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Before building the mathematical model of the system, all assumptions, limitations and concepts used were explained and summarised. The production- inventory control problem of perishable products under random demand and periodic review policy was considered. The SC model consists of two stages, namely supply and production. It is assumed that decision makers exist in every stage and has the responsibility of managing inventory at that stage.

The raw materials necessary for producing the product are obtained from one or more external suppliers via single sourcing for each item which is in consistent with JIT practice. The inventory of products is produced according to the production plan under a JIT schedule with suppliers. For that reason, suppliers have agreed to carry all of the material inventories and to instantly deliver those following requests as shown in Figure 6.1. This instantaneous delivery just in time for the production is possible when a strategic partnership is developed among them. The production operates on a day-to-day basis as cleaning is required at the end of the production day for hygiene reasons. Energy is consumed as input in the production and warehousing activities to condition and preserve perishable food which gives CO2 emissions as the output from the process as shown in Figure 6.1.

Multiple products are produced here in the production plant. It is possible that different products may require the same type of raw materials. In a situation where products scheduled to produce in a period can be used instantaneously to satisfy demand in the same period (i.e. zero production lead time), it was assumed no restrictions on manpower and machine, and each product is manufactured by a dedicated production facility/line because the problem of product scheduling can be disregarded. Only the data taken from bill of materials of the products is needed. Otherwise, a production at the producer is assumed the planned production lead time. This planned lead time is interpreted as a time, determined by experience, in which it may be expected that any reasonable work order can be produced. The production lead time is

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regarded in the major setup cost and the average environmental impacts for setting up the production run (i.e. the major set up cost reflects the total cost incurred in setting up the productions for a number of lead time periods and the same applies to environmental impacts). By adopting planned lead time, the reliability between SC operations planning and scheduling as regards the feasibility of the planning is increased (Spitter et al., 2005). It also reduces system nervousness. Demand variability during this planned production lead time can be taken care of with safety stock.

Materials and products are assumed to have a fixed lifetime. After the end of the lifetime, if they are not consumed by demand, units are spoiled and have to be discarded as food waste as shown in Figure 6.1. It was assumed that the ageing of the materials and products begins just at the time replenishment order is delivered to supplier and production run at producer is completed and there is no decrease in the value during their usable lifetime which they are of satisfactory quality and functionality for both suppliers and producer. An implication of fixed lifetime of raw materials is that the supplier supplies fresh items upon their delivery. An implication of fixed lifetime of products is that the producer satisfies customers’ requirement regarding remaining shelf life of products. It is assumed that there is known material replenishment transportation lead time for every raw material to each supplier similar to the planned lead time by Spitter et al. (2005). The author introduces the maximum planned transport lead time at the supply stage. For that reason, the planning interval begins in period . There is assumed to be quick sorting and packaging activities going on in the supply stage which can be done on the same day materials are delivered to the supplier and then the delivery is continued to the producer just in time for the production run.

The inventory of products can be depleted randomly or according to a FIFO issuing policy or a LIFO issuing policy and all unmet demands are backlogged. The demand for product per unit time , facing the producer is a nonnegative random variable following a distribution with probability density function and cumulative distribution function . In order to account for the different remaining shelf life requirements from retailers is introduced. The demand for product from retailer per unit time , represents a

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fraction of forecast demand that may request products with different remaining shelf life (i.e. may require longer remaining shelf life). Producer might get this data from his forecast or as an assumption that a certain percentage of forecast demand will have different requirements. The demand considered here is non- stationary, meaning that it can vary from period to period, and it is also assumed that demands in different periods are independent. Therefore, it is appealing to investigate the use of order policy with time-dependent order-up-to levels and replenishment cycle lengths (Rn, Sn) in this case.

In this particular setting, the problem considered in this study is to determine production and inventory decisions with the aim of finding trade-off among total associated costs, total spoiled food wastes and total emissions in the SC. This problem can be categorised as deterministic (deterministic equivalent modelling approach employed for stochastic demand), constrained (required service level), mix-integer (order-up-to levels) and with multiple performance measures (economic and environmental objectives).

In this section, a quantitative model that can manage economic issues along with sustainability concerns is developed in response to the need in practice. The model of the cost function per unit time for producer, suppliers and the SC which is the sum of all companies’ cost function is built. In addition to this economic assessment, waste and emissions estimations are modelled for environmental judgment. Especially, the author extends and develops other studies that have been carried out by Rau et al. (2003), Tarim and Kingsman (2004), Jaber and Goyal (2008), Pauls-Worm et al. (2010), Rossi et al. (2010a), Rossi et al. (2010b), Rong et al. (2011), Abdallah et al. (2012), Bouchery et al. (2012), Chaabane et al. (2012), Zanoni and Zavanella (2012), Absi et al. (2013), Chen et al. (2013), Kouki et al. (2013), Pauls-Worm et al. (2014), Soysal et al. (2014), Bozorgi et al. (2014) and Govindan et al. (2014). The mathematical model will be developed using notations listed comprehensively below to capture the above described operations.