Manufacturer’s Optimal Auditing Policy of Supplier's Capacity

TABLE OF CONTENTS

ABSTRACT

1 INTRODUCTION
1.1 Intention and Context of the Research
1.2 Motivation and Academic Purpose

2 LITERATURE REVIEW
2.1 Newsvendor Model: Optimal Inventory and Production Capacity Policy
2.2 Competition in the Supply Chain
2.3 Principal-Agent Models and Asymmetric Information
2.4 Optimal Equilibrium Procurement Strategies
2.5 Supplier Auditing

3 ORGANIZATION OF THE ARTICLE

4 MATHEMATICAL MODEL DEVELOPMENT
4.1 Pre-Assumptions and External Contracting Conditions in the Manufacturing Context
4.2 Application-Implicit Transformation of Action into Reasonable Variables for the Mathematical Model
4.3 Mathematical Model Formulation
4.4 Mathematical Model Specification
4.4.1 Assumption of Two Types of Suppliers and Two Different Production Schedules
4.4.2 Simplification Approach: Reduction of Decision Variables
4.4.3 Individual and Overall Welfare Considerations
4.5 Optimality Analysis
4.5.1 Conversion of Min / Max Functions
4.5.2 Optimal Capacity Auditing Probability of the Inefficient Supplier Type
4.5.3 Optimal Order Quantities for the Efficient and Inefficient Supplier Type

5 NUMERICAL STUDY AND SENSITIVITY ANALYSIS
5.1 Transfer of the General Model into MATLAB
5.2 MATLAB Solver for Optimal Selection of Decision Variables
5.3 Sensitivity Analysis
5.3.1 Impact of Efficient Supplier Occurrence Probability on Manufacturer’s Profit
5.3.2 Impact of Marginal Retailing Price Level on Manufacturer’s Profit
5.3.3 Impact of Efficient Supplier Occurrence Probability on Inefficient Supplier Capacity Audit Probability
5.3.4 Impact of Inefficient Supplier’s Untrustworthiness Penalty on Inefficient Supplier Capacity Audit Probability
5.3.5 Impact of Market Demand Variation on Optimal Order Quantity
5.3.6 Impact of Efficient Supplier Occurrence Probability on Optimal Order Quantity

6 SUMMARY AND FUTURE OUTLOOK
6.1 Summary of Results and Research Process
6.2 Limitations and Future Outlook

LIST OF NOMENCLATURE

TABLE OF TABLES

TABLE OF FIGURES

REFERENCE

APPENDIX

Reformulation as a Linear Program

ABSTRACT

This research investigates a manufacturer’s optimal capacity auditing policy in the principal-agent context facing asymmetric information. Supply chain coordination through supplier capacity auditing has the ability to affect both supply chain efficiency and risk management challenges. So far not a lot of research on principal-agent relationships in the manufacturing context has been performed – specifically not on supplier capacity auditing with one party facing asymmetric information. The basic assumption of this model is the contracting relationship of one manufacturer and one supplier. A price-quantity schedule is offered by manufacturer and the supplier selects one single-period fixed-price contract without any further communication and negotiation. The manufacturer’s decision according to the price-quantity schedules offered is upon order quantity, transfer payment and the audit application probability. The supplier’s production capacity is known to the supplier at all times but unknown to the manufacturer. In addition to that, the demand is stochastically distributed and concretizes after both parties agreed about the contract.

Basic profit functions of the manufacturer and the supplier are adjusted to match the requirements of the manufacturing context. In order to derive optimal closed form solutions, the equations are modified following an approach of [1]. An optimal capacity audit mechanism for a manufacturer who procures components from a supplier possessing private capacity information is pointed out. Welfare gains can be found for the manufacturer and the overall supply chain – originating from the supplier’s altered incentive compatibility constraints in response to the audit threat.

Key words: Supply Chain Efficiency; Supply Chain Coordination; Newsvendor Model; Principal-Agent; Asymmetric Information

1 INTRODUCTION

1.1 Intention and Context of the Research

A continuously increasing price pressure and more specific production orders are two indicators of the structural change the manufacturing industry is facing during the last years. Intensified customer and market expectations force companies to focus on their core competencies and outsource more general work processes to specialized partners. The inter-company cooperation logically becomes a very important piece of production planning and control. The network of related companies today is the modern and most important organizational form.

Due to the reasons just mentioned and others, companies are facing an increasing competitive pressure [2]. A consequent and iterative alignment with customer requirements is an important prerequisite of sustainable long-term success [3]. According to an AT Kearny study count the companies’ logistic capabilities as a crucial purchase criterion for customers [4]. Differentiation only through product features is no longer enough to survive the tough competition [5; 6]. The logistic indicators delivery time and delivery reliability developed as purchase criteria next to product quality and price. To fulfill the customer requirements economically efficient – facing high endogenous and exogenous uncertainty in this complex network structures – a rather holistically approach is required [7].

Several reasons for limited growth potential of the profit margins in companies can be found. First, the main trends of globalization and digitalization lead to transparency of cost, diverse procurement channels and therefore to international competition. Second, a higher volatility in all sections of the market has to be noticed. Technical developments or exogenous shocks possibly lead to a highly variable market price and demand at any time. This new level of uncertainty has to be addressed with appropriate risk management. Third, technical products’ optimization potential decreases with every attempt on that subject. By now for many products only incremental product development is possible. Respectively high cost savings are hard to realize when changing the technical product itself and without loss of product quality.

Other product related processes on the other hand offer potential for optimization and promise high impact. Supporting processes need to be investigated in order to save cost wherever possible and increase companies’ profit margins. Due to the growing relevance of goods exchange in the network of related companies, opportunities for huge scaling effects exist. Only little developments have a strongly positive effect to a company’s profit. One of these opportunities e.g. can be found in the upstream supply chain. Procurement has been identified to be one of the major drivers in a company’s ability to compete in the global market place [8]. Mainly due to the fact that the final product costs are directly affected by the procurement costs [9]. A suitable and efficient procurement strategy therefore has to be developed. When a customer product manufacturer purchases parts of the final product from a supplier the situation can be used for implementation of an efficient procurement strategy. Usually both parties agree at least for a share of the expected quantity needed about the terms and conditions of the cooperation and fix the result in a contract. The coordination of the supply of key components with the product demand furthermore is a challenge to all market players.

Unexpected variation of product demand, component cost, and availability uncertainties can have a significant impact on a manufacturing company’s profits. Uncertainty in product demand, component cost, and component availability results e.g. occurs form external shocks such as natural hazards, terrorism or political instability and results in significant procurement risks [10]. E.g. Dell needed to pay higher-than-expected memory prices in 1999 and the company’s stock price thus dropped about 7%. In 2001 Ford suffered a $1 billion loss because of forward-contract agreements on the precious metal palladium. Cisco Systems had to take a $2.5 billion inventory write-down due to low demand in the same year [10]. In modern supply chains information seems to replace inventory. There is no doubt that information changes the way supply chains should be managed effectively which maybe leads to lower inventories. Having accurate and instant information should not make the managing process less effective than if this information is not available. Possessing and constantly generating this information provides a tremendous opportunity to improve the way the supply chain is designed and managed. It will help to reduce variability in the supply chain, to make better forecasts and enables a better coordination of supplier-manufacturer relationships. In strategic partnerships information can be shared and inventory can be managed efficiently within the supply chain [11]. The following two figures illustrate the main threads just mentioned and show how to react in a reasonable manner. In order to adapt to the changed conditions an efficient supply chain strategy and an appropriate risk management are of very high value.

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Figure 1.1 Supportive Environment for Disruptive Changes in Supply Chain Applications

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Figure 1.2 Supportive Environment for Disruptive Changes in Risk Management Applications

One attempt to reduce the threats caused by uncertainty is optimal option contracting. Manufacturers generally have the choice of purchasing goods from suppliers or from the spot market. Option contracts assure manufacturers a future supply at a pre-negotiated price, a practice widely used in high-tech industries. In customized contracts manufacturers pay suppliers a premium for the option to buy a fixed quantity at a capped price. “ If the spot price increases above forecasted levels, the option is exercised at the fixed strike price for the agreed-upon quantity. Conversely, if the spot price falls below the strike price, the procurement teams can let the option lapse and buy in the open market” [8]. A manufacturer who procures an agreed amount of products from his supplier using ex-ante agreed option contracts takes a lot of pressure from his supplier and pools demand risk. This aspect and the enhanced flexibility in ordering from suppliers enables the manufacturer to more appropriately manage the trade-off of price and inventory risk. In the traditional long-term contracts both parties agreed about a fixed quantity rewarded with a fixed price in the future. This type of contract implicitly incurs a huge inventory risk for the manufacturer. This risk can be reduced with option contracts due to their flexibility somewhere in the middle between fixed price contracts and purchasing from the open spot market [9]. The supplier has the advantages of no longer facing demand uncertainties – allowing him to better plan his own capacities, procurement and production. Receiving the option premium even if the manufacturer does not purchase, in case spot market price or market demand drops, is his security and incentive to do so. These advantages are rewarded with a lower product price paid by the manufacturer. The utility of both parties simultaneously increases.

Another attempt to reduce the threats caused by increasing cost transparency, decreasing optimization potential of technical products and a growing volatility in market demand and uncertainty in general is supplier capacity auditing technology. A manufacturer and a supplier agree about the delivery of a specific quantity for a specific price. Risks are pooled at the manufacturer’s side and the supplier can fully exploit the effects of scale without facing any uncertainty in his production. In consequence of that he is able to offer the products for a lower price. The overall supply chain network is able to significantly leverage its efficiency, increases its capacity and both parties of the supply chain profit at the end. This research work therefore will focus on an optimal contractual policy under audit uncertainty from a manufacturer’s perspective.

1.2 Motivation and Academic Purpose

Supply chain management can be understood as a set of approaches utilized to efficiently minimize system wide costs while satisfying service level requirements at the same time [12]. A supply chain strategy determines the nature of procurement of raw materials, transportation of materials to and from the company. In addition to that, all manufacturing operations or activities to provide the service. Moreover, distribution of the product to the customer and follow up service activities. “Given that firms are rarely completely vertically integrated, it is important to recognize that the supply chain strategy defines not only what processes within the firm should do well but also what the role played by each supply chain entity is” [13] . Supply chain strategy includes a specification of the structure and what many traditionally call "supplier strategy," "operations strategy," and "logistics strategy" [13]. The relation of supplier and manufacturer – determined by the contracts both parties agree about – therefore is part of the supply chain strategy of a company. As will be shown in the following, to further optimize this relationship several scholars recently researched on many different contract types. E.g. buy-back contracts, revenue-sharing contracts, quantity flexibility contracts, advanced-purchase discount contracts and option contracts. As already mentioned above this research work will focus on an optimal contractual policy under audit uncertainty from a manufacturer’s perspective and its value to corporate goals. Easily the connection between effective contracting and business success can be found in the application of a reasonable supply chain strategy.

Target figures in the supply chain management are delivery time, delivery reliability and delivery cost. Effective contracting offers the power to positively vary all of them. Reacting instantly is a crucial asset to fulfil important customer requirements. Contracts using audit technology offer the possibility to take the manufacturer-supplier relationship several levels up. Due to an advanced capacity management through option contracts, the manufacturer knows what quantities for sure to purchase from his supplier and when to target the spot market. Option contracts guarantee fixed prices for an agreed amount of parts. In case excess capacity is needed the manufacturer targets the spot market. In case the capacity finally needed is lower than the optional agreed quantity or spot market prices drop heavily, the lapse of the option is the better choice. The manufacturer has to pay the option premium. Complementary the striking price is lower and due to the flexibility of the contract inventories can be kept at low level causing only little holding or other overage costs. Ex-ante knowledge allows better planning and appropriate reactions in case of market demand or market price changes. Possession of information needs not strictly to be an advantage but it for sure will not harm. More likely sharing valuable information throughout the whole supply chain will help to minimize process risks and increase the net value added of all supply chain participants. With option contracts it is possible to combine the security of optional pre-contractual agreed purchasing and the advantage of a favourable price. Hewlett-Packard reported cumulative cost savings of $425 million from their procurement risk management approach [10]. This practice case clearly shows the advantage of efficient and appropriate risk management, resulting in a system-wide optimality. It proves that the utility of both contract partners can be leveraged through well-balanced option contract design.

Assuming that a supplier and a customer product manufacturer never exchange only one single contract offers for the production of one kind of part, but a whole schedule of possible price-quantity-contracts among to make a decision. E.g. due to positive scale effects of the supplier and negative inventory risk of the manufacturer, a higher optional procurement quantity will be rewarded by the manufacturer with a lower marginal price only. Moreover, taking into account that every supplier has restrictions in his own procurement and production system. Obviously this restrictive area just described is rather complex and the one optimal contract out of the set of contracts is probably not observable for the manufacturer. Assuming further that a manufacturer purchases every part of the final product from only one supplier. All these reasons just described lead to an information gap between the contracting parties and open the space for untruthful behaviour of the supplier. The manufacturer might be interested whether the supplier works efficient or not efficient and out of this reason maybe asks an excessive price or disposes a rather small capacity. To check the efficiency and capacity of suppliers – with the intention to maximize own profits or simply contract the best and most reliable suppliers that expend highest efforts to develop further – supplier audits could be the method of choice. Audit technology helps to detect the supplier’s untruthful report and imposes the supplier with some punishment when false report is detected. At a specific cost the manufacturer is able to identify the true nature of the supplier with an audit. These cost of audits may hinder a systematic use of audits [1]. In practice, audits can be considered a special form of test. They investigate whether an investigated unit fulfills all requirements or not. Audits are performed systematically meaning that a reasonable, planned and consequent procedure is applied. Very important for the conduction of audits is the independence of the auditing institution [14].

As a basis of this academic thesis an attempt of optimal response strategies will be used. The equilibrium of an incomplete information game consisting of two parties: a capacity-constrained supplier and a manufacturer, will be analyzed. The supplier has a specific capacity level which is his private information. Conditioned on these capacities, he selects – without any cooperation or further negotiations with the manufacturer – one out of the offered quantity-price schedules. At the equilibrium the goal of each manufacturer is to design a quantity-price schedule that maximizes his expected profit. This research intends to investigate from a manufacturer’s perspective how to appropriately address the risk of audits and audit caused penalties in a supplier-manufacturer relationship. Existing research will be mathematically extended in the sense of additionally taking into account supplier capacity audit risk. A sensitivity analysis will be performed in order to extract valuable understanding upon which to buyer-utility-efficiently design option contract offers in the future.

2 LITERATURE REVIEW

A lot of different literature can be found related to the topics inventory policy, supplier production capacity, competition in supply chains, equilibrium procurement strategies, different attempts of option contracting, principal-agent models and asymmetric information or auditing. This chapter of the thesis intends to give a short overview about all different kinds of literature related to a supplier’s optimal order quantity when he has the lead in the contracting process. Supply chains can be coordinated by using multiple channels. In this chapter some of them and the theories behind them will be investigated.

2.1 Newsvendor Model: Optimal Inventory and Production Capacity Policy

In the context of finding a supplier’s optimal order quantity, the optimal inventory policy plays a great role. [15] discussed the value of sharing supplier’s capacity information in a relationship of one single manufacturer and two suppliers. The research offers great contributions concerning capacity information sharing but on the other hand does not fully investigate how other parameters affect the performance of supply chains and how several other parameters interact. A production planning model with uncertain capacity and demand was studied by [16]. Facing increasing product complexity, manufacturing environment complexity and increased emphasis on product quality are only a few uncertainties in today’s production processes. An adequate model must incorporate these uncertainties in production planning. In their work they analyze an aggregate planning problem for a single product with a random demand and a random capacity. They indicate optimal cost policies for a manufacturer with respect to different time horizons and come to the conclusion that in single-period problems random capacity does not affect the optimal procurement policy and in multiple-period problems order-up-to policies – that are dependent on the distribution of capacity – are optimal. Moreover, production planning decisions must incorporate the effects of internal uncertainties on costs.

Every participant in a production network has definite and in the short term even deterministic capacity constraints which have to be considered in contract agreements with partners. [17] show in their research an interesting approach on supplier-capacity. Basically a manufacturer needs a set of components for his own production, each produced by and procured from different suppliers. The firm needs to determine their individual production capacities before observing the actual demand. Both manufacturer and supplier are allowed to decide on supplier’s capacity in different scenarios. In this research, a centralized and a decentralized equilibrium is observed. The authors suppose two different situations in which manufacturer and supplier separately can decide upon supplier’s production capacity level by leveraging the price of products in the contract. For the decentralized setting, the situation when suppliers set the price, the authors come to the conclusion that the performance degrades in both the manufacturer’s share of production capacity and the number of suppliers. The centralized setting dominates the decentralized setting if the manufacturer’s share of capacity cost is larger than the reciprocal of the number of firms involved.

2.2 Competition in the Supply Chain

[18] identified the problem of “double marginalization”. There exists a coordination failure in this serial supply chain because there are two margins and neither firm considers the entire supply chain’s margin when making a decision. [19] or [20] show possibilities to coordinate supply chains through the usage of contracts. As already mentioned above a reasonable and well-adjusted supply chain strategy is an important instrument in the organization of a production network. Furthermore, the supply chain strategy is a crucial asset in satisfying customer requirements. Different research found evidence that the information structure of a supply chain plays an important role in its performance. One could also assume that role may become even more critical in the presence of competition [21–23]. Vertical information sharing is investigated by [24] who research on contracting and information sharing in two competing supply chains. Two competing supply chains are compared in order to detect a difference triggered by an information sharing culture. Both supply chains are equal besides the possibility of different investment cost for information sharing. A two-stage approach is applied. At the first stage a player has to decide whether to invest in information sharing and if yes, how much to invest in it. At the second stage the sharing information structure requested at the first stage is implemented. The value of information sharing in a supply chain is studied regarding two different contract types: contract menus and linear price contracts. Facing contract menus on the one hand a supply chain that does not have information sharing will lower its selling quantities because of negative quantity distortions. The dominant strategy therefore is to invest in information sharing infrastructure when investment costs are low. Facing linear price contracts on the other hand the value of information sharing becomes negative to a supply chain and the dominant strategy is not to invest in information sharing infrastructure.

2.3 Principal-Agent Models and Asymmetric Information

The underlying case of this research can easily be interpreted as a principal-agent model. The supplier (agent) has private information and tries to sell a quantity of products to the manufacturer (principal) who possesses no private information. The agent selects one option out of a price-quantity schedule – a set of combinations of quantity and product price. The principal makes sure to satisfy the market demand with his offer. Expending a specific effort allows the principal to reveal an agent’s true character [1].

[25] consider an informed principal model. They study the reliability of signaling in after-sales service contracts. [26] consider the existence of two suppliers and a manufacturer who possesses private information regarding his valuation function. Asymmetric information often exists in principal-agent situations because the agent very likely possesses private information and knowledge which is not available to the principal. This information gap for sure has fundamental implications for the contract design ([1]. To reach the optimal for both contract parties it is important to overcome this existing information gap. [27] suggests maximizing supplier’s profit by designing a contract in the way that the supplier knows the manufacturer’s marginal cost. The value of demand information is researched by [28]. They show that pre-obtained demand knowledge gives a manufacturer some advantages in a single stage model – a model with one time-period only. [29] show the value of pre-possessing demand information. [30] investigate how to demand forecasts can be shared trustworthy with the help of contracts consisting e.g. of commitments and options. [31] sudies a model with firms possessing private information where each supplier gets a signal that consisting of private information about own and other suppliers’ costs. [32] investigate the equilibrium of an incomplete information game. Two capacity-constrained suppliers and a one manufacturer are the players in that game. The supplier’s capacity is their own private information. Simultaneously, the suppliers offer quantity-price schedules to the manufacturer without any cooperation and the manufacturer decides what quantity to purchase. The existence of a Nash equilibrium can be found. The cost of every marginal quantity does not dependent of the production capacities and only depends on the manufacturers’ evaluation function and the distribution of capacity. Another important finding is that the supplier possessing the larger production capacity sells all his potential supply in case the manufacturer asks a higher quantity than every single supplier offers.

2.4 Optimal Equilibrium Procurement Strategies

Supply function equilibria under various scenarios are widely studied by e.g. [33] or [34]. A single period procurement problem is discussed by [35]. A manufacturer can influence a supplier’s capacity with an investment. He finds his expected profit correlated with the supplier’s capacity. [36] investigates the upper and lower bounds of optimal order quantity in an inventory model with uncertain demand and production capacity. The influence of random supplier capacity is shown by [37] in an EOQ-model under random supplier capacity. Capacity constrained supply-function equilibria are investigated e.g. by [38] and [39]. This research mainly is motivated by applications in electricity markets. [40] investigates the profit gain, optimal reorder point and target safety stock when demand and capacity are uncertain. [41] additionally consider outsourcing of capacity to a third party. [42] integrates parameters like holding cost or shortage cost and shows how they affect the optimal capacity level. [43] also study a procurement problem with the alternative of using either an option contract or the instant purchasing option of the spot market taking into account price uncertainty. [44] compare in a multi-period environment a fixed-price long-term contract including price certainty with a flexible short-term contract facing price uncertainty. [45] assume a manufacturer takes into account not only the expected profit but also the associated risk and apply a mean-variance analysis to the procurement contracts. [46] furthermore develop an approach for a supplier’s portfolios to optimally manage demand risk. In this research work a competition model for procurement of short-life-cycle products is investigated. A manufacturer facilitates a specific production capacity at the supplier’s sites before the actual demand realizes. Final production orders are set after the market demand concretizes. The manufacturer does not want to take all capacity and inventory risk and therefore rather signs flexible contracts with several suppliers. The authors design the interaction between suppliers and the manufacturer as a game in in which the suppliers are leading. The suppliers then compete about providing supply capacity and the manufacturer optimizes his expected profit by selection of suppliers. It can be found that supply chain inefficiencies, respectively the loss of profit due to competition, are at most 25% of the profit of a centralized supply chain.

Option contracting has been widely studied during the last years as it promises profit and efficiency gains mainly for the manufacturer. [47] study bidding and contracting actions in option pricing. They investigate a two-part contract fee structure consisting of a reserve capacity cost and a cost to supply product – an option and a strike price. The supplier acts as the leader. They derive both the supplier’s optimal bidding and the manufacturers’ optimal contracting strategies. [48] extend this basic model. They take into account both multiple suppliers and the possibility to purchase from a spot market with stochastically distributed price. [49] illustrate that an option contract provides flexibility for the manufacturer and develop such conditions on the cost parameters that channel coordination can be achieved. [50] develop an option model to quantify a flexible supply contracts. More specifically it can be differentiated between single-period supplier-models and multi-period multi-supplier models. [51] researches on the single-period, single-supplier case. She extends existing approaches for a limited supplier capacity and additionally forces the manufacturer to fulfil the stochastic demand. [30] investigate the effect of information applying a game theory approach. [8] also analyze a single-period model and for two different cases – one taking into account correlated market demand and spot price and another uncorrelated situation. They indicate the benefits of exploitation a portfolio of option contracts in contrast to purchasing from one fixed price contract. The authors find that two carefully selected option suppliers are enough for the manufacturer to nearly achieve an optimal performance in procurement. [9] on top of that work additionally take into account setup cost and capacity constraints. They evaluate portfolio procurement in supply chains assuming to either procure parts for future demand from a seller using fixed price contracts, option contracts or the spot market. In a single-period model with random but possibly correlated product demand and spot price they develop an optimal procurement strategy for the manufacturer. A path algorithm is used to derive the optimal procurement solution and the minimum procurement cost. In addition to the benefit of portfolio contract procurement which is studied, the authors find that in case demand and spot price are independent the procurement cost function offers a rather intuitive possibility for interpretation and therefore different applicable managerial insights. [52] study single-period models showing that a mix of long-term procurement and spot market procurement is optimal. [53] also address this scenario in their single-period models and showed that a mixed procurement strategy consisting of both a long-term contract and spot market procurement is optimal. [54] consider the problem of a manufacturer procuring quantity options from multiple suppliers under demand uncertainty. On top of that [55] analyze a multi-supplier model using portfolios of option contracts. As they additionally assume multiple periods, they moreover take into account replenishment policies.

2.5 Supplier Auditing

“Risk is a threat to an organization (or a project) that reduces the likelihood that the organization (or the project manager) will achieve one or more of its objectives” [56]. Therefore, expediting – following up on actual deliveries made by subcontractors and suppliers – is a typical job performed by independent auditors in procurement. Primary, audits are performed to prevent late deliveries and quality problems from suppliers. The method is usually performed in advance. [57] and [58] postulated that only relatively little attention is paid to non-cost factors such as quality and delivery performance. Resulting in evolved manufacturing systems which do not satisfactorily contribute to the competitive position of the company and in manufacturing strategies which are not coherent with overall business strategies. Audits are an important part in the strategy formation process. The authors indicate that identification, collection and structuring of information is a critical activity. They furthermore introduce the concept of efficiency audits that investigate whether resources are being utilized as optimal as it is practical. [59] induces seven main reasons – including changing external demands and the power of information technology – why business performance measurements are so topical. To help customers to decide with which suppliers they should concentrate their business in order to rationalize their supply base, supplier audits are the method of choice. [60] examine the association between agency cost and audit-procurement practices and also the combination of audit-procurement practices, audit quality and audit fees. They find that well-developed audit-procurement practices are directly related with the hiring of auditors possessing a higher level of industry experience and that well-developed audit-procurement practices have a positive effect on audit fees. [61] develop a model of audit production and find that audits are more efficient for clients that are larger and are highly automated. [62] investigate the efficiency of audit production and research whether any inefficiencies in the production process are correlated with audit pricing. The work proves that moderate inefficiencies are economically costly for the firm.

[1] in their famous work come to the conclusion that a situation might be imaginable when it is of higher utility to an efficient agent to pretend that he is inefficient. The combination of transfer received for an agreed quantity produced satisfies his utility function better with inefficient than with efficient parameters. However, the principal on the other side prefers to contract an efficient agent because of his higher generated output per marginal transfer unit. The authors intended to relax the efficient type’s incentive constraint and thus make it costly for him to lie and reveal his true character. They assume that besides the threat of termination in long-term contracting relationships a principal has the opportunity to check his agent with audit technology. The probability of an audit and the monetary punishment connected with an agent’s non-truthful report has – due to the fact that audit application costs and hypothetical punishment costs going hand in hand with auditing – significant impact on the principal’s profit function and the probability of the agent reporting truthfully. [1]

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Figure 2.1 Specification - Supplier Capacity Audits in Manufacturing Context

3 ORGANIZATION OF THE ARTICLE

This research work is supposed to merge key research findings of both supply chain efficiency theory and uncertainty theory in order to close the research gap of an optimal contractual policy under audit uncertainty form a manufacturer’s perspective. Therefore, literature and attempts concerning optimal order quantity research, capacity and inventory theory need to be evaluated and applied. The equilibrium of an incomplete information game consisting of two parties: a capacity-constrained supplier with private knowledge and a manufacturer without any private knowledge, will be analyzed. The supplier has a specific capacity level which is his private information. Conditioned on this capacity he selects one out of the offered quantity-price schedules. This happens without negotiation or cooperation with the contractor. The goal of the manufacturer is to design a price-quantity schedule that allows with high probability the maximization of his utility function. The supplier also aims to maximize his utility function and selects the most-optimal quantity-price schedule.

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Figure 3.1 Assumed Framework for Supplier-Manufacturer Contract Model Development

Starting from the research on the value of supplier’s capacity information [63] and from the basic economic concept newsvendor problem and principal-agent model including the asymmetric information problem [1], this research intends to investigate from a manufacturer’s perspective how to appropriately address the risk of audits and audit caused penalties in a supplier-manufacturer relationship with the utility-maximal launch of efficient price-quantity schedules. Existing research will be mathematically extended in the sense of additionally taking the possibility of supplier capacity audits into account. It will be shown how the existence of audits, the probability of performing an audit and the penalty connected with an untruthful report detected by auditing affect the utility of a manufacturer and the decision of a supplier. Further research on existing attempts to describe the problem will be performed. A mathematical closed form equation will be derived and as an extension attached to an existing model. An analytic approach to the problem is the scientific necessity. The adjustment of existing attempts to a manufacturing context with its very specific general conditions will be important. Existing research so far did not take these special conditions into account and therefore existing attempts are not decidedly useful in manufacturing context. Last a sensitivity analysis will be performed in order to extract valuable understanding upon which to manufacturer-utility-efficiently design option contract offers in the future.

4 MATHEMATICAL MODEL DEVELOPMENT

In the following we in terms of time assume that the manufacturer offers a set of price-quantity contracts to his supplier (t = 0). The supplier knows his own unique production capacity (t = 1) before he accepts or rejects the contract offers (t = 2). The manufacturer has the possibility to audit the supplier’s capacity and can force him to produce up to his maximum production capacity (t = 4). In case the supplier accepts one price-quantity contract out of the whole set, he produces and delivers to the retailer before the selling season (t = 3). Right after the manufacturer receives the quantity of products and reordering is not possible, the season demand occurs (t = 5). Finally, the manufacturer pays the transfer to his supplier (t = 6).

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Figure 4.1 Assumed Timeline of Actions

4.1 Pre-Assumptions and External Contracting Conditions in the Manufacturing Context

The situation assumed in this research work can be also described in a game theory context. It is widely known as a Stackelberg game or Stackelber duopoly (in case of exactly two market players: one market leader and one market follower). The Stackelberg game is a strategic game with two market players playing. The main characteristic of this game is that the market leader moves first and market followers move after the market leader finished his move. Both players compete for units of quantity. Several conditions need to be fulfilled for a Stackelberg game. First, the market leader knows that the market follower observes his move. Second, the market leader knows for sure that the market follow does not have the opportunity to select his action before the market leader made his move. Third, when the market leader conducted his move it is binding and there is no chance for him to undo what he just did. An example for a Stackelberg game in reality is the situation when a market leader has a monopoly and a market follower enters this new market. [64]

In the following we assume one manufacturer and one supplier agreeing about a fixed-price contract for a single period. The manufacturer needs to satisfy a stochastic market demand and offers price-quantity schedules to the supplier. The supplier has no information about the specific demand but knows the mean and other characteristics of the demand distribution. The demand distribution takes the shape of a standard normal distribution.

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Figure 4.2 Stochastic Distribution of Market Demand

The manufacturer’s expectation of the market demand is represented by the variable X. F(X) is the cumulative distribution function and f(X) the distribution function of the market demand. The manufacturer’s profit represents compensation for bearing the demand and the supplier-capacity-uncertainty risks.

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