Optimal delivery service strategy for high-tech product manufacturers with time and spatial dependent demand
6.2 Model formulations
6.2.1 Discriminating service strategy
The decision maker in this study is a manufacturer who operates multiple high-tech product manufacturing plants around the world and serves customers in different regions. This study proposes discriminating service strategy, which differs from the traditional and typical uniform service strategy where customers at different regions are served according to the same delivery cycle. Periods with considerable demands suggest that frequent and short service cycles are suitable and may stimulate customer demand for products and reduce logistics cost due to shipping economies.
Consequently, long service cycles are suggested when demand is very low. In addition, delay in receiving products is determined here as time span between customer purchasing and receiving products, and depends on service cycles and average distances between plants and customers.
Let s and k denote a specific customer and a manufacturing plant, respectively.
And let i and s T(is) be a specific service cycle and duration of service cycle i for s customer s under discriminating service strategy during the study period,
)) ( ), ( ( )
(is t0 is tm is
T = , i =1,2,.., s I , where s I is the total service cycle for customer s s during the study period and t0(is) and tm(is) represent the start and end times of
service cycle i , respectively. Note that the unit of s T(is) is day and s T i T
s
I
i s =
∑
=1)
( ,
where T denotes the study period. At each tm(is), i =1,2,.., s I , the operator begins s to deliver products to customer s accumulated during service cycle i . Figure 6.2 s illustrates the profile of service cycles, where
is
f represents total demand of customer
s for the manufacturer during service cycle i . s
time item
) (is T
)
0(is
t tm(is)
(1) (2)
is
f
Figure 6.2 The profile of service cycles
Consider two curves in Figure 6.2 representing cumulative number of products, which have been: (1) purchased by customer s; (2) delivered by the manufacture. To satisfy customer demand accumulated during service cycles, the manufacturer must assign which manufacturing plants as well as their production to serve customers in different regions. Let fk(is) represent product amount produced by plant k and deliver to customer s for service cycle i . To meet customer demand during service cycle s i , s the following equation must be satisfied:
∑
∀=
k k s k s
i f i i
fs ( )β ( ) ∀ is ∀s (6.1)
where )βk(is is an indicator variable, if plant k serves and delivers products to customer s for service cycle i , then s βk(is)=1 and the delivered product amount is
) (s
k i
f ; otherwise, βk(is)=0 . Total product demand of customer s for the manufacturer can then be formulated as:
)
Regarding products delivery, total amount of products delivered by plant k during the study period can be obtained by summing up total amount delivering to different customers at various service cycles. Let f represent total amount delivered by plant k k during the study period and f can be formulated as: k
In this study, the cost components incurring to serve customers in different regions is defined as including production cost and logistics cost. The production cost incorporates both capital and variable production cost. The capital cost includes costs related to the purchasing and installation of related equipments, plant construction, land rents, etc. as well as differences among manufacturing plants due to different locations and size. The variable production cost includes those paid for input factors other than key-component, such as labor, utility and insurance, etc. The monthly production cost of plant k, L , can be formulated as follows: k increasing with the increase in monthly production amount, the average production cost per unit product at plant k decreases with it, ( )
)
production cost per unit product for the manufacturer is obtained as:
∑
Due to production restriction, some plants will not be assigned to serve customers with large demands where demands might exceed production amount. Let t be a kj specific delivery time of plant k during the study period, j=1, 2,..,m , where k m k represents delivery frequency of plant k. The duration of a specific delivery cycle of plant k is (tkj−1 ,tkj), j=2, 3,..,m , which is composed of two consecutive delivery times, k
( . Assume a continuous production, the relationship between product delivered and production amount can be formulated as:
)
(6.6) express total production amount of plant k accumulated during delivery cycle (tkj ,tkj+1) and the right hand side is product amount delivered at time tkj+1. Eq. (6.6) implies that it requires a considerable amount of time for a plant with small capacity producing sufficient products so as to serve customer with large demand, resulting in a long duration of service cycle. A relationship exits among delivery times of a plant, customers the plant served and the service cycles of those customers. For βk(is)=1, plant k is assigned to serve and satisfy the demand accumulated during service cycle is; consequently, the delivery time coincides with the end time of service cycle,
)
The logistics cost includes transportation cost and inventory cost. The transportation cost involves both fixed and variable transportation costs. Fixed
transportation costs are costs attributable to each shipment regardless of shipment volume, while variable transportation costs involve things such as the fuel and handling fees, among others. The transportation cost per shipment for delivering fk(is) units of product from plant k to customer s can be shown as Gsk+tskfk(is), where Gsk and
k
ts represent, respectively, the fixed transportation cost and the variable transportation cost, both depending on distance between plant k and customer s. The transportation cost resulting from serving customer s with service cycle is can be formulated as:
∑
∀Eq. (6.7) shows that assigning more plants to serve a specific customer, i.e. more shipment incurred, leads to a high transportation cost and the cost will be higher if two locations are distant apart. The average transportation cost per unit product during the study period can be shown as:
∑∑ ∑∑∑
∀ = ∀Eq. (6.8) shows that the average transportation cost depends not only the assignment of plants to serve customers but also service cycles for customers during the study period.
A frequent shipment may incur high fixed transportation cost, thereby leading to an increased transportation cost.
The inventory cost, also called waiting cost or opportunity cost, is the cost associated with delay of goods, including the opportunity cost tied up in storage, any value lost while waiting, etc. The delivery and service cycles influence inventory level of both customers and the manufacturer. A frequent delivery strategy may lead to a reduced inventory cost for the manufacturer since it involves a small storage space at plants. In addition, this strategy may serve customer with a reduced waiting time for products, thereby a high satisfaction for the manufacturer. However, the benefits
brought by frequent delivery in inventory cost will be offset by the increased transportation cost if product amount delivered per shipment fall short of economical shipment size. Since the goal of delivery and service strategies is to reduce logistics cost and satisfy customer needs, the number and duration of service cycles should be determined in accordance with the time-dependent demand of customers in various regions. To avoid double counting, inventory cost discussed here reflects the relationship between the production and delivery cycles of plants. The average waiting time can be measured by one half of the headway between two consecutive shipments under a continuous production. The inventory cost of plant k during the study period can then be formulated as follows:
∑
=where g represents the inventory cost per unit product per month. Eq. (6.9) shows that a frequent shipment, short duration of service cycles, leads to a decreased inventory cost.
The average inventory cost per unit product during the study period can be shown as:
∑∑ ∑
=Eqs. (6.8) and (6.10) show the relationship between transportation cost and inventory cost with respect to shipment size, delivery frequency and the assignment of plants to customers during the study period. There are two sided effects on transportation cost if the manufacturer assigns frequent delivery cycles to serve customers whose periods are featured with large demand. The average transportation cost would be low only if large shipment could be accumulated and shipping economies exists. Otherwise, frequent shipment leads to high transportation cost because of large fixed transportation cost. The transportation cost will be even higher if customers are served by multiple plants whose distance are far apart. The above perspective also implies that long
service cycles are suitable when demand is very low. From Eqs. (6.5), (6.8) and (6.10), the total average cost per unit product are:
d r
h + + (6.11)
The discussions so far demonstrate how service cycles and the assignments of plants to customers influence manufacturer’s cost. This study further deals with dynamic and time-sensitive customer demand, and investigate how service cycle durations affects customer demand for manufacturer products. This study applies a binary logit model to determine customer choice probabilities for manufacturer products.
Let o be the objective manufacturer and r be a representative of other manufacturers in the market. Let Ux(s,t) represent the total utility of customer s who purchase products from manufacturer x at time t, Ux(s,t)=Vx(s,t)+εx, x=o, r, where Vx(s,t) and εx represent, respectively, the deterministic component and unobservable or immeasurable factors of Ux(s,t) . Supposing that all εx are independent and identically distributed as a Gumbel distribution, then customer choice probability of purchasing products from manufacturer o at time t, )Pro(s,t , can be estimated as:
)
The difference in utility values of customer purchasing from manufacturer o and r, )
v = o − r , determines the choice probability, which can be rewritten as
) represent delay in receiving products from manufacturer o and r at time t, respectively.
β0 reflects alternative specific constant and β1 and β2 are parameters, respectively.
Delay in receiving products depends on service cycles and average transportation time to customers in different regions. The demand of products during a specific service cycle may be received separately since it may involve various plants serve a specific customer, thereby resulting in an inconsistent product receiving time. Let
is
H be the average transportation time from plants to customer s, and
is
H can be represented as:
∑
∑
∀
= ∀
k k s
k
s k k s
i i
i T
Hs ( )
) ( β
β
(6.14)
where T represents average transportation time from locations of plant k to customer sk s, depending on their spatial distance and shipping mode and
∑
∀k βk(i )s reflects plant number. Eq. (6.14) shows that a close proximity of customers and plants results in less transportation time. Consequently, delay in receiving products when customer s purchases products from manufacturer o at time t can then be given by:
t H i t
Tst m s is
o, = ( )+ − t∈(t0(is),tm(is)) (6.15) From Eq. (6.15), delay in receiving products depends not only on transportation time, but also on customer purchasing time. As customer purchasing time approaches dispatching date, the customer perceives a decreased delay in receiving products.
Normally, total customer demand for manufacturer product can be estimated by multiplying customer demand for products and the choice probability of purchasing products from the manufacturer. Assume total demand for products of customer s at time t is exogenous and denoted as q , then the time-dependent demand for st manufacturer product of customer s at time t can be expressed as qtsPro(s,t). Furthermore, total demand for manufacturer product of customer s during service cycle
i , s f , can be represented as: is
Eq. (6.16) shows that total demand for manufacturer product increases with increasing value of choice probability of purchasing products from the manufacturer, Pro(s,t). In addition to the choice probability, total demand accumulated during service cycle i s will be large if the service cycle is periods with numerous demands for products. Total customer demand for manufacturer product during the study period can be expressed as
∑∑
∀ ∀s i is
fs .
Profit throughout the entire study period can be calculated based on the product price, average cost per unit product, average key-component purchase cost per item and total customer demand for manufacturer product, such as
∑∑
∀ ∀where l and τ represent the average key-component purchase cost per item and profit throughout the study period, respectively. A increased profit can be realized be an increased customer demand and a decreased cost per unit product.
However, interactive relationships exist among profit, average cost per unit product and total customer demand for manufacturer product. Customer demand will increase with a frequent service cycle since it involves a less delay in receiving products, which further results in a reduced production cost; conversely, logistics cost might increase due to frequent shipments without transportation cost economies. On the other hand, the cost saving by assigning least service cycles will be offset if customer demand is little and customer intention to purchase manufacturer product shrinks combined. The above least service cycle strategy also yields a high production cost since there is
insufficient demand to realize the production cost economies. Only if a delivery and service strategy with considerations of time-dependent customer demand, demand-supply interaction and spatial spreads of plants and customers, will maximized profit be realized.
A nonlinear mixed integer programming model is formulated here for determining the optimal number and duration of service cycles for different customers and their plant assignment by maximizing profit subject to demand-supply equality. From Eqs.
(6.11), (6.16) and (6.17) and the discussions above, the nonlinear MIP problem is as follows:
Eq. (6.18a) represents the objective function that maximizes profit throughout the study period. Eq. (6.18b) states relationships among customer demand, product delivered by plants and plant assignment with respective to delivery cycles. Eq. (6.18c) defines customer demand for manufacturer product during service cycles. Eq. (6.18d) constrains the relationship between product delivered and production amount by plants.
Eq. (6.18e) constrains that the summation of the duration of all service cycle for a customer must be equal to the study period. Eqs. (6.18f), (6.18g) and (6.18h) define the decision variables g(vk),vk, tm(is), t0(is) and fk(is) to be nonnegative. Eq.
(6.18i) defines the decision variable βk(is) to be binary. The decision variables are
k
k v
v
g( ), , tm(is), t0(is) , )fk(is and βk(is) . That is, the high-tech product manufacturer can apply the model to optimally decide the capacity as well as the monthly production amount for all manufacturing plants, which manufacturing plants should produce and deliver how much of the production to serve customer in different regions. Moreover, the optimal duration and service cycles for different combinations of the manufacturing plants and customers can also be determined.