340project Final

Optimizing Production for Golf-

Sport Plants in Arizona to

Maximize Profit

By

Erik Baker

Clayton Jeanette

Nicole Grisamore

December 10, 2014

Abstract

In this project, we studied the problem of optimizing the production of golf parts and sets to

maximize profits by the linear programming method. First, we formulated this problem as a LP,

and then used a simplex algorithm to solve the problem. We also included sensitivity analysis

and duality analysis. Our results show that the company achieved a max profit of $202,127.10

based on the constraints of production for month one. After a 12% increase in production costs

for the parts and sets, the maximum profit decreased to $159,652.00 in the second month.

1. Problem Description

Golf-Sport is a company that produces golf components for people to build their own clubs

as well as pre-constructed golf club sets. The components are produced in three locations,

Chandler, Glendale, and Tucson. Five components are produced: steel shafts, graphite shafts,

forged iron heads, metal wood heads, and metal wood heads with titanium inserts. All of the

locations can produce the same components; however, each has its own set of constraints and

unit costs (resource costs). The constraints for each location consist of labor and packaging

machine time. In addition to selling individual components, Golf-Sport assembles golf club sets

based on a fixed component amount of 13 shafts, 10 iron heads, and 3 wood heads. Each set

requires that the shafts be the same type and the wood heads be the same type. The products

are then sold at the factory retail store for each location. The goal is to effectively find a

recommendation for the golf company in terms of production and sales over a two month

period. Additionally, there are a few other questions to solve regarding sensitivity analysis. We

want to find out if you got more graphite or advertising cash, how much you would want, how

much would you use, and would you be willing to pay? Taking into account how adding (at

certain locations) extra packing machine hours, assembly hours, or extra labor hours would

affect the company. Also think about the labor costs in how much you would be willing to pay

per hour and what the needed extra hours are. Finally, consider if the company’s operation is

still sustainable if it adds an advertising program that promises increased demand (50% max

increase) and how the additional demand will affect production costs (how much more).

2. Modeling and Formulation of the Problem

Problem data

Tables 1-3 represent the availability of resources (labor, packing, and advertising) for each

component in the three different plant locations.

Table 1: Product-Resource Constraints: Chandler

Products Labor

(Minutes/Unit) Packing

(Minutes/Unit) Advertising

($/Unit)

Steel Shafts 1 4 1.0

Graphite shafts 1.5 4 1.5

Forged iron heads 1.5 5 1.1

Metal wood heads 3 6 1.5

Titanium insert heads 4 6 1.9

Monthly availability (minutes)

12,000 20,000 -

Table 2: Product-Resource Constraints: Glendale

Products Labor



($/Unit)

Steel Shafts 3.5 7 1.1

Graphite shafts 3.5 7 1.1

Forged iron heads 4.5 8 1.1

Metal wood heads 4.5 9 1.2

Titanium insert heads 5.0 7 1.9


15,000 40,000 -

Table 3: Product-Resource Constraints: Tucson

Products Labor



($/Unit)

Steel Shafts 3 7.5 1.3

Graphite shafts 3.5 7.5 1.3

Forged iron heads 4 8.5 1.3

Metal wood heads 4.5 9.5 1.3

Titanium insert heads 5.5 8.0 1.9


22,000 20,000 -

Table 4:

Plant Time

(Minutes per set) Total Time Available

(Minutes)

Chandler 65 5,500

Glendale 60 5,000

Tucson 65 6,000

Table 4 shows the minutes needed and available for each plant location.

Table 5: Minimum and Maximum Product Demand per Month

Products Chandler Glendale Tucson Steel shafts [0, 2,000] [0, 2,000] [0, 2,000] Graphite shafts [100, 2,000] [100, 2,000] [50, 2,000] Forged Iron Heads [200, 2,000] [200, 2,000] [100, 2,000] Metal wood heads [30, 2,000] [30, 2,000] [15, 2,000] Titanium insert

heads [100, 2,000] [100, 2,000] [100, 2,000] Set: Steel, metal [0, 200] [0, 200] [0, 200] Set: Steel, insert [0, 100] [0,100] [0, 100] Set: Graphite, metal [0, 300] [0, 300] [0, 300] Set: Graphite, insert [0, 400] [0, 400] [0, 400] Table 5 displays the range in demand for each component and golf set in all three locations.

Table 6:

Material, Production, and Assembly Costs ($) per Part or Set

Products Chandler Glendale Tucson

Steel shafts 6 5 7

Graphite shafts 19 18 20

Forged Iron Heads 4 5 5

Metal wood heads 10 11 12 Titanium insert heads 26 24 27

Set: Steel, metal 178 175 180

Set: Steel, insert 228 220 240

Set: Graphite, metal 350 360 370

Set: Graphite, insert 420 435 450

Table 6 represents the cost of materials, production, and assembly per part or set for each

location.

Table 7: Revenue per Part or Set ($)

Products Chandler Glendale Tucson

Steel shafts 10 10 12

Graphite shafts 25 25 30

Forged Iron Heads 8 8 10

Metal wood heads 18 18 22

Titanium insert heads 40 40 45

Set: Steel, metal 290 290 310

Set: Steel, insert 380 380 420

Set: Graphite, metal 560 560 640

Set: Graphite, insert 650 650 720

Table 7 gives the revenue received for each part or set sold in each location.

Decision variables

We chose the decision variables based on parts that the plants produce. The decision variables

make up the objective function which is to maximize profit and to find the optimal combination

of part production across the three locations to achieve the goal.

First subscript- plant

1- Chandler

2- Glendale

3- Tucson

Second subscript- product

1- Steel Shaft

2- Graphite Shaft

3- Forged Iron Heads

4- Metal Wood Heads

5- Titanium Insert heads

6- Steel Metal Set

7- Steel Insert Set

8- Graphite Metal Set

9- Graphite Insert Set

X11 - Steel Shaft produced in Chandler

X12 – Graphite Shaft produced in Chandler

X13 – Forged Shaft produced in Chandler

X14 –Metal Wood Heads produced in Chandler

X15 –Titanium Insert produced in Chandler

X16- Steel Metal Set Chandler

X17- Steal Insert Set Chandler

X18- Graphite Metal Set Chandler

X19- Graphite Insert Set Chandler

X21 – Steel Shaft produced in Glendale

X22 –Graphite Shaft produced in Glendale

X23 –Forged Iron Heads produced in Glendale

X24 –Metal Wood Heads produced in Glendale

X25 –Titanium Insert Heads produced in Glendale

X26- Steel Metal Set Glendale

X27- Steal Insert Set Glendale

X28- Graphite Metal Set Glendale

X29- Graphite Insert Set Glendale

X31 - Steel Shaft produced in Tucson

X32 - Graphite Shaft produced in Tucson

X33 - Forged Iron Heads produced in Tucson

X34 - Metal Wood Heads produced in Tucson

X35 - Titanium Insert Heads produced in Tucson

X36- Steel Metal Set Tucson

X37- Steal Insert Set Tucson

X38- Graphite Metal Set Tucson

X39- Graphite Insert Set Tucson

Objective function represents profit which is the revenue minus the material, production, and

assembly part for every golf part and set produced at each location.

Max z=

(10-6)X11 + (25-19)X12 + (8-4)X13 + (18-10)X14 + (40-26)X15 + (290-178)X16 + (380-228)X17 +

(560-350)X18 + (650-420)X19 + (10-5)X21 + (25-18)X22 + (8-5)X23 + (18-11)X24 + (40-24)X25 +

(290-175)X26 + (380-220)X27 + (560-360)X28 + (650-435)X29 (12-7)X31 + (30-20)X32 + (10-5)X33 +

(22-12)X34 + (45-27)X35 + (310-180)X36 + (420-240)X37 + (640-370)X38 + (720-450)X39

Max z =

4X11 + 6X12 + 4X13 + 8X14 + 14X15 + 112X16 + 152X17 + 210X18 + 230X19 + 5X21 + 7X22 + 3X23 + 7X24 +

16X25 + 115X26 + 160X27 + 200X28 + 215X29 + 5X31 + 10X32 + 5X33 + 10X34 + 18X35 + 130X36 + 180X37 +

270X38 + 270X39

2.1 Constraint

Product-Resource Constraints: Chandler

Labor: X11 + 1.5 X12 +1.5 X13 + 3X14 +4 X15 <=12000 (2)

Packing: 4X11 + 4X12 + 5X13 + 6X14 + 6X15 <=20000 (3)

Product-Resource Constraints: Glendale

Labor: 3.5X21 + 3.5X22 + 4.5X23 + 4.5X24 + 5.0 X25 <= 15000 (4)

Packing: 7X21 + 7X22 + 8X23 + 9X24 + 7X25 <= 40000 (5)

Product-Resource Constraints: Tucson

Labor: 3X31 + 3.5X32 + 4X33 + 4.5X34 + 5.5X35 <= 22000 (6)

Packing: 7.5X31 + 7.5X32 + 8.5X33 + 9.5X34 + 8.0X35 <= 35000 (7)

Advertising

X11 +1.5 X12 +1.1 X13 +1.5 X14 +1.9 X15 +1.1 X21 + 1.1 X22 + 1.1 X23 + 1.2 X24 + 1.9 X25 + 1.3 X31

+1.3 X32 + 1.3 X33 + 1.3 X34 + 1.9 X35 <= 20000 (8)

Time Constraints for the minutes of labor it takes to assemble sets at each plant

65(X16 + X17 + X18 + X19 ) <= 5500 (9)

60(X26 + X27 + X28 + X29 ) <= 5000 (10)

65(X36 + X37 + X38 + X39 ) <= 6000 (11)

Graphite Constraint: the corporation limits the amount of graphite used per month

0.25(X12 + X22 + X32 ) >= 1000 (12)

Demand Constraints: The minimum and maximum amount each store must stock of each part

and set to meet demand.

0<= X11 <= 2000 (13)

100<= X12 <= 2000 (14)

200<= X13 <=2000 (15)

30<= X14 <=2000 (16)

100<= X15 <=2000 (17)

0<= X16 <=200 (18)

0<= X17 <=100 (19)

0<= X18 <=300 (20)

0<= X19 <=400 (21)

0<= X21 <=2000 (22)

100<= X22 <=2000 (23)

200<= X23 <=2000 (24)

30<= X24 <=2000 (25)

100<= X25 <=2000 (26)

0<= X26 <=200 (27)

0<= X27 <=100 (28)

0<= X28 <=300 (29)

0<= X29 <=400 (30)

0<= X31 <=2000 (31)

50<= X32 <=2000 (32)

100<= X33 <=2000 (33)

15<= X34 <=2000 (34)

100<= X35 <=2000 (35)

0<= X36 <=200 (36)

0<= X37 <=100 (37)

0<= X38 <=300 (38)

0<= X39 <=400 (39)

Whole Objective

Max z =

4X11 + 6X12 + 4X13 + 8X14 + 14X15 + 112X16 + 152X17 + 210X18 + 230X19 + 5X21 + 7X22 + 3X23 + 7X24 +

16X25 + 115X26 + 160X27 + 200X28 + 215X29 + 5X31 + 10X32 + 5X33 + 10X34 + 18X35 + 130X36 + 180X37 +

270X38 + 270X39

X11 + 1.5 X12 +1.5 X13 + 3X14 +4 X15 <=12000

4X11 + 4X12 + 5X13 + 6X14 + 6X15 <=20000

3.5X21 + 3.5X22 + 4.5X23 + 4.5X24 + 5.0 X25 <= 15000

7X21 + 7X22 + 8X23 + 9X24 + 7X25 <= 40000

3X31 + 3.5X32 + 4X33 + 4.5X34 + 5.5X35 <= 22000

7.5X31 + 7.5X32 + 8.5X33 + 9.5X34 + 8.0X35 <= 35000

X11 +1.5 X12 +1.1 X13 +1.5 X14 +1.9 X15 +1.1 X21 + 1.1 X22 + 1.1 X23 + 1.2 X24 + 1.9 X25 + 1.3 X31 +1.3 X32 + 1.3 X33 + 1.3 X34 + 1.9 X35 <= 20000

65(X16 + X17 + X18 + X19 ) <= 5500

60(X26 + X27 + X28 + X29 ) <= 5000

65(X36 + X37 + X38 + X39 ) <= 6000

0.25(X12 + X22 + X32 ) >= 1000

0<= X11 <= 2000

100<= X12 <= 2000

200<= X13 <=2000

30<= X14 <=2000

100<= X15 <=2000

0<= X16 <=200

0<= X17 <=100

0<= X18 <=300

0<= X19 <=400

0<= X21 <=2000

100<= X22 <=2000

200<= X23 <=2000

30<= X24 <=2000

100<= X25 <=2000

0<= X26 <=200

0<= X27 <=100

0<= X28 <=300

0<= X29 <=400

0<= X31 <=2000

50<= X32 <=2000

100<= X33 <=2000

15<= X34 <=2000

100<= X35 <=2000

0<= X36 <=200

0<= X37 <=100

0<= X38 <=300

0<= X39 <=400

4. Solving the Problem 1. Input objective function and constraints into Lindo.

2. Modify LP to match Lindo syntax.

3. Solve.

4. View results.

5. Results and Analysis

Tables 8 and 9 show the optimal production amount of each golf part and sets in Chandler,

Glendale, and Tucson for the first and second month respectively. After solving the LP in Lindo,

the max profit was found to be $202,127.10 in the first month and $159,652.00 for the second.

These values were found by imputing the variable values into the objective function, which was

made by subtracting assembly costs from sales prices. The assembly costs increased by 12% in

month 2 while the production times remained the same, as a result of this change the total

profit decreases. This profit loss was caused by the increased cost to the plants (production

costs) with no change in revenue.

Table 8:

Variable Part Value

X11 Steel shafts in Chandler 0.00

X12 Graphite shafts in Chandler 1705.00

X13 Forged iron heads in Chandler 200.00

X14 Metal wood Heads in Chandler 30.00

X15 Titanium insert heads in Chandler 2000.00

X16 Set: Steel, metal in Chandler 0.00

X17 Set: Steel, insert in Chandler 0.00

X18 Set: Graphite, metal in Chandler 0.00

X19 Set: Graphite, insert in Chandler 84.62

X21 Steel shafts in Glendale 0.00

X22 Graphite shafts in Glendale 1132.86

X23 Forged iron heads in Glendale 200.00

X24 Metal wood Heads in Glendale 30.00

X25 Titanium insert heads in Glendale 2000.00

X26 Set: Steel, metal in Glendale 0.00

X27 Set: Steel, insert in Glendale 0.00

X28 Set: Graphite, metal in Glendale 0.00

X29 Set: Graphite, insert in Glendale 83.33

X31 Steel shafts in Tucson 0.00

X32 Graphite shafts in Tucson 2000.00

X33 Forged iron heads in Tucson 100.00

X34 Metal wood heads in Tucson 331.58

X35 Titanium insert heads in Tucson 2000.00

X36 Set: Steel, metal in Tucson 0.00

X37 Set: Steel, insert in Tucson 0.00

X38 Set: Graphite, metal in Tucson 0.00

X39 Set: Graphite, insert in Tucson 92.31

Table 9:

Variable Part Value

X11 Steel shafts in Chandler 0.00

X12 Graphite shafts in Chandler 867.14

X13 Forged iron heads in Chandler 200.00

X14 Metal wood Heads in Chandler 588.57

X15 Titanium insert heads in Chandler 2000.00

X16 Set: Steel, metal in Chandler 0.00

X17 Set: Steel, insert in Chandler 0.00

X18 Set: Graphite, metal in Chandler 0.00

X19 Set: Graphite, insert in Chandler 84.62

X21 Steel shafts in Glendale 0.00

X22 Graphite shafts in Glendale 1132.86

X23 Forged iron heads in Glendale 200.00

X24 Metal wood Heads in Glendale 30.00

X25 Titanium insert heads in Glendale 2000.00

X26 Set: Steel, metal in Glendale 0.00

X27 Set: Steel, insert in Glendale 0.00

X28 Set: Graphite, metal in Glendale 0.00

X29 Set: Graphite, insert in Glendale 83.33

X31 Steel shafts in Tucson 0.00

X32 Graphite shafts in Tucson 2000.00

X33 Forged iron heads in Tucson 100.00

X34 Metal wood heads in Tucson 331.58

X35 Titanium insert heads in Tucson 2000.00

X36 Set: Steel, metal in Tucson 0.00

X37 Set: Steel, insert in Tucson 0.00

X38 Set: Graphite, metal in Tucson 92.31

X39 Set: Graphite, insert in Tucson 0.00

6. Sensitivity Analysis and Duality

In Table 12, the dual prices for the corresponding constraints (Rows 2-12) represent the

minimum resources that can be used to achieve maximum profit. The values correspond to

the Dual LP (yi) values and the shadow prices to show how changing resource constraints

can affect achievable profit. Rows 13-51 corresponds to fixed demand constraint and

cannot be minimized. The reduced cost corresponds to the dual excess variables for that

given Xij value. In table 14 the allowable increase and decrease represents respectively the

maximum change in coefficient value that can occur for the current LP to remain optimal. If

the change in an objective function coefficient falls outside this range, the current LP would

no longer be optimal. Table 15 the allowable increase and decrease represents respectively

the maximum change that can occur for the right hand side of the constraints. If the

constraint limitation is beyond this, the current LP would not be optimal anymore. When we

changed two coefficients in the objective function, the profit for Titanium insert heads in

Chandler was decreased and the profit of Set: Steel, metal made in Chandler was increased,

we solved again in Lindo and observed the results; we found that the maximum profit

decreased making this not a wise choice. Changing the constraints for labor and packing in

Glendale however proved to benefit the maximum profit when the monthly available labor

was increased and packing decreased.

Table 12 shows the dual prices of each constraint that are equal to the shadow prices

for the max problem. Row 8 is the advertising constraint which has a shadow price of 0, this

represents that changing the advertising budget in either the negative or positive direction

will not have to direct effect on the total profits. This makes sense because based on the

data given there is no correlation between advertising and sales of parts. Row 11

represents the time constraint to assemble sets in the Tucson plant. The shadow price is

4.15, this value represents the dollar amount the total profits will change when the total

time available for the Tucson plant is increased or decreased a minute.

Table 10:

Variable Part Value Reduced Cost

X11 Steel shafts in Chandler 0.00 2.00

X12 Graphite shafts in Chandler 1705.00 0.00

X13 Forged iron heads in Chandler 200.00 0.00

X14 Metal wood Heads in Chandler 30.00 0.00

X15 Titanium insert heads in Chandler 2000.00 0.00

X16 Set: Steel, metal in Chandler 0.00 118.00

X17 Set: Steel, insert in Chandler 0.00 78.00

X18 Set: Graphite, metal in Chandler 0.00 20.00

X19 Set: Graphite, insert in Chandler 84.62 0.00

X21 Steel shafts in Glendale 0.00 2.00

X22 Graphite shafts in Glendale 1132.86 0.00

X23 Forged iron heads in Glendale 200.00 0.00

X24 Metal wood Heads in Glendale 30.00 0.00

X25 Titanium insert heads in Glendale 2000.00 0.00

X26 Set: Steel, metal in Glendale 0.00 100.00

X27 Set: Steel, insert in Glendale 0.00 55.00

X28 Set: Graphite, metal in Glendale 0.00 15.00

X29 Set: Graphite, insert in Glendale 83.33 0.00

X31 Steel shafts in Tucson 0.00 2.89

X32 Graphite shafts in Tucson 2000.00 0.00

X33 Forged iron heads in Tucson 100.00 0.00

X34 Metal wood heads in Tucson 331.58 0.00

X35 Titanium insert heads in Tucson 2000.00 0.00

X36 Set: Steel, metal in Tucson 0.00 140.00

X37 Set: Steel, insert in Tucson 0.00 90.00

X38 Set: Graphite, metal in Tucson 0.00 0.00

X39 Set: Graphite, insert in Tucson 92.31 0.00

Table 11:

Variable Part Value Reduced Cost

X11 Steel shafts in Chandler 0.00 1.25

X12 Graphite shafts in Chandler 867.14 0.00

X13 Forged iron heads in Chandler 200.00 0.00

X14 Metal wood Heads in Chandler 588.57 0.00

X15 Titanium insert heads in Chandler 2000.00 0.00

X16 Set: Steel, metal in Chandler 0.00 88.96

X17 Set: Steel, insert in Chandler 0.00 54.96

X18 Set: Graphite, metal in Chandler 0.00 11.60

X19 Set: Graphite, insert in Chandler 84.62 0.00

X21 Steel shafts in Glendale 0.00 1.25

X22 Graphite shafts in Glendale 1132.86 0.00

X23 Forged iron heads in Glendale 200.00 0.00

X24 Metal wood Heads in Glendale 30.00 0.00

X25 Titanium insert heads in Glendale 2000.00 0.00

X26 Set: Steel, metal in Glendale 0.00 68.80

X27 Set: Steel, insert in Glendale 0.00 29.20

X28 Set: Graphite, metal in Glendale 0.00 6.00

X29 Set: Graphite, insert in Glendale 83.33 0.00

X31 Steel shafts in Tucson 0.00 2.60

X32 Graphite shafts in Tucson 2000.00 0.00

X33 Forged iron heads in Tucson 100.00 0.00

X34 Metal wood heads in Tucson 331.58 0.00

X35 Titanium insert heads in Tucson 2000.00 0.00

X36 Set: Steel, metal in Tucson 0.00 117.20

X37 Set: Steel, insert in Tucson 0.00 74.40

X38 Set: Graphite, metal in Tucson 92.31 0.00

X39 Set: Graphite, insert in Tucson 0.00 9.60

Table 12: Month 1

Row Slack or Surplus Dual Prices

2 1052.50 0.00

3 0.00 1.50

4 0.00 2.00

5 16200.00 0.00

6 2107.89 0.00

7 0.00 1.05

8 1114.30 0.00

9 0.00 3.54

10 0.00 3.58

11 0.00 4.15

12 209.46 0.00

13 2000.00 0.00

14 295.00 0.00

15 1605.00 0.00

16 1800.00 0.00

17 0.00 -3.50

18 1970.00 0.00

19 0.00 -1.00

20 0.00 5.00

21 1900.00 0.00

22 200.00 0.00

23 100.00 0.00

24 300.00 0.00

25 315.38 0.00

26 2000.00 0.00

27 867.14 0.00

28 1032.86 0.00

29 1800.00 0.00

30 0.00 -6.00

31 1970.00 0.00

32 0.00 -2.00

33 0.00 6.00

34 1900.00 0.00

35 200.00 0.00

36 100.00 0.00

37 300.00 0.00

38 316.67 0.00

39 20000.00 0.00

40 0.00 2.11

41 1950.00 0.00

42 1900.00 0.00

43 0.00 -3.95

44 1668.42 0.00

45 316.58 0.00

46 0.00 9.58

47 1900.00 0.00

48 200.00 0.00

49 100.00 0.00

50 300.00 0.00

51 307.69 0.00

Table 13: Month 2

Row Slack or Surplus Dual Prices

2 633.57 0.00

3 0.00 1.13

4 0.00 1.62

5 16200.00 0.00

6 2107.89 0.00

7 0.00 0.90

8 1533.23 0.00

9 0.00 2.76

10 0.00 2.71

11 0.00 3.47

12 209.46 -3.25

13 2000.00 0.00

14 1132.86 0.00

15 767.14 0.00

16 1800.00 0.00

17 0.00 -2.17

18 1411.13 0.00

19 558.57 0.00

20 0.00 4.08

21 1900.00 0.00

22 200.00 0.00

23 100.00 0.00

24 300.00 0.00

25 315.38 0.00

26 2000.00 0.00

27 867.14 0.00

28 1032.86 0.00

29 1800.00 0.00

30 0.00 -4.87

31 1970.00 0.00

32 0.00 -1.59

33 0.00 5.04

34 1900.00 0.00

35 200.00 0.00

36 100.00 0.00

37 300.00 0.00

38 316.67 0.00

39 20000.00 0.00

40 0.00 1.66

41 1950.00 0.00

42 1900.00 0.00

43 0.00 -3.26

44 1668.42 0.00

45 316.58 0.00

46 0.00 7.55

47 1900.00 0.00

48 200.00 0.00

49 100.00 0.00

50 207.69 0.00

51 400.00 0.00

Table 14:

OBJ COEFFICIENT RANGES

VARIABLE CURRENT ALLOWABLE ALLOWABLE

COEF INCREASE DECREASE

X11 4.00 2.00 INFINITY

X12 6.00 3.33 0.666667



X15 14.00 INFINITY 5.00

X16 112.00 117.99 INFINITY

X17 152.00 77.99 INFINITY

X18 210.00 19.99 INFINITY

X19 230.00 INFINITY 19.99


X22 7.00 4.20 1.56


X24 7.000000 2.000000 INFINITY

X25 16.000000 INFINITY 6.000000

X26 115.000000 99.999992 INFINITY

X27 160.000000 54.999996 INFINITY

X28 200.000000 14.999995 INFINITY

X29 215.000000 INFINITY 15.000000

X31 5.000000 2.894737 INFINITY

X32 10.000000 INFINITY 2.105263

X33 5.000000 3.947368 INFINITY

X34 10.000000 2.666667 3.666667

X35 18.000000 INFINITY 9.578947

X36 130.000000 140.000000 INFINITY

X37 180.000000 90.000008 INFINITY

X38 270.000000 0.000007 INFINITY

X39 270.000000 INFINITY 0.000007

Table 15:

RIGHTHAND SIDE RANGES

ROW CURRENT ALLOWABLE ALLOWABLE

RHS INCREASE DECREASE

2 12000.000000 INFINITY 1052.500000

3 20000.000000 1180.000000 3351.428467

4 15000.000000 3035.000000 2932.499756

5 40000.000000 INFINITY 16200.000000

6 22000.000000 INFINITY 2107.894775

7 35000.000000 4450.000000 3007.500000

8 20000.000000 INFINITY 1114.304565

9 5500.000000 20500.000000 5500.000000

10 5000.000000 18999.998047 5000.000000

11 6000.000000 20000.000000 6000.000000

12 1000.000000 209.464279 INFINITY

13 2000.000000 INFINITY 2000.000000

14 2000.000000 INFINITY 295.000000

15 100.000000 1605.000000 INFINITY

16 2000.000000 INFINITY 1800.000000

17 200.000000 670.285706 200.000000

18 2000.000000 INFINITY 1970.000000

19 30.000000 558.571411 30.000000

20 2000.000000 558.571411 196.666672

21 100.000000 1900.000000 INFINITY

22 200.000000 INFINITY 200.000000

23 100.000000 INFINITY 100.000000

24 300.000000 INFINITY 300.000000

25 400.000000 INFINITY 315.384613

26 2000.000000 INFINITY 2000.000000

27 2000.000000 INFINITY 867.142883

28 100.000000 1032.857178 INFINITY

29 2000.000000 INFINITY 1800.000000

30 200.000000 651.666626 200.000000

31 2000.000000 INFINITY 1970.000000

32 30.000000 651.666626 30.000000

33 2000.000000 586.499939 607.000000

34 100.000000 1900.000000 INFINITY

35 200.000000 INFINITY 200.000000

36 100.000000 INFINITY 100.000000

37 300.000000 INFINITY 300.000000

38 400.000000 INFINITY 316.666656

39 2000.000000 INFINITY 2000.000000

40 2000.000000 401.000000 837.857117

41 50.000000 1950.000000 INFINITY

42 2000.000000 INFINITY 1900.000000

43 100.000000 353.823547 100.000000

44 2000.000000 INFINITY 1668.421021

45 15.000000 316.578949 INFINITY

46 2000.000000 375.937500 1900.000000

47 100.000000 1900.00000 INFINITY

48 200.000000 INFINITY 200.000000

49 100.000000 INFINITY 100.000000

50 300.000000 INFINITY 300.000000

51 400.000000 INFINITY 307.692322

7. Conclusions

Once we picked a case to optimize, we identified the pertinent data given in the

problem description and thought of several ways to approach optimization. After

considering our options we agreed that formulating an LP and inputting it into Lindo

would be the best solution. Then we looked through the data tables to determine all of

our decision variables that will make up the objective function and constraints. After

establishing the variables into categories of the different golf parts and sets, we were

able to formulate an objective function to represent the maximum profit from each

plant production (revenue-cost). The data tables also provided most of the constraints

for the problem, including resources, assembly time, production costs, revenue, and

demand. Once this was all put together in Lindo, the optimal values were found by

solving the LP.

In the future several aspects of the problem could be changed to find an even

more optimal solution. For example, the plants manufacturing capabilities can be

manipulated by changing labor sources, plant capacity, or number of plants. We could

also study how advertising effects demand of the products and the potential profit

increase that would come along with it. Expanding the type of products produced could

also benefit Golf-Sport by increasing profits, depending on costs and demand of these

new products. Then finding new distributors for the raw materials could also lower the

costs of production and entering into additional markets where these products could be

sold at a higher price (due to higher demand) in hopes to increase the total profit margin.

References: [1] W.L. Winston, M. Venkataramanan, Introduction to Mathematical Programming, 4th

edition, Publisher: Duxbury Press, 2003.

Appendix:

Max

4X11+6X12+4X13+8X14+14X15+112X16+152X17+210X18+230X19+5X21+7X22+3X23+7X24+16X25+115X26+160X2

7+200X28+215X29+5X31+10X32+5X33+10X34+18X35+130X36+180X37+270X38+270X39

st

X11 + 1.5 X12 +1.5 X13 + 3X14 +4 X15 <=12000

4X11 + 4X12 + 5X13 + 6X14 + 6X15 <=20000

3.5X21 + 3.5X22 + 4.5X23 + 4.5X24 + 5.0 X25 <= 15000

7X21 + 7X22 + 8X23 + 9X24 + 7X25 <= 40000

3X31 + 3.5X32 + 4X33 + 4.5X34 + 5.5X35 <= 22000

7.5X31 + 7.5X32 + 8.5X33 + 9.5X34 + 8.0X35 <= 35000

1.0X11 +1.5 X12 +1.1 X13 +1.5 X14 +1.9 X15 +1.1 X21 + 1.1 X22 + 1.1 X23 + 1.2 X24 + 1.9 X25 + 1.3 X31 +1.3 X32 +

1.3 X33 + 1.3 X34 + 1.9 X35 <= 20000

65X16 + 65X17 + 65X18 + 65X19 <= 5500

60X26 + 60X27 + 60X28 + 60X29 <= 5000

65X36 + 65X37 + 65X38 + 65X39 <= 6000

0.25X12 + 0.25X22 + 0.25X32 >= 1000

X11<=2000

X12<=2000

X12>=100

X13<=2000

X13>=200

X14<=2000

X14>=30

X15<=2000

X15>=100

X16 <=200

X17 <=100

X18 <=300

X19 <=400

X21<=2000

X22<=2000

X22>=100

X23<=2000

X23>=200

X24<=2000

X24>=30

X25<=2000

X25>=100

X26<=200

X27<=100

X28<=300

X29<=400

X31<=2000

X32<=2000

X32>=50

X33<=2000

X33>=100

X34<=2000

X34>=15

X35<=2000

X35>=100

X36<=200

X37<=100

X38<=300

X39<=400

End

LP OPTIMUM FOUND AT STEP 21

OBJECTIVE FUNCTION VALUE

1) 202127.1

VARIABLE VALUE REDUCED COST

X11 0.000000 2.000000

X12 1705.000000 0.000000

X13 200.000000 0.000000

X14 30.000000 0.000000

X15 2000.000000 0.000000

X16 0.000000 118.000000

X17 0.000000 78.000000

X18 0.000000 20.000000

X19 84.615387 0.000000

X21 0.000000 2.000000

X22 1132.857178 0.000000

X23 200.000000 0.000000

X24 30.000000 0.000000

X25 2000.000000 0.000000

X26 0.000000 100.000000

X27 0.000000 55.000000

X28 0.000000 15.000000

X29 83.333336 0.000000

X31 0.000000 2.894737

X32 2000.000000 0.000000

X33 100.000000 0.000000

X34 331.578949 0.000000

X35 2000.000000 0.000000

X36 0.000000 140.000000

X37 0.000000 90.000000

X38 0.000000 0.000000

X39 92.307693 0.000000

ROW SLACK OR SURPLUS DUAL PRICES

2) 1052.500000 0.000000

3) 0.000000 1.500000

4) 0.000000 2.000000

5) 16200.000000 0.000000

6) 2107.894775 0.000000

7) 0.000000 1.052632

8) 1114.304565 0.000000

9) 0.000000 3.538461

10) 0.000000 3.583333

11) 0.000000 4.153846

12) 209.464279 0.000000

13) 2000.000000 0.000000

14) 295.000000 0.000000

15) 1605.000000 0.000000

16) 1800.000000 0.000000

17) 0.000000 -3.500000

18) 1970.000000 0.000000

19) 0.000000 -1.000000

20) 0.000000 5.000000

21) 1900.000000 0.000000

22) 200.000000 0.000000

23) 100.000000 0.000000

24) 300.000000 0.000000

25) 315.384613 0.000000

26) 2000.000000 0.000000

27) 867.142883 0.000000

28) 1032.857178 0.000000

29) 1800.000000 0.000000

30) 0.000000 -6.000000

31) 1970.000000 0.000000

32) 0.000000 -2.000000

33) 0.000000 6.000000

34) 1900.000000 0.000000

35) 200.000000 0.000000

36) 100.000000 0.000000

37) 300.000000 0.000000

38) 316.666656 0.000000

39) 2000.000000 0.000000

40) 0.000000 2.105263

41) 1950.000000 0.000000

42) 1900.000000 0.000000

43) 0.000000 -3.947368

44) 1668.421021 0.000000

45) 316.578949 0.000000

46) 0.000000 9.578947

47) 1900.000000 0.000000

48) 200.000000 0.000000

49) 100.000000 0.000000

50) 300.000000 0.000000

51) 307.692322 0.000000

NO. ITERATIONS= 21

RANGES IN WHICH THE BASIS IS UNCHANGED:





X12 6.00 3.33 0.666667



X15 14.00 INFINITY 5.00

X16 112.00 117.99 INFINITY

X17 152.00 77.99 INFINITY

X18 210.00 19.99 INFINITY

X19 230.00 INFINITY 19.99


X22 7.00 4.20 1.56


X24 7.000000 2.000000 INFINITY

X25 16.000000 INFINITY 6.000000

X26 115.000000 99.999992 INFINITY

X27 160.000000 54.999996 INFINITY

X28 200.000000 14.999995 INFINITY

X29 215.000000 INFINITY 15.000000

X31 5.000000 2.894737 INFINITY

X32 10.000000 INFINITY 2.105263

X33 5.000000 3.947368 INFINITY

X34 10.000000 2.666667 3.666667

X35 18.000000 INFINITY 9.578947

X36 130.000000 140.000000 INFINITY

X37 180.000000 90.000008 INFINITY

X38 270.000000 0.000007 INFINITY

X39 270.000000 INFINITY 0.000007




2 12000.000000 INFINITY 1052.500000

3 20000.000000 1180.000000 3351.428467

4 15000.000000 3035.000000 2932.499756

5 40000.000000 INFINITY 16200.000000

6 22000.000000 INFINITY 2107.894775

7 35000.000000 4450.000000 3007.500000

8 20000.000000 INFINITY 1114.304565

9 5500.000000 20500.000000 5500.000000

10 5000.000000 18999.998047 5000.000000

11 6000.000000 20000.000000 6000.000000

12 1000.000000 209.464279 INFINITY

13 2000.000000 INFINITY 2000.000000

14 2000.000000 INFINITY 295.000000

15 100.000000 1605.000000 INFINITY

16 2000.000000 INFINITY 1800.000000

17 200.000000 670.285706 200.000000

18 2000.000000 INFINITY 1970.000000

19 30.000000 558.571411 30.000000

20 2000.000000 558.571411 196.666672

21 100.000000 1900.000000 INFINITY

22 200.000000 INFINITY 200.000000

23 100.000000 INFINITY 100.000000

24 300.000000 INFINITY 300.000000

25 400.000000 INFINITY 315.384613

26 2000.000000 INFINITY 2000.000000

27 2000.000000 INFINITY 867.142883

28 100.000000 1032.857178 INFINITY

29 2000.000000 INFINITY 1800.000000

30 200.000000 651.666626 200.000000

31 2000.000000 INFINITY 1970.000000

32 30.000000 651.666626 30.000000

33 2000.000000 586.499939 607.000000

34 100.000000 1900.000000 INFINITY

35 200.000000 INFINITY 200.000000

36 100.000000 INFINITY 100.000000

37 300.000000 INFINITY 300.000000

38 400.000000 INFINITY 316.666656

39 2000.000000 INFINITY 2000.000000

40 2000.000000 401.000000 837.857117

41 50.000000 1950.000000 INFINITY

42 2000.000000 INFINITY 1900.000000

43 100.000000 353.823547 100.000000

44 2000.000000 INFINITY 1668.421021

45 15.000000 316.578949 INFINITY

46 2000.000000 375.937500 1900.000000

47 100.000000 1900.000000 INFINITY

48 200.000000 INFINITY 200.000000

49 100.000000 INFINITY 100.000000

50 300.000000 INFINITY 300.000000

51 400.000000 INFINITY 307.692322

Month 2:

Max

3.28X11+3.72X12+3.52X13+6.8X14+10.88X15+90.64X16+124.64X17+168X18+179.6X19+4.4X21+4.84X22+2.4X23+

5.68X24+13.12X25+94X26+133.6X27+156.8X28+162.8X29+4.16X31+7.6X32+4.4X33+8.56X34+14.76X35+108.4X36

+151.2X37+225.6X38+216X39

st

X11 + 1.5 X12 +1.5 X13 + 3X14 +4 X15 <=12000

4X11 + 4X12 + 5X13 + 6X14 + 6X15 <=20000

3.5X21 + 3.5X22 + 4.5X23 + 4.5X24 + 5.0 X25 <= 15000

7X21 + 7X22 + 8X23 + 9X24 + 7X25 <= 40000 3X31 + 3.5X32 + 4X33 + 4.5X34 + 5.5X35 <= 22000

7.5X31 + 7.5X32 + 8.5X33 + 9.5X34 + 8.0X35 <= 35000

1.0X11 +1.5 X12 +1.1 X13 +1.5 X14 +1.9 X15 +1.1 X21 + 1.1 X22 + 1.1 X23 + 1.2 X24 + 1.9 X25 + 1.3 X31 +1.3 X32 +

1.3 X33 + 1.3 X34 + 1.9 X35 <= 20000

65X16 + 65X17 + 65X18 + 65X19 <= 5500

60X26 + 60X27 + 60X28 + 60X29 <= 5000

65X36 + 65X37 + 65X38 + 65X39 <= 6000

0.25X12 + 0.25X22 + 0.25X32 >= 1000

X11<=2000

X12<=2000

X12>=100

X13<=2000

X13>=200

X14<=2000

X14>=30

X15<=2000

X15>=100

X16 <=200

X17 <=100

X18 <=300

X19 <=400

X21<=2000

X22<=2000

X22>=100

X23<=2000

X23>=200

X24<=2000

X24>=30

X25<=2000

X25>=100

X26<=200

X27<=100

X28<=300

X29<=400

X31<=2000

X32<=2000

X32>=50

X33<=2000

X33>=100

X34<=2000

X34>=15

X35<=2000

X35>=100

X36<=200

X37<=100

X38<=300

X39<=400

End



1) 159652.0


X11 0.000000 1.253333

X12 867.142883 0.000000

X13 200.000000 0.000000

X14 588.571411 0.000000

X15 2000.000000 0.000000

X16 0.000000 88.959999

X17 0.000000 54.959999

X18 0.000000 11.600000

X19 84.615387 0.000000

X21 0.000000 1.253333

X22 1132.857178 0.000000

X23 200.000000 0.000000

X24 30.000000 0.000000

X25 2000.000000 0.000000

X26 0.000000 68.800003

X27 0.000000 29.199993

X28 0.000000 5.999997

X29 83.333336 0.000000

X31 0.000000 2.597895

X32 2000.000000 0.000000

X33 100.000000 0.000000

X34 331.578949 0.000000

X35 2000.000000 0.000000

X36 0.000000 117.199997

X37 0.000000 74.400002

X38 92.307693 0.000000

X39 0.000000 9.600000

ROW SLACK OR SURPLUS DUAL PRICES

2) 633.571411 0.000000

3) 0.000000 1.133333

4) 0.000000 1.615238

5) 16200.000000 0.000000

6) 2107.894775 0.000000

7) 0.000000 0.901053

8) 1533.233032 0.000000

9) 0.000000 2.763077

10) 0.000000 2.713333

11) 0.000000 3.470769

12) 0.000000 -3.253333

13) 2000.000000 0.000000

14) 1132.857178 0.000000

15) 767.142883 0.000000

16) 1800.000000 0.000000

17) 0.000000 -2.146667

18) 1411.428589 0.000000

19) 558.571411 0.000000

20) 0.000000 4.080000

21) 1900.000000 0.000000

22) 200.000000 0.000000

23) 100.000000 0.000000

24) 300.000000 0.000000

25) 315.384613 0.000000

26) 2000.000000 0.000000

27) 867.142883 0.000000

28) 1032.857178 0.000000

29) 1800.000000 0.000000

30) 0.000000 -4.868571

31) 1970.000000 0.000000

32) 0.000000 -1.588571

33) 0.000000 5.043809

34) 1900.000000 0.000000

35) 200.000000 0.000000

36) 100.000000 0.000000

37) 300.000000 0.000000

38) 316.666656 0.000000

39) 2000.000000 0.000000

40) 0.000000 1.655439

41) 1950.000000 0.000000

42) 1900.000000 0.000000

43) 0.000000 -3.258947

44) 1668.421021 0.000000

45) 316.578949 0.000000

46) 0.000000 7.551579

47) 1900.000000 0.000000

48) 200.000000 0.000000

49) 100.000000 0.000000

50) 207.692307 0.000000

51) 400.000000 0.000000

NO. ITERATIONS= 2

RANGES IN WHICH THE BASIS IS UNCHANGED:




X11 3.280000 1.253333 INFINITY

X12 3.720000 0.813333 3.530667

X13 3.520000 2.146667 INFINITY

X14 6.800000 4.080000 1.220000

X15 10.880000 INFINITY 4.080000

X16 90.639999 88.960007 INFINITY

X17 124.639999 54.960007 INFINITY

X18 168.000000 11.600006 INFINITY

X19 179.600006 INFINITY 11.600006

X21 4.400000 1.253333 INFINITY

X22 4.840000 3.530667 1.235556

X23 2.400000 4.868571 INFINITY

X24 5.680000 1.588571 INFINITY

X25 13.120000 INFINITY 5.043809

X26 94.000000 68.800003 INFINITY

X27 133.600006 29.199997 INFINITY

X28 156.800003 5.999999 INFINITY

X29 162.800003 INFINITY 6.000000

X31 4.160000 2.597895 INFINITY

X32 7.600000 INFINITY 1.655439

X33 4.400000 3.258947 INFINITY

X34 8.560000 2.096889 3.290667

X35 14.760000 INFINITY 7.551579

X36 108.400002 117.199997 INFINITY

X37 151.199997 74.400002 INFINITY

X38 225.600006 INFINITY 9.599996

X39 216.000000 9.599996 INFINITY




2 12000.000000 INFINITY 633.571411

3 20000.000000 1267.142822 3351.428467

4 15000.000000 2685.000000 2932.499756

5 40000.000000 INFINITY 16200.000000

6 22000.000000 INFINITY 2107.894775

7 35000.000000 4450.000000 3007.500000

8 20000.000000 INFINITY 1533.233032

9 5500.000000 20500.000000 5500.000000

10 5000.000000 18999.998047 5000.000000

11 6000.000000 13500.000000 6000.000000

12 1000.000000 209.464279 191.785721

13 2000.000000 INFINITY 2000.000000

14 2000.000000 INFINITY 1132.857178

15 100.000000 767.142883 INFINITY

16 2000.000000 INFINITY 1800.000000

17 200.000000 670.285706 200.000000

18 2000.000000 INFINITY 1411.428589

19 30.000000 558.571411 INFINITY

20 2000.000000 558.571411 1411.428589

21 100.000000 1900.000000 INFINITY

22 200.000000 INFINITY 200.000000

23 100.000000 INFINITY 100.000000

24 300.000000 INFINITY 300.000000

25 400.000000 INFINITY 315.384613

26 2000.000000 INFINITY 2000.000000

27 2000.000000 INFINITY 867.142883

28 100.000000 1032.857178 INFINITY

29 2000.000000 INFINITY 1800.000000

30 200.000000 651.666626 200.000000

31 2000.000000 INFINITY 1970.000000

32 30.000000 651.666626 30.000000

33 2000.000000 586.500000 537.000000

34 100.000000 1900.000000 INFINITY

35 200.000000 INFINITY 200.000000

36 100.000000 INFINITY 100.000000

37 300.000000 INFINITY 300.000000

38 400.000000 INFINITY 316.666656

39 2000.000000 INFINITY 2000.000000

40 2000.000000 401.000000 837.857117

41 50.000000 1950.000000 INFINITY

42 2000.000000 INFINITY 1900.000000

43 100.000000 353.823547 100.000000

44 2000.000000 INFINITY 1668.421021

45 15.000000 316.578949 INFINITY

46 2000.000000 375.937500 1900.000000

47 100.000000 1900.000000 INFINITY

48 200.000000 INFINITY 200.000000

49 100.000000 INFINITY 100.000000

50 300.000000 INFINITY 207.692307

51 400.000000 INFINITY 400.000000

Changed Coefficients (Decreased X15 and increased X16)

Max

4X11+6X12+4X13+8X14+4X15+212X16+152X17+210X18+230X19+5X21+7X22+3X23+7X24+16

X25+115X26+160X27+200X28+215X29+5X31+10X32+5X33+10X34+18X35+130X36+180X37+27

0X38+270X39

st

X11 + 1.5 X12 +1.5 X13 + 3X14 +4 X15 <=12000

4X11 + 4X12 + 5X13 + 6X14 + 6X15 <=20000

3.5X21 + 3.5X22 + 4.5X23 + 4.5X24 + 5.0 X25 <= 15000

7X21 + 7X22 + 8X23 + 9X24 + 7X25 <= 40000 3X31 + 3.5X32 + 4X33 + 4.5X34 + 5.5X35 <= 22000

7.5X31 + 7.5X32 + 8.5X33 + 9.5X34 + 8.0X35 <= 35000

1.0X11 +1.5 X12 +1.1 X13 +1.5 X14 +1.9 X15 +1.1 X21 + 1.1 X22 + 1.1 X23 + 1.2 X24 + 1.9 X25 +

1.3 X31 +1.3 X32 + 1.3 X33 + 1.3 X34 + 1.9 X35 <= 20000

65X16 + 65X17 + 65X18 + 65X19 <= 5500

60X26 + 60X27 + 60X28 + 60X29 <= 5000

65X36 + 65X37 + 65X38 + 65X39 <= 6000

0.25X12 + 0.25X22 + 0.25X32 >= 1000

X11<=2000

X12<=2000

X12>=100

X13<=2000

X13>=200

X14<=2000

X14>=30

X15<=2000

X15>=100

X16 <=200

X17 <=100

X18 <=300

X19 <=400

X21<=2000

X22<=2000

X22>=100

X23<=2000

X23>=200

X24<=2000

X24>=30

X25<=2000

X25>=100

X26<=200

X27<=100

X28<=300

X29<=400

X31<=2000

X32<=2000

X32>=50

X33<=2000

X33>=100

X34<=2000

X34>=15

X35<=2000

X35>=100

X36<=200

X37<=100

X38<=300

X39<=400

End



1) 189923.7


X11 0.000000 1.333333

X12 2000.000000 0.000000

X13 200.000000 0.000000

X14 1733.333374 0.000000

X15 100.000000 0.000000

X16 0.000000 18.000000

X17 0.000000 78.000000

X18 0.000000 20.000000

X19 84.615387 0.000000

X21 0.000000 2.000000

X22 1132.857178 0.000000

X23 200.000000 0.000000

X24 30.000000 0.000000

X25 2000.000000 0.000000

X26 0.000000 100.000000

X27 0.000000 55.000000

X28 0.000000 15.000000

X29 83.333336 0.000000

X31 0.000000 2.894737

X32 2000.000000 0.000000

X33 100.000000 0.000000

X34 331.578949 0.000000

X35 2000.000000 0.000000

X36 0.000000 140.000000

X37 0.000000 90.000000

X38 0.000000 0.000000

X39 92.307693 0.000000

Changed Constraints

Max

4X11+6X12+4X13+8X14+14X15+112X16+152X17+210X18+230X19+5X21+7X22+3X23+7X24+16X25+115X26+160X2

7+200X28+215X29+5X31+10X32+5X33+10X34+18X35+130X36+180X37+270X38+270X39

s.t.

X11 + 1.5 X12 +1.5 X13 + 3X14 +4 X15 <=12000

4X11 + 4X12 + 5X13 + 6X14 + 6X15 <=20000

3.5X21 + 3.5X22 + 4.5X23 + 4.5X24 + 5.0 X25 <= 17000

7X21 + 7X22 + 8X23 + 9X24 + 7X25 <= 40000 3X31 + 3.5X32 + 4X33 + 4.5X34 + 5.5X35 <= 25000

7.5X31 + 7.5X32 + 8.5X33 + 9.5X34 + 8.0X35 <= 35000

1.0X11 +1.5 X12 +1.1 X13 +1.5 X14 +1.9 X15 +1.1 X21 + 1.1 X22 + 1.1 X23 + 1.2 X24 + 1.9 X25 + 1.3 X31 +1.3 X32 +

1.3 X33 + 1.3 X34 + 1.9 X35 <= 20000

65X16 + 65X17 + 65X18 + 65X19 <= 5500

60X26 + 60X27 + 60X28 + 60X29 <= 5000

65X36 + 65X37 + 65X38 + 65X39 <= 6000

0.25X12 + 0.25X22 + 0.25X32 >= 1000

X11<=2000

X12<=2000

X12>=100

X13<=2000

X13>=200

X14<=2000

X14>=30

X15<=2000

X15>=100

X16 <=200

X17 <=100

X18 <=300

X19 <=400

X21<=2000

X22<=2000

X22>=100

X23<=2000

X23>=200

X24<=2000

X24>=30

X25<=2000

X25>=100

X26<=200

X27<=100

X28<=300

X29<=400

X31<=2000

X32<=2000

X32>=50

X33<=2000

X33>=100

X34<=2000

X34>=15

X35<=2000

X35>=100

X36<=200

X37<=100

X38<=300

X39<=400

End



1) 206127.1


X11 0.000000 2.000000

X12 1705.000000 0.000000

X13 200.000000 0.000000

X14 30.000000 0.000000

X15 2000.000000 0.000000

X16 0.000000 118.000000

X17 0.000000 78.000000

X18 0.000000 20.000000

X19 84.615387 0.000000

X21 0.000000 2.000000

X22 1704.285767 0.000000

X23 200.000000 0.000000

X24 30.000000 0.000000

X25 2000.000000 0.000000

X26 0.000000 100.000000

X27 0.000000 55.000000

X28 0.000000 15.000000

X29 83.333336 0.000000

X31 0.000000 2.894737

X32 2000.000000 0.000000

X33 100.000000 0.000000

X34 331.578949 0.000000

X35 2000.000000 0.000000

X36 0.000000 140.000000

X37 0.000000 90.000000

X38 0.000000 0.000000

X39 92.307693 0.000000

340project Final

Documents

Transcript of 340project Final