Problems on Theory of constraint
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Company produces two products, Y and Z that are processed in four
departments, A, B, C and D. Product Y requires three types of
materials, M1, M2 and M4. Product Z requires two types of materials,
M2 and M3
The requirements for each product are summarized in the table below.
Resource
Required per unit of
Product Y
Required per unit of
Product Z
Material 1
Material 2
Material 3
Material 4
Department A
₹100
₹100
₹15
15 minutes
₹100
₹100
10 minutes
Department B
Department C
15 minutes
15 minutes
30 minutes
5 minutes
Department D
15 minutes
5 minutes
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Each department has 2,400 minutes of available time per week. The
Company's operating expenses are ₹30,000 per week. Based on current
demand, the company can sell 100 units of product Y and 50 units of
product Z per week. Sales prices are ₹450 for product Y and ₹500 for
product Z. All four materials are available in sufficient quantities. The
needed workers are also available.
Required:
1. Determine the company's constraint.
2. Determine the throughput per unit for each product.
3. Determine the throughput per minute of the constrained resource for
each product.
4. Determine the product mix needed to maximize throughput, i.e., the
number of units of Y and Z that should be produced per week.
5. Determine the maximum net income per week for the Company.
6. Suppose the company broke the current constraint by doubling the
capacity of that resource. What would become the new constraint?
Solution
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The following approach is useful when there are only two products and there is
only one binding constraint in addition to demand. However, a different
approach is needed when there are overlapping constraints, i.e., more than one
binding constraint. Linear programming is needed to solve the more difficult
problems involving multiple products with multiple binding constraints.
1. Determine the company's constraint.
Time requirements to meet demand for each department are calculated as
follows:
Department Product Y
A
B
C
D
(15 min)(100 units)
(15 min)(100 units)
(15 min)(100 units)
(15 min)(100 units)
Product Z
(10 min)(50 units)
(30 min)(50 units)
(5 min)(50 units)
(5 min)(50 units)
Total Time Required Per Week
2,000 minutes
3,000 minutes
1,750 minutes
1,750 minutes
Each machine center has only 2,400 minutes of available time per week. B is the
constraint because it does not have enough capacity to process 100 units of Y and
50 units of Z per week.
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2. Determine the throughput per unit for each product
Throughput per unit for each product is needed so that we can
determine how to use the constraint to maximize throughput.
Throughput per unit is as follows:
Throughput Per Unit = Sales price - Materials Cost
Product
Y
Z
Sales
price
Materials Cost
₹450 – 215
₹500 – 200
- Throughput Per Unit
₹235
₹300
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3. Determine the throughput per minute of the constrained
resource for each product.
Throughput/ Minute (constrained resource ) =
Product
Y
Z
Throughput/Unit
Minutes required in B
₹235 ÷ 15
₹300 ÷ 30
Throughput/Unit
Minutes required in B
Throughput/ Minute
₹15.67
₹10.00
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4. Determine the product mix needed to maximize throughput,
i.e., the number of units of Y and Z that should be produced per
week.
• Maximizing throughput requires producing as much of the
product with the highest throughput per minute of the
constrained resource as needed to meet demand.
• So the company should produce 100 units of product Y.
• This requires (100 units)(15 minutes) = 1,500 minutes of
time in the constraint department B and leaves 2,400 - 1,500
= 900 minutes for the production of 30 units of product Z,
i.e., 900 minutes ÷ 30 minutes per unit = 30 units of Z.
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5. Determine the maximum net income per week for TOC
Company.
Sales:
100 units of Y = (100)(₹450)
30 units of Z = (30)(₹500)
COGS:
100 units of Y = (100)(₹215)
30 units of Z = (30)(₹200)
Throughput
Less Operating expense
Net income
₹45,000
15,000
₹21,500
6,000
₹60,000
27,500
32,500
30,000
₹2,500
Note: An assumption in this illustration is that there are no beginning or
ending inventories of work in process or finished goods.
COGS: Cost of Goods Sold
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6. Suppose the company broke the current constraint by
doubling the capacity of that resource. What would
become the new constraint?
Doubling the capacity of Department B would allow the
company to produce 60 units of product Z in addition to
the 100 units of product Y.
Since only 50 units of Z are demanded, the company would
have unused capacity in all departments.
Therefore, external demand would become the constraint.
Of course the new solution would be at the intersection of
the demand constraints, i.e., 100 Ys and 50 Zs.
Graphic Analysis
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The following graphic analysis provides a general approach
for solving simple product mix problems that is also
applicable when there are overlapping constraints.
First, plot the constraints to find the feasible solution
space. The B constraint is 15Y + 30Z = 2,400 so Department
B could produce 160 Y's (i.e., 2,400/15) or 80 Z's (i.e.,
2,400/30), or some combination of Y and Z indicated by the
constraint line connecting those two points on the graph.
The department B constraint and demand constraints
define the feasible solution space indicated by the mustard
colored area on the graph.
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2. Check the solution at each corner point, or plot the objective function and
move it to the right as far as possible in the feasible solution space. The
objective function is to maximize throughput where Throughput = 235Y +
300Z. Checking the corner points we find that 100 Y and 30 Z provides the
greatest amount of throughput.
Corner Point
100 Y and zero Z
100 Y and 30 Z
60 Y and 50 Z
Zero Y and 50 Z
Throughput
(100)(235) = 23,500
(100)(235) + (30)(300) = 32,500
(60)(235) + (50)(300) = 29,100
0 + (50)(300) = 15,000
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• If we plot the objective function 235/300 (i.e., it takes .7833 of a Z to
produce as much throughput as 1 Y),
• we can locate the solution by moving it to the outer most point in the
feasible solution space as illustrated below.
• The first objective function shows that 100 Ys would produce the same
throughput as 78.3333 Zs.
• This is an iso-throughput line indicating that any point on the line
represents a combination of Y and Z that produces ₹23,500 of
throughput.
• The point indicated by 100 Y and 30 Z is the last point the objective
function touches in the feasible solution space as we move it up and to
the right.
• This point indicates the solution to our product mix problem.
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Q2. A Furniture Company XYZ produces two products, End Tables and Sofas that are
processed in five departments, Saw Lumber, Cut Fabric, Sand, Stain, and Assemble. End
tables are produced from raw lumber. Sofas require lumber and fabric. Glue and thread
are plentiful and represent a relatively insignificant cost that is included in operating
expense. The specific requirements for each product are provided in the table below.
Resource
or
Activity
(Quantity available per month)
Lumber (4,300 board feet)
Fabric (2,500 yards)
Saw Lumber (280 hours)
Cut Fabric (140 hours)
Sand (280 hours)
Stain (140 hours)
Assemble (700 hours)
& Required per End Table
10 board ft @ ₹10 = ₹100
30 minutes
30 minutes
20 minutes
60 minutes
Required per Sofa
7.5 board ft @ ₹10 = ₹75
10 yards @ 17.50 = ₹175
20 minutes
20 minutes
10 minutes
30 minutes
90 minutes
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The Company's operating expenses are ₹75,000 per month. Based on
current demand, the company can sell 300 End Tables and 180 Sofas per
month. Sales prices are ₹300 for End Tables and ₹500 for Sofas.
Required:
1. Determine the XYZ Company's constraint.
2. Determine the throughput per minute of the constrained resource for
each product.
3. Determine the product mix needed to maximize throughput, i.e., the
number of End Tables and Sofas that should be produced per month.
4. Determine the maximum net income per month for the XYZ Company.
5. Suppose XYZ Company broke the current constraint resource. What
would become the new constraint?
6. Solve the XYZ Company product mix problem assuming that only
3,000 board feet of lumber can be obtained rather than 4,300 board
feet.
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Q3. A Company produces two products, X and Y that are processed in two
departments, A and B. The requirements for each product are provided in the
table below.
Resource
Department A
Department B
Required per unit of
Product X
2 hours
3 hours
Required per unit of
Product Y
4 hours
2 hours
Weekly capacity is 80 hours for Department A and 60 hours for Department B.
Throughput per unit is ₹60 for X and ₹50 for Y.
Required:
1. Assume that the company can sell as much of X and Y as it can produce.
Determine the number of units of X and Y that should be produced per week to
maximize throughput.
2. Now assume that Weekly demand is 12 for Product X and 14 for Product Y. What
would be the new product mix needed to maximize throughput?