period $F(\cdot)$= the cumulative distribution function (CDF) of demand in each period
 Demand Information $\xi_t$ = the random observation of demand in period $t$, $t = 1, 2,\dots, T$ $f(\cdot)$= the probability density function (PDF) of demand in each period $F(\cdot)$= the cumulative distribution function (CDF) of demand in each period
 Decision Variables $s_t$ = optimal reorder level in period $t$ $S_t$ = optimal order-up-to level in period $t$
 Decision Variables $s_t$ = optimal reorder level in period $t$ $S_t$ = optimal order-up-to level in period $t$
 Cost Functions $L(\cdot)$ = one period inventory holding and shortage penalty cost function $V_t(x)$ = total minimal expected cost from period $t$ onwards ($t-1,\dots, 2, 1$), given that the on-hand inventory at the beginning of period $t$ is $x$ $G_t(y)$ = total expected cost from period t onwards after inventory level is increased to $y$
 Cost Functions $L(\cdot)$ = one period inventory holding and shortage penalty cost function $V_t(x)$ = total minimal expected cost from period $t$ onwards ($t-1,\dots, 2, 1$), given that the on-hand inventory at the beginning of period $t$ is $x$ $G_t(y)$ = total expected cost from period t onwards after inventory level is increased to $y$
 Other Useful Functions $\delta(z) = \left\{ \begin{array}{lc} 1&\textrm{if } z > 0 \\ 0& \textrm{if } z = 0 \end{array} \right.$, the indicator function for ordering decisions $x^+$ = $\max\{ x, 0 \}$
 Other Useful Functions $\delta(z) = \left\{ \begin{array}{lc} 1&\textrm{if } z > 0 \\ 0& \textrm{if } z = 0 \end{array} \right.$, the indicator function for ordering decisions $x^+$ = $\max\{ x, 0 \}$
Optimal Solutions for the Case with Unit Lost Sales Cost l = 2
 $K$ $t$ Optimal Reoder Point($s_t$) Optimal Order-up-to Level($s_t$) $\Delta_t=S_t-s_t$ Optimal Cost $G_t(S_t)$ 0.1 1 1.401973349 2.045088007 0.643114721 2.892765731 2 2.279867486 3.144352495 0.845991005 5.161645538 3 2.298361490 3.453500000 1.155138510 7.213139600 4 2.284635504 3.447230000 1.162594496 9.062221220 5 2.287417275 3.449250000 1.161832725 10.725835800 0.5 1 0.666145602 2.045088007 1.478942404 2.892765731 2 1.631133899 3.434334720 1.803200821 5.280907719 3 1.589211420 4.215151800 2.632884666 7.499670690 4 1.522671338 4.399540000 2.876868662 9.526311140 5 1.541934791 4.351150000 2.809215209 11.341494260 1 1 0.107558637 2.045088007 1.937529370 2.892765731 2 1.245066678 3.623702424 2.378635746 5.361056167 3 1.242995200 4.699700000 3.456704800 7.499670690 4 1.118285145 5.263110000 4.144824855 9.873385400 5 1.109158173 5.243850000 4.134691827 11.838495150
 $K$ $t$ Optimal Reoder Point($s_t$) Optimal Order-up-to Level($s_t$) $\Delta_t=S_t-s_t$ Optimal Cost $G_t(S_t)$ 0.1 1 1.401973349 2.045088007 0.643114721 2.892765731 2 2.279867486 3.144352495 0.845991005 5.161645538 3 2.298361490 3.453500000 1.155138510 7.213139600 4 2.284635504 3.447230000 1.162594496 9.062221220 5 2.287417275 3.449250000 1.161832725 10.725835800 0.5 1 0.666145602 2.045088007 1.478942404 2.892765731 2 1.631133899 3.434334720 1.803200821 5.280907719 3 1.589211420 4.215151800 2.632884666 7.499670690 4 1.522671338 4.399540000 2.876868662 9.526311140 5 1.541934791 4.351150000 2.809215209 11.341494260 1 1 0.107558637 2.045088007 1.937529370 2.892765731 2 1.245066678 3.623702424 2.378635746 5.361056167 3 1.242995200 4.699700000 3.456704800 7.499670690 4 1.118285145 5.263110000 4.144824855 9.873385400 5 1.109158173 5.243850000 4.134691827 11.838495150
Optimal Solutions for the Case with Unit Lost Sales Cost l = 10
 $K$ $t$ Optimal Reoder Point($s_t$) Optimal Order-up-to Level($s_t$) $\Delta_t=S_t-s_t$ Optimal Cost $G_t(S_t)$ 0.1 1 3.810323346 4.380527192 0.570203846 4.318148680 2 4.797805509 5.568997000 0.771191491 6.934754800 3 4.864161753 5.928412500 1.064250747 9.291513170 4 4.844958018 5.924312000 1.079353982 11.416826560 5 4.848185556 5.926906000 1.078720444 13.328890410 5 1 1.477580090 4.380527192 2.902947102 4.318148680 2 2.835586519 6.555841000 3.720254481 7.428294760 3 3.147720304 8.310725000 5.163004696 10.437563500 4 3.019171696 9.780020000 6.760848304 13.352972400 5 2.912465707 10.980260000 8.067794293 16.156118600 10 1 0.670234159 4.380527192 3.710293006 4.318148680 2 2.211120742 6.755695000 4.544571258 7.538793306 3 2.723958450 8.738920000 6.014961550 10.701657500 4 2.637428687 10.476020000 7.838591313 13.811397880 5 2.530633770 12.015940000 9.485306230 16.854614980 15 1 0.075836309 4.380527192 4.304690883 4.318148680 2 1.775579517 6.851150000 5.075570483 7.592848600 3 2.417147303 8.949725000 6.532577697 10.836869060 4 2.382954891 10.818600000 8.435645109 14.050696010 5 2.282913956 12.513120000 10.230206040 17.220767220
 $K$ $t$ Optimal Reoder Point($s_t$) Optimal Order-up-to Level($s_t$) $\Delta_t=S_t-s_t$ Optimal Cost $G_t(S_t)$ 0.1 1 3.810323346 4.380527192 0.570203846 4.318148680 2 4.797805509 5.568997000 0.771191491 6.934754800 3 4.864161753 5.928412500 1.064250747 9.291513170 4 4.844958018 5.924312000 1.079353982 11.416826560 5 4.848185556 5.926906000 1.078720444 13.328890410 5 1 1.477580090 4.380527192 2.902947102 4.318148680 2 2.835586519 6.555841000 3.720254481 7.428294760 3 3.147720304 8.310725000 5.163004696 10.437563500 4 3.019171696 9.780020000 6.760848304 13.352972400 5 2.912465707 10.980260000 8.067794293 16.156118600 10 1 0.670234159 4.380527192 3.710293006 4.318148680 2 2.211120742 6.755695000 4.544571258 7.538793306 3 2.723958450 8.738920000 6.014961550 10.701657500 4 2.637428687 10.476020000 7.838591313 13.811397880 5 2.530633770 12.015940000 9.485306230 16.854614980 15 1 0.075836309 4.380527192 4.304690883 4.318148680 2 1.775579517 6.851150000 5.075570483 7.592848600 3 2.417147303 8.949725000 6.532577697 10.836869060 4 2.382954891 10.818600000 8.435645109 14.050696010 5 2.282913956 12.513120000 10.230206040 17.220767220
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