Flexibility analysis on a supply chain contract: deterministic and stochastic settings
Student thesis: Doctoral Thesis
This thesis is based on the application of ﬂexibility analysis on a supply chain contract. The work was implemented using a practical case exploring a relationship between a car manufacturer (buyer) and a parts supplying company (supplier). Commonly, such contracts are ratiﬁed for a ﬁxed duration  typically three years. A nominal order quantity (or initial capacity reservation) and a variation rate controlling the potential adjustments with respect to the nominal quantity for each period are imposed on the supplier when signing the contract. The supplier guarantees to meet the ﬁrm order should it fall within the range agreed upon, and charges a unit price of the product linear in the variation rate in order to protect himself from risks. The buyer in return is required to order the minimum quantity in each period deﬁned by the nominal quantity and variation rate in the contract.
The overall goal throughout the course of this PhD was to analyse this Quantity Flexibility (QF) contract at the strategic (or contracting) level. The prime focus  from the buyer’s perspective  was to develop a policy that determines the optimal nominal order quantity (Q) and variation rate (β) underpinning the contract that ensures the actual order quantity satisﬁes the actual demand as much as possible in each period and the total cost, including purchasing cost, inventory holding and backlogging costs, is minimised over the contract length. The approach taken in this study is aimed at solving the problem in two diﬀerent settings. One is called the deterministic setting, where the demands are considered as deterministic and the other is called stochastic setting, where the demands are stochastic and stationary.
For the deterministic setting, a parametric Linear Programming (pLP) model is developed from the buyer/retailer’s perspective to help analyse the optimal combination of values of β and Q. In the pLP model, the decision variables are the actual order quantity in each period, represented by vector x and β and Q are treated as the parameters. For each combination of values of β and Q, the optimal value of the vector x can be found by solving the corresponding Linear Programming (LP) model to optimality. However, the number of the combinations of the values of β and Q could be unlimited. To explore the optimal combination of values of β and Q, the convexity of the optimal value of the pLP model has been examined. Due to the fact that the optimal combination of values of β and Q cannot be analytically found due to mathematical intractability, this thesis numerically evaluates the best combination of β and Q to draw some managerial insights based on the ﬁndings.
For the stochastic setting, this thesis analyses the longrun behaviour of the system when the signed contract is executed and calculates the mathematical expectation of perperiod total purchasing, inventory holding and backlogging costs, as a function of the contracting parameters β and Q. The optimal values of these parameters are calculated through simulation of various demand patterns. For this purpose, we consider the basic case with zero lead time and a very simple order policy during the execution of the contract. These assumptions are nevertheless reasonable in the context of a car manufacturer and a supplier delivering JustInTime (JIT) parts using a QF contract. The evolution of the inventory position can be modelled with a Markov chain and the longrun behaviour of the system can then be analysed by considering the steadystate. Due to mathematical intractability, the steadystate is estimated through simulation.Our models diﬀer from the previous similar works found in the literature where the QF mechanism is implemented in a way that in the models used in these works the nominal quantity (Q) and the ﬂexibility parameter (β) are:
ˆ  analysed separately,
ˆ  coupled with other forms of coordinative drives,
 or computed using approximate methods and heuristics that are unable to ﬁrmly guarantee global optimum solutions.
The overall goal throughout the course of this PhD was to analyse this Quantity Flexibility (QF) contract at the strategic (or contracting) level. The prime focus  from the buyer’s perspective  was to develop a policy that determines the optimal nominal order quantity (Q) and variation rate (β) underpinning the contract that ensures the actual order quantity satisﬁes the actual demand as much as possible in each period and the total cost, including purchasing cost, inventory holding and backlogging costs, is minimised over the contract length. The approach taken in this study is aimed at solving the problem in two diﬀerent settings. One is called the deterministic setting, where the demands are considered as deterministic and the other is called stochastic setting, where the demands are stochastic and stationary.
For the deterministic setting, a parametric Linear Programming (pLP) model is developed from the buyer/retailer’s perspective to help analyse the optimal combination of values of β and Q. In the pLP model, the decision variables are the actual order quantity in each period, represented by vector x and β and Q are treated as the parameters. For each combination of values of β and Q, the optimal value of the vector x can be found by solving the corresponding Linear Programming (LP) model to optimality. However, the number of the combinations of the values of β and Q could be unlimited. To explore the optimal combination of values of β and Q, the convexity of the optimal value of the pLP model has been examined. Due to the fact that the optimal combination of values of β and Q cannot be analytically found due to mathematical intractability, this thesis numerically evaluates the best combination of β and Q to draw some managerial insights based on the ﬁndings.
For the stochastic setting, this thesis analyses the longrun behaviour of the system when the signed contract is executed and calculates the mathematical expectation of perperiod total purchasing, inventory holding and backlogging costs, as a function of the contracting parameters β and Q. The optimal values of these parameters are calculated through simulation of various demand patterns. For this purpose, we consider the basic case with zero lead time and a very simple order policy during the execution of the contract. These assumptions are nevertheless reasonable in the context of a car manufacturer and a supplier delivering JustInTime (JIT) parts using a QF contract. The evolution of the inventory position can be modelled with a Markov chain and the longrun behaviour of the system can then be analysed by considering the steadystate. Due to mathematical intractability, the steadystate is estimated through simulation.Our models diﬀer from the previous similar works found in the literature where the QF mechanism is implemented in a way that in the models used in these works the nominal quantity (Q) and the ﬂexibility parameter (β) are:
ˆ  analysed separately,
ˆ  coupled with other forms of coordinative drives,
 or computed using approximate methods and heuristics that are unable to ﬁrmly guarantee global optimum solutions.
Original language  English 

Awarding Institution  
Supervisors/Advisors 

Award date  Sep 2017 
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