managerial report€¦ · are planning to sell these products to their authorized dealers. ......

Workload Balancing

Production Strategy

Hart Venture Capital

Investment Strategy

Truck Leasing Strategy

MANAGERIAL REPORT

1303041015

Uğur Altuntaş

MIS

CONTENTS

EXECUTIVE SUMMARY ............................................................................................................................. 1

WORKLOAD BALANCING ......................................................................................................................... 3

1-) Decision Variables .......................................................................................................................... 3

2-) Objective Function Definition ........................................................................................................ 3

3-) Solutions ......................................................................................................................................... 4

PRODUCTION STRATEGY ....................................................................................................................... 10

1-) Decision Variables ........................................................................................................................ 12

2-) Objective Function Definition ...................................................................................................... 12

3-) Solutions ....................................................................................................................................... 13

HART VENTURE CAPITAL ....................................................................................................................... 16



3-) Solutions ....................................................................................................................................... 17

INVESTMENT STRATEGY ........................................................................................................................ 20

1. Decision Variables ......................................................................................................................... 20

2. Objective Function Definition ........................................................................................................ 20

3. Solutions ........................................................................................................................................ 21

TRUCK LEASING STRATEGY .................................................................................................................... 25



3-) Solutions ....................................................................................................................................... 27

1

EXECUTIVE SUMMARY

This report represent the solution of case problems and there are 5 cases and each case

has problems in different perspective. In these report, linear programming was used to

solve problems. The purpose of the report is to facilitate management's decision making,

achieve best results, reduce costs and increase profits by finding optimal solutions.

In these report there are 5 case;

1-) Workload Balancing Case : DIGITAL IMAGING

Digital Imaging (DI) produces photo printers for both professional and consumer

markets. The company recently introduced two new models of color photo printers

namely DI-910 model and DI-950 model.

In this problem, management wants to use their assembly line in most effective

way and maximize the total profit. Solution provide these conditions and

manament’s constraints by using linear programming.

2-) Production Strategy Case : BETTER FITNESS INC.

Better Fitness, Inc. (BFI), manufactures exercise equipment and it recently

designed two universal weight machines : BodyPlus 100 and BodyPlus200. They

are planning to sell these products to their authorized dealers. They can purchase

machines for 70% of the suggested retail price.

In this problem, management wants to know how many units to produce each

weight machine that will maximize the profit. Solution provide the numbers to the

management.

3-) Hart Venture Capital Case : HVC

Hart Venture Capital (HVC) is a company of providing venture capital for software

development and Internet application. Currently, there are two companies:

Security Systems and Market Analysis need additional capital. Management wants

to find the optimal percentage of each project that HVC should fund in order to

yield the maximum net present value of the total investment.

The problem the firms faces is a maximization problem. The net present value

generated by the sum of the net present value multiplied by the recommended

2

percentage of two investments will be the objective function.Solution provide the

optimal percentage of each project by using linear programming.

4-) Investment Strategy Case : J.D. WILLIAMS INC.

J.D. Williams Inc. has a client who wishes to invest $800.000 with the firm in order

to maximize his yield after a period of one year. The firm wants to allocate the funds

while accommodating some requirements related to portfolio composition of the

investment. The portfolio must have investments in 3 funds: growth stock fund,

income fund and money market fund.

Based on the knowledge of J.D. Williams Inc. linear programming was used to

determine the best investment portfolio that will minimize risk and maximize

annual yield. This solution provide the management an asset allocation model that

provides investment strategy to obtain optimal investment combinations.

5-) Truck Leasing Strategy Case : REEP CONSTRUCTION

In this case, Bob Reep the owner and president of Reep Construction. Bob

recently won a contract for the excavation and site preparation of a new rest area

on the Pennsylvania Turnpike.To complete his new project, he need to lease new

trucks. So, he need to figure out a way to accommodate a short term leasing plan

for trucks within budget to complete his new project. So, Bob wants to know

optimal leasing plan that minimize his cost.

According to the required number of trucks and fuel cost, linear programming

was used that minimize total cost. Solution provide optimal leasing plan and

additional costs associated with optimal leasing plan to Bob Reep.

The recommendations in this report are based on the linear programming, which is an

optimal solution method to achieve the best outcome. So, managers can apply the

recommendations in this report.

3

WORKLOAD BALANCING

Digital Imaging (DI) produces color printers for both the professional and consumer

markets. The company introduced two new color printers and company wants to determine

how many units of each printer to produce to maximize profit.

DI – 910 DI – 950 Time Available

(minutes)

Line 1 3 mins 6 mins 480

Line 2 4 mins 2 mins 480

Profit/unit $42 $87

Table 1.0 : Workload Balancing Variables

1-) Decision Variables

X : Number of Units to Produce of DI-910 printer

Y : Number of Units to Produce of DI-950 printer

2-) Objective Function Definition

Objective function is maximizing the profit by determining how many units of each printer

to produce.

Max Z = 42X + 87Y

Subject to

3X + 6Y <= 480

4X + 2Y <= 480

4

3-) Solutions

1-) The recommended number of units of each printer to produce to maximize total

contribution to profit for an 8 hour shift as follow:

X = 0, Y = 80

42 * (0) + 87 * (80) = $6.960,00

Figure 1.0 : Workload Balancing Question 1 Excel Solution

Figure 1.1 : Workload Balancing Question 1 Graphical Solution.

5

Figure 1.2: Workload Balancing Question 1 Feasible area points

The values of point A and C at the feasible area are known. I need to find my point B. To

find the point B, i got the equations of intersecting lines. Then i use MMulti and Minverse

formulas of excel. After that i put the points in Z function then i got automatically the

result of Z.

As a result i choose the Point A because it has highest Z value.

But management may not accept the solution because of;

• Supply of DI-950 model is more than market’s demand

• May be there is a demand for DI-910 in the market

• There may be a problem for 320 minutes that is not used on Line 2

2-) In question 2, management wants to number of DI-910 printers produced must be at

least as great as the number of DI-950 units produced.So we just add X-Y =>0 constraint

function and follow the steps as in question 1 solution.

Figure 1.3: Workload Balancing Question 2 Excel Solution

6

Figure 1.4: Workload Balancing Question 2 Graphical Solution

The new constraint function changed the our graph and feasible area.So now, we need to

find out coordinates of point D.

Figure 1.5 : Workload Balancing Question 2 Feasible Area Points

3-) In line 2 we use 320 minutes so there is a 480-320=160 minutes time that we not used.

So line 2 is non-operational more than 2 and a half hours. The reason for this lack may

concern the management is as follow;

7

Figure 1.6 : Workload Balancing Question 3 Used Shift

Management should concern about this lack because it may some effects on the efficiency

because as the plant is automated the time lag between supply and demand of lines may

lead to inefficiency and operating cost will also be incurred for time lag even without

operation. Also, unused resources may lose money to company

4-) In question 4 management wants to better balance between total time on line 1 and

line 2. So management set the new balance as the difference between two lines 30 minutes

or less. So this is a new constraint and we need new constraint function. This function will

be ;

(3X+6Y) – (4X+2Y) <=30 ➔ -X + 4Y <= 30

We add the new constraints function to excel and follow the same steps as before.


With a new constraints our profit and lack times have changed. The difference between

profits is $6880 - $6815 = $65 ,

And the lack time difference is 30 minutes.

8

Figure 1.8 : Workload Balancing Question 4 Graphical Solution


We choose E because its more profitable.

5-) In question 5 management set the new objective from maximizing profits to

maximizing production. This give us new Z function as;

Max Z = X+Y

9


After the new Z function our profit is reduced $160 according to part(1) solution and line

2 balanced with line 1.

Figure 2.1 : Workload Balancing Question 5 Graphical Solution

10


We choose point B because it has the highest Z value.

PRODUCTION STRATEGY

Better Fitness, Inc. (BFI), manufactures exercise equipment.It recently designed two

universal weight machines for the home exercise market. At a recent trade show,

demonstrations of the machines resulted in significant dealer interest and authorized BFI

dealers can purchase machines for 70% of the suggested price. So company want to know

number of machines to produce that maximizes total profit.

Body Plus 100 Body Plus 200

• Frame Unit • Press Station • Pec-dec Station

• Frame Unit • Press Station • Pec-dec Station • Leg-press Station

Table 1.1 : Production Strategy Components of Machines

11

Body Plus 100

Machining & Welding Machine (hours)

Painting & Finishing (hours)

Assembling & Testing (hours)

Raw Material Cost ($)

Packaging Cost ($)

Frame Unit 4 4 450

Press Station 2 1 2 350 50

Pec-dec Station

2 2 250

Total 8 5 2 $1000 $50

Table 1.2 : Production Strategy BodyPlus 100

Body Plus 200

Machining & Welding Machine (hours)

Painting & Finishing (hours)

Assembling & Testing (hours)

Raw Material Cost ($)

Packaging Cost ($)

Frame Unit 5 4 650

Press Station 3 2 2 400 75

Pec-dec Station

2 2 250

Leg-press Station

2 2 200

Total 12 10 2 $1500 $75

Table 1.3 : Production Strategy: BodyPlus 200

12

Time Capacity (hours)

Cost/hours

Machining & Welding Machine

600 20

Painting & Finishing 450 15

Assembling & Testing & Packaging

140 12

Table 1.4 : Production Strategy Machine Capacity and Cost


X : Number of Units to Produce of BodyPlus 100

Y : Number of Units to Produce of BodyPlus 200


Table 1.5 : Production Strategy Profits/units

Objective function is maximize the total profit by determining how many units to produce

of each machine

Max Z = 371X + 461Y

Body Plus 100

Body Plus 200

Retail Price $2400

Labor Cost = (20*8)+(15*5)+(12*2) =$259

Raw Material Cost = $1000 + $50 = $1050

Dealer price = 0.70* $2400 =$1680

Profit = $371

Retail Price $3500

Labor Cost = (20*12)+(15*10)+(12*2) =$414

Raw Material Cost = $1500 + $75 = $1575

Dealer price = 0.70* $3500 =$2450

Profit = $461

13

Subject to

1-) 8X + 12Y <= 600

2-) 5X + 10Y <= 450

3-) 2X + 2Y <=140

4-) X – 3Y <=0

3-) Solutions

1-) Recommended number of BodyPlus 100 and BodyPlus200 machines to produce as

follows;

X = 50, Y = 16

371 * (50) + 461* (16) = $25.926

Figure 2.3 : Production Strategy Question 1 Excel Solution

14

Figure 2.4 : Production Strategy Question 1 Graphical Solution

Figure 2.5 : Production Strategy Question 1 Feasible Area Points

We choose point C because it has highest Z value.

2-) In question 2 the management wants to know the affect that if there is no constraint

that the number of units of the Body Plus 200 produced be at least 25% of the total

production. So the C4 constraint function ( X – 3Y <=0 ) is removed

15

Figure 2.6 : Production Strategy Question 2 Excel Solution

Figure 2.7 : Production Strategy Question 2 Graphical Solution

16

Figure 2.8 : Production Strategy Question 2 Feasible Area Points

After the new solution our profit increased by

$26.870,00 - $26.233,33 = $636,67

3-) In question 3 management wants to know that where should they focus on to increase

the profit.

I think, if we increase the machine and welding time we will succesful. Because there is

slack in painting & finishing and assembling & testing & and packaging. So, by increasing

the machine and welding time, we can increase the profit by utilizing the slack in painting

& finishing and assembling, testing & packaging.

HART VENTURE CAPITAL

Hart Venture Capital (HVC) is an investment company that provide venture capital for

companies. In this problem HVC has two company to invest. First is Security Systems

Company that needs additional capital to develop an internet security software package

and the second is Market Analysis Company that needs additional capital to develop a

software package for conducting customer satisfaction surveys.

HVC wants to know how to make most profitable investments on two companies and

maximize their net present value(NPV).

17

Years Security System Market Analysis Maximum Investment

1 $600.000 $500.000 $800.000

2 $600.000 $350.000 $700.000

3 $250.000 $400.000 $500.000

Table 1.6 : Hart Venture Capital: Investment oppurtunities of HVC

Table 1.7 : Hart Venture Capital: NPV of stocks if investment will be %100


X : Percentage of Fund Invested in Security Systems

Y : Percentage of Fund Invested in Market Analysis


Objective function is maximize the net present value of the total investment in Security

Systems and Market Analysis

Max Z = X/100(1.800.000) + Y/100(1.600.000)

Subject to

1-) (X/100) * (600000) + (Y/100) * (500000) <= 800000

2-) (X/100) * (600000) + (Y/100) * (350000) <= 700000

3-) (X/100) * (250000) + (Y/100) * (400000) <= 500000

4-) X, Y <= 100

3-) Solutions

1-) HVC should invest %60,869 on Security System and %86,956 on Market Analysis so

net present value will become $2.486.956,52

Security System Market Analysis

Net Present Value $1.800.000 $1.600.000

18

Figure 2.9 : Hart Venture Capital Question 1 Excel Solution

Figure 3.0 : Hart Venture Capital Question 1 Graphical Solution

Figure 3.1 : Hart Venture Capital Question 1 Feasible Area Points

19

2-) The recommendation for capital allocation plan for Security Systems and Market

Analysis for the coming three year period is as follow;

Figure 3.2 Hart Venture Capital Question 2 Capital Allocation

3-) HVC decides to invest $ 100,000 more in the first year. It will change the percentage

and NPV of these investments as;

• % invested funded for Security Systems will become 68,852%

• % invested funded for Market Analysis will become 81,967%

• Net Present Value will become $2.550.819,67

Figure 3.3 : Hart Venture Capital Question 3 New Investment Fund

4-) If an additional $100.000 is available, the capital allocation plan that i recommend is

as follow;

Figure 3.4 : Hart Venture Capital Question 4 Capital Allocation

20

5-) With additional $100.000 , NPV value increased $63.863,15 with $77.049,18 slack. So

it is profitable investment and slack can be used in another profitable investments of

HVC.

INVESTMENT STRATEGY

J.D. Williams is an investment advisory firm that manages $120 million in funds for its

clients. The company utilizes several financial approaches in advising their clients how to achieve

optimal portfolio returns.

J.D. Williams has recently contracted with a new client and would like to determine

what is the best way to allocate the client's $800,000 in available funds for optimal

growth. The subsequent sections of this report provide an outline of the investment

recommendation provided to the client.

.

Amount Invested of Total Portfolio (%)

Risk of the Money Invested in (%)

Annual Yield (%)

Growth Stock Fund 0.20 – 0.40 0.10 0.18

Income Fund 0.20 – 0.50 0.07 0.01

Money Market Fund >= (at least) 0.30 0.01 0.075

Table 1.8 : Investment Strategy Variables

1. Decision Variables

G = $ amount of investment in growth stock fund

I = $ amount of investment in income fund

M = $ amount of investment in money market fund

2. Objective Function Definition

Objective function is maximize the total return of the portfolio.

Z = 0.18G + 0.125I + 0.075M

21

Subject to

1G + 1I + 1M <= 800.000

0.10G + 0.7I + 0.01M <= 0.05*(G+I+M)

0.20 <= G <= 0.40

0.20 <= I <= 0.50

0.30 <= M

3. Solutions

1-) The optimal portfolio allocation J.D. Williams recommends is as follows:

Growth Stock Fund = $248.889

Income Fund = $160.000

Money Market Fund = $391.111

Total = $800.000

The anticipated annual yield is :

Growth Stock Fund = $248.889 * 0,18 = $44.800

Income Fund = $160.000 * 0,125 = $20.000

Money Market Fund = $391.111 * 0.075 = $29.333

Total = $94.133

Total Anticipated Annual Yield = $94.133 / $800.000 = 11.77%

22

Figure 3.5 : Investment Strategy Question 1 Excel Solution

2-) In terms of the risk tolerance index, if the client's index were increased from 0,05 to

0,055 the annual yield on investment would increase by $4,667 from the original optimal

estimation of $94,133 to a new projection of $98,800.

The modified asset allocation recommendation and its corresponding projected annual

return are as follows:

Fund Allocation Projected Annual Yield

Growth Fund = $293.333 Growth Fund = $293.333 * 0,18 = $52.800

Income Fund = $160.000 Income Fund = $160.000 * 0,125 = $20.000

Market Fund = $346.667 Market Fund=$346.667 * 0,075 =$26.000

Total = $ 800.000 Total = $98.800

Total Anticipated Annual Yield = $98.800 / $800.000 = 12.35%

23

Figure 3.6: Investment Strategy Question 2 Excel Solution

3-) I would not any change in investment.Since 0,16 is in the objective coefficient range

for the growth fund return, there would be no change in allocation. However, the return

would decrease by $4978.

Figure 3.7 : Investment Strategy Question 3 Excel Solution

However, a decrease to 0,14 is outside the objective function coefficient range. So, we

have to resolve the problem by changing objective function:

Z = 0.14G + 0.125I + 0.075M

24


4-) Current optimal solution has more invested in the growth stock fund than the income

fund and our client has concern about this situation. Our client’s new request is amount

invested in growth stock fund should not exceed the amount invested in income fund.

So, for adding the this request we have to resolve the problem with a new constraint:

G – I <= 0

With a new constraint, the value of the solution has decreased from $94.133 to $93.067.

This is only a decrease of 0.2% in yield. Since the yield decrease is so small, we may

prefer this portfolio.


25

5-) I would recommend the model only when the situation of the new client is similar. If

new clients have similar objectives and constraints, this model can be use. However, J.D.

Williams is an advisory firm that works for the individual investors according to their

specific needs. So, i would not recommend the use of this asset allocation model as a

general guide to financial investment.

TRUCK LEASING STRATEGY

In this problem, Bob Reep the president of the Reep Construction recently won a contract

for a new rest area for the Pennsylvania. Bob Reep estimated that it would take four

months to perform the new project and that 10, 12, 14 and 8 trucks would be needed in

months 1 through 4. The firm currently has 20 trucks but some of these trucks will used

by existing projects. These trucks were obtained last year when Bob signed a long term

lease with PennState Leasing. Thus, to complete the Project, Bob will have to lease

additional trucks.

Bob learned that he can obtain short-term leases for 1 – 4 months. Short term leases differ

from long-term leases in that the short term leasing plans include the cost of both a truck

and a driver. But, Bob will use his own drivers even if costs higher. So, Bob would like the

acquire a lease that would minimize the cost of meeting the monthly trucking

requirements for his new project.

Required Truck(s) for the Project

(unit)

Available Truck(s) on Hand (unit)

Need to Lease

Truck(s) (unit)

Month 1

10

1

9

Month 2

12

2

10

Month 3

14

3

11

Month 4

8

1

7

Table 1.9 : Truck Leasing Strategy Variables

26

Daily Fuel Cost = $100 / truck

Monthly fuel costs for short term and long term leased trucks

Monthly fuel cost per truck = 20 * $100 = $2.000


Xij = Number of trucks from a short term lease in month i for period of J month(s)

For example X23 means, number of trucks leased for 3 months on the 2nd month of

the project.

Yi = Number of trucks from a long term lease used in month i


Decision Variables

Cost for Leasing Period

X11, X21, X31, X41

X12, X22, X32

X13, X23

X14

For 1 month $4.000 + $2.000 = $6.000 For 2 month 2 ($3.700 + $2.000) = $11.400 For 3 month 3 ($3.225 + $2.000) = $15.675 For 4 month 4 ($3.040 + $2.000) = $20.160

Min Z = 6.000X11 + 11.400X12 + 15.675X13 + 20.160X14 + 6.000X21 + 11.400X22 + 15.675X23

+ 6.000X31 + 11.400X32 + 6.000X41 + 2.000Y1 + 2.000Y2 + 2000Y3 + 2000Y4

Subject to

1) X11 + X12 + X13 + X14 + Y1 = 10

2) X12 + X13 + X14 + X21 + X22 + X23 + Y2 = 12

3) X13 + X14 + X22 + X23 + X31 + X32 + Y3 = 14

27

4) X14 + X23 + X32 + X41 + Y4 = 8

5) Y1 <= 1

6) Y2 <= 2

7) Y3 <= 3

8) Y4 <= 1

3-) Solutions

1-) The optimal leasing plan as follow:

The number of leased trucks in Xij and Yi ;

X13 X14 X23 X31 Y1 Y2 Y3 Y4

3 6 1 1 1 2 3 1

15.675X13 + 20.160X14 + 15.675X23 + 6.000X31 + 2.000Y1 + 2.000Y2 + 2000Y3 +

2000Y4

Optimal Leasing Cost = $203.660

Figure 4.0 : Truck Leasing Strategy Question 1 Excel Solution

28

2-) The cost associated with the optimal leasing plan as follow:

Our short term leased truck includes also driver costs but Bob Reep want to use our own

drivers. So, in addition to our optimal leasing plan, the driver costs of short term and long

term leased trucks will be add.

Driver Cost = $20 / hour

Monthly Driver Cost = $20 * Shift (hours ) * Number of Used Trucks

Used Truck in Project Shift

(hours)

Driver Cost

($)

Month 1 10 160 32.000

Month 2 12 160 38.400

Month 3 14 160 44.800

Month 4 8 160 25.600

Table 2.0 : Truck Leasing Strategy Question 2 Driver Costs

Total Driver Cost = $140.800

The Cost Associated with the Optimal Leasing Plan = $140.800

Total Cost for the Project = $203.660 + $140.800 = $344.460

3-) The cost for Reep Construction to maintain its current policy of no layoffs:

In short term leasing plan, the PennState company gives the driver along with the trucks.

But, Bob Reep chose to use his own drivers and pay them $20 an hour. So, the cost of

maintain no layoffs policy is;

29

Total Used Truck

in Project

Used for

Long Term

Leased

Truck

Used for

Short Term

Leased

Truck

Shift

(hours)

Driver Cost

for Short

Term

leased($)

Month 1 10 1 9 160 28.800

Month 2 12 2 10 160 32.000

Month 3 14 3 11 160 35.200

Month 4 8 1 7 160 22.400

Table 2.1 : Truck Leasing Strategy Question 3 Driver Costs

The cost of maintaining no layoffs policy = $118.400

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