Practice: A Production System Manufacturing
Two Products, P and Q
$90 / unit
P: 110 units / week
Q:
$100 / unit
60 units / week
D
5 min.
D
10 min.
Purchased Part
$5 / unit
C
10 min.
C
5 min.
B
25 min.
A
15 min.
B
10 min.
A
10 min.
RM1
$20 per
unit
RM2
$20 per
unit
RM3
$25 per
unit
Time available at each work center: 2,400 minutes per week.
Operating expenses per week: $6,000. All the resources cost the same.
Theory of Constraints 1- Basics
Ardavan Asef-Vaziri
Nov-2010
1
1. Identify The Constraint(s)
Product A
P
15
Q
10
B
10
35
C
15
5
D Contribution Margin: P($45), Q($55)
10 Market Demand: P(110), Q(60)
5
Can we satisfy the demand?
Resource requirements for 110 P’s and 60 Q’s:
 Resource A: 110 (15) + 60 (10) = 2250
minutes
 Resource B: 110(10) + 60(35) = 3200
minutes
 Resource C: 110(15) + 60(5) = 1950
minutes
 Resource D: 110(10) + 60(5) = 1400
minutes
Theory of Constraints 1- Basics
Ardavan Asef-Vaziri
Nov-2010
2
2. Exploit the Constraint : Find the Throughput
World’s Best Solution
Resource B is Constrained - Bottleneck
Product
P
Profit $
45
Resource B needed (min)
10
Profit per min of Bottleneck
45/10 =4.5
Q
55
35
55/35 =1.6
Per unit of bottleneck Product P creates more profit than
Product Q
Produce as much as P, then Q
Theory of Constraints 1- Basics
Ardavan Asef-Vaziri
Nov-2010
3
2. Exploit the Constraint : Find the
World’s Best Solution to Throughput
For 110 units of P, need 110 (10) = 1100 min. on B,
leaving 1300 min. on B, for product Q.
Each unit of Q requires 35 minutes on B. So, we can
produce 1300/35 = 37.14 units of Q.
We get 110(45) +37.14(55) = 6993 per week.
After factoring in operating expense ($6,000), we make
$993 profit.
Theory of Constraints 1- Basics
Ardavan Asef-Vaziri
Nov-2010
4
2. Exploit the Constraint : Find the
World’s Best Solution to Throughput
 How much additional profit can we make if market
for P increases from 110 to 111; by 1 unit.
 We need 1(10) = 10 more minutes of resource B.
 We need to subtract 10 min of the time allocated to Q
and allocate it to P.
 For each unit of Q we need 35 min of resource B.
 Our Q production is reduced by 10/35 = 0.29 unit.
 One unit increase in P generates $45. But $55 is lost for
each unit reduction in Q. Therefore if market for P is
111 our profit will increase by 45(1)-55(0.29) = $29.
Theory of Constraints 1- Basics
Ardavan Asef-Vaziri
Nov-2010
5
Practice: LP Formulation
Decision Variables
x1 : Volume of Product P
x2 : Volume of Product Q
Product
A
B
C
D
Profit Margin
Demand
P
Q
Capacity
15
10
10
35
15
5
10
5
45
55
110
60
2400 2400 2400 2400
Resource A
15 x1 + 10 x2  2400
Market for P
x1  110
Resource B
10 x1 + 35 x2  2400
Market for Q
x2  60
Resource C
15 x1 + 5 x2  2400
Objective Function
Maximize Z = 45 x1 +55 x2 -6000
Resource D
10 x1 + 5 x2  2400
Nonnegativity
x1  0, x2  0
Theory of Constraints 1- Basics
Ardavan Asef-Vaziri
Nov-2010
6
Practice: Optimal Solution
Product P Product Q LHS (Needed)
RHS (Available)
Resource A
15
10
2021.43
<=
2400
Resource B
10
35
2400.00
<=
2400
Resource C
15
5
1835.71
<=
2400
Resource D
10
5
1285.71
<=
2400
Market P
1
110.00
<=
110
Market Q
1
37.14
<=
60
Profit/Unit
45
55
993 Total Profit
Product Mix 110.00
37.14
Continue solving the problem, by assuming the same
assumptions of 20% discount for the Japanese market.
Theory of Constraints 1- Basics
Ardavan Asef-Vaziri
Nov-2010
7
A Practice on Sensitivity Analysis
Cell
Name
$B$10 Product Mix Product P
$C$10 Product Mix Product Q
Final
Reduced Objective
Allowable
Allowable
Value
Cost
Coefficient Increase
Decrease
??????
0
45
1E+30 29.28571429
37.14
0
55
102.5
55
Constraints
Cell
$D$3
$D$4
$D$5
$D$6
$D$7
$D$8
Name
Resource A LHS (Needed)
Resource B LHS (Needed)
Resource C LHS (Needed)
Resource D LHS (Needed)
Market P LHS (Needed)
Market Q LHS (Needed)
Final
Shadow Constraint Allowable
Value
Price
R.H. Side
Increase
2021.43
0.000
2400
1E+30
2400.00
1.571
2400
800
1835.71
0.000
2400
1E+30
1285.71
0.000
2400
1E+30
110.00
29.286
110 31.17647059
37.14
0.000
60
1E+30
Allowable
Decrease
378.5714286
1300
564.2857143
1114.285714
80
22.85714286
What is the value of the objective function? Z= 45(?) + 55(37.14)-6000!
2400(0)+ 2400(1.571)+2400(0) +2400(0)+110(29.286)+ 60(0) =6993
Is the objective function Z = 6993?
6993-6000 = 993
Theory of Constraints 1- Basics
Ardavan Asef-Vaziri
Nov-2010
8
A Practice on Sensitivity Analysis
How many units of product P?
What is the value of the objective function?
Z= 45(???) + 55(37.14)-6000 = 993.
45X1= 4950
Final Reduced Objective
X1 = 110 Cell
Name
Value
Cost Coefficient
$B$10 Product Mix Product P
$C$10 Product Mix Product Q
??????
37.14
0
0
Allowable Allowable
Increase
Decrease
45
1E+30 29.28571429
55
102.5
55
Constraints
Cell
$D$3
$D$4
$D$5
$D$6
$D$7
$D$8
Name
Resource A LHS (Needed)
Resource B LHS (Needed)
Resource C LHS (Needed)
Resource D LHS (Needed)
Market P LHS (Needed)
Market Q LHS (Needed)
Theory of Constraints 1- Basics
Final Shadow Constraint Allowable Allowable
Value
Price
R.H. Side
Increase
Decrease
2021.43
0.00
2400
1E+30 378.5714286
2400.00
1.57
2400
800
1300
1835.71
0.00
2400
1E+30 564.2857143
1285.71
0.00
2400
1E+30 1114.285714
110.00
29.29
110 31.17647059
80
37.14
0.00
60
1E+30 22.85714286
Ardavan Asef-Vaziri
Nov-2010
9
Step 4 : Elevate the Constraint(s). Do We Try To
Sell In Japan?
Processing Times
A
C
B
15
15
10
10
5
35
Product
P
Q
D
10
5
Product Costs and Profits
Product
Selling
Price
P (domestic)
90
Q (domestic) 100
P (Japan)
72
Q (Japan)
80
Manufg.
Cost
45
45
45
45
Profit per $/Constraint
unit
Minute
45
4.5
55
1.57
27
2.7
35
1
Even without increasing capacity of B, we can increase our profit.
Theory of Constraints 1- Basics
Ardavan Asef-Vaziri
Nov-2010
10
2. Exploit the Constraint : Find the
World’s Best Solution to Throughput
For 110 units of P, need 110 (10) = 1100 min. on B,
leaving 1300 min. on B, for product P in Japan.
Each unit of PJ requires 10 minutes on B. So, we can
produce 1300/10 = 130 units of PJ.
We get 110(45) +130(27) = $8460 - $6000 = $2460 profit.
Check if there is another constraint that would not allow
us to collect that much profit. Let’s see.
Theory of Constraints 1- Basics
Ardavan Asef-Vaziri
Nov-2010
11
1. Identify The Constraint(s)
Product A
P
15
PJ
15
B
10
10
C
15
15
D Contribution Margin: P($45), PJ($27)
10 Market Demand: P(110), PJ(infinity)
10
Can we satisfy the demand?
Resource requirements for 110 P’s and 130 PJ’s:
 Resource A: 110 (15) + 130 (15) = 3600 minutes
 Resource B: 110(10) + 130(10) = 2400 minutes
 Resource C: 110(15) + 130(15) = 3600 minutes
 Resource D: 110(10) + 130(10) = 2400 minutes
 We need to use LP to find the optimal Solution.
Theory of Constraints 1- Basics
Ardavan Asef-Vaziri
Nov-2010
12
Step 4 : Exploit the Constraint(s).
Product
P
Resource A
15
Resource B
10
Resource C
15
Resource D
10
Market P
1
Market Q
Profit/Unit
45
Product Mix 110.00
Q
10
35
5
5
PJ
15
10
15
10
QJ
10
35
5
5
1
55
28.24
27
31.18
40
0.00
2400
<=
2400
2400
<=
2400
2258.8
<=
2400
1552.9
<=
2400
110
<=
110
28.2353
<=
60
1344.71 Total Profit
Not $2460 profit, but $1345. The $6000 is included.
Let’s buy another machine B at investment cost of
$100,000, and operating cost of $400 per week. Weekly
operating expense $6400. How soon do we recover
investment?
Theory of Constraints 1- Basics
Ardavan Asef-Vaziri
Nov-2010
13
Step 4 : Elevate the Constraint(s). New
Constraint
Product
Resource A
Resource B
Resource C
Resource D
Market P
Market Q
Profit/Unit
Product Mix
P
15
10
15
10
1
45
84.71
Q
10
35
5
5
PJ
15
10
15
10
QJ
10
35
5
5
1
55
60.00
27
0.00
40
52.94
2400
<=
2400
4800
<=
4800
1835.3
<=
2400
1411.8
<=
2400
84.7059
<=
110
60
<=
60
2829.41 Total Profit
Original Profit: $993
No Machine but going to Japan: $1345 profit.
Buy a machine B: $2829 profit. The $6400 is included.
Going to Japan has no additional cost. Buying additional machine
has initial investment and weekly operating costs.
$2829-$1345 = $1484  $100,000/$1484 = 67.4 weeks
Theory of Constraints 1- Basics
Ardavan Asef-Vaziri
Nov-2010
14
Buying a machine A at the same cost
Also add one machine A. Initial investment 100,000. Operating
cost $400/week.
Product
Resource A
Resource B
Resource C
Resource D
Market P
Market Q
Profit/Unit
Product Mix
P
15
10
15
10
1
45
110.00
Q
10
35
5
5
PJ
15
10
15
10
QJ
10
35
5
5
1
55
60.00
27
16.32
40
41.05
2905.3
<=
4800
4800
<=
4800
2400
<=
2400
1768.4
<=
2400
110
<=
110
60
<=
60
3532.63 Total Profit
From $2829 to $3533 = $3533 - $2829 = $704. The $6800 included..
$100,000/$704 = 142 weeks
Now B & C are a bottleneck
Theory of Constraints 1- Basics
Ardavan Asef-Vaziri
Nov-2010
15