On-time probability

For obtaining the on-time probability (η), we compute the value function V_t⁰(s) recursively from the following relationship.

V_t⁰(s) =















1_{r1≥1}+P

s⁰∈Sp⁰_s,s0V_t−1⁰ (s⁰), ∀s ∈S⁰ 1{r_k∗≥1}+P

s⁰∈Sp¹_s,s0(k^∗)V_t−1⁰ (s⁰), ∀s ∈S¹ 1{r₁≥1}+Pⁱ(k^∗) P

s⁰∈Sp²_s,s0(k^∗|a= 1)V_t−1⁰ (s⁰) + (1−Pⁱ(k^∗)) P

s⁰∈Sp²_s,s0(k^∗|a= 0)V_t−1⁰ (s⁰)

, ∀s ∈Sⁱ, i=LS, P S (4.20) where

1{r1≥1} =

(1 r₁ ≥1 in state s 0 otherwise

because, given that an order is completed at an arbitrary period t, the completion is on-time if the remaining time of the order until the due-date is one or larger at period t−1.

The solution of the value-iteration algorithm gives the steady-state probability that, if an order is completed in an arbitrary period, it is on-time. Since a completion can only occur when the system is not idle (with probability 1−π₀), the relationship between η and equation (4.20) is:

mins∈S{V_t⁰(s)−V_t−1⁰ (s)} ≤η·(1−π₀)≤max

s∈S {V_t⁰(s)−V_t−1⁰ (s)} (4.21)

(4.14) also applies as the stopping criterion for iterations described in this section.

4.4. Numerical study

In the numerical study, we first analyze the decisions made by marketing under a sequential approach. Secondly, we answer the following research questions: (1) How much can the profitability be increased? (2) How are the system utilization, the service level of the firm and the selection of different customer types affected when the simultaneous optimization approach, rather than one of the sequential approaches, is considered? (3) How do the tardiness penalty structure and the market-related characteristics affect the answers to the first two questions?

We take K = 5 and set parameters that define the arrival and service processes respectively as γ = 0.5 and β = 0.4. We select an arrival probability that is larger than the order completion probability to generate a high frequency of prospective customer arrivals so that the system has the chance to obtain more customers if better price/leadtime quotes are made. The set of quotable prices and leadtimes are p ∈ {p_min, p_min + 1, ..., pmax} (i.e. integer values from p_min to p_max), L ∈ {Lmin, Lmin+ 1, ..., Lmax+ 1} with pmin = Lmin = [E[Y]] = [1/β] = 3 and pmax = L_max = 10. E[Y] denotes the expected processing time of an order.

We use tardiness penalty structures, in which the tardiness penalty involves: only a fixed cost, only a variable cost and both a fixed and a variable cost. Since K = 5 and E[Y] = 2.5, the expected production leadtime of the K^th arrived pending order is 12.5 under a FCFS processing. If this order is quoted a leadtime of L_max= 10, the expected tardiness is 2.5. We take (fixed cost, unit variable cost) pairs ∈ {(6,0),(3,1),(0,2)} all of which result in the same tardiness cost when an order is late by 3 periods. The processing cost per time unit isc= 0. The stopping criterion of the value-iteration algorithm is set to = 10⁻⁵.

We assume that there is a common price sensitivity parameter (ξp) as well as a com-mon leadtime sensitivity parameter (ξL) defined for the system. We then consider different sensitivities of two customer types to be specified based on a parameter (κ∈[0,1]) that determine the heterogeneity between the two customer types:

ξ_L^LS =ξ_L+κ·ξ_L ξ_p^LS =ξ_p−κ·ξ_p

ξ_L^{P S} =ξ_L−κ·ξ_L

ξ_p^{P S} =ξ_p+κ·ξ_p (4.22) whereκ = 0 means that customer types do not differ in their sensitivity to price and leadtime, i.e. that there is a single customer type, whileκ= 1 makesLS customers only sensitive to leadtime and P S customers only sensitive to price. We consider

ξ₀ = 0.1, ξ_L = 0.75 and ξ_p = 0.75 as a base case for customer sensitivity (Case 1). These values are selected to be similar to the ones considered by Easton and Moodie (1999) and Watanapa and Techanitisawad (2005). Both of these studies take ξ₀ = 0.1. Easton and Moodie (1999) take ξ_L = 0.5 and ξ_p = 0.75 while Watanapa and Techanitisawad (2005) take ξ_L values between 0.3 and 0.9 and ξ_p between 0.3 and 0.75. Then we double the leadtime sensitivity (Case 2), the price sensitivity (Case 3) and finally both of them (Case 4), for numerically exercising the impact of customer sensitivity parameters. In each case, we also vary the proportion of LS customer arrivalsζ ∈ {0.05,0.5,0.95}and the contrast parameter κ∈ {0.2,0.6,1} respectively. These variations result in nine sub-cases.

The sequential approaches and their computation are explained in the following:

• Marketing (Step 1): Marketing makes (p, L) quotes to arriving prospective cus-tomers with the objective of maximizing the long-run average profit per time unit. It uses information about the type of a prospective customer, the current number of orders in the manufacturing system and the upper bound on the num-ber of pending orders. On the other hand, it has no information regarding the sequence in which the orders are processed, i.e. about the remaining times until the due-date for orders that are currently in the system. Marketing assumes FCFS in order processing and keeps track of the remaining time of orders until the due-date under this assumption for evaluating the tardiness penalties. This problem is a special case of the optimization problem described in Section 4.2.

Hence, in order to compute the optimal solution to this problem, we run our MDP together with the constraint that K(s) ={1} in all states with ind= 1.

• Manufacturing (Step 2):

Approach 1. Manufacturing processes orders based on FCFS. The optimal solu-tion to this approach is the one obtained in the first step because the order processing policy is in accordance with the assumption of the marketing department.

Approach 2. Manufacturing processes orders based on EDD. The EDD rule is a simple and effective priority rule that is often used in practice (Keskinocak and Tayur, 2004). The optimal solution to this approach is computed by running the MDP in which the (p, L) pairs taken from the first step and the priority dispatching decisions are fixed in accordance with the EDD rule.

This means that for any state s = (1, n, r1, ..., r_n) K(s) = {k}, where k is chosen such that r_k = min{r1, r₂, ..., r_n}. For orders with equal remaining

time until the due-date and for orders with negative remaining time until the due-date, FCFS applies.

Approach 3. Manufacturing processes orders according to the optimal priority dispatching policy. The optimal solution to this approach is also obtained by running the MDP with the (p, L) pairs taken from Step 1. In this case there is no restriction on the dispatching decisions.

Note that all three approaches involve simultaneous optimization of dynamic price and leadtime quotation and coordination of marketing with manufacturing. Their sub-optimality, if any, comes from not harmonizing these decisions with dispatching decisions, i.e. from not collaborating for a joint policy.