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This paper analyzes the implementation of transshipments among two independent locations which belong to one parent firm and the impact of transshipments, when implemented between these two locations. We examine how the possibility of such transshipments affects the optimal inventory orders at each location and find that, if each location aims to maximize its own profit, their inventory choices will not in general maximize joint profits. However, we find transshipment prices which if implemented by the parent firm as incentive design will make each location behave in order to optimize aggregate profit.

Inventory management is a critical aspect of enterprise-wide optimization. Inventories are used in production and logistics networks to coordinate supply cycles and to mitigate the risk associated with uncertainty. The importance of inventory in industrial applications derives from the effect of stockouts in the levels of customer satisfaction and the impact of stock out in the economic balance of companies.

Remarkably, the value of U.S inventories was estimated to be over $1707 billion in December 2013 [

Inventory plays a variety of roles. It can be used to buffer tempered mismat- ches between supply availability, processing rates, and demand in integrated production networks. It is also useful to satisfy safety requirements, in cases where the availability of hazardous raw materials can be replaced with inventory of its products.

Inventory also plays an important role in the coordination of maintenance turnaround between various facilities across a site. In general, the benefits of inventories derive from their ability to reduce the interdependence of processing units, to mitigate the effects of bottlenecks, and to facilitate capacity utilization.

Many inventory management strategies have been proposed since Harris introduced the Economic Order Quantity (EOQ) model in 1913. Before the 1930s, merchants used the skills of experience and intuition in their shop keeping writing down purchases or looking at how many units were gone at the end of the day and then do their best to forecast for the future needs. However, efficiency and mass production became the main objectives of business after the industrial revolution, along with an improved customer experience at the point of sale.

In the 1960s, the affordable laser technology which allowed for smaller, faster and cheaper readers or scanners as well as the Universal Product Code (UPC) was developed. These systems boosted the efficiency of the computing power which helped in trading and managing inventories better.

In recent years, more flexible inventory systems also allow lateral transshipments within an echelon, thus, between wholesalers or retailers [

As a risk-pooling strategy, transshipment provides an efficient and effective approach for firms to reduce inventory carrying costs without negative effects on customer service. For example, Saturn has achieved impressive success with after- sales service through the practice of sharing automobile parts among its dealers [

The study of multi-location models with transshipment is an important contribution for mathematical inventory theory as well as for inventory practice. The idea of lateral transshipments is not new. The first study dates back to the sixties, the two-location-one-period case with linear cost functions was considered by Aggarwal [

Krishnan [

The effect of lateral transshipment on the service levels in a two-location- one-period model was studies by Tagaras [

Our major contribution is to determine the demand variations facing the two locations under which a transshipment policy is likely to increase performance of the parent firm. In particular, we examine how the variance in demand affects inventory levels, revenues and profits of the firm.

This paper is organized as follows. The next section provides a review of literature on inventory management and transshipments. Section 3 sets up the basic framework that allows transshipment between two locations. We look at the classic newsvendor problem as a special case within this framework. Section 4 examines how the possibility of transshipments between these locations affects inventory orders in the case where they are centrally determined by the parent firm. To illustrate the model, we give a numerical analysis in this for normal distribution and compare to the corresponding results of the classical newsvendor model and finally, Section 5 presents the conclusions and future research.

Inventory can be described as the stock of any item or resource that is used in an organization [

Inventory management under uncertainty originated in the 1950s and the first significant contribution to this field was provided in 1951 by Arrow [

The work of Arrow et al. [

Since then, many papers on inventory management and modeling under uncertainty have been published as seen in Chu [

A technique often used in the literature to deal with the complexity of inventory management and modeling under uncertainty is to assume that the uncertain parameter, for instance demand, is random and that its probability distribution is known with certainty. In such cases, stochastic programming can be used to incorporate uncertainty into the model. Similar examples may be found in Maiti et al. [

In the field of supply chain management, one of the most important components is meeting customer demand in an effective and efficient manner. Although, there are a number of possible solutions, transshipment is among the most practical approach.

Lateral transshipments within an inventory system are stock movements between locations of the same echelon [

In the periodic review system, Krishnan et al. [

Building on Krishnan et al. [

Robinson [

Other recent related transshipment studies can be found in the work of Yu, Tang and Niederloff [

Our basic framework follows closely the work of Krishnan et al. [

Firm i obtains revenue y i for every unit sold and if it cannot meet the entire demand, it is penalized P i for every unit of unmet demand. Unsold stock has a salvage value, S i and we also define the marginal value of additional sales at location i as M i = y i + P i .

From

According to the complete pooling assumptions defined by Tagaras [

Notation | Description |
---|---|

Unit price of goods sold by Location | |

Procurement cost paid by Location | |

Average inventory carrying/holding cost | |

Salvage value for each unit of unsold stock | |

Penalty cost for each unit of unmet demand | |

Cost incurred in transshipping a unit from Location | |

Transshipment price charged by | |

Optimal inventory levels for | |

Quantity of transshipments from | |

Actual sales in unit by | |

Amount of unsold stock at | |

Amount of unmet demand of | |

Level of demand at locations |

Source: Li Zou [

Following the constraints above, we define the transshipment quantity from location i to location j as: X i j = min [ ( D j − Q j ) + , ( Q i − D i ) + ] , where

x + = max ( x , 0 ) . In this expression, the ( D j − Q j ) + represents excess demand at location j while surplus stock at location i is represented by ( Q i − D i ) + .

The transshipment level takes the lower value between the overstock at one location and the under stock at the other location. Transshipment level from location j to location i can similarly be written as:

X j i = min [ ( Q j − D j ) + , ( D i − Q i ) + ] .

Given the transshipment quantity above, sales for location i can be written as R i = min ( D i , Q i ) + X j i units while unsold stock at location i is

U i = ( Q i − D i − X i j ) + and unmet demand is Z i = ( D i − Q i − X i j ) + .

Similarly, the same expressions can be written for location j as follows:

R j = min ( D j , Q j ) + X i j ; U j = ( Q j − D j − X j i ) + ; Z j = ( D j − Q j − X i j ) +

The expressions above depend on D i , D j , Q i and Q j ; therefore, expected pro- fit at location will be:

е i ( Q i , Q j ) = E [ y i R i + ( C i j − t i j ) X i j − C j i X j i + s i U i − p i Z i ] − w i Q i (a)

Next, we develop a classic newsvendor problem using equation (a) above. This newsvendor has no transshipments but provides very useful benchmark. By setting X i j = X j i = 0 and eliminating the subscripts, we get the expected profit for the newsvendor as:

е n ( Q ) = E [ y R + s U − p Z ] − w Q = E [ y min ( D , Q ) + s ( Q − D ) + − p ( D − Q ) + ] − w Q (b)

Let the probability of stockouts be denoted by Prob ( D > Q ) , the marginal profitability of increasing the stock level is given by:

δ е n δ Q = y Prob ( D > Q ) + s Prob ( D < Q ) + p Prob ( D > Q ) − w (c)

The optimal inventory choice trades off the expected marginal benefit with its marginal cost, w . There exists a unique order quantity Q n that maximizes the newsvendor’s profit if the distribution over demand is continuous and strictly increasing. Setting Equation (c) to zero and rearranging, we arrive at a similar solution for the newsvendor problem:

Prob ( D < Q ) = ( M − w M − s ) (d)

The expected total profits for the two locations will therefore be:

е t ( Q 1 , Q 2 ) = E [ y 1 R 1 + y 2 R 2 − t 21 X 21 − t 12 X 12 + s 1 U 1 + s 2 U 2 − p 1 Z 1 − p 2 Z 2 ] − w 1 Q 1 − w 2 Q 2 (e)

Next, we calculate the derivative of expression (e) above to obtain the marginal unit of inventory at location i as follows:

δ е t δ Q i = ( y i + p i ) Prob ( D i > Q i + ( Q j − D j ) + ) + ( y j + p j ) Prob ( D j > Q j + Q i − D i , Q i > D i ) + t j i Prob ( Q i < D i < Q i + Q j − D j ) − t i j Prob ( Q i + Q j − D j < D i < Q i ) + s i Prob ( Q i > D i + ( D j − Q j ) + ) + s j Prob ( Q j > D j + D i − Q i , D i > Q i ) − w i

Collecting terms, we have:

δ е t δ Q i = M i ( 1 − Prob ( D i < Q i ) − Prob ( Q i < D i < Q i + Q j − D j ) ) + ( t j i + s j ) Prob ( Q i < D i < Q i + Q j − D j ) + ( M j − t i j ) Prob ( Q j + Q i − D j < D i < Q i ) + s i ( Prob ( D i < Q i ) − Prob ( Q i + Q j − D j < D i < Q i ) ) − w i (f)

Expression (f) above has three components:

1) The additional inventory at location i will result in incremental sales at location i which will yield M i per unit except when there is surplus inventory at location i or the effective excess supply at location j could have enabled transshipments from j to i . In this case, the marginal unit of Q i is t j i + s j .

2) If there is no stockout at location i but demand exceeds effective supply at location j , the marginal unit can be transshipped to j , which will yield M j − t i j .

3) If demand at location j does not exceed effective supply, and there is no stockout at i , the marginal unit is worth only its salvage value s i . We can subtract the marginal procurement cost w i from the above to get the net marginal benefit.

Event I

In this event, location i has demand D 1 , which is greater than the stock quantity Q 1 ; whereas location j has demand D 2 less than its stock quantity Q 2 . This event is parallel to event V.

Event II

Location i is out of stock whereas location j is overstocked. This event is parallel to event IV.

Event III

Both location i and j are overstocked in this event and therefore, no tra- nsshipments occur in this event.

Event IV

Location j is out of stock, whereas location i is overstocked and the joint stock quantities for both Location i and j is more than the aggregate demand at the two locations.

Event V

In this event, Location j has demand D 2 , which is greater than its stock quantity Q 2 , whereas Location i has demand D 1 less than its stock quantity Q 1 . Hence, Location i has a surplus stock ( Q 1 − D 1 ) and Location j is out of stock by the amount ( D 2 − Q 2 ) . Also, in this event, the total demand of Location i and j denoted as ( D 1 + D 2 ) , is greater than their aggregate stock ( Q 1 + Q 2 ) .

Event VI

Both Location i and j are out of stock in this event and no transshipments occur.

The probability functions of the various events are given in

The work of Robinson [

Rearranging expression ( f ) , the conditions characterizing the optimal inventory order are:

α i ( Q i ) − β i ( Q i , Q j ) ( M j − s i − t i j M i − s i ) + γ i ( Q i , Q j ) ( M i − s j − t j i M i − s i ) = ( M i − w i M i − s i ) (g)

Event | Description | Probability | Notation | Transshipments |
---|---|---|---|---|

III | Both ations | |||

V | Location | |||

II | Location |

Source: Li Zou [

The right-hand side of Equation (g) above is the adjustment of the newsvendor solution. The left-hand side adjusts Q i up due to the possibility of transshipment from location i to j ; and the third on the left-hand side adjusts Q i down due to the possibility of transshipments from location j to i .

Next, we continue to calculate the transshipment prices and we ask the following question: does there exist a pair of transshipment prices that maximizes the total profit? If yes, then this can be implemented in many practical situations by the parent firm as incentive design to make each location behave in order to optimize aggregate profit.

There exist a unique set of transshipment prices C i j for i , j = 1 , 2 that yields the joint optimal solution given by:

C i j * = ( M j β i β j + ( s j + t j i − M i ) β j γ i − ( s i + t i j ) γ i γ j β i β j − γ i γ j ) (h)

It is sufficient to look for transshipment prices which induce the locations to choose optimal order quantities.

In this section, we compute explicit numerical example to illustrate the analytical results. Assuming demand realizations at the two locations are independent and distributed normally, with mean 100 and standard deviation 50. Thus,

D i ~ N ( 100 , 50 ) . We truncate the distribution at 0 and redistribute this proportionally to the positive part of the distribution in order to rule out negative realizations of the demand variable. It should be noted that, this yields a slightly higher mean and slightly lower variance.

As provided in

The optimal order quantities and average profits are illustrated in

Cost parameter | Notation | Assumed values |
---|---|---|

Procurement cost | 20.00 | |

Selling price | 40.00 | |

Salvage value | 10.00 | |

Penalty cost | 0 | |

Transshipment cost | 2 |

Source: Li Zou [

Optimal Inventory Q | Average Profit at each location, | ||
---|---|---|---|

Without transshipment | Newsvendor | 122.5 | 1530 |

With transshipments, thus | 12 | 107.0 | 1660 |

18 | 112.3 | 1672 | |

20 | 114.1 | 1674 | |

22 | 115.9 | 1675 | |

23.3 | 117.1 | 1676 | |

26 | 119.4 | 1675 | |

35 | 127.0 | 1661 | |

40 | 130.9 | 1648 |

Source: Rudi, Kapur, and Pyke [

As the numbers in

In conclusion, this study analyses the role that transshipments play in modern inventory management and its benefits between two independent locations belonging to one parent firm. It is found in this study that, the use of an appropriate transshipment price is an effective tool for these locations to maximize their profits and optimize inventory level decisions. There exists a unique transshipment price that is optimal for both locations.

We show in the study how this also affects the optimal inventory orders at each location and found that, if each location aims to maximize its own profit, their inventory choices will not in general, maximize joint profits. In the case where inventory choice varies with transshipment prices, we find transshipment prices that induce the locations to choose inventory levels consistent with joint- profit maximization.

For future research, it is worthwhile to consider larger system consisting of more than two firms since our study only considered a two-location newsvendor problem with and without transshipments for a single replenishment period.

Anthony, B. and Augustin, D.S. (2017) Intra-Firm Inventory Management Model with Transshipments. American Journal of Industrial and Business Management, 7, 15-26. http://dx.doi.org/10.4236/ajibm.2017.71002