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In this paper, we consider the multi-product multi-constraint newsboy problem with random fuzzy demands and total discount. The demand of the products is often stochastic in the real word but the estimation of the parameters of distribution function may be done by fuzzy manner. So an appropriate option to modeling the demand of products is using the random fuzzy variable. The objective function of proposed model is to maximize the expected profit of newsboy. We consider the constraints such as warehouse space and restriction on quantity order for products, and restriction on budget. We also consider the batch size for products order. Finally we introduce a random fuzzy multi-product... 

In this paper, a multi-product multi-chance constraint joint single-vendor multi-buyers inventory problem is considered in which the demand follows a uniform distribution, the lead-time is assumed to vary linearly with respect to the lot size, and the shortage in combination of backorder and lost-sale is assumed. Furthermore, the orders are placed in multiple of packets, there is a limited space available for the vendor, there are chance constraints on the vendor service rate to supply the products, and there is a limited budget for each buyer to purchase the products. While the elements of the buyers' cost function are holding, shortage, order and transportation costs, the set up and... 

In this paper, an Economic Production Quantity (EPQ) model is studied, in which the production defective-rate follows either a uniform or a normal probability distribution. Shortages are allowed and take a backorder state, and the existence of only one machine causes a limited production capacity for the common cycle length of all products. The aim of this research is to determine the optimal production quantity of each product, such that the expected total cost including holding, shortage, production, setup and defective items cost is minimized. The mathematical model of the problem is derived, for which the objective function is proved to be convex. Then, a derivative approach is utilized... 

In this paper, a multiproduct single-machine production system under economic production quantity (EPQ) model is studied in which the existence of only one machine causes a limited production capacity for the common cycle length of all products, the production defective rates are random variables, shortages are allowed and take a combination of backorder and lost sale, and there is a service rate constraint for the company. The aim of this research is to determine the optimal production quantity, the allowable shortage level, and the period length of each product such that the expected total cost, including holding, shortage, production, setup and defective items costs, is minimized. The... 

In this paper, a multi-buyer multi-vendor supply chain problem is considered in which there are several products, each buyer has limited capacity to purchase products, and each vendor has warehouse limitation to store products. In this chain, the demand of each product is stochastic and follows a uniform distribution. The lead-time of receiving products from a vendor to a buyer is assumed to vary linearly with respect to the order quantity of the buyer and the production rate of the vendor. For each product, a fraction of the shortage is backordered and the rest are lost. The ordered product quantities are placed in multiple of pre-defined packets and there are service rate constraints for... 

An uncertain economic order quantity (UEOQ) model with payment in advance is developed to purchase high-price raw materials. A joint policy of replenishments and pre-payments is employed to supply the materials. The rate of demand is considered LR-fuzzy variables, lead-time is taken to be constant, and it is assumed that shortage does not occur in the cycles. The cycle is divided into three parts; the first part is the time between the previous replenishment-time to the next order-time (t0), the second part is the period between t0 to a payment-time (tk), and the third part is the period between tk to the next replenishment-time. At the start of the second part (t0), α% of the purchasing... 

In this paper, a multiproduct multi-chance constraint stochastic inventory control problem is considered, in which the time-periods between two replenishments are assumed independent and identically distributed random variables. For the problem at hand, the decision variables are of integer-type, the service-level is a chance constraint for each product, and the space limitation is another constraint of the problem. Furthermore, shortages are allowed in the forms of fuzzy random quantities of lost sale that are backordered. The developed mathematical formulation of the problem is shown to be a fuzzy random integer-nonlinear programming model. The aim is to determine the maximum level of... 

This study solves a chance-constraint supply chain problem with stochastic demand which follows a uniform distribution. Fuzzy delay times (moving, waiting and setup time) are assumed to be lot size dependent and shortage is partially backordered. The buyer is responsible for the costs incurred in ordering, holding, shortage and transportation, while the vendor is responsible for setup and holding costs. The service rate of each product has a chance constraint and the buyer has a budget constraint. Our objective is to determine the re-order point and the order quantity of the products such that the total cost is minimized. Since the problem is uncertain integer-nonlinear, two hybrid... 

Production systems with scrapped and rework items have recently become an interesting subject of research. While most attempts have been focused on finding the optimal production quantity in a simple production system, little work appears on a joint production environment. In this research, two joint production systems in a form of multiproduct single machine with and without rework are studied where shortage is allowed and backordered. For each system, the optimal cycle length, the backordered and production quantities of each product are determined such that the cost function is minimised. Proof of the convexity of the involved objective functions of each model is provided and numerical... 

An economic order quantity (EOQ) model with payment in advance is developed to purchase highprice raw materials. A joint policy of replenishments and pre-payments is employed to supply the materials. The rate of demand and lead time are taken to be constant and it is assumed that shortage does not occur in the cycles. The cycle is divided into three parts; the first part is the time between the earlier replenishment-time to the next order-time (to), the second part is the period between to to a payment-time (tk) and the third part is the period between tk to the next replenishment-time. At the start of the second part (to), α% of the purchasing cost is paid. The (1-α)% remaining purchasing... 

In this paper, we develop the multi-product, multi-constraint, Single Period Problem (SPP) with uncertain demands, considering an incremental discount situation. Three new models are presented for multi-product, multi-constraint SPP in fuzzy, stochastic and rough environments. We consider constraints, such as service rate, restriction on order quantity and restrictions on warehouse space and budget. We also consider that the order quantity is a multiplier of predefined batch size. Furthermore, three kinds of solution algorithm, (1) harmony search, (2) hybrid intelligent based on harmony search and fuzzy simulation and (3) hybrid intelligent based on harmony search and rough simulation, are... 

In this paper, we develop the multi-products multi-constraints newsboyproblem with fuzzy demands considering the total discount situation. Since thedemands of products are often fuzzy in the real world, the profit of the newsboyis fuzzy too. A new model is presented for multi-products multi-constraintsnewsboy problem in fuzzy Environment (FM-PM-CNP). The objective function ofproposed model is to maximize the expected profit of newsboy. We consider theconstraints such as service rate, restriction on quantity order, and restrictionon warehouse space and budget. We also consider batch size for products order.Furthermore, a hybrid intelligent algorithm based on simulated annealing andfuzzy... 

In this paper, a multiproduct inventory control problem is considered in which the periods between two replenishments of the products are assumed independent random variables, and increasing and decreasing functions are assumed to model the dynamic demands of each product. Furthermore, the quantities of the orders are assumed integer-type, space and budget are constraints, the service-level is a chance-constraint, and that the partial back-ordering policy is taken into account for the shortages. The costs of the problem are holding, purchasing, and shortage. We show the model of this problem is an integer nonlinear programming type and to solve it, a harmony search approach is used. At the... 

While the usual assumptions in multiperiodic inventory control problems are that the orders are placed either at the beginning of each period (periodic review) or depending on the inventory level they can happen at any time (continuous review), in this paper we assume the periods between two replenishments of several products are identical and independent random variables. For the problem at hand, the order quantities (decision variables) are of integertype and there are two kinds of space and service level constraints for each product. A model of the problem is first developed in which a combination of backorder and emergency orders is considered for the shortages, and the costs are... 

Multi-periodic inventory control problems are mainly studied employing one of two assumptions. The first is the continuous review, where depending on the inventory level, orders can be placed at any time, and the other is the periodic review, where orders can be placed only at the beginning of each period. In this paper, we relax these assumptions and assume that the time-periods between two replenishments are random fuzzy variables. While in the model of the problem at hand the decision variables are of integer type and there are space and service level constraints, for the shortages we consider a combination of back-order and lost-sales. We show the model of this problem to be an... 

This article points out the real-world prevalence of the multiple-product, multiple-constraint newsboy problem with two objectivesthe 'newsstand problem'in which there are total and incremental discounts on purchasing prices. The constraints are the warehouse capacity and the batch forms of the order placements. The first objective of this problem is to find the order quantities that maximize expected profit, and the second objective is maximizing the service rate. It is assumed that the holding and shortage costs, modelled by a quadratic function, occur at the end of the period. Moreover, the decision variables are integer. A formulation of the problem is presented and shown to be an... 

Multi-periodic inventory control problems are mainly studied by employing one of two assumptions. First, the continuous review, where depending on the inventory level, orders can happen at any time, and next the periodic review, where orders can only be placed at the beginning of each period. In this paper, we relax these assumptions and assume the times between two replenishments are independent random variables. For the problem at hand, the decision variables (the maximum inventory of several products) are of integer-type and there is a single space-constraint. While demands are treated as fuzzy numbers, a combination of back-order and lost-sales is considered for the shortages. We... 

There are two main assumptions in multiperiodic inventory control problems. The first is the continuous review, where, depending on the inventory level, orders can happen at any time, and the other is the periodic review, where orders can only happen at the beginning of each period. In this paper, these assumptions are relaxed, and the periods between two replenishments are assumed independent and identically distributed random variables. Furthermore, the decision variables are assumed integer-type and that there are two kinds of space and budget constraints. The incremental discounts to purchase products are considered, and a combination of backorder and lost sales are taken into account... 

Increasing rates of environmental and social concerns involved in various communities have recently attracted researchers’ attention to consider them in proper designs of sustainable supply chains (SSCs). In this paper, the planning problem of a multi-period multi-echelon sustainable closed loop supply chain (SCLSC) is investigated. A comprehensive model that seriously takes into account the social and environmental aspects of the supply chain (SC) decisions and measures their social and environmental effects using the GRI (Global Reporting Initiative) guideline indicators is proposed. The proposed planning model includes tactical decisions comprising the price of the product and logistic... 

While the usual assumptions in multi-periodic inventory control problems are that the orders are placed at the beginning of each period (periodic review) or depending on the inventory level they can happen at any time (continuous review), in this article, we relax these assumptions and assume that the periods between two replenishments of the products are independent and identically distributed random variables. Furthermore, assuming that the purchasing price are triangular fuzzy variables, the quantities of the orders are of integer-type and that there are space and service level constraints, total discount are considered to purchase products and a combination of back-order and lost-sales...