1. Optimizing Production for Golf-
Sport Plants in Arizona to
Maximize Profit
By
Erik Baker
Clayton Jeanette
Nicole Grisamore
December 10, 2014
2. 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.
4. 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: MinimumandMaximumProductDemandper 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]
ForgedIronHeads [200, 2,000] [200, 2,000] [100, 2,000]
Metal woodheads [30, 2,000] [30, 2,000] [15, 2,000]
Titaniuminsert
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,andAssembly
Costs($) perPart or Set
Products Chandler Glendale Tucson
Steel shafts 6 5 7
Graphite shafts 19 18 20
ForgedIronHeads 4 5 5
Metal woodheads 10 11 12
Titaniuminsert
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.
5. Table 7: Revenue perPartor Set($)
Products Chandler Glendale Tucson
Steel shafts 10 10 12
Graphite shafts 25 25 30
ForgedIronHeads 8 8 10
Metal woodheads 18 18 22
Titaniuminsert
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
6. 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
10. 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
11. 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 shaftsinChandler 0.00
X12 Graphite shaftsinChandler 1705.00
X13 ForgedironheadsinChandler 200.00
X14 Metal woodHeads inChandler 30.00
X15 TitaniuminsertheadsinChandler 2000.00
X16 Set:Steel,metal inChandler 0.00
X17 Set:Steel,insertinChandler 0.00
X18 Set:Graphite,metal inChandler 0.00
X19 Set:Graphite,insertinChandler 84.62
X21 Steel shaftsinGlendale 0.00
X22 Graphite shaftsinGlendale 1132.86
X23 ForgedironheadsinGlendale 200.00
X24 Metal woodHeads inGlendale 30.00
X25 TitaniuminsertheadsinGlendale 2000.00
X26 Set:Steel,metal inGlendale 0.00
X27 Set:Steel,insertinGlendale 0.00
X28 Set:Graphite,metal inGlendale 0.00
X29 Set:Graphite,insertinGlendale 83.33
X31 Steel shaftsinTucson 0.00
X32 Graphite shaftsinTucson 2000.00
X33 ForgedironheadsinTucson 100.00
X34 Metal woodheadsin Tucson 331.58
X35 TitaniuminsertheadsinTucson 2000.00
X36 Set:Steel,metal inTucson 0.00
X37 Set:Steel,insertinTucson 0.00
X38 Set:Graphite,metal inTucson 0.00
X39 Set:Graphite,insertinTucson 92.31
12. Table 9:
Variable Part Value
X11 Steel shaftsinChandler 0.00
X12 Graphite shaftsinChandler 867.14
X13 ForgedironheadsinChandler 200.00
X14 Metal woodHeads inChandler 588.57
X15 TitaniuminsertheadsinChandler 2000.00
X16 Set:Steel,metal inChandler 0.00
X17 Set:Steel,insertinChandler 0.00
X18 Set:Graphite,metal inChandler 0.00
X19 Set:Graphite,insertinChandler 84.62
X21 Steel shaftsinGlendale 0.00
X22 Graphite shaftsinGlendale 1132.86
X23 ForgedironheadsinGlendale 200.00
X24 Metal woodHeads inGlendale 30.00
X25 TitaniuminsertheadsinGlendale 2000.00
X26 Set:Steel,metal inGlendale 0.00
X27 Set:Steel,insertinGlendale 0.00
X28 Set:Graphite,metal inGlendale 0.00
X29 Set:Graphite,insertinGlendale 83.33
X31 Steel shaftsinTucson 0.00
X32 Graphite shaftsin Tucson 2000.00
X33 ForgedironheadsinTucson 100.00
X34 Metal woodheadsin Tucson 331.58
X35 TitaniuminsertheadsinTucson 2000.00
X36 Set:Steel,metal inTucson 0.00
X37 Set:Steel,insertinTucson 0.00
X38 Set:Graphite,metal inTucson 92.31
X39 Set:Graphite,insertinTucson 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
13. 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 shaftsinChandler 0.00 2.00
X12 Graphite shaftsinChandler 1705.00 0.00
X13 ForgedironheadsinChandler 200.00 0.00
X14 Metal woodHeads inChandler 30.00 0.00
X15 TitaniuminsertheadsinChandler 2000.00 0.00
X16 Set:Steel,metal inChandler 0.00 118.00
X17 Set:Steel,insertinChandler 0.00 78.00
X18 Set:Graphite, metal inChandler 0.00 20.00
X19 Set:Graphite,insertinChandler 84.62 0.00
X21 Steel shaftsinGlendale 0.00 2.00
X22 Graphite shaftsinGlendale 1132.86 0.00
X23 ForgedironheadsinGlendale 200.00 0.00
X24 Metal woodHeads inGlendale 30.00 0.00
X25 TitaniuminsertheadsinGlendale 2000.00 0.00
21. 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
22. 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
