# 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 & Management Optimization, 2017, 13 (2) : 931-945. doi: 10.3934/jimo.2016054
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##### References:
 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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