Desired Supply Line Value Calculation
for Multi-Supplier Systems’
Ahmet Mutallip and Hakan Yasarcan
Industrial Engineering Department
Bogazici University
Bebek — Istanbul 34342 — Turkey
ahmetmutallip@hotmail.com; hakan.yasarcan@boun.edu.tr
Abstract
Desired Supply Line is the product of expected acquisition lag and desired acquisition
rate. This calculation ensures that supply line would produce the desired acquisition rate
given that it is at this desired level. A wrongly calculated Desired Supply Line value leads
to a steady-state error preventing stock approach its goal. Therefore, correct calculation
of Desired Supply Line values is crucial. Desired acquisition rate is equal to the expected
loss flow in a single-supplier system. However, it is not easy to decide on the desired
acquisition rates for a multi-supplier system. We give a general formula for the
calculation of Desired Supply Line values based on the supplier utilization priorities and
supplier production/shipment capacities.
Keywords: acquisition lag; constant loss; multi-supplier system; steady-state error;
stochastic loss; stock management; supply line.
1. Introduction
The sourcing success of a manufacturer does not only depend on the ordering
strategies, but it also depends on supplier selection. A firm should use multiple-sourcing
strategy instead of a single-sourcing strategy in order to reduce procurement risk (Arda
and Hennet, 2006; Chiang, 2001; Chiang and Benton, 1994; Jokar and Sajadich, 2008;
Minner, 2003; Ramasesh, 1991; Sculli and Shum, 1990; Sculli and Wu, 1981; Thomas
and Tyworth, 2006). In the presence of stochastic lead times, multiple-sourcing strategy
reduces the effective lead time (Minner, 2003). Multiple sourcing also reduces the
dependency on a single supplier, thus the power of supplier over the buyer (Burke,
' This research is supported by a Marie Curie International Reintegration Grant within the 7th European
Ce ity Fi k P (grant number: PIRGO7-GA-2010-268272) and also by
Bogazici University Research Fund (grant no: 6924-13A03P 1).
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Carrillo and Vakharia, 2007; Newman, 1989). Some firms give scores to its suppliers,
and decide on their priority levels. They give orders according to the priority levels of its
suppliers (Burke, Carrillo and Vakharia, 2007).
Determining the target value for a supply line, which is called Desired Supply Line, is
important when anchor-and-adjust heuristic is used. Desired Supply Line is calculated by
multiplying expected acquisition lag and desired acquisition rate (Sterman, 2000). In a
single-supplier stock management problem, desired acquisition rate for a stock is equal to
the expected loss flow from that stock. Therefore, Desired Supply Line is equal to
expected acquisition lag times expected loss flow. However, determining desired
acquisition rate, thus, calculating Desired Supply Line, is not that straightforward in the
presence of multiple suppliers.
A stock having multiple supply lines can be seen in inventory management, human
resource I a capacity 1 and personnel training. For example, firms
have different human resources management processes. Some firms use an internal
human resources department. Some outsource their human resources needs to private
agents. Additionally, some of them use both an internal department and private agents.
Each entity who deals with human resources management has a different operational
mechanism which has its own working capacity and hiring lead time. The hiring/firing
process of each entity corresponds to a different supply line for the human workforce of a
firm. Firms using multi-sourcing strategies need to decide on the utilization level of each
supplier. They should also determine Desired Supply Line values for each of those supply
lines, which is the main issue examined in this work.
In a stock management task, the goal is maintaining a stock at a desired level. This is
achieved by adjusting for the supply line and the corresponding stock at the same time.
To adjust for the supply line, its desired level should be chosen appropriately. We
developed formulations for Desired Supply Line value calculation for multi-supplier
systems in the presence of constant Loss Flow and stochastic Loss Flow with a known
stationary mean.
2. Generic Formulations of Desired Supply Line
The general formula for Desired Supply Line is seen in Equation 1:
Desired Supply Line = Acquisition Delay Time x Loss Flow qd)
Equation 1| is valid for a stock with a single supply line. It needs to be adjusted when
a stock has multiple suppliers and, thus, multiple supply lines. When there are n supply
lines attached to a stock, each supply line needs to have its own Desired Supply Line
value. As an example, a stock management system having 2 supply lines attached to a
stock is presented in Figure 1.
Desired Supply Acquisition Delay
Line 1 Time |
Control Flow 1 L_Line 1_ Acquisition
Stock Stock
Adjustment Time Adj 7
a, 7
Weight Of Supply Line -
Supply Line Adjustment Control Flow ay Salk
a7 Loss Flow
\ Yeas Supp!
Line 2
Desired Supply
Line 2
Control Flow 2 Acquisition
Acquisition Delay
Time 2
Figure 1. Stock-Flow Diagram of the Stock Management Task with 2 Supply Lines.
If there is an error in determining the desired supply line values, there will be a
steady-state error. Therefore, desired supply line values should correctly be selected. If
the selected values are proper, the average value of Stock subtracted from its desired level
will be equal to zero in the long run.
In order to balance the inflows to and outflow from the stock, the total average
acquisition flow should be equal to the average loss flow:
y El Acquisition Flow, ] =. E[Loss Flow] (2)
i=l
Average value of an inflow (i.e. control flow) attached to a supply line should balance
the outflow (i.e. acquisition flow) from that supply line. This is needed to maintain each
supply line around its desired value.
[Control Flow, ] = El Acquisition Flow, | for i=1,2,++n (3)
Equations 2 and 3 yield Equation 4:
> E[Control Flow, ] = E[Loss Flow] (4)
i=l
According to Equation 4, the total average control flow should also be equal to the
average loss flow in the long run. Desired Supply Line values should be selected so as to
satisfy Equations 2 and 4.
It is known that the expected outflow from a supply line (i.e. expected acquisition
flow) is equal to the average value of the supply line (i.e. desired supply line) divided by
the delay time of that supply line (i.e. acquisition delay time).
EA , Flow |= Desired Supply Line, (5)
Aquisition Delay Time,
Equations 2 and 5 yield Equation 6:
Desired Supply Li
E[Control Flow, ] 7 LER OE SURE ENG: (6)
Aquisition Delay Time,
From Equation 6, Equation 7 can be obtained:
Desired Supply Line, = Aquisition Delay Time, x E[Control Flow, | (7)
The expected value of a control flow can be obtained using the priority assigned to
the related supplier, and the probability distribution function of the loss flow. Once the
expected control flow values are obtained, Desired Supply Line values can be obtained
using Equation 7. Although Equation 7 is always valid, its application may not be that
straightforward due to the difficulties in obtaining expected Control Flow values.
3. Multiple Supplier Examples
In this part, we give the applications of proposed Desired Supply Line calculation
method for constant Loss Flow case and stochastic Loss Flow with a known stationary
mean case. The distribution of Loss Flow (i.e. demand), supplier capacity limitations, and
supplier priorities are the factors to be considered in control decisions. Handling
stochastic demand is more problematic than handling constant demand in supply chain
management (Nahmias, 2009; Jokar and Sajadieh, 2008, Schmitt, 2007). One other
important concern in supply chain management is the capacity of suppliers. Production
and shipment capacity constraints of suppliers lead to more oscillatory stock behaviors
(Goncalves and Arango, 2010; Minner, 2003; Schmitt, 2007; Springer and Kim, 2010).
The following control flow equation is used in both of the examples:
(8)
Expected Loss Flow + Stock Adjustment
Total Control Flow =
+ Supply Line Adjustment
Note that, our examples assume three-supplier stock management system. The
following individual orders to the three suppliers are calculated as given below. The
priority of a supplier is represented by the index assigned to that supplier (low index
represents high priority level).
Capacity of
Total Control Flow, Total Control Flow < ‘i
Control \ _ Supplier, ®)
Flow, . . Capacity of
Capacity of Supplier,, Total Control Flow =
Supplier,
Total Control Capacity of
Flow Total Supplier,
max - ROHR Control |< +
Capacity of Flow Capacity of
(om _ Supplier, Supplier, (10)
Flow, Capacity of
Total Supplier,
Capacity of Supplier,, Control |= +
Flow Capacity of
Suppplier,
Capacity of Supplier,
) = makx| Total Control Flow — + 0 (11)
Control
Capacity of Supplier,
Flow,
3.1. Three Supplier System with a Constant Loss Flow
In a single-supplier system, Desired Supply Line is calculated by using Equation 1
when Loss Flow is constant. Desired acquisition rate of the supply line is equal to Loss
Flow given that supply line reaches its desired level.
In a multi-supplier system, desired acquisition rate of each supplier is selected by the
decision maker depending on their priority levels and production/shipment capacities.
The sum of the desired acquisition rates must be equal to Loss Flow in order to prevent a
steady-state error. To calculate Desired Supply Line of a supplier, desired acquisition rate
of that supplier must be multiplied by Acquisition Delay Time of the same supplier.
Let’s assume there is a three-supplier system. The stock to be managed has a constant
Loss Flow equal to 60. Acquisition delay times of the suppliers are 8, 12, and 16 in order.
Target level of Stock is 0. The decision maker wants to receive 28 units from the first
supplier, 22 units from the second supplier and 10 units from the third supplier. The
desired value of each supply line is found by Equation 7. So, desired values of supply
lines become 224, 264 and 160 in order. Notice that desired acquisition rate of a supplier
must also be equal to expected control flow of that supplier for supply line stability.
As it can be seen from Figure 2, Stock stays on its desired level when Stock and its
supply lines start at their desired levels. It is also observed from Figure 3 that even
though Stock does not start at its desired level (starts at 250), both Stock and its supply
lines seek their desired levels.
400
300
200
100
0
0 10 20 30 40 50 60. 7 80 9 100
Time (Month)
Stock —+——4++_+— Supply Line2. —3——3——3—.
Supply Line). —-2——2——2— Supply Line3]_. ——4——4——4—
Figure 2. Stock and Supply Line Behaviors in a Three-Supplier System when Loss Flow
is Constant.
400
0 10 20 30 40 50 60 70 80 90 100
Time (Month)
Stock —+——+—__+—+— Supply Line2. —3——-3—3—
Supply Line]. ——2——2——2— Supply Lines, ——4——4——4—
Figure 3. Stock and Supply Line Behaviors in a Three-Supplier System when Loss Flow
is Constant and Stock does not start at its Desired Level.
3.2. Three Supplier System with a Stochastic Loss Flow
We have a three-supplier stock management model which has a stochastic Loss Flow.
Loss Flow has a normal probability distribution with mean 60 and standard deviation 12.
Simulation runs are obtained in discrete time; unity is used as the simulation time step. In
this example, Stock Adjustment Time and Weight of Supply Line are taken as one. Under
these assumptions, the distribution of Control Flow is equal to the distribution of Loss
Flow (see Appendix). In our model, the first supplier has priority over the second supplier
and the second supplier has priority over the third. First and second suppliers have
limited shipment, their capacity limits are 40 and 25 in order. The decision maker gives
the orders up to 40 from the first supplier, orders between 40 and 65 from the second
supplier, and orders above 65 from the second supplier. If order exceeds 40, first supplier
provides 40 units and, if order exceeds 65, second supplier provides 25 units while first
supplier still provides 40 units. Desired Supply Line depends on desired acquisition rate
and acquisition lead time. Desired acquisition rates do not depend on acquisition lead
time or the order of the supply line. However, they are affected by the capacity
limitations of the suppliers. The upper limit of Control Flow of a supplier is its
production/shipment capacity.
; ro)
(() = ——xe ?\"? 12
So) =—om—xe (12)
40
E[Control Flow, ] = fx x fx) x dx + 40 x fx) x dx (13)
E[Control Flow, |= jf (x-40)x f(x)xdx+ fps x f(x) x dx (14)
E[Control Flow, |= fr-65)x f (x)xdx (15)
Equation 12 shows the probability distribution function of normal distribution.
Equations 13, 14, and 15 are valid because the distribution of Control Flow is equal to the
distribution of Loss Flow (see Appendix). According to Equations 13, 14, and 15, desired
acquisition rates are consecutively equal to 39.76207, 17.54096, and 2.696963. Desired
Supply Line values become 318.0966, 210.4915, and 43.15141 consecutively for the first,
second, and third suppliers (see Equation 7). If Desired Supply Line values were
calculated assuming constant Loss Flow (i.e. equal to mean of Loss Flow which is 60),
they would be 320, 240, and 0 instead. This would lead to steady state error causing
higher penalties.
In Figure 4, “DSL” on the x-axis corresponds to our base run which uses the
calculated Desired Supply Line values of 318.0966, 210.4915, and 43.15141. Penalty
values are generated by using Equations 16 and 17 and the average of five different seeds
is taken. The length of the simulations is 250. There are three lines in Figure 4 for the
three suppliers. As one moves to the right on a line, Desired Supply Line value
corresponding to that line increases while the other two Desired Supply Line values
remain at their base levels. As one moves to the left on a line, Desired Supply Line value
corresponding to that line decreases while the other two Desired Supply Line values
remain at their base levels. An increase or a decrease in the proposed Desired Supply Line
values results in an increase in the total penalties according to Figure 4. These results
approve the appropriateness of our desired supply line value calculation method:
Total Penalty, =0 [item : time] (16)
Desired Stock Level|
x DT (17)
— Stock
Total Penalty,,, = Total Penalty, +
30000
— Desired Supply Linet
— Desired Supply Line2|
Total Penalty
~~ Desired Supply Line3|
5000
o
ge ges aS
FS KX SX ¥ £
Desired Supply Line
Figure 4. Total Penalty vs. Desired Supply Line Values.
4. Conclusion
A wrongly calculated Desired Supply Line value leads to a steady-state error
preventing stock approach its goal. Therefore, desired supply line values should correctly
be selected. In this work, the calculation of Desired Supply Line values in multi-supplier
systems is examined. Using multiple suppliers instead of a single supplier reduces the
procurement risks in stock management. However, determining Desired Supply Line
values in a multi-supplier system is not that straightforward compared to a single-supplier
system.
In steady state, inflow (Control Flow) to a supply line has to be equal to the outflow
(Acquisition Flow) from that supply line. Also, the total inflow (sum of all acquisition
flows) to a stock has to be equal to the outflow (Loss Flow) from that stock. Note that,
outflow from a supply line is, at the same time, an inflow to the corresponding stock.
Eventually, this brings the deduction that the sum of all inflows to the supply lines in a
stock management system (i.e. control flows) has to be equal to the outflow (i.e. Loss
Flow) from the main stock of that system. Therefore, the selection of Desired Supply Line
values must ensure that different supply lines in total produce the total desired acquisition
rate. We give a general approach in obtaining proper Desired Supply Line values in a
multi-supplier stock management system. The desired values obtained by using this
approach make the average value of Stock subtracted from its desired level equal to zero
in the long run.
According to the general approach in determining the Desired Supply Line values in a
multi-supplier stock management system, once the expected control flow values are
obtained, Desired Supply Line values can be obtained using Equation 7. Although
Equation 7 is always valid, its application may not be that straightforward due to the
difficulties in obtaining expected Control Flow values. In this study, this approach is
applied to two cases: one under constant Loss Flow assumption and the other one under
stochastic Loss Flow (normally distributed with known mean and variance) assumption.
As a continuation of this study, we are planning first to extend the application of this
approach to a case under stochastic Loss Flow (normally distributed with unknown mean
and variance) assumption with exponential smoothing heuristic used in expectation
formation. Secondly, the generality of the results obtained from the first extension of the
study will be discussed for other Loss Flow distributions.
References
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Appendix
Stock Adjustment Time and Weight of Supply Line are both chosen as 1. With these
settings, two consecutive control flows are calculated by Equations 18 and 19.
Desired Stock Desired Supply Line
Control Flow, = Expected Loss Flow + - + - (18)
Stock, SupplyLine,
Desired Stock Desired Supply Line
Control Flow,,, = Expected Loss Flow + - + - (19)
Stock, SupplyLine,,,
Stock formulation is seen in Equation 20 and Supply Line formulation is seen in
Equation 21 with these settings.
Stock,,, = Stock, + Acquisition Flow, — Loss Flow, (20)
Supply Line,,, = Supply Line, + Control Flow, — Acquisition Flow, (21)
Difference of two consecutive control flows is shown in Equation 22.
Control Flow,,, — Control Flow, = Stock, — Stock,,, + Supply Line, — Supply Line, ,, (22)
When Equations 20 and 21 are plugged in to Equation 21, Equation 23 is obtained.
Control Flow,,, - Control Flow, = Loss Flow, — Control Flow, (23)
Equation 23 yields Equation 24. Equation 24 shows that Control Flow follows Loss
Flow from 1 time unit behind with these settings. This means that their probability
distributions are exactly the same. Therefore, our expected control flows can be
calculated by using the probability distribution of Loss Flow.
Control Flow,,, = Loss Flow, (24)
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