# Applications in Inventory Control and Supply Chain Management

## Introduction

The newsvendor problem refers to a mathematical model used in solving problems that are especially concerned with inventory control or supply chain management. Porteus (2008) highlights that the newsvendor problem has numerous applications including decision making in the manufacturing as well as the service industry. Often it is used in making purchase or order decisions when a couple of variables are known while others are unknown. A common scenario is the newspaper seller having to make an order of newspapers to sell everyday without knowing the actual number of papers needed to optimize a single business day. The dilemma is how many newspapers to buy from the wholesaler to sell at retail prices knowing full well that any left newspapers are economically shorthanded at the end of the day.

Porteus (2008) explains that the newsvendor problem occurs within the supply chain at a point when the amount needed of a given resource is random and a decision has to be made with regards to the amount of the resource to have prior to finding out how much is needed, in addition to keeping in mind the economic consequences of either having too much or too less of the resource. Upon such a problem the cost and price, lead time, the uncertainty and consequent assumption thereof as well as the value of the potential leftovers once the market is closed include among the factors to be considered and used in the development of the model. While the ultimate goal is to maximize profits, over ambition in the purchase may lead to unexpected loses given the dynamic nature of the market. Newsvendor problem is mostly applied in inventory and supply chain management of perishable products and services such as meals in a cafeteria or dairy meat products as well as seasonal or periodic products or services such as season and fashion clothing’s, newspapers and others.

### Mathematical Model

In instances where the perishable products not sold at the end of the day can be sold back to the manufacturer for a salvage value, the general model is determination of the highest possible profit from an inventory purchase thereby determining the value of products to buy for maximum profitability. Snyder (2016) points out that the general form of the problem uses the underage and overage cost instead of purchase cost, revenue and salvage value. The underage cost, 𝑝, is the cost incurred per unit of unmet demand: it equals the lost profit, 𝑟−𝑐. The overage cost, ℎ, is the cost incurred per unit of unused inventory: it equals the purchase cost less the salvage value,

If the demand is normally distributed with a mean of μ and a standard deviation of σ, then the equation above simplifies to:

Where the α-quantile of the standard normal distribution is

### Example 1

Suppose the newsvendor buys newspapers at a purchase cost of 𝑐=\$0.35, sells them for 𝑟=\$1.25, and salvages them for 𝑣=\$0.20.

The demand in each day is normally distributed with a mean of 120 and a standard deviation of 30.

𝑝=1.25−0.35=0.90

ℎ=0.35−0.20=0.15

### Therefore

α=0.90/(0.90+0.15)=0.86 and Φ⁻¹(α)=1.07.

### Therefore,

In other words, the newsvendor should order 120 units, plus an additional 32 newspapers (or 1.07 standard deviations’ worth of inventory) to protect against the uncertainty

### Example 2

A business individual buys products for a cost price c of \$80

All the cells within the demand supply framework are worked out to determine the total profit derived from different quantities of supply and demand,

Total Profit Derived = Minimum between the values of product demanded or supplied x Marginal profit –Maximum between the values of product demanded or supplied x marginal loss

Eventually the expected profits for every amount of product within the demand range are calculated to derive the expected profit from which the highest determines the effective value for supply into the market.

In this case based on the excel results 23 is the maximum value of products to be supplied for a maximum profit of \$1551.5

### Literature review

The newsvendor problem is a fundamental building block for models of inventory management in the face of stochastic demand (Porteus 1990), and at a broader level for models of supply chain systems (Cachon 2002). It is crucial in ensuring controlled purchases of products and services to be used in business so as to maximize on profits regardless of a variety of crucial unknown variables including the level of demand. Shah (2014) highlights that the news vendor problems describe a pattern of deciding how much to order/inventory management when the demand is dynamic, but the pattern of demand is not. The investors problem is that he is obliged to stock his/her entire inventory prior to the selling season knowing only an estimate distribution from which the demand will be drawn: ordering too less leads to lost business while ordering too many leads to excess stock and thus a loss. As such the kinds of problems in the supply chain to be fixed by the newsvendor problem include issues of expected demand and intended supply. Quantifying the amount of products to be procured in lee of the sales intended provide a problem within the supply chain which can be effectively solved by the newsvendor problem.

Four out of the five articles evaluated in the development of the literature review highlight that while the general idea is that organizations involved in prescriptions that flow from newsvendor based models typically assume an optimal stock in correspondence to an averaged market value over numerous trade periods, the actual practice from behavioral studies of newsvendor problem find that people often make suboptimal and Biased choices. According to Schweitzer and Cachon (2000), this is actually true even for those who have been exposed to the optimal solution in an MBA classroom. They further point out a possibility that this is due to the desire to optimize profits due to market forecasting that stems from everyday market encounters and experience. In addition, a wide range of factors are considered by different individuals which require new techniques if the system is to be optimized.

The opportunity costs as well as inventory holding cost are cost considerations that are crucial to the newsvendor problem (Areerakulkan, 2017). Opportunity cost informs the cost that will be forgone in a scenario where the demand is in excess of the actual stock in the inventory, while the inventory holding cost concerns the cost lost in the inventory being in excess of the demand. The optimal combination of all the costs plays an important role in business cost that must be subsequently minimized to consequently maximize on the profits. While maximization of the profits is the key goal in the visions and missions of businesses, the newsvendor problem solves the problem through use of a median solution which also minimizes the chances of loses significantly thereby describing its suitability of use in the solution of problems within the supply chain of products and services. Sinha (2015) highlights sometimes, though not often, median solution may be better than solutions maximizing expected profit, or minimizing losses.

Wang, Tang and Huang (2013) highlights an experiment that deals with a two period newsvendor setting in which the second period is a product extension of the first period product, the newsvendor setting and newsvendor problem provide a perfect scenario with which the demand and supply in the economy can be controlled and successful influence of the consumer base carried out highlighting another suitability of its use in the solution of supply chain problems as well as the enhancement of supply chain and inventory management. According to Wang, Tang and Huang (2013) an intentionally made shortage on the first item creates stimulation in the sales of the second item. The stochastic demand of these two items is assumed to be a linear-additive pattern comprising a deterministic demand and an error demand, where the deterministic demand consists of a primary demand and consumer price elasticity, and the error demand is hypothesized to be exponentially distributed.

Despite these suitability factors, however, the Newsvendor problem also possess a few flaws and limitations when it comes to their application in the solution of supply chain problems. A study by Merzifonluoglu and Feng (2013) points out that newsvendor problem as a solution based model comes in short in a supply chain that consist of a multiple variety of unreliable and non identical suppliers. A problem in a supply chain of unique products coming from different unreliable sources may not be suitably solved by the newsvendor problem given the need for a suitable range in both demand and supply within a particular period. With the use of newsvendor problem model, forecast updating for improved profitability can only be executed at an increased cost while a retailer can postpone order placement to obtain a better demand forecast with a shorter supply lead time. However, the manufacturer would charge the retailer a higher cost for a shorter lead time and set restrictions on the ordering times and quantities. This prevents retailers from taking full advantage of demand forecast updating to improve profits (Zheng, Wu and Shu, 2016).

Another limitation of the use of newsvendor problem solution model involves in the supply chain and inventory management of new and innovative products. Gaspers- Wieloch (2017) clarifies that new and innovative products presents a complication when it comes to defining probabilities or even probability like quantities given the absence of data for forecasting upcoming demand through statistical analysis.

### Summary of Findings

The newsvendor problem provides a model for the determination of amount of inventory quantities to buy when the demand is dynamic but the pattern with which the demand occurs is fixed. This enables the maximization of profits and significant reduction of any losses. This makes the newsvendor problem quite an effective model in the solution of inventory management and supply chain problems of perishable products within a dynamic demand market place. Given it capitalizes on the use of median quantities rather than leaning on the maximization of profits and minimization of losses, the opportunity cost on inventory management is highly implemented while the probability of losses from low demands and high supplies are greatly and significantly reduced. Further the newsvendor problem is very effective in the management of inventory and sale of substitute products within specific respective time periods. However in supply chains that consist a wide range of products which are provided by different none identical suppliers, development of different chains will be crucial if the use of news vendor problem is going to be useful as a decision making model for inventory procurement. Further a critical update in the forecasting strategies is critical if the newsvendor problem will be used in the solution of supply chain problems that involve new and innovative products given the complication when it comes to defining probabilities or even probability like quantities given the absence of data for forecasting upcoming demand through statistical analysis.

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## References

Cachon, G.P. (2002), Supply chain coordination with contracts. S. Graves and Ton de Kok (Eds.). Handbook in OR & MS, Supply Chain Management. Elsevier, North-Holland

Gaspars-Wieloch, H. (2016). Newsvendor problem under complete uncertainty: a case of innovative products. Central European Journal of Operations Research, 25(3), pp.561-585.

Porteus, E.L. (1990). Stochastic inventory theory. D.P. Heyman and M.J. Sobel (Eds.). Handbook in OR & MS, Vol 2. Elsevier, North-Holland, 605-652.

Schweitzer, M.E. and Cachon, G.P (2000), Decision bias in the newsvendor problem with known demand distribution: experimental evidence. Management Science, Vol. 46, pp. 404-420.

Wang, K., Tung, C. and Huang, Y. (2013). A Two-Period Newsvendor Model with Product Extension and Shortage-Making Strategy. ISRN Operations Research, 2013, pp.1-4.

Zheng, M., Wu, K. and Shu, Y. (2016). Newsvendor problems with demand forecast updating and supply constraints. Computers & Operations Research, 67, pp.193-206.

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