An Optimal Algorithm for Finding (r, Q) Policy in a Price-Dependent Order Quantity Inventory System with Soft Budget Constraint
This paper is concerned with the single-item
continuous review inventory system in which demand is stochastic
and discrete. The budget consumed for purchasing the ordered items
is not restricted but it incurs extra cost when exceeding specific
value. The unit purchasing price depends on the quantity ordered
under the all-units discounts cost structure. In many actual systems,
the budget as a resource which is occupied by the purchased items is
limited and the system is able to confront the resource shortage by
charging more costs. Thus, considering the resource shortage costs as
a part of system costs, especially when the amount of resource
occupied by the purchased item is influenced by quantity discounts,
is well motivated by practical concerns. In this paper, an optimization
problem is formulated for finding the optimal (r, Q) policy, when the
system is influenced by the budget limitation and a discount pricing
simultaneously. Properties of the cost function are investigated and
then an algorithm based on a one-dimensional search procedure is
proposed for finding an optimal (r, Q) policy which minimizes the
expected system costs.
[1] H. Galliher, P.M. Morse and M. Simond, “Dynamics of two classes of
continuous-review inventory systems” Operations Research, Vol. 7, pp.
362-384, 1959.
[2] A. Federgruen, and Y.S. Zheng, “An efficient algorithm for computing
an optimal (r, Q) policy in continuous review stochastic inventory
systems” Operations research, Vol. 40, pp. 808-813, 1992.
[3] R.M. Hill, and S.G. Johansen, “Optimal and near-optimal policies for
lost sales inventory models with at most one replenishment order
outstanding” European journal of operational research, Vol. 169, pp.
111-132, 2006.
[4] A. Federgruen and M. Wang, “Monotonicity properties of a class of
stochastic inventory systems” Annals of Operations Research, Vol. 208,
pp. 155-186, 2013.
[5] F. Olsson, “Analysis of inventory policies for perishable items with
fixed leadtimes and lifetimes” Annals of Operations Research, Vol. 217,
pp. 399-423, 2014.
[6] S. Minner and E.A. Silver, “Multi-product batch replenishment
strategies under stochastic demand and a joint capacity constraint” IIE
Transactions, Vol. 37, pp. 469-479, 2005.
[7] X. Zhao, M. Qiu, J. Xie and Q. He, “Computing (r, Q) policy for an
inventory system with limited sharable resource” Computers &
Operations Research, Vol. 39, pp. 2368-2379, 2012.
[8] M. Ang, J. S. Song, M. Wang and H. Zhang, “On properties of discrete
(r, q) and (s, T) inventory systems”, European Journal of Operational
Research, Vol. 229, pp. 95-105, 2013.
[9] Y.S. Zheng, “On properties of stochastic inventory systems”,
Management science, Vol. 38, pp. 87-103, 1992.
[10] B. Ghalebsaz-Jeddi, B.C. Shultes and R. Haji, “A multi-product
continuous review inventory system with stochastic demand, backorders,
and a budget constraint” European Journal of Operational Research, Vol.
158, pp. 456-469, 2004.
[11] M.A. Hariga, “A single-item continuous review inventory problem with
space restriction” International Journal of Production Economics, Vol.
128, pp. 153-158, 2010.
[12] J.M. Betts and R.B. Johnston, “Determining the optimal constrained
multi-item (Q, r) inventory policy by maximising risk-adjusted profit”
IMA Journal of Management Mathematics, Vol. 16, pp. 317-338, 2005.
[13] A. Kundu and T. Chakrabarti, “A multi-product continuous review
inventory system in stochastic environment with budget constraint”
Optimization Letters, Vol. 6, pp. 299-313, 2012.
[14] X. Zhao, F. Fan, X. Liu and J. Xie, “Storage-space capacitated inventory
system with (r, Q) policies” Operations research, Vol. 55, pp. 854-865,
2007.
[15] M.N. Katehakis and L.C. Smit, “On computing optimal (Q, r)
replenishment policies under quantity discounts” Annals of Operations
Research, Vol. 200, pp. 279-298, 2012. [16] Y. Feng and J. Sun, “Computing the optimal replenishment policy for
inventory systems with random discount opportunities” Operations
Research, Vol. 49, pp. 790-795, 2001.
[17] H. Kawakatsu, “Optimal Quantity Discount Strategy for an Inventory
Model with Deteriorating Items” Electrical Engineering and Intelligent
Systems, Springer, 2013, pp. 375-388.
[18] S. Papachristos and K. Skouri, “An inventory model with deteriorating
items, quantity discount, pricing and time-dependent partial
backlogging” International Journal of Production Economics, Vol. 83,
pp. 247-256, 2003.
[19] J. Moussourakis and C. Haksever, “Models for Ordering Multiple
Products Subject to Multiple Constraints, Quantity and Freight
Discounts” American Journal of Operations Research, Vol. 3, pp. 521,
2013.
[20] W. Benton, “Quantity discount decisions under conditions of multiple
items, multiple suppliers and resource limitations”, The International
Journal of Production Research, Vol. 29, pp. 1953-1961, 1991.
[21] S.M. Mousavi and S.H. Pasandideh, “A Multi-Periodic Multi-Product
Inventory Control Problem with Discount: GA Optimization Algorithm”
Journal of Optimization in Industrial Engineering, Vol., pp. 37-44, 2011.
[22] A.A. Taleizadeh, S.T.A. Niaki, M.B. Aryanezhad and A.F. Tafti, “A
genetic algorithm to optimize multiproduct multiconstraint inventory
control systems with stochastic replenishment intervals and discount”
The International Journal of Advanced Manufacturing Technology, Vol.
51, pp. 311-323, 2010.
[23] G.W. Hadley, “TM, 1963. Analysis of Inventory systems”, Englewood
Cliffs, NJ, Vol., 1963.
[1] H. Galliher, P.M. Morse and M. Simond, “Dynamics of two classes of
continuous-review inventory systems” Operations Research, Vol. 7, pp.
362-384, 1959.
[2] A. Federgruen, and Y.S. Zheng, “An efficient algorithm for computing
an optimal (r, Q) policy in continuous review stochastic inventory
systems” Operations research, Vol. 40, pp. 808-813, 1992.
[3] R.M. Hill, and S.G. Johansen, “Optimal and near-optimal policies for
lost sales inventory models with at most one replenishment order
outstanding” European journal of operational research, Vol. 169, pp.
111-132, 2006.
[4] A. Federgruen and M. Wang, “Monotonicity properties of a class of
stochastic inventory systems” Annals of Operations Research, Vol. 208,
pp. 155-186, 2013.
[5] F. Olsson, “Analysis of inventory policies for perishable items with
fixed leadtimes and lifetimes” Annals of Operations Research, Vol. 217,
pp. 399-423, 2014.
[6] S. Minner and E.A. Silver, “Multi-product batch replenishment
strategies under stochastic demand and a joint capacity constraint” IIE
Transactions, Vol. 37, pp. 469-479, 2005.
[7] X. Zhao, M. Qiu, J. Xie and Q. He, “Computing (r, Q) policy for an
inventory system with limited sharable resource” Computers &
Operations Research, Vol. 39, pp. 2368-2379, 2012.
[8] M. Ang, J. S. Song, M. Wang and H. Zhang, “On properties of discrete
(r, q) and (s, T) inventory systems”, European Journal of Operational
Research, Vol. 229, pp. 95-105, 2013.
[9] Y.S. Zheng, “On properties of stochastic inventory systems”,
Management science, Vol. 38, pp. 87-103, 1992.
[10] B. Ghalebsaz-Jeddi, B.C. Shultes and R. Haji, “A multi-product
continuous review inventory system with stochastic demand, backorders,
and a budget constraint” European Journal of Operational Research, Vol.
158, pp. 456-469, 2004.
[11] M.A. Hariga, “A single-item continuous review inventory problem with
space restriction” International Journal of Production Economics, Vol.
128, pp. 153-158, 2010.
[12] J.M. Betts and R.B. Johnston, “Determining the optimal constrained
multi-item (Q, r) inventory policy by maximising risk-adjusted profit”
IMA Journal of Management Mathematics, Vol. 16, pp. 317-338, 2005.
[13] A. Kundu and T. Chakrabarti, “A multi-product continuous review
inventory system in stochastic environment with budget constraint”
Optimization Letters, Vol. 6, pp. 299-313, 2012.
[14] X. Zhao, F. Fan, X. Liu and J. Xie, “Storage-space capacitated inventory
system with (r, Q) policies” Operations research, Vol. 55, pp. 854-865,
2007.
[15] M.N. Katehakis and L.C. Smit, “On computing optimal (Q, r)
replenishment policies under quantity discounts” Annals of Operations
Research, Vol. 200, pp. 279-298, 2012. [16] Y. Feng and J. Sun, “Computing the optimal replenishment policy for
inventory systems with random discount opportunities” Operations
Research, Vol. 49, pp. 790-795, 2001.
[17] H. Kawakatsu, “Optimal Quantity Discount Strategy for an Inventory
Model with Deteriorating Items” Electrical Engineering and Intelligent
Systems, Springer, 2013, pp. 375-388.
[18] S. Papachristos and K. Skouri, “An inventory model with deteriorating
items, quantity discount, pricing and time-dependent partial
backlogging” International Journal of Production Economics, Vol. 83,
pp. 247-256, 2003.
[19] J. Moussourakis and C. Haksever, “Models for Ordering Multiple
Products Subject to Multiple Constraints, Quantity and Freight
Discounts” American Journal of Operations Research, Vol. 3, pp. 521,
2013.
[20] W. Benton, “Quantity discount decisions under conditions of multiple
items, multiple suppliers and resource limitations”, The International
Journal of Production Research, Vol. 29, pp. 1953-1961, 1991.
[21] S.M. Mousavi and S.H. Pasandideh, “A Multi-Periodic Multi-Product
Inventory Control Problem with Discount: GA Optimization Algorithm”
Journal of Optimization in Industrial Engineering, Vol., pp. 37-44, 2011.
[22] A.A. Taleizadeh, S.T.A. Niaki, M.B. Aryanezhad and A.F. Tafti, “A
genetic algorithm to optimize multiproduct multiconstraint inventory
control systems with stochastic replenishment intervals and discount”
The International Journal of Advanced Manufacturing Technology, Vol.
51, pp. 311-323, 2010.
[23] G.W. Hadley, “TM, 1963. Analysis of Inventory systems”, Englewood
Cliffs, NJ, Vol., 1963.
@article{"International Journal of Business, Human and Social Sciences:70713", author = "S. Hamid Mirmohammadi and Shahrazad Tamjidzad", title = "An Optimal Algorithm for Finding (r, Q) Policy in a Price-Dependent Order Quantity Inventory System with Soft Budget Constraint", abstract = "This paper is concerned with the single-item
continuous review inventory system in which demand is stochastic
and discrete. The budget consumed for purchasing the ordered items
is not restricted but it incurs extra cost when exceeding specific
value. The unit purchasing price depends on the quantity ordered
under the all-units discounts cost structure. In many actual systems,
the budget as a resource which is occupied by the purchased items is
limited and the system is able to confront the resource shortage by
charging more costs. Thus, considering the resource shortage costs as
a part of system costs, especially when the amount of resource
occupied by the purchased item is influenced by quantity discounts,
is well motivated by practical concerns. In this paper, an optimization
problem is formulated for finding the optimal (r, Q) policy, when the
system is influenced by the budget limitation and a discount pricing
simultaneously. Properties of the cost function are investigated and
then an algorithm based on a one-dimensional search procedure is
proposed for finding an optimal (r, Q) policy which minimizes the
expected system costs.", keywords = "(r, Q) policy, Stochastic demand, backorders, limited resource, quantity discounts.", volume = "9", number = "7", pages = "2462-7", }