Computers and Mathematics with Applications. An EOQ model of homogeneous products while demand is salesmen s initiatives and stock sensitive

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Computers and Mathematics with Applications 6 (011) 577 587 Contents lists available at ScienceDirect Computers and Mathematics with Applications journal homepage: www.elsevier.

Bhangar Pin-74350 4PGS (South) West Bengal India a r t i c l e i n f o a b s t r a c t Article histor: Received 31 March 011 Received in revised form 16 Ma 011 Accepted 17 Ma 011 Kewords: Stock

1 Computers and Mathematics with Applications 6 (011) Contents lists available at ScienceDirect Computers and Mathematics with Applications journal homepage: An EOQ model of homogeneous products while demand is salesmen s initiatives and stock sensitive Shib Sankar Sana Department of Mathematics Bhangar Mahavidalaa Universit of Calcutta Bhangar Pin PGS (South) West Bengal India a r t i c l e i n f o a b s t r a c t Article histor: Received 31 March 011 Received in revised form 16 Ma 011 Accepted 17 Ma 011 Kewords: Stock sensitive Salesmen s initiatives Control theor Equilibrium Effort Stabilit The author develops an inventor model to determine the retailer s optimal order quantit for homogeneous products. It is assumed that the amount of displa space is limited and the demand of the products is dependent on the displa stock level where a huge stock of one product has a negative effect on the other product. Also the replenishment rate depends on the level of stock of the items. The objective of the model is to maimize the profit function considering the effect of inflation and time value of mone b Pontragin s Maimal Principles. The stabilit analsis of the concerned dnamical sstem has been carried out analticall as well as numericall. 011 Elsevier Ltd. All rights reserved. 1. Introduction Most researchers and practitioners have recognized salesmen s initiatives and the influence of displaed stock level on customers demand. According to Levin et al. 1 large piles of consumer goods displaed in a supermarket will lead the customer to bu more. Silver and Peterson also noted that sales at the retail level tend to be proportional to the amount of inventor displaed. Baker and Urban 3 assumed a power-form inventor-level-dependent demand rate that would decline along with the stock level throughout the entire ccle. Datta and Pal 4 revised the model of Baker and Urban 3 b assuming the demand rate that depleted to a given level of inventor beond which it was a constant. B their assumption not all customers are motivated to purchase goods b a huge stock. Some of the customers ma arrive to purchase goods because of its goodwill good qualit or facilities in spite of low stock level. Zhou et al. 5 formulated a decentralized twoechelon suppl chain where the demand of the product is dependent on the inventor level on displa. Soni and Shah 6 developed an optimal ordering polic for retailers in a trade credit environment where demand is partiall constant and partiall dependent on the stock. Goal and Chang 7 investigated an ordering transfer inventor model when the storage capacit is limited and the demand rate depends on the displa stock level. The obtained the retailer s optimal ordering quantit and the number of transfers per order from the warehouse to the displa area for maimizing the average profit per unit ield b the retailer. Hsieh et al. 8 developed Datta and Pal s 4 model allowing partial backlogging. Hsieh and De 9 provided some useful properties for finding the optimal replenishment schedule with stock-dependent demand under eponential partial backlogging. Min et al. 10 investigated an EOQ model for perishable items under permissible dela in paments where demand of products varies linearl with the level of stock. Sajadieh et al. 11 developed an integrated vendor buer model for a two-stage suppl chain assuming demand of the products as a positive power function of the displaed inventor. As regards the research articles related to the stock-dependent demand pattern mention should Fa: address: shib_sankar@ahoo.com /$ see front matter 011 Elsevier Ltd. All rights reserved. doi: /j.camwa

2 578 S.S. Sana / Computers and Mathematics with Applications 6 (011) be made of the works of Urban 1 Pal et al. 13 Goh 14 Padmanabhan and Vrat 15 Sarker et al. 16 Datta and Paul 17 Balkhi and Benkherouf 18 Chang 19 Hou and Lin 0 Min and Zhou 1 Ghosh et al. among others. In recent ears man tpes of business have been involved in various forms of promotional effort to boost market demand. Goal and Gunasekaran 3 investigated an integrated production inventor marketing model for determining the EPQ (economic production quantit) and EOQ (economic order quantit) for raw materials in a multi-stage production sstem. The incorporated the effect of different marketing policies such as the price per unit product and the advertisement frequenc on the demand of a perishable item. Man studies have focused on the effect of promotions on sales using store or market level data 4 6. Sun 7 made relation of the customers behavior with different tpes of promotions and identified that promotions had a strong impact on stronger brands. Based on the work of Divakar et al. 6 Ramanathan and Muldermans 8 applied structural equation modeling to investigate the impact of promotions and other factors on the sales of soft drinks. Sana and Chaudhuri 9 investigated an inventor model for stock with advertising sensitive demand. Sana 30 developed an interesting multi-item EOQ model for deteriorated and ameliorating items when the time varing demand is influenced b enterprises initiatives like advertising media and salesmen effort. The monetar situation in each countries has changed to such an etent due to large scale inflation and consequent sharp declines in the purchasing power of mone. So it has not been possible to ignore the effects of inflation and time value of mone. Buzacott 31 was the first in this direction who derived epressions for the optimal order quantit considering inflation and time value of mone. In this direction the works of Biermann and Thomas 3 Brahmbatt 33 Datta and Pal 34 Bose et al. 35 Chung et al. 36 Chen 37 Chung and Lin 38 are worth mentioning among others. In this paper a dnamical sstem of differential equation of two tpes of homogeneous products has been considered when the demand rate of each product depends on the salesmen initiatives and level of stock of the products. The replenishment rates of the products are also dependent on the on-hand inventor. The huge stock of one product decreases the demand of the another product. The stabilit analsis of the sstem has been done. Finall a profit function incorporating inventor cost selling price cost of effort has been optimized b Pontragin s Maimal Principles.. Notation The following notation are considered to develop the model: Notation: X(t) on-hand stock of item 1 at time t. Y(t) on-hand stock of item at time t. D demand rate of item 1. D demand rate of item. R replenishment rate of item 1. R replenishment rate of item. L maimum storage capacit of item 1. L maimum storage capacit of item. E(t) joint effort function of the salesmen s initiatives at time t. p purchasing cost per unit item 1. p purchasing cost per unit item. h inventor cost per unit item 1 per unit time. h inventor cost per unit item per unit time. γ cost per unit effort. s selling price per unit item 1. s selling price per unit item. δ = (r i) r and i are rate of interest and inflation per unit currenc respectivel. 3. Formulation of the model The author considers a stock-dependent inventor model of two similar products such that the demand of the products depends on the on-hand inventor (X(t) Y(t)) and effort (E(t)) b advertising or salesmen initiatives. As the demand of the products are dependent on the level of stock the replenishment rate varies with the stock level of the products. Here the demand rates of items 1 and are as follows: D = D = C EX l 1 E l X C EY l 3 E l 4 Y X 1 a a Y 1 a a X Y where C and C are the positive coefficients of the demand of items 1 and respectivel. L and L are the storage capacities of the items 1 and respectivel. It is noticed that (D D ) (C X/l 1 (1 a )X C a Y Y/l 3 (1 a a Y)) as E for X (1) ()

3 S.S. Sana / Computers and Mathematics with Applications 6 (011) fied values of (X Y). The parameters (l 1 l 3 ) are proportional to the ratios of the stock level to the demand rates at higher level of effort and the parameters (l l 4 ) are proportional to the ratios of the effort level to the demand rates at higher stock levels. The replenishment rates of items 1 and are dependent on the level of on-hand inventor (X(t) Y(t)) at time t those are given below: R = r (1 X/L )X R = r (1 Y/L )Y where (r r ) are positive replenishment coefficients respectivel. Then the governing differential equations are as follows: ẊẎ r (1 X/L )X C EX = l 1 E l X 1 a X a Y r (1 Y/L )Y C EY l 3 E l 4 Y 1 a = F (X Y). (5) G (X Y) Y a X For non-zero critical points ( ȳ) the following conditions are satisfied: r (1 /L ) = C E l 1 E l 1 a a ȳ r (1 ȳ/l ) = C E l 3 E l 4 ȳ 1 a a solving the Eqs. (6) and (7) we have got the critical points ( ȳ) of the dnamical sstem. Now differentiating F (X Y) and G (X Y) partiall with respect to X and Y we have X (F (X Y)) = r (1 X/L ) C E l 1 E l X 1 a X r C l E a Y L (l 1 E l X) Y (F a (X Y)) = X (a Y) X (G a (X Y)) = Y (a X) Y (G (X Y)) = r (1 Y/L ) C E l 3 E l 4 Y 1 a Y r C l 4 E. a X L (l 3 E l 4 Y) At the non-zero critical point ( ȳ) the above partial derivatives are as follows: X (F ( ȳ)) = r C l E L (l 1 E l ) Y (F a ( Ȳ)) = (a ȳ) X (G a ( ȳ)) = ȳ (a ) Y (G ( ȳ)) = ȳ r C l 4 E. L (l 3 E l 4 ȳ) (3) (4) (6) (7) 3.1. Boundedness of the sstem Lemma. All solutions of the sstem of Eq. (5) which start in R Proof. Let us consider the function are uniforml bounded. U(X Y) = X 1 l Y (8) where l is a positive constant. The time derivative of Eq. (8) is U = Ẋ 1 l Ẏ = X r 1 XL C E l 1 E l X 1 a 1l r a Y Y 1 YL C E l 3 E l 4 Y 1 a. a X

4 580 S.S. Sana / Computers and Mathematics with Applications 6 (011) For each s > 0 we obtain Therefore U su = Ẋ 1 l Ẏ = X r 1 XL C E l 1 E l X 1 a a Y 1 l Y r 1 Y C E L l 3 E l 4 Y 1 a s X 1l a X Y < X r C 1 s 1 l 1 l Y r C 1 s l 3 < L r C 1 s 1 l 1 l L r C 1 s. l 3 U su < K where K/ = MaL {r C l 1 1 s} 1 l L {r C l 3 1 s}. Appling the theor of differential inequalit we have 0 < U(X Y) < (K/s)(1 e st ) U(X(0) Y(0))e st. (9) When t the above ields 0 < U < K/s. Therefore all the solution of Eq. (5) that start in R where confined to the region R R = {(X Y)ϵR : U = (K/s) ϵ for an ϵ > 0}. Hence the proof. 3.. Local stabilit analsis We shall now investigate the local behavior of critical points of the dnamical sstem in Eq. (5). The variational matri of the sstem of Eq. (5) is V( ȳ) = X (F ( ȳ)) X (G ( ȳ)) Y (F ( ȳ)). (10) Y (G ( ȳ)) The characteristic equation of V( ȳ) is λ λψ 1 ( ȳ) ψ ( ȳ) = 0. Now ( ȳ) to be a stable node if both the eigenvalues of the above is negative i.e. ψ 1 ( ȳ) < 0 and ψ ( ȳ) > 0 (11) are satisfied where ψ 1 ( ȳ) = X (F ( ȳ)) Y (G ( ȳ)) = r C l E ȳ r C l 4 E L (l 1 E l ) L (l 3 E l 4 ȳ) and ψ ( ȳ) = X (F ( ȳ)) Y (G ( ȳ)) = r C l E ȳ L (l 1 E l ) Y (F ( ȳ)) r L C l 4 E (l 3 E l 4 ȳ) X (G ( ȳ)) ȳ a (a ) a (a ȳ).

5 S.S. Sana / Computers and Mathematics with Applications 6 (011) Global stabilit analsis We shall stud the global stabilit of the sstem of Eq. (5) b considering a suitable Lapunov function F(X Y) = (X ) ln(x/ ) h(y ȳ) ȳ ln(y/ȳ) where h is a suitable constant to be determined later. F( ȳ) is zero at the equilibrium point ( ȳ) and is positive for all other values of (X Y) ϵ R. The time derivative of F along the trajectories of Eq. (5) is X Y Ḟ = Ẋ h Ẏ X Y = (X ) r (1 X/L ) C E l 1 E l X 1 a a Y h(y ȳ) r (1 Y/L ) C E l 3 E l 4 Y 1 a a X X l (X ) = (X ) r C E L (l 1 E l X)(l 1 E l a (Y ȳ) ) (a Y)(a ȳ) Y ȳ l 4 (Y ȳ) h(y ȳ) r C E L (l 3 E l 4 Y)(l 3 E l 4 ȳ) a (X ) (a X)(a ) = (X ) r C El a (X )(Y ȳ) L (l 1 E l X)(l 1 E l ) (a Y)(a ȳ) ha (a X)(a ) (Y ȳ) h r C El 4 L (l 3 E l 4 Y)(l 3 E l 4 ȳ) = X Y ȳ T PX Y ȳ where a11 a P = 1 a 11 = a 1 a 1 = 1 a = r a C El L (l 1 E l X)(l 1 E l ) a h (a Y)(a ȳ) r L C El 4 (l 3 E l 4 Y)(l 3 E l 4 ȳ) a = a 1 (a X)(a ) h. The eigenvalues of the characteristic equation of the above matri are both negative if a 11 a < 0 and a 11 a (a 1 ) > 0 are satisfied. Therefore the interior equilibrium point ( ȳ) is globall asmptoticall stable if the above inequalities hold simultaneousl Optimal goal of the polic The net profit of the project including inflation and time value of mone is J = π(x Y E)e δt 0 where C EX π(x Y E) = (s p ) l 1 E l X 1 a X a Y h X h Y γ E π X = (s C l 1 E p ) (l 1 E l X) 1 a a Y π Y = (s C l 3 E p ) (l 3 E l 4 Y) 1 a a X (s p ) (s p ) (s p ) π E = (s C l X p ) (l 1 E l X) (s C l 4 Y p ) (l 3 E l 4 Y) γ. a Y (a X) a X (a Y) C EY l 3 E l 4 Y h h 1 a a X Y (1)

6 58 S.S. Sana / Computers and Mathematics with Applications 6 (011) Now our objective is to maimize J subject to the state of Eq. (5) using Pontragin s Maimum Principles. The control variable E(t) is subject to the constraint 0 E(t) E ma E ma is a feasible upper limit for the harvesting effort. The Hamiltonian of the problem is H = π(x Y E)e δt λ 1 (t)f (X Y E) λ (t)g (X Y E) (13) where λ 1 (t) and λ (t) are adjoint variables. The optimal control E(t) which maimizes H must satisf the following conditions: H E = 0 dλ 1 dλ = H X = H Y. Now H/E = 0 at ( ȳ Ē) gives us C l X (s p ) (l 1 E l X) (s C l 4 Y p ) (l 3 E l 4 Y) γ e δt C l λ 1 (t) (l 1 Ē λ C l 4 ȳ (t) = 0. (14) l ) (l 3 Ē l 4 ȳ) Now dλ 1 dλ 1 Similarl dλ = H at ( ȳ Ē) gives us X π = e δt F λ 1 X ( ȳē) X π = e δt F λ 1 Y ( ȳē) Y From Eq. (15) we have dλ1 π λ = e δt F λ 1 X X Substituting the above in Eq. (16) we have G1 G d λ 1 dλ 1 π = δ X = Q 1 e δt X G X Y π Y ( ȳē) ( ȳē) G G λ. (15) X ( ȳē) G λ. (16) Y ( ȳē) X F λ 1 X. (17) G Y π e δt Y X G F Y G X i.e. d λ 1 ψ dλ 1 1 ψ λ 1 = Q 1 e δt π G π G π Q 1 = δ X X Y Y X ( ȳē) r C l 4 Ē C l 1 Ē = δ ȳ (s p ) L (l 3 Ē l 4 ȳ) (l 1 Ē 1 a a ȳ (s p ) l ) a ȳ (a ) h a ȳ a (s (a ) p ) (a ȳ) C (s l 3 Ē p ) (l 3 Ē l 4 ȳ) 1 a h a (18) and (ψ 1 ψ ) are as before. The auiliar equation of Eq. (18) is µ ψ 1 µ ψ = 0. (19) The roots (µ 1 µ ) of Eq. (19) are positive b virtue of Eq. (11). Therefore the solution of λ 1 (t) is λ 1 (t) = Ae µ 1t Be µ t Q 1 δ ψ 1 δ ψ e δt.

S.S. Sana / Computers and Mathematics with Applications 6 (011) 577 587 583 Fig. 1. Phase portrait of item 1 () and item ().

7 S.S. Sana / Computers and Mathematics with Applications 6 (011) Fig. 1. Phase portrait of item 1 () and item () Fig.. Global attractor of eample 1. The shadow price λ 1 (t)e δt remains bounded as t if and onl if A = 0 = B and then λ 1 (t) = Similarl we have where λ (t) = Q = = Q 1 δ ψ 1 δ ψ e δt. Q δ ψ 1 δ ψ e δt δ π δ Y r L F π F π Y X X Y ( ȳē) C l Ē C l 3 Ē (s p ) (l 1 Ē l ) (l 3 Ē l ȳ) 1 a a a (s p ) (a ȳ) h (0) (1)

8 584 S.S. Sana / Computers and Mathematics with Applications 6 (011) a a ȳ (s (a ȳ) p ) (a ) C (s l 1 Ē p ) (l 1 Ē 1 a h. l ) a ȳ Substituting λ 1 (t) and λ (t) in Eq. (14) we have C l (s p ) (l 1 Ē l (s C l 4 ȳ ) p ) (l 3 Ē l 4 ȳ) γ Q 1 C l = δ ψ 1 δ ψ (l 1 Ē Q C l 4 ȳ l ) δ ψ 1 δ ψ (l 3 Ē l 4 ȳ). () Now solving Eqs. (6) (7) and () we have the optimal equilibrium solution ( ȳ Ē). Let us consider a numerical eample as follows: Fig. 3. On-hand inventor of item 1 versus time. Fig. 4. On-hand inventor of item versus time. Fig. 5. Replenishment rate of item 1 versus time.

9 S.S. Sana / Computers and Mathematics with Applications 6 (011) Fig. 6. Demand rate of item 1 versus time. Fig. 7. Replenishment rate of item versus time. Fig. 8. Demand rate of item versus time. Eample 1. The values of the parameters are considered in appropriate units as follows: s = $50 p = $5 s = $60 p = $30 h = $0.5 h = $0.5 r = 8 r = 8 C = $0.8 C = $0.6 L = 700 units L = 600 units γ = $100 a = 000 a = 500 l 1 = 0.3 l = 0. l 3 = 0. l 4 = 0.3 r = 16% i = 11% δ = Then the optimal solution is ( = ȳ = Ē = 160.6). This solution is a stable node (see Fig. 1) because the eigenvalues are negative ( ). This is locall as well as globall stable (see Fig. ). The on-hand inventories replenishment rates and demand rates of items 1 and are showed in Figs. 3 8 as follows. 4. Conclusion The salesmen effort (advertising promotional effort etc.) plas an important role to boost sale the items that results in more profit in oligopolistic marketing management. Determination of demands and costs due to salesmen effort is quite difficult. The author has formulated the new demand functions which are increasing function of common effort (E) and

10 586 S.S. Sana / Computers and Mathematics with Applications 6 (011) level of stock displaed of the same product. But the stock displaed of one product decreases the demand of other product those stock level are smaller than the other. Frankl speaking not all customers are attracted or motivated to purchase goods b the huge stock. Some of the customers purchase goods because of its goodwill good qualit or facilities and brand images of the product. The replenishment rates of each item depend on the on-hand inventor level of the concerned item. It is also considered the effect of inflation and time value of mone on the profit function. The approach in this paper is to concentrate on investment for the purpose of salesmen initiatives and replenishment rates of the homogeneous products which maimize the profit. The model provides a major new contribution the effect of salesmen effort and level of stocks of homogeneous products on demand in a dnamical sstem to operations in management practice. Acknowledgements The author epresses his gratitude to the editor and referees for useful comments that were ver helpful in improving the presentation of the article. References 1 R.I. Levin C.P. McLaughlin R.P. Lamone J.F. Kottas Productions Operations Management: Contemporar Polic for Managing Operating Sstems McGraw-Hill New York 197 p E.A. Silver R. Peterson Decision Sstems for Inventor Management and Production Planning 3rd ed. Wile New York R.C. Baker T.L. Urban Single-period inventor dependent demand models Omega 16 (1988) T.K. Datta A.K. Pal Deterministic inventor sstems for deteriorating items with inventor level dependent demand rate and shortages Opsearch 7 (1990) Y.W. Zhou J. Min S.K. Goal Suppl chain coordination under an inventor-level-dependent demand rate International Journal of Production Economics 113 (008) H. Soni N.H. Shah Optimal ordering polic for stock-dependent demand under progressive pament scheme European Journal of Operational Research 184 (008) S.K. Goal C.T. Chang Optimal ordering and transfer polic for an inventor with stock dependent demand European Journal of Operational Research 196 (009) T.P. Hsieh C.Y. De L.Y. Ouang Optimal lot size for an item with partial backlogging rate when demand is stimulated b inventor above a certain stock level Mathematical and Computer Modelling 51 (010) T.P. Hsieh C.Y. De Optimal replenishment polic for perishable items with stock-dependent selling rate and capacit constraint Computers & Industrial Engineering 59 (010) J. Min Y.W. Zhou J. Zhao An inventor model for deteriorating items under stock-dependent demand and two-level trade credit Applied Mathematical Modelling 34 (010) M.S. Sajadieh A. Thorstenson M.R.K. Jokar An integrated vendor buer model with stock-dependent demand Transportation Research Part E 46 (010) T.L. Urban An inventor model with an inventor-level-dependent demand rate and relaed terminal conditions Journal of the Operational Research Societ 43 (199) S. Pal A. Goswami K.S. Chaudhuri A deterministic inventor model for deteriorating items with stock-dependent demand rate International Journal of Production Economics 3 (1993) M. Goh EOQ models with general demand and holding cost functions European Journal of Operational Research 73 (1994) G. Padmanabhan P. Vrat EOQ models for perishable items under stock dependent selling rate European Journal of Operational Research 86 (1995) B.R. Sarker S. Mukherjee C.V. Balan An order-level lot size inventor model with inventor level dependent demand and deterioration International Journal of Production Economics 48 (1997) T.K. Datta K. Paul An inventor sstem with stock-dependent price-sensitive demand rate Production Planning & Control 1 (001) Z.T. Balkhi L. Benkherouf On an inventor model for deteriorating items with stock dependent and time-varing demand Computers and Operations Research 31 (004) C.T. Chang Inventor models with stock-dependent demand and nonlinear holding costs for deteriorating items Asia-Pacific Journal of Operational Research 1 (004) K.L. Hou L.C. Lin An EOQ model for deteriorating items with price-and stock-dependent selling rate under inflation and time value of mone International Journal of Sstems Science 37 (006) J. Min Y.W. Zhou A perishable inventor model under stock-dependent selling rate and shortage-dependent partial backlogging with capacit constraint International Journal of Sstems Science 40 (009) S.K. Ghosh S. Khanra K.S. Chaudhuri Optimal price and lot size determination for a perishable product under conditions of finite production partial backordering and lost sale Applied Mathematics and Computation 17 (011) S.K. Goal A. Gunasekaran An integrated production inventor-marketing model for deteriorating items Computers & Industrial Engineering 8 (1995) J.P. Dube Multiple discreteness and product differentiation: demand for carbonated soft drinks Marketing Science 3 (004) P.A. Naik K. Raman R.S. Winer Planning marketing-mi strategies in the presence of interaction effects Marketing Science 4 (005) S. Divakar B.T. Ratchford V. Shankar CHAN4CAST: a multichannel multi-region sales forecasting model and decision support sstem for consumer packaged goods Marketing Science 4 (005) B. Sun Promotion effect on endogenous consumption Marketing Science 4 (005) U. Ramanathan L. Muldermans Identifing demand factors for promotional planning and forecasting: a case of a soft drink compan in the UK International Journal of Production Economics 18 (010) S.S. Sana K. Chaudhuri An inventor model for stock with advertising sensitive demand IMA J Management Mathematics 19 (008) S.S. Sana Demand influenced b enterprises initiatives a multi-item EOQ model of deteriorating and ameliorating items Mathematical and Computer Modelling 5 (010) J.A. Buzacott Economic order quantit with inflation Operations Research Quarterl 6 (1975) H. Beirman J. Thomas Inventor decision under inflationar condition Decision Science 8 (1977) A. Brahmbatt Economic order quantit under variable rate of inflation and mark-up prices Productivit 3 (198) T.K. Datta A.K. Pal Effects of inflation and time-value of mone on inventor model with linear time-dependent demand rate and shortages European Journal of Operational Research 5 (1991) S. Bose A. Goswami K.S. Chaudhuri An EOQ model for deteriorating items with linear time-dependent demand rate and shortages under inflation and time discounting Journal of the Operational Research Societ 46 (1995)

11 S.S. Sana / Computers and Mathematics with Applications 6 (011) K. Chung J. Liu S. Tsai Inventor sstems for deteriorating items taking account of time value Engineering Optimization 7 (1997) J. Chen An inventor model for deteriorating items with time-proportional demand and shortages under inflation and time discounting International Journal of Production Economics 55 (1998) K. Chung C. Lin Optimal inventor replenishment models for deteriorating items taking account of time value of mone Computers & Operations Research 8 (001)

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