Slide mô hình ra quyết địnhCASE 4

Vision is a large company that produces videocapturing devices for military applications such as missiles, longrange cameras, and aerial drones. Four different types of cameras (differing mainly by lens type) are made in the three plants in the system. Each plant can produce any of the four camera types, although each plant has its own individual constraints and unit costs. These constraints cover labor and ma chining restrictions, and the specific values are given in Tables 8–10. Note that even though the products are identical in the three plants, different production processes are used and thus the products use different amounts of resources in different plants. The corpo ration controls the material that goes into the lenses; the material requirements for each product are given in the last column of Tables 8–10. A total of 3,500

Production

Planning and

Shipping

CASE 4

MR.DANG VU TUNG

SIE - HUST

2 GROUP 5

MODEL FORMULATION

SOLUTION

SENSITIVITY ANALYSIS

MODEL FORMULATION

SOLUTION

SENSITIVITY ANALYSIS

Three Plants:

Asland (Plant 1)

Huntington (Plant 2)

Johnson City (Plant 3)

Three Customers:

RAYco

HONco

MMco

The potential issues:

- Get more material

- Get more inspection capacity

- Add extra machine hours

- Handle RAYco’s demand increases

For each customer:

- Product sales price

- Shipping cost

- Maximum product sales Inspection Capacity (Shipping from Plant 1 and 2 to RAYco and HONco)

Four Types of Products:

Small

Medium

Large

Precision

Using Liner Programming to create

a plan for production and shipping

For each plant:

- Labor hours available

- Machine hours available

- Production cost

- Total available material

Assumption: Produce less than or equal to

customer’s demand

MODEL FORMULATION

SOLUTION

SENSITIVITY ANALYSIS

DATA DECISION

VARIABLES

OBJECTIVE

FUNCTION CONSTRAINTS

Data + Liner program

= Model

DATA

x_1 = small from Ashland to Rayco x_2 = small from Ashland to Honco x_3 = small from Ashland to MMco x_4 = medium from Ashland to Rayco x_5 = medium from Ashland to Honco x_6 = medium from Ashland to MMco x_7 = large from Ashland to Rayco x_8 = large from Ashland to Honco x_9 = large from Ashland to MMco x_10 = precision from Ashland to Rayco x_11 = precision from Ashland to Honco x_12 = precision from Ashland to Mmco x_13 = small from Huntington to Rayco x_14 = small from Huntington to Honco x_15 = small from Huntington to MMco x_16 = medium from Huntington to Rayco x_17 = medium from Huntington to Honco x_18 = medium from Huntington to MMco

x_19 = large from Huntington to Rayco x_20 = large from Huntington to Honco x_21 = large from Huntington to MMco x_22 = precision from Huntington to Rayco x_23 = precision from Huntington to Honco x_24 = precision from Huntington to MMco x_25 = small from Johnson City to Rayco x_26 = small from Johnson City to Honco x_27 = small from Johnson City to MMco x_28 = medium from Johnson City to Rayco x_29 = medium from Johnson City to Honco x_30 = medium from Johnson City to MMco x_31 = large from Johnson City to Rayco x_32 = large from Johnson City to Honco x_33 = large from Johnson City to MMco x_34 = precision from Johnson City to Rayco x_35 = precision from Johnson City to Honco x_36 = precision from Johnson City to MMco

DECISION

VAR

Objective: Maximize Total Profit

Max Z = Profit Per Unit (Revenue − Cost) * xj for j = (1,2,3, ,36),

Where Revenue = Sales price/Unit, and Cost = Shipping cost/Unit + Production cost/Unit.

Therefore our objective function is

z = 2x_1 + 0.4x_2 + 0.9x_3 + x_4 + 0.4x_5 − 0.1x_6 +3x_7 + 2.4x_8 + 3.9x_9 + 2x_10 − 1.6x_11 − 0.1x_12 + 2.8x_13 + 1.5x_14 + 2x_15 − 0.2x_16 − 0.5x_17 − x_18 + 0.8x_19 + 0.5x_20 + 2x_21 + 3.8x_22 + 0.5x_23 + 2x_24 + 1.6x_25 + 0.5x_26 + 0.7x_27 + 1.6x_28 + 1.5x_29 + 0.7x_30 +

1.6x_31 + 1.5x_32 + 2.7x_33 + 4.6x_34 + 1.5x_35 + 2.7x_36

OBJECTIVE FUNCTION

+ Resources Constraints:

Labor Hours for Ashland Plant : 3x_1 +3x_2 +3x_3 +3x_4 +3x_5 +3x_6 +4x_7 +4x_8 +4x_9 + 4x_10 + 4x_11 + 4x_12 ≤ 6000

Machine Hours for Ashland Plant : 8x_1 + 8x_2 + 8x_3 + 8.5x_4 + 8.5x_5 + 8.5x_6 + 9x_7 + 9x_8 +9x_9 +9x_10 +9x_11 +9x_12 ≤ 10000

Labor Hours for Huntington Plant : 3.5x_13 + 3.5x_14 + 3.5x_15 + 3.5x_16 + 3.5x_17 + 3.5x_18 + 4.5x_19 + 4.5x_20 + 4.5x_21 + 4.5x_22 + 4.5x_23 + 4.5x_24 ≤ 5000

Machine Hours for Huntington Plant : 7_x13 + 7x_14 + 7x15 + 7x1_6 + 7x_17 + 7x_18 + 8x_19 + 8x_20 + 8x_21 + 9x_22 + 9x_23 + 9x_24 ≤ 12500

Labor Hours for Johnson City Plant : 3x_25 + 3x_26 + 3x_27 + 3.5x_28 + 3.5x_29 + 3.5x_30 + 4x_31 + 4x_32 + 4x_33 + 4.5x_34 + 4.5x_35 + 4.5x_36 ≤ 3000

Machine Hours for Johnson City Plant : 7.5x_25 + 7.5x_26 + 7.5x_27 + 7.5x_28 + 7.5x_29 + 7.5x_30 + 8.5x_31 + 8.5x_32 + 8.5x_33 + 8.5x_34 + 8.5x_35 + 8.5x_36 ≤ 6000

Total Materials Used by Each Plant : 1x_1 + 1x_2 + 1x_3 + 1.1x_4 + 1.1x_5 + 1.1x_6 + 1.2x_7 + 1.2x_8 + 1.2x_9 + 1.3x_10 + 1.3x_11 + 1.3x_12 + 1.1x_13 + 1.1x_14 + 1.1x_15 + 1x_16 + 1x_17 + 1x_18 + 1.1x_19 + 1.1x_20 + 1.1x_21 + 1.4x_22 + 1.423x_23 + 1.4x_24 + 1.1x_25 + 1.1x_26 + 1.1x_27 + 1.1x_28 + 1.1x_29 + 1.1x_30 + 1.3x_31 + 1.3x_32 + 1.3x_33 + 1.3x_34 + 1.3x_35 + 1.3X_36 ≤ 3500

+ Sales and shipping constraints

Maximum Small Product Sales to RAYco : 17(x_1 + x_13 + x_25) ≤ 200

Maximum Medium Product Sales to RAYco : 18(x_4 + x_16 + x_28) ≤ 300

Maximum Large Product Sales to RAYco : 22(x_7 + x_19 + x_31) ≤ 500

Maximum Precision Product Sales to RAYco : 29(x_10 + x_22 + x_34)≤ 200

Maximum Small Product Sales to HONco : 16(x_2 + x_14 + x_26) ≤ 400

Maximum Medium Product Sales to HONco : 18(x_5 + x_17 + x_29) ≤ 300

Maximum Large Product Sales to HONco : 22(x_8 + x_20 + x_32) ≤ 200

Maximum Precision Product Sales to HONco : 26(x_11 + x_23 + x_35) ≤ 400

Maximum Small Product Sales to MMco : 16(x_3 + x_15 + x_27) ≤ 200

Maximum Medium Product Sales to MMco : 17(x_6 + x_18 + x_30) ≤ 400

Maximum Large Product Sales to MMco : 23(x_9 + x_21 + x_33) ≤ 300

Maximum Precision Product Sales to MMco : 27(x_12 + x_24 + x_36) ≤ 300

Inspection Capacity: x_1 +x_2 +x_4 +x_5 +x_7 +x_8 +x_10 +x_11 +x_13 +x_14

+x_16 + x_17 +x_19 +x_20 +x_22 +x_23 ≤1500

CONSTRAINTS

MODEL FORMULATION

SOLUTION

SENSITIVITY ANALYSIS

OPTIMAIL SOLUTION OPTIMAL VALUE

X1 = 11.76 (rounded up 12) X4 = 16.67 (rounded up 17) X7 = 22.73 (rounded up 23) X10 = 6.897 (rounded up 7)

X15 = 12.5 X21 = 13.04 (rounded up 13) X24 =11.11 (rounded up 11)

Others equal 0

The maximum profit is Z = 195,48 Number of unit sales is 94.71

Result: Would not meet small, medium, large, precision demand for HONco

and medium for MMco.

- Production & Shipping

Assuming that we round up the products to integer values,

we get the following results:

12 small products from Ashland to RAYco

17 medium products from Ashland to RAYco

23 large products from Huntington to RAYco

7 precision products from Ashland to RAYco

12.5 small products from Huntington to MMco

13 large products from Huntington to MMco

11 precision products from Huntington to MMco

- Cost & Revenue

We have calculated the total cost (=production costs + shipping costs) and the revenue that Vision company will receive

according to each plant.

For the Ashland plant, the total cost is $1095 and the total revenue is $1219.

For the Huntington plant, the total cost is $722.45 and the total revenue is $796.

For the Johnson City plant, the total cost is $0 and the total revenue is $0.

Suggestion

MODEL FORMUALTION

SOLUTION

SENSITIVITY ANALYSIS

"If you could get more material, how much would you like? What would you be willing to pay for it?”

No, it is not necessary to get more material because the shadow price of the toal material constraint is 0.

This is also because we have a lack of 3500 − 11.764706 = 3488.235294 units for the total material constraint.

Inspection Capacity

"If you could get more inspection capacity, how much would you like? How would you use it? What would you be willing to pay for it?”

No, it is not necessary to get more inspection capacity because the shadow price of the inspection constraint is 0.

This is also because we have a lack of 1500 − 58.055197 = 1441.944803 units for the inspection capacity

constraint.

Machine Hours

"At what plant(s) would you like to add extra machine hours? How much would you be willing to pay per hour? How many extra hours would you like?”

No, it is not necessary to add extra machine hours at any plants because the shadow price of the machine hours for each plant constraint is 0.

This is also because we have a lack of 10000 − 360.73207 = 9639.26793 units for the machine hours for plant 1 constraint and 12500 − 291.84783 = 12208.15217 units for the machine hours for plant 2.

RAYco's Demand +50%

"Marketing is trying to get RAYco to consider a 50% increase in its demand Can we handle this with the current system or do we need more resources? How much more money can we make if we take on the additional

demand?”

Increase our profit to $257.48, which is an increase of $62 by selling 124.21 the number of units (including: small, medium, large and precision products).

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