Supply chain analysis and design individual case study

Supply Chain Analysis and Design

OMGT 2277 SUPPLY CHAIN ANALYSIS AND DESIGN
ASSIGNMENT 1 – INDIVIDUAL CASE STUDY

COVER SHEET
Term

Title of Assignment

Names and student ID

B 2020
Individual Case Study

Huynh Kim Son – s3694699

Location

SGS Campus

Lecturer

Hiep P

Word Count
(Main content without list of
references, cover page, etc.)

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Supply Chain Analysis and Design

QUESTION 1
INPUTS
DEMAND FOR FANCY BAGS
Month 1
3000

Month 2
5000

Month 3
2000

Month 4
1000

WORKERS INVOLVED
-

Initial number of workers: 20

-

Regular wage (per worker): $1,500

-

Overtime wage (per hour/ worker): $13

-

Maximum standard working time (per worker): 160 hours

-

Maximum overtime working hours (per worker): 20 hours

-

Worker hired cost (per worker): $1600

-

Woker fired cost (per worker): $2000

PRODUCTION INVOLVED
-

Initial stock: 500 bags

-

Number of hours required to produce 1 bag: 4 hours

-

Raw material cost (per bag): $15

-

Inventory holding cost (per bag): $3

DECISION VARIABLES
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Supply Chain Analysis and Design

TOTAL COST = Change-In-Workforce-Level costs + Total Regular-Time Wages + Total
Overtime Wages + Total Raw Material cost + Total Inventory Holding cost

Change-In-Workforce-Level Costs
Renova estimates that the cost associated with increasing the workforce level for any month
is $1,600 per worker hired. A similar cost associated with decreasing the workforce level for
any month is $2,000 per worker fired.
Decision Variables
Let Im denote the number of workers hired in month m
Let Dm denote the number of workers fired in month m
m=1, 2, 3, 4; m=1 refers to month 1, m=2 refers to month 2, m=3 refers to month 3, and m=4
refers to month 4.
Change-In-Workforce-Level Costs = 1600I1 + 1600I2 + 1600I3 + 1600I4 + 2000D1 + 2000D2 +
2000D3 + 2000D4

EXTRA: Total Number of Workers (This element depends on the change-in-workforcelevel and NOT considered as a DECISION VARIABLE in Linear Programming.
However, it is needed in order to conduct Linear Programming in the case given)
Renova informs that currently they are having 20 workers available.
Let Nm denote the total number of workers the company needs in month m
N0 = 20

N1 = 20 + I1 – D1
N 2 = N 1 + I2 – D 2
N 3 = N 2 + I3 – D 3
N 4 = N 3 + I4 – D 4
Regular Wages Costs
Renova stated that the regular wage per worker of the company is $1,600 per month.
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Supply Chain Analysis and Design

Total Regular Wages = 1500N1 + 1500N2 + 1500N3 + 1500N4

Overtime Wages Costs
Renova pays workers $13 per overtime hour they perform.
Decision Variables
Let Om denote the amount of overtime hours required to sufficiently produce fancy bags to
satisfy the demand in month m
Total Overtime Wages = 13O1 + 13O2 + 13O3 + 13O4

Raw Material Costs
Renova determined that for each fancy bags produced it needs $15 cost of raw material.
Decision Variables
Let Xm denote the production volume in units for fancy bags in month m
Total Raw Material Cost = 15X1 + 15X2 + 15X3 + 15X4

Inventory Holding Costs
Renova determined that the holding cost for each unit of inventory is $3.
Decision Variables
Let Sm denote the inventory level for fancy bags in month m

Total Inventory Holding Cost = 3S1 + 3S2 + 3S3 + 3S4

OBJECTIVE FUNCTION
Minimum Total Cost or
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Supply Chain Analysis and Design

z = 1600I1 + 1600I2 + 1600I3 + 1600I4 + 2000D1 + 2000D2 + 2000D3 + 2000D4 + 1500N1 + 1500N2 +
1500N3 + 1500N4 + 13O1 + 13O2 + 13O3 + 13O4 + 15X1 + 15X2 + 15X3 + 15X4 + 3S1 + 3S2 + 3S3 + 3S4

CONSTRAINTS
1. Inventory after Production meets Customer Demand
We must guarantee that the production schedule meets customer demand. Fancy bags that
reach to the customer can come from both this month’s production or last month’s stock.
Therefore, the demand fulfilment requirement takes the form of:
(Ending Inventory from the previous month) + (Production Volume this month) – (Ending
Inventory for this month) ≥ This month’s Demand


Given that the Inventory Level at the beginning of Month 1 is 500 units.



There are no requirements regarding the compulsory inventory level at the end of any
months.



The demand for Month 1, Month 2, Month 3 and Month 4 are 3000, 5000, 2000, 1000
respectively.

Take all elements into consideration, using the production volume variable and inventory
level variable, we are able to form the constraints below:
 Month 1: 500 + X1 – S1 ≥ 3000 -> X1 – S1 ≥ 2500
 Month 2: S1 + X2 – S2 ≥ 5000
 Month 3: S2 + X3 – S3 ≥ 2000
 Month 4: S3 + X4 – S4 ≥ 1000

2. Production Capacity meets Production Volume
The key element for the whole Linear Programming in the case of Renova is the appropriate
workforce level in order to make the number of bags in demand. Therefore, the production
capacity from the total available working time of workers must be larger or equal to the
production volume. It could be larger or equal, NOT equal, due to the firing cost occurred in
the change-in-workforce level. For instance, supposed we are having 50 workers in April but
in May we only need 40 workers to produce the required production volume. However, if the
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Supply Chain Analysis and Design

firing cost of 10 workers are higher than the wages we pay for those 10 excessive workers, it
is better that we maintain the same workforce level despite of the excessive capacity.
Production capacity is calculated by the total amount of available working time, including
regular-time and overtime, divided by the time required to make 1 bag.
Due to the demonstration above, we have the formula:
[(Number of Workers this month)*(Maximum Standard Working Time per Worker) + (Total
Number of Overtime Hours this month)]/(Time required to make 1 bag) ≥ (Production
Volume this month)



Given that Maximum standard working time per worker is 160 hours.



The amount of time required to make 1 bag is 4 hours

Using these given data, number of worker variable, amount of overtime variable, production
volume variable, we have the constraints:
 Month 1: (160N1 + O1)/4 ≥ X1
 Month 2: (160N2 + O2)/4 ≥ X2
 Month 3: (160N3 + O3)/4 ≥X3
 Month 4: (160N4 + O4)/4 ≥ X4

3. Overtime Capacity
The total amount of overtime needed in fancy bags’ production schedule is a decision
variable. Renova has overtime policy in which each worker can only work overtime for
maximum 20 hours per month. Hence, the total amount of overtime conducted cannot exceed
the maximum available overtime of all workers. We have:
(Amount of Overtime conducted) ≤ (Number of Workers)*(Maximum Overtime per Worker)


Given that the maximum amount of overtime each worker could conduct is 20 hours

Using the data given, number of workers variable and amount of overtime variable, we have
the constraints:
 Month 1: O1 ≤ 20N1
 Month 2: O2 ≤ 20N2
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Supply Chain Analysis and Design

 Month 3: O3 ≤ 20N3
 Month 4: O4 ≤ 20N4

4. Decision Variables non-negativity
We must guarantee that all of the decision variables, including Number of Workers Hired,
Number of Workers Fired, Production Volume, Inventory Level and the amount of overtime,
are non-negative because negativity in any of these decision variables would be ineffective
and unrealistic.
Therefore, we have:
I1, I2, I3, I4, D1, D2, D3, D4, X1, X2, X3, X4, S1, S2, S3, S4, O1, O2, O3, O4 ≥ 0

5. Decision Variables Integer
There are specific variables that we must guaranteed that their results are integer number,
including Number of Workers Hired, Number of Workers Fired, Production Volume and
Inventory Level.

Figure 1: Decision Variables without Integer Constraint
As an illustration, without the integer constraint, the results are as Figure 1. Although the
result without the integer constraint may have a better objective function, but it is not
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Supply Chain Analysis and Design

applicable in real world cases. Integer constraint is necessary in this case because we cannot
hire 0,33 worker or produced 0,5 product.

Therefore, we have:
I1, I2, I3, I4, D1, D2, D3, D4, X1, X2, X3, X4, S1, S2, S3, S4 = int

(REPRODUCE THE MODEL AND SOLVING THE PROBLEM ARE INCLUDED IN
THE EXCEL FILE)

INTERPRETING THE RESULT

QUESTION 3
1. Investigating the relationship between initial number of workers and the total
cost
Due to the hiring and firing cost occurred in the change-in-workforce level process of
Renova, certainly the initial number of workers has impacts towards the total cost.
Specifically, supposed the company needs 84 workers to produce enough fancy bags in
demand. In the situation 1 which they are having 20 workers, they may have to hire 84-20=64
more workers to operate the required production plan, and this is where hiring cost of 64
workers occurs. On the other hand, if they already have 80 workers, they only need to hire
84-80=4 more workers, and 60 (64-4) workers hiring cost is saved compared to situation 1.
Throughout the example above, we could observe that when the initial number of workers
increases, the hiring cost of the company will decrease. Hiring cost is accounted in the total
cost of Renova (Total Cost Formula in LP model section above), hence a decrease in hiring
cost would result in a reduction in the total cost. Take all into consideration, it is appropriate
to state that the relationship between “Initial Number of Workers” and “Total Cost” is
negative, in which an element increases while the other one decreases and vice versa.
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Supply Chain Analysis and Design

2. Reproducing Linear Programming and Interpreting the Result

The CEO of Renova indicates that “there is an opportunity to begin the process by selecting
any initial number of workers between 0 and 200”.
In the previous Linear Programming provided to Renova, the initial number of Workers play
the role of an input (20 initial workers). However, due to the company’s new additional
information, we considered it as a decision variable because we do not know what the best
solution is if we could have any initial number of workers between 0 and 200. The 0 to 200
range now becomes our constraint for the initial number of workers. Everything else in the
previous Linear Programming does not change.
STEP 1: CONDUCTING LINEAR PROGRAMMING TO FIND THE BEST
SOLUTION
Initial Number of Workers
Renova addressed that this can be selected between 0 and 200.
Decision Variable
Let No denote the Initial Number of Workers of the whole period.
Constraint
No ≤ 200
No non-negativity
No integer
(Everything else in the previous Linear Programming Model stays the same)

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Supply Chain Analysis and Design

The integer constraint for the Initial Number of Workers is sill necessary in this case due to
the same reason stated in the LP model above – we could not have, for instance, 80.7 initial
workers. Therefore, in order to figure out the optimal applicable solution when the range of
initial workers is between 0 and 200, Integer constraint is required. After reproducing the LP
model

in

Excel,

we

have

the

results:

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Supply Chain Analysis and Design

Figure 2: Optimal Initial Number of Workers in Excel Linear Programming Model
As expected, the optimal initial number of workers is 84. This could be explained as the
required workforce level for month 1 production is 84. By having 84 initial workers, the
Hiring Cost would be 0 and that is the point where Total Cost would be reduced significantly.
The comparison is below:

Figure 3: Costs when Initial Number of Workers is 20

Figure 4: Total Cost when Initial Number of workers is optimal
Due to the cost comparison between Figure 3 (20 initial workers) and Figure 4 (84 initial
workers), it is observable that every other cost stays the same. The only cost that change is
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Supply Chain Analysis and Design

the change-in-workforce level cost, which is subjected to firing cost because hiring cost
turned to 0. By this, Renova saves $102,400 (774,120-671,720) by eliminating the hiring
cost. Another perspective to be considered is the Minimum Total Cost between Normal LP
and Integer LP. Normal LP provides the lower minimum total cost but again, LP in this
situation suggest starting with 83.33 workers and certainly it is not applicable in real world
circumstance. Therefore, it is pertinent to conclude that the optimal applicable solution for
Renova is starting the production plan with 84 workers.

To specifically investigate how the variation in the initial number of workers affects the
decision variable and the total cost, we need to observe the sensitivity report provided by
Excel solver. Nevertheless, when an integer constraint is being set, Excel refuses to provide a
sensitivity report. Therefore, supposed we already know the optimal initial number of
workers is 84 as above, we now assume the company will start with 84 initial workers. This
action removes the needs for the integer constraint, all others decision variable and objective
function maintain the same because it is already optimal, but now we are able to see the
sensitivity report and observe the variation.
STEP 2: ASSUMING THE INITIAL NUMBER OF WORKERS IS THE RESULT OF
THE LINEAR PROGRAMMING ABOVE
Constraint:
-

No = 84

-

Remove all integer constraint

STEP 3: ADDRESSING THE SENSITIVITY REPORT CONDUCTED

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Supply Chain Analysis and Design

After

conducting

the

steps

above,

we

get

the

sensitivity

report

as

below:

Subjected to cells $H$80 (last cell in Figure), the allowable increase and allowable decrease
of the initial number of workers is ~4.24 and ~0.67 units respectively. The shadow price is a
positive number, which indicates that there is a positive relationship between the variation of
initial workers and the total minimum cost. To be clarified, for each unit increase or decrease
in the initial number of workers, the minimum total cost would be increase by $705. As long
as the constraint R.H.S stays between the allowable range, the shadow price of $705 is
applicable. If the constraint R.H.S exceeds the allowable range, the whole LP are adjusted,
and this sensitivity report cannot be used to assess the variation in the minimum total cost
associated with the initial number of workers anymore.
Allowable range:
Lower bound: 84 - 0.67 = 83.33
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Supply Chain Analysis and Design

Upper bound: 84 + 4.24 = 88.24
83.33 ≤ N0 ≤88.24
To be applicable in realistic circumstances, the range should follow an integer value:
84 ≤ N0 ≤ 88
In conclusion, we recommend that Renova should start their production line with the Initial
Number of Workers between 84 and 88 in order to get Minimum Total Cost under control
with a shadow price of $705. If possible, the optimal applicable solution they could have is
starting with 84 workers, in which the Total Cost is minimum with the value of $671,720.

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