2. Background on Sustainable Supply Chains
2.1 Background on Discrete Flow Model
2.2.1 Notation of the DFM
t A specific point in time
t Unit time period
s(t) Inventory level at manufacturing warehouse (MW) S Manufacturing warehouse (MW) inventory capacity g(t) Number of products shipping from MW to ELC (Laakdal) r(t) Inventory level of end-of-life products returned
D Demand from Nike Store (customer) at time t y(t) Satisfied demand from ELC to Store at time t l(t) Unsatisfied demand from ELC to Store at time t w(t) Inventory level of ELC at time t
z(t) Number of end-of-life products returned at time t p Percentage of returned products to the sold products
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p* Optimal percentage of returned products to the sold products q(t) Number of products being transported from MW to ELC
Lead time between MW and ELC (this value will be the multiple of t)
Useful life cycle of a new product to the customer X European Logistics Center’s (ELC’s) capacity
V Capacity of vehicles
α(t) Production availability of Manufacturing Facility 1 (F1) at time t β(t) Production availability of Remanufacturing Facility 1 (F2) at time t U1 Maximum production rate of F1
u1(t) Production rate of F1 at time t U2 Maximum production rate of F2 u2(t) Production rate of F2 at time t MTBF1 Mean time between failures of F1 MTBF2 Mean time between failures of F2 MTTR1 Mean time to repair F1
MTTR2 Mean time to repair F2
cs Unit inventory holding cost for the Manufacturing warehouse cl Unit cost for lost sale
cr Unit cost for holding returned product’s inventory ct Unit cost for transporting a product from MW to ELC cw Unit inventory holding cost for ELC
cu1 Unit manufacturing cost of a product from raw materials at F1 cu2 Unit remanufacturing cost of a product from returned product at F2 cz Unit return cost of used product
T Total time span of the model
f(t) Cost function of the system at time t F(T) Total cost function (Objective function)
Decision variable: p*
14 Figure 2.2. Representation of the System
As demonstrated in figure 2.2. we have two manufacturing facilities. First one (F1) sources raw materials from the supplier, then manufactures the products. It is assumed that supplier has ample capacity. The second facility (F2) re-manufactures the end of life products that are coming from customers. We assume that remanufactured Nike products have the same quality level as the products manufactured from supplier’s raw material. In our model,
(re)manufactured goods consolidate in a manufacturing warehouse (MW). Consequently, goods are shipped to the European Logistics Center (ELC) in Belgium. From ELC, products are shipped to the Nike Store where they are sold to the customers. Daily customer demand are be represented by random numbers of Negative Binomial distribution (see section 3.3). If a product finishes its life cycle in customer use, and is returned, that product can be
remanufactured. End-of-Life products are consolidated in a separate center, then shipped to the re-manufacturing facility (F2). Our aim is to understand the most beneficial flow; most advantageous percentage of products that should return to manufacturing. We aim to optimize the flow by finding the returned percentage that minimizes the cost/emission. We assume 1 day as a unit time period (t=1), which is reasonable for many supply chains, as lead time is calculated with a unit of ‘a day’. Unsatisfied demand is assumed to be lost sale. Service level is highly important for Nike; therefore, we will assume that maximum production rate is always higher than demand values (U>D). After analyzing the financial management aspect,
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we extend our model into environmental impact (e.g. carbon emission). By this way, we are able to assess the result of supply chain decisions in terms of environmental concerns.
At a single point in time, t, customer demand (Nike Stores), D, occurs following a Negative Binomial distribution with parameters R (7.91391) and P (4.0812e-05), which is satisfied from the ELC in Laakdal-Belgium. Demand assumption regarding Negative Binomial distribution is validated in the section 3.3. In our model, both manufacturing facilities replenish the inventory amount of manufacturing warehouse (MW) in Asia. Manufacturing warehouse replenishes the products in ELC with vehicles, in our case via maritime
transportation with transit time . Transport vehicles have maximum total capacity of V, transporting q(t) number of products at time ‘t’ to ELC. ELC has a maximum capacity of ‘X’.
For returned end of life product’s inventory ‘r’, we have a consolidation center that is assumed to have no lead time to remanufacturing facility and to Nike stores in order to simplify the model. This consolidation center is also assumed to have no capacity constraint either. Returned products are calculated at every time point ‘t’. The time horizon subject to this study is represented by ‘T’ (1000 days).
Figure 2.3. Time horizon of model
In order to realistically design the model; at every time point ‘t’, there are certain events that may occur. These can be: the failure of manufacturing facilities F1 and F2, repair of F1 and F2, ELC being full, ELC being out of stock, losing sales, and return of end of life products.
All these events are reflected in the mathematical model.
We assume that supplier for F1 has ample capacity, therefore, F1 will never have issues regarding sourcing raw material. However, F1 may be running or down for the point in time.
α(t) =
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Time between failures are exponentially distributed, with a mean of 30 days. On the other hand, time to repair is set to 1 day deterministically to simplify the model. Maximum production rate is set to the parameter U1 (i.e. 0 ≤ u1(t) ≤ U1). If F1 is not running, then u1(t)=0. Same approach for the F2, when F2 is running, maximum production rate of F2 is U2, (i.e. 0 ≤ u1(t) ≤ U1), and when F2 is down, u2(t)=0. In parallel with (Turki et al., 2017), we make the following assumptions:
1) We assume that time per t=1 day.
2) We assume that is the transportation time, and is a multiple of t.
3) Unlike Turki et al. (2017) deploying deterministic demand, we assume the demand ‘D’ to follow negative binomial distribution. (see section 3.3)
4) Remanufactured and manufactured products have the same quality levels, thus same prices.
5) We assume that in the first (useful life cycle of FTW) period of time, ELC has enough inventory to fulfill the demand. This periods can be seen as a warm up period for the model.
(i.e. w(0) ≥ D)
Demand ‘D’ is met with the on-hand inventory at the beginning of each period, suggesting that we use the ELC inventory level from previous period w(t-t). Thus, the
value of fulfilled demand at time t is represented in the following equation:
y(t)= ( 3)
The ELC in Belgium, which fulfills Nike Stores’ demand, gets replenished with products coming from the manufacturing warehouse. Thus, the ELC inventory level at a specific time
‘t’ is w(t) is the sum of the products in this ELC left from the previous period (t − ∆t), and the number of products that were shipped from manufacturing warehouse at time (t − τ), minus the number of products that were sold to the Nike Stores at time t:
w(t) = w(t − ∆t) − y(t) + g(t − τ) (4)
D If D ≤ w(t-t)
w(t-t) otherwise
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The ELC inventory level after the delivery and the fulfillment of the Nike stores’ demands during period τ, is represented as follows:
w(t + τ) = w(t) − τ * D + g(t) (5) Shipped units from manufacturing warehouse MW at time ‘t’ require τ units of time to be delivered to ELC. The number of units outgoing from manufacturing warehouse inventory ‘S’
at time t (i.e. g(t)) is built upon the ELC inventory level at time t + τ and its total capacity, manufacturing stock level at time w(t − ∆t), transport vehicle capacity ‘V’, and the demand from Nike stores.
g(t)= (6)
The units produced in F1 and F2 replenish the manufacturing warehouse inventory ‘S’. Total production rate from F1 and F2 is:
u(t) = u1(t) + u2(t) (7)
The manufacturing warehouse inventory level at time ‘t’ consists of the level of this inventory left from the previous period (t − ∆t), plus the units produced during the previous period, minus the units being shipped from manufacturing warehouse S at time t:
s(t) = s(t − ∆t) + u(t − ∆t) − g(t) (8)
When the demand is not satisfied from the ELC inventory, lost sales occur. Number of lost sales is defined in the following equation:
l(t) = D − y(t) (9)
We assume that the number of end-of-life products returning (z) to be remanufactured, are the percentage ‘p’ (0 < p < 1) of the satisfied demand quantity at time (t − ). As mentioned before parameter represents the useful life cycle of the product. Obviously, when (t − ) < 0, there is no returned products.
V If s(t − ∆t) ≥ V and w(t + τ) ≤ X-V+ τ * D τ * D If s(t − ∆t) ≥ τ * D and w(t + τ)=X
s(t − ∆t) If s(t − ∆t) ≥ τ * D
0 otherwise
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z(t)= (10)
Figure 2.4. Product’s useful life cycle to the customer
As mentioned, F2 is supplied from the inventory center that consolidates the returned end-of-life products z(t). Following equation represents the number of returned products:
r(t) = r(t − ∆t) + z(t) – u2(t − ∆t) (11)
To ensure that manufactured products do not exceed the Manufacturing warehouse inventory level, we deployed a hedging point policy for production rates of F1 and F2. This policy is applied when both Manufacturing facilities are up and running (Kumar, 1986). Portioning of unit production while both facilities are in active state is represented by the following
functions:
FPU1 = (12)
FPU2 =
(13) p* y(t − ) if (t − ) ≥ 0
0 otherwise
[y/2] + 1 if y is odd
(y/2) if y is even
[y/2] if y is odd
(y/2) if y is even
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Production control for both F1 and F2 are described in the above page based on inventory levels and random events and states of the parameters. The following rules ought to assist in controlling the outcome of manufacturing facilities by incorporating returned units,
production capacity of F1, and inventory levels.
Equation 16 represents the cost function at time t, which comprises of inventor costs, lost sales costs, transportation cost, returned products costs and manufacturing/remanufacturing costs.
f(t)= cs*s(t) + cr*r(t)+cw*w(t) +ct*q(t)+cl*l(t)+cu1*u1(t)+cu2*u2(t)+cz*z(t) (16)
We define the total cost function in equation 17, where T represents the model horizon.
𝐹(𝑇) = ∑𝑡=𝑇f(t)
𝑡=0 (17)