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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

Yanyi Xu 1, , Arnab Bisi 2,, and  Maqbool Dada 2,

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.

Keywords: Inventory, (s, S) policies, P Fn distribution, lost sales, backorder.
Mathematics Subject Classification: Primary: 90B05, 91B70; Secondary: 97K60.
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. 

Table  . 
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
Table Options

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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
Table  . 
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
Table Options

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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
Table  . 
Decision Variables
$s_t$ = optimal reorder level in period $t$
$S_t$ = optimal order-up-to level in period $t$
Table Options

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Decision Variables
$s_t$ = optimal reorder level in period $t$
$S_t$ = optimal order-up-to level in period $t$
Table  . 
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 $y$
Table Options

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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 $y$
Table  . 
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
$x^+$ = $\max\{ x, 0 \}$
Table Options

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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
$x^+$ = $\max\{ x, 0 \}$
Table 1.  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 $G_t(S_t)$
0.111.4019733492.0450880070.6431147212.892765731
22.2798674863.1443524950.8459910055.161645538
32.2983614903.4535000001.1551385107.213139600
42.2846355043.4472300001.1625944969.062221220
52.2874172753.4492500001.16183272510.725835800
0.510.6661456022.0450880071.4789424042.892765731
21.6311338993.4343347201.8032008215.280907719
31.5892114204.2151518002.6328846667.499670690
41.5226713384.3995400002.8768686629.526311140
51.5419347914.3511500002.80921520911.341494260
110.1075586372.0450880071.9375293702.892765731
21.2450666783.6237024242.3786357465.361056167
31.2429952004.6997000003.4567048007.499670690
41.1182851455.2631100004.1448248559.873385400
51.1091581735.2438500004.13469182711.838495150
Table Options

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$K$ $t$ Optimal Reoder Point($s_t$) Optimal Order-up-to Level($s_t$) $\Delta_t=S_t-s_t$ Optimal Cost $G_t(S_t)$
0.111.4019733492.0450880070.6431147212.892765731
22.2798674863.1443524950.8459910055.161645538
32.2983614903.4535000001.1551385107.213139600
42.2846355043.4472300001.1625944969.062221220
52.2874172753.4492500001.16183272510.725835800
0.510.6661456022.0450880071.4789424042.892765731
21.6311338993.4343347201.8032008215.280907719
31.5892114204.2151518002.6328846667.499670690
41.5226713384.3995400002.8768686629.526311140
51.5419347914.3511500002.80921520911.341494260
110.1075586372.0450880071.9375293702.892765731
21.2450666783.6237024242.3786357465.361056167
31.2429952004.6997000003.4567048007.499670690
41.1182851455.2631100004.1448248559.873385400
51.1091581735.2438500004.13469182711.838495150
Table 2.  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 $G_t(S_t)$
0.113.8103233464.3805271920.5702038464.318148680
24.7978055095.5689970000.7711914916.934754800
34.8641617535.9284125001.0642507479.291513170
44.8449580185.9243120001.07935398211.416826560
54.8481855565.9269060001.07872044413.328890410
511.4775800904.3805271922.9029471024.318148680
22.8355865196.5558410003.7202544817.428294760
33.1477203048.3107250005.16300469610.437563500
43.0191716969.7800200006.76084830413.352972400
52.91246570710.9802600008.06779429316.156118600
1010.6702341594.3805271923.7102930064.318148680
22.2111207426.7556950004.5445712587.538793306
32.7239584508.7389200006.01496155010.701657500
42.63742868710.4760200007.83859131313.811397880
52.53063377012.0159400009.48530623016.854614980
1510.0758363094.3805271924.3046908834.318148680
21.7755795176.8511500005.0755704837.592848600
32.4171473038.9497250006.53257769710.836869060
42.38295489110.8186000008.43564510914.050696010
52.28291395612.51312000010.23020604017.220767220
Table Options

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$K$ $t$ Optimal Reoder Point($s_t$) Optimal Order-up-to Level($s_t$) $\Delta_t=S_t-s_t$ Optimal Cost $G_t(S_t)$
0.113.8103233464.3805271920.5702038464.318148680
24.7978055095.5689970000.7711914916.934754800
34.8641617535.9284125001.0642507479.291513170
44.8449580185.9243120001.07935398211.416826560
54.8481855565.9269060001.07872044413.328890410
511.4775800904.3805271922.9029471024.318148680
22.8355865196.5558410003.7202544817.428294760
33.1477203048.3107250005.16300469610.437563500
43.0191716969.7800200006.76084830413.352972400
52.91246570710.9802600008.06779429316.156118600
1010.6702341594.3805271923.7102930064.318148680
22.2111207426.7556950004.5445712587.538793306
32.7239584508.7389200006.01496155010.701657500
42.63742868710.4760200007.83859131313.811397880
52.53063377012.0159400009.48530623016.854614980
1510.0758363094.3805271924.3046908834.318148680
21.7755795176.8511500005.0755704837.592848600
32.4171473038.9497250006.53257769710.836869060
42.38295489110.8186000008.43564510914.050696010
52.28291395612.51312000010.23020604017.220767220
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