Lease Contract in Usage Based Remanufactured Equipment Service System

In this paper we study a usage based lease contract for remanufactured equipment as an implementation of product service system. Under this lease contract, the equipment is leased for a period of \( \Gamma \) with a maximum usage, U. If the usage of the equipment exceeds U at time \( \Gamma \), then lessee will be charged some additional cost. Otherwise there will be no additional cost. The price of the lease contract for the remanufactured equipment is much cheaper than that of a new one. As a result, the lease contract for the remanufactured equipment would be a more attractive option to the lessee. The decision problem for the lessee is to select the best option suitable to its requirement, and the decision problem for the lessor is find the optimal maintenance policy and the price for each length of periods offered. We provide numerical examples for illustrating the optimal decisions for the lessee, and the lessor, which maximizes the expected profit for each party.


Introduction
In recent global economy, a manufacturing company cannot just provide products alone to the customers, in order to remain competitive, but it needs to offer solutions for their businesses. An innovative way to achieve this is to offer a package of product and services called as a product-service system (PSS). [1] defines PSS as "an integrated product and service offering that delivers value in use". As a result, under PSS, there is a significant shift from a "traditional" product oriented model which provides products alone (or selling products) to a "service oriented" model (e.g. selling either product usage or performance) that will give the opportunity for the manufacturing company to gain competitive advantage [2)]. This shift also contributes to reducing associated environmental impacts (i.e. refurbishment, remanufacturing and recycling of durable products, which save more energy and reduce waste through the product's life) and the volume of goods in the economy to the sufficiency strategies and establish long-term relations with customers. The others effect of the PSS are reducing lifecycle impacts of products and services through product servicing, remanufacturing and recycling [3]. Research of lease contracts in PSS have been studied by many researchers (See [4], [5], [6], [7]). Since leasing equipment (rather than purchase it) becomes a common practice in company that functioning equipment as generating revenue. It also has many positive points i.e saving on initial investment, flexibility on equipment upgrading, and cost reduction in maintenance and inventory [8]. Moreover study lease contract (LC) that involves both lessee and lessor has been attracted by many researchers. A comprehensive review of LC from the lessor's or lessee's perspective can be found in [9]. For the case where the study is done from the lessor and lessee point of views then a game theory formulation is needed to modelling the decision problems (See [10]). The LC for the new lease item, can be found in [8], [11] to name a few, whilst [12], [13] examined the lease contract for used items. Finally, [14] considered lease options which include a remanufactured equipment. When the equipment is used intensively (or with high usage) per unit of time, the usage experienced affects significantly the deterioration of the equipment. This indicates the need to consider age and usage in modelling the failure and also defining the lease contract which involves two parameters -i.e. age and usage limits (called a two dimensional lease contract). We are aware only the works by [15] and [16] belong to this group. In [15] the period of the contract is always the same with a maximum usage rate whilst [16] consider a two dimensional lease contract for maximum age or usage.
In this paper, we study a usage based lease contract for remanufactured equipment as an implementation of product service system. We consider a multi-period LC in which each period has a time limit but no usage limit. However, if the usage exceeds the maximum usage allowed in the contract, then the lessee has to pay some additional cost. In general, Original Equipment Manufacturer or OEM (as a lessor) offers not only a LC for a brand new equipment but also a LC for a remanufactured one. As the price of the LC for the remanufactured equipment is much lower than the price of a new one, and hence it would be a more attractive option to the companies. This paper deals with a multi period lease contract for a remanufactured equipment (such as dump trucks) in which the price scheme of LC gives some incentive for the lessee when the equipment is leased for more than one periods.
The paper is organised as follows. In section 2 we give model formulation for the two dimensional lease contract studied. Sections 3 and 4 deal with model analysis and the optimal decisions for the lessor and the lessee. In section 5, we provide with a numerical example. Finally, we conclude with topics for further research in Section 6.

Model Formulation
In this section, we first define a new LC, describe failure model, formulate a preventive maintenance policy and its effect on reliability, and then obtain the expected profit for a lessor and a lessee. Notations:

Modelling Failure
In general, most products at the end of the first life or end-of-use have a low reliability (or their reliability is below the threshold value of reliability R*). Remanufacturing involves disassembly, cleaning, and refurbishment or replacement of parts to improve the reliability of the equipment to a like-new one or it improve the reliability of the product to at least the same level of the threshold reliability. Let , then the reliability of the remanufactured product is We model failure of the remanufactured product as follows. It is considered that failure is not only influenced by age but also usage. Let Y be the constant usage rate for a given customer (e.g. y=120 km/day for a dump truck). For a given customer (or usage rate, y ), let () y rtbe the conditional hazard function which is a non-decreasing function of t (the age of the truck) and y. An accelerated failure time (AFT) model is proposed to model the effect of age and usage rate on degradation of the truck. In AFT model the distribution function for Ty is given by F(t, αy), with a scale parameter given by  

Analysis
We carry out the analysis to obtain the expected profit for the lessor, and the lessee.

Lessor's Expected Profit
The lessor's expected total cost consists of preventive maintenance cost and corrective maintenance cost . If 1 ( , ) After simplification for m =1, we have the expected total cost of the lessor given by Expected total revenue: The expected total revenue is the sum of the price of LC and some additional revenues due to the total usage of the equipment is greater than max U (the maximum usage). The price of LC is dependent on the usage rate and the number of LC periods () m given by where d C is the additional cost charged (e.g. $/km or $/page copied). ( , ) my  is viewed as some additional revenues for the lessor. As a result, the expected profit is given by ( ) ( , ) , where r b C is the annual cost of the remanufactured equipment.

Lessee's Expected Profit
The lessee's expected profit is equal to the expected total revenue minus the total expected cost, that will be given as follows.

Case 1: [Joint Optimization]
We consider that the lessee and lessor would like to work jointly to obtain a joint optimal profit. Then, the strategy set of the lessor and the lessee is given by

Numerical Example
The  , and the profit for the lessee is less than that of lessor started at y=3.0 (as the additional cost charged to the customer is much bigger as y is larger (y>2.5)). In contrast, the profit resulting from the Nash bargaining solution (Case 2) is always the same for both the lessor and the lessee for each y, and it decreases as y increases. This is due to the total profit generated by both parties decreasing as y increases (or more failures occur and hence bigger downtimes as the usage is larger). The joint optimization is favor the lessor. This is expected as the bargaining strategy shares equally the total profit generated.

Managerial implementation
The results shows that implementation of the leasing concept connected to a product remanufactured can undoubtedly deliver economic benefits to the lessor and lessee.
As it is not a new product, and hence a thorough preparation of a special marketing strategy that results in a customer acceptance is required. This is agreed with [3].

Conclusion
In this paper we have studied a usage based lease contract for remanufactured equipment such as dump trucks. Under this lease contract, the equipment is leased for a period of  and a maximum usage, U. We find the optimal solution jointly (joint optimization) for both parties, and then seek the optimal decisions using a Nash game theory formulation. One can model the decision problems for the lessor and the lessee using a Stackelberg game theory formulation, and consider a subsequent of LC periods in which the usage pattern may change significantly from period to period (this is due to a different lessee leases the equipment). These two topics are currently under investigation.