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Practice: A Production System Manufacturing Two Products, P and Q

$90 / unit

$100 / unit

Q:

P:

60 units / week

110 units / week

D

D

Purchased Part

10 min.

5 min.

$5 / unit

C

B

C

10 min.

5 min.

25 min.

B

A

A

15 min.

10 min.

10 min.

RM1

RM2

RM3

$20 per

$20 per

$25 per

unit

unit

unit

Time available at each work center: 2,400 minutes per week.

Operating expenses per week: $6,000. All the resources cost the same.

Contribution Margin: P($45), Q($55)

Market Demand: P(110), Q(60)

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) = 1950minutes
- Resource D: 110(10) + 60(5) = 1400 minutes

2. Exploit the Constraint : Find the Throughput World’s Best Solution

Resource B is Constrained - Bottleneck

Product P Q

Profit $ 45 55

Resource B needed (min) 10 35

Profit per min of Bottleneck 45/10 =4.555/35 =1.6

Per unit of bottleneck Product P creates more profit than Product Q

Produce as much as P, then Q

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.

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.

Decision Variables

x1 : Volume of Product P

x2 : Volume of Product Q

Resource A

15 x1 + 10 x2 2400

Resource B

10 x1 + 35 x2 2400

Resource C

15 x1 + 5 x2 2400

Resource D

10 x1 + 5 x2 2400

Market for P

x1 110

Market for Q

x2 60

Objective Function

Maximize Z = 45 x1 +55 x2 -6000

Nonnegativity

x1 0, x2 0

Continue solving the problem, by assuming the same assumptions of 20% discount for the Japanese market.

A Practice on Sensitivity Analysis

- 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

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
- X1 = 110

Step 4 : Elevate the Constraint(s). Do We Try To Sell In Japan?

Even without increasing capacity of B, we can increase our profit.

$/Constraint

Minute

4.5

1.57

2.7

1

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.

Contribution Margin: P($45), PJ($27)

Market Demand: P(110), PJ(infinity)

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) = 3600minutes
- Resource D: 110(10) + 130(10) = 2400 minutes
- We need to use LP to find the optimal Solution.

Step 4 : Exploit the Constraint(s).

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?

Step 4 : Elevate the Constraint(s). New Constraint

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

Buying a machine A at the same cost

Also add one machine A. Initial investment 100,000. Operating cost $400/week.

From $2829 to $3533 = $3533 - $2829 = $704. The $6800 included..

$100,000/$704 = 142 weeks

Now B & C are a bottleneck

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