LP Formulation problems 1. Product-mix problems: The Style and Comfort Furniture Manufacturing Company wishes to determine its production schedule for the next quarter. The company produces four types of furniture, including sofas, love seats, recliners, and coffee tables. The profit contribution from selling one sofa is $120, one love seat is $105, one recliner is $150, and one coffee table is $73. The quarterly production budget is set at $180,000. Each unit of a sofa, love seat, recliner, and coffee table costs $400, $300, $500, and $150, respectively. The sales forecasts indicate that the potential sales volume is limited to 200 units of sofas, 150 units of love seats, 100 units of recliners, and 400 units of coffee tables. There are an aggregate of 800 machine hours available and 1,200 labor hours available. Table 1 summarizes the number of machine hours and the number of labor hours required per unit of each product. Table 1: Per Unit Machine and Labor Hour Required for Each Product Per unit Machine and Labour hours required for each product ProductMachine hours/Unit
Labour hours/Unit
Sofa
2.0
2.5
Love seat
1.0
2.0
Recliner
2.2
3.0
Coffee table
0.75
1.0
The management also imposed the following policy constraints: i.
At least 40% of all production costs must be incurred for the sofas.
ii. At least 25% of all production costs must be allocated to the recliners. iii. There must be at least 30 love seats manufactured. 2. Blending problems: TUNACO Oil company produces three grades of gasoline (regular, premium and super) by blending three types of crude oil. All three types of crude oil contain two important ingredients (X & Y) required to produce the three gasoline grades. The percentage of these ingredients differs in each type of crude oil. These percentages are given in the following table.
Crude oil type 1
2
3
Ingredient X
40
20
45
Ingredient Y
35
30
40
Cost of each type of crude oil is $0.80, $0.60 and $0.40 per gallon for crude oil types 1, 2 & 3 respectively. Each gallon of regular gasoline must contain at least 35% of ingredient X. Each gallon of premium gasoline can contain at most 45% of ingredient Y. Each gallon of super gasoline can contain at most 40% of ingredient Y. Daily demand for regular, premium and super grades of gasoline is 400,000, 150,000 and 150,000 gallons respectively. Daily production quantities are 350,000 gallons of crude oil 1; 250,000 gallons for crude oil 2; and 200,000 gallons for crude oil 3. How many gallons of each type of crude oil should be used in the three grades of gasoline to satisfy demand at minimum cost? 3. Marketing Applications (Selection of media-mix): The Long Last Appliance Sales Company is in the business of selling appliances such as microwave ovens, traditional ovens, refrigerators, dishwashers, washers, dryers, and the like. The company has stores in the greater Chicago-land area and has a monthly advertising budget of $90,000. Among its options are radio advertising, advertising in the cable TV channels, newspaper advertising, and direct-mail advertising. A 30-second advertising spot on the local cable channel costs $1,800, a 30second radio ad costs $350, a half-page ad in the local newspaper costs $700, and a single mailing of direct-mail insertion for the entire region costs $1,200 per mailing. The number of potential buying customers reached per advertising medium usage is as follows: Radio TV Newspaper Direct mail
7,000 50,000 18,000 34,000
Due to company restrictions and availability of media, the maximum number of usages of each medium is limited to the following: Radio TV Newspaper Direct mail
35 25 30 18
The management of the company has met and decided that in order to ensure a balanced utilization of different types of media and to portray a positive image of the company, at least 10 percent of the advertisements must be on TV. No more than 40 percent of the advertisements must be on radio. The cost of advertising allocated to TV and direct mail cannot exceed 60 percent of the total advertising budget. Formulate the above problem as a LP problem with a view to maximizing the reach of potential buying customers.
4. Market Research: Market Facts Inc. is a marketing research firm that works with client companies to determine consumer reaction toward various products and services. A client company requested that Market Facts investigate the consumer reaction to a recently developed electronic device. Market Facts and the client company agreed that a combination of telephone interviews and direct-mail questionnaires would be used to obtain the information from different type of households. The households are divided into six categories: i.
Households containing a single person under 40 years old and without children under 18 years of age.
ii. Households containing married people under 40 years old and without children under 18 years of age. iii. Households containing single parents with children under 18 years of age. iv. Households containing married families with children under 18 years of age. v.
Households containing single people over 40 years old without children under 18 years of age.
vi. 6. Households containing married people over 40 years old and without children under 18 years of age. The client company has requested that the total people contacted via direct mail and phone interviews be 50,000. There can be no more than 2000 phone interviews and no more than 48,000 mail-in questionnaires. The cost of a direct-mail questionnaire including the cost of a self-stamped, mail-back envelope, is $1.00 for a household without children under 18 years of age and $1.50 for households with children. The cost of the phone interview also differs depending on whether the household contains children less than 18 years of age and if it contains a married couple. The interview costs more for the household with children because the interviewer has to ask more questions. The cost of a phone interview of a household with a married couple with children is $15. The cost of a phone interview of a household containing single adults under 40 years old without children is $10; for married adults under 40 years old without children, it is $11. The cost of a phone interview of a household with a single
person over 40 years old without children is $7. The cost of a phone interview of a household with a married couple over 40 years old without children is $9. The cost of a phone interview of a household with a single parent with children is $12. Discussions between Market Facts and the client company have resulted in the following restrictions: Restrictions: •
At least 60 percent of the phone interviews must be conducted at households with children.
•
At least 50 percent of the direct-mail questionnaires must be mailed to households with children.
•
No more than 30 percent of the phone interviews and mail-in questionnaires must be conducted at households with single people.
•
At least 25 percent of the phone interviews and mail-in questionnaires must be conducted at households that contain married couples.
How many phone interviews will be conducted with different types of households and how many direct mail-questionnaire will be mailed to different types of households to minimize the total cost of this market survey? 5. Financial applications: First American Bank is in the process of devising a loan policy that involves a maximum of $12 million. The following table provides the pertinent data about available types of loans. Type of loan
Interest rate
Bad-debt ratio
Personal
0.140
0.10
Car
0.130
0.07
Home
0.120
0.03
Farm
0.125
0.05
Commercial
0.100
0.02
Bad debts are unrecoverable and produce no interest revenue. Competition with other financial institutions requires that the bank allocate at least 40% of the funds to farm and commercial loans. To assist the housing industry in the region, home loans must equal at least 50% of the personal, car and home loans. The bank also has stated policy of not allowing the overall ratio of bad debts on all loans to exceed 4%. 6. Multi-period Production Scheduling:
Morton and Monson Inc. is a small manufacturer of parts for the aerospace industry. The production capacity for the next four months is given as follows: Production Capacity in Units Month
Regular Production
Overtime Production
January
3,000
500
February
2,000
400
March
3,000
600
April
3,500
800
The regular cost of production is $500 per unit and the cost of overtime production is $150 per unit in addition to the regular cost of production. The company can utilize inventories to reduce fluctuations in production, but carrying one unit of inventory costs the company $40 per unit per month. Currently there are no units in inventory. However, the company wants to maintain a minimum safety stock of 100 units of inventory during the months of January, February, and March. The estimated demand for the next four months is as follows: Month
January
February
March
April
Demand
2,800
3,000
3,500
3,000
The production manager is in the process of preparing a 4-month production schedule. What is the schedule that minimizes total cost, if the company wants to have 300 units in inventory at the end of April? 7. Workforce Scheduling: Lincoln General Hospital is trying to determine the nursing schedule of the pediatric department. The nursing staff consists of full-time nurses who work eight-hour shifts and part-time nurses who work four-hour shifts. The supervisor of nurses divides the day into six four-hour periods. In each period, a different level of demand (No of cases to be treated) is expected, which requires different number of nurses to be hired. The required number of nurses for each time period is given in the following table: Time period index
Time period
Required no. of nurses
1
7-11a.m.
7
2
11a.m-3 p.m.
9
3
3-7 p.m.
12
4
7-11 p.m.
5
5
11 p.m.-3 a.m.
4
6
3-7 a.m.
3
It is also required that there must be at least two full-time nurses at each time period and the number of part-time nurses cannot exceed the number of full-time nurses in any time period. The full-time nurses get paid @ $160 per shift, while the part-time nurses get paid @ $50 per shift. The normal shifts can begin at the start of any of the four-hour periods. 8. Financial Planning: First American Bank issues five types of loans. In addition, to diversify its portfolio, and to minimize risk, the bank invests in risk-free securities. The loans and the risk-free securities with their annual rate of return are given in the following Table: Table 1: Rates of Return for Financial Planning Problem Type of Loan or Security
Annual Rate of Return (%)
Home mortgage (first)
6
Home mortgage (second)
8
Commercial loan
11
Automobile loan
9
Home improvement loan
10
Risk-free securities
4
The bank’s objective is to maximize the annual rate of return on investments subject to the following policies, restrictions, and regulations: i.
The bank has $90 million in available funds.
ii. Risk-free securities must contain at least 10 percent of the total funds available for investments. iii. Home improvement loans cannot exceed $8,000,000. iv. The investment in mortgage loans must be at least 60 percent of all the funds invested in loans. v. The investment in first mortgage loans must be at least twice as much as the investment in second mortgage loans. vi. Home improvement loans cannot exceed 40 percent of the funds invested in first mortgage loans. vii. Automobile loans and home improvement loans together may not exceed the commercial loans.
viii. Commercial loans cannot exceed 50 percent of the total funds invested in mortgage loans. 9. Agriculture Applications: A farm owner is interested in determining how to divide the farmland among four different types of crops. The farmer owns two farms in separate locations and has decided to plant the following four types of crops in these farms: corn, wheat, bean, and cotton. The first farm consists of 1,450 acres of land, while the second farm consists of 850 acres of land. Any of the four crops may be planted on either farm. However, after a survey of the land, based on the characteristics of the farmlands, the following Table shows the maximum acreage restrictions the farmer has placed for each crop. Table 1: Max Acreage restrictions for Agricultural Problem Crop Farm
Corn
Wheat Bean
Cotton
Farm1 550
450
350
400
Farm2 250
300
200
350
The revenue per acre for each crop is estimated as follows: Revenue/acre Crop
Revenue/acre Crop
Corn
$500
Bean
$300
Wheat
$400
Cotton
$350
In determining the optimal cultivation of land, the farmer has to account for the cost of fertilizer estimated for each acre of land. Due to the different terrain and soil, the two farms have different costs of fertilizers per acre. Farm
Cost of Fertilizer/Acre
Farm 1
$100
Farm 2
$70
Seasonal demand for the four crops is given in table 2: Crop
Seasonal demand (Acres’ worth)
Corn
450
Wheat
550
Bean
400
Cotton
600
The farmer has a storage facility that can store 100 acres’ worth of the excess supply of different types of crops. In addition, the farmer wants to ensure that total wheat and bean cultivation must be proportionally equal to the maximum acreage restriction of both farms. In other words, the farm owner wants the same proportion of wheat and beans in both farms. The farmer’s objective is to determine how much of each crop to plant on each farm in order to maximize profit and satisfy seasonal demand. 10. Investment Problem: Investor Doe has $10,000 to invest in four projects. The following table gives the cash flow for the four investments. Cash flow ($1000) at the start of Project
Year 1 Year 2 Year 3 Year 4 Year 5
1
-1.00
0.50
0.30
1.80
1.20
2
-1.00
0.60
0.20
1.50
1.30
3
0.0
-1.00
0.80
1.90
0.80
4
-1.00
0.40
0.60
1.80
0.95
The information given in the table can be interpreted as follows: For project 1, $1.00 invested at the start of year 1 will yield $0.50 at the start of year 2, $0.30 at the start of year 3, $1.80 at the start of year 4 and $1.20 at the start of year 5. The remaining entries can be interpreted similarly. The entry 0.0 indicates that no transaction is taking place. Doe has the additional option of investing in a bank account that earns 6.5% annually. All funds accumulated at the end of one year can be reinvested in the following year. Formulate the problem as a linear program to determine the optimal allocation of funds to investment opportunities. 11. Investment Problem: HiRise Construction can bid on two 1-year projects. The following table provides the quarterly cash flow (in millions of dollars) for the two projects. Cash flow (in millions of $) at Project
1/1/08 4/1/08 7/1/08 10/1/08
12/31/08
I
-1.0
-3.1
-1.5
1.8
5.0
II
-3.0
-2.5
1.5
1.8
2.8
HiRise has cash funds $1 million at the beginning of each quarter and may borrow at most $1 million at a 10% nominal annual interest rate. Any borrowed money must be returned at the end of the quarter.
Surplus cash can earn quarterly interest at an 8% nominal annual rate. Net accumulation at the end of one quarter is invested in the next quarter. (i)
Assume that HiRise is allowed partial or full participation in the two projects. Determine the level of participation that will maximize the net cash accumulated on 12/31/2008.
(ii)
Is it possible in any quarter to borrow money and simultaneously end up with surplus funds? Explain.
