Submit an Excel file with two sheets (one for problem#1, one for problem#2), or more if needed.

1.  (from Winston, pg 648)
This problem is a very simple NLP--essentially, only one variable.
 
Each morning during rush hour, 10,000 people want to travel from
New Jersey to New York City.  If a person takes the subway, the
trip lasts 40 minutes.  If x thousand people per morning drive to
NYC, it takes 20+5x minutes to make the trip.  This problem illustrates
a basic fact of life: if people are left to their own devices, they will
cause more congestion than need actually occur!
        a) Show that if people are left to their own devices, an average
        of 4,000 people will travel by road from NJ to NYC.  Here you should
        assume that people will divide up between subway and road in a way
        that makes the average travel time by road equal to the average
        travel time by subway.  When this "equilibrium" occurs, nobody
        has an incentive to switch from road to subway or vice versa.
        This is known as the "user optimal" scenario.
        b) Show that the average travel time per person is minimized
        if 2,000 people travel by road.
        This is known as the "system optimal" scenario.
        c) What percent worse than system optimal is the user-optimal scenario?
        d) (optional) is 10,000 people a reasonable number for NJ-to-NYC? Do some
         research on the internet.


2. Pick one of the following problems:
* Fli-Bi-Nite airlines (see below); also called "tankering"
* Shotspotter (see the electronic coursepack)
* the transportation problem (see below)
* the curve-fitting problem (see below)
* the stock market optimization problem called "Quadratic programming example" at
http://people.brunel.ac.uk/~mastjjb/jeb/or/nlp.html
* Selected problems from
Optimization modeling with spreadsheets, second edition [electronic resource] / Kenneth R. Baker. 
in the EMU Library (online), and see below for some examples
* or any other optimization problem that you find interesting--let me know before you get started on it.


--------------------------------------------------------------
Fli-Bi-Nite airlines has a flight that starts in LA, flies to Houston,
        then NYC, then Miami.  The length of each segment (in miles) is
        1500, 1700, and 1300 respectively.
        At each stop, the plane may purchase up to 10,000 gallons of fuel.
        The prices in each city (in cents/gal) are: 88, 15, 105, and 95.
        The plane's tank can hold at most 12,000 gallons.  To allow for
        the possibility of circling while waiting to land, we require that
        the ending fuel level for each leg of the flight be 600 gallons.
        We'll suppose that the airplane gets 0.3 miles per gallon.
        The plane starts in LA with 4933.3333 gallons before buying fuel 
        and taking off.
        
        Formulate and solve an optimization that can be used to minimize the fuel cost
        incurred in completing the schedule.
--------------------------------------------------------------

"Transportation Problems" are a subclass of network flow problems.
You have a set of source cities and a set of demand cities, the
amount of supply in each source city, the amount of demand in each
demand city, and the shipping costs for each (source,demand) city pair.
You must exactly meet demand in each demand city, and each source city
must ship out exactly what it produces (no more or less).

a) Formulate and solve a transportation problem as follows:
Source cities: San Francisco and Los Angeles (SF and LA)
Demand cities: Las Vegas and Phoenix (LV and Ph)

Costs:                to:        LV        Ph
from        SF                50        300
        LA                320        60
(costs are per item shipped).
Supply: 5 in SF, 10 in LA
Demand: 7 in LV, 8 in Ph

Find the optimal shipping amounts and the total cost.

b) Add one unit of supply in SF and one unit of demand in Ph, re-solve,
and report the new optimal decisions and total cost.  Comment as appropriate.

d) In both (a) and (b), total supply and total demand matched each other
perfectly.  What would you do if supply exceeded demand? 

e) What would you do if demand exceeded supply?

---------------------------------------------------------------
Curve fitting
Consider this data on the growth of a batch of yeast:
Time(hours)        Biomass
0        9.6
1        18.3
2        29.0
3        47.2
4        71.1
5        119.1
6        174.6
7        247.3

a) Using excel's usual trendline feature, fit an Exponential model to it.
b) Compute the residuals using that exponential model, and the sum of
squared residuals.
c) Write down (using math notation) a nonlinear program that will choose
the coefficients in the exponential model to minimize the sum of
squared residuals.
c) Set up a nonlinear program to choose the coefficients in the
exponential model, and optimize it using Solver.  Report your optimal
coefficients and the sum of squared residuals.
d) Compare the two results (those of parts a&b versus part c).  Use
graphs and sentences as appropriate.  If there is a difference, explain why.
-----------------------------------------------
Here are a bunch of problems from
"Optimization modeling with spreadsheets, second edition" [electronic resource] / Kenneth R. Baker. 
I know the formatting is not great. But if one of them looks interesting, you could go look up the original version
of it in the book, since the EMU library has an electronic copy.

1.1  
Determining an Optimal Price A firm's Marketing Department has estimated the demand curve of a product as y = 1100 — 7x, where y represents demand and x represents the unit selling price (in dollars) for the relevant decision period. The unit cost is known to be $24. What price maximizes net income from sales of the product?
 
1.2  
Pricing in Two Markets Global Products, Inc. has been making an electronic appliance for the domestic market. Demand for the appliance is price sensitive, and the demand curve is known to follow the linear function D = 4000— 5P, where D represents annual demand and P represents selling price in the home currency, which is the Frank (F). The cost of manufacturing the appliance is 100F.
For the coming year, Global is planning to sell the same product in a foreign market, where the currency is the Marc (M). From surveys, the demand curve in the foreign country is estimated to follow a different linear function, D = 2000— 2P, where the price is denominated in Marcs.
All production will be carried out at Global's domestic plant, with the expectation that the unit cost will remain unchanged. The exchange rate is 1.5 M/F, and Global plans to offer an equivalent price in both markets.
If Global were to operate exclusively in its domestic market, what would be its profit-maximizing price and its annual profit?
When Global sells in both markets at one equivalent price, what is its profit-maximizing price and its annual profit?
 
1.3  
Locating a Distribution Center Northeast Parts Supply is a wholesale distributor of components for printers, fax machines, scanners, and related equipment. Northeast stocks expensive spare parts, which dealers prefer not to hold, and offers same-day delivery on any order. The firm now serves eight dealers in the New England area and wishes to locate its distribution facility at a central point. In particular, its dealers have each been assigned a location on an x-y grid, and Northeast would like to find the best location for the distribution facility.
The eight dealers and their grid locations are shown in the following table:
 Open table as spreadsheet
Dealer
1
2
3
4
5
6
7
8
x-location
25
82
10
27
93
14
68
147
y-location
32
36
71
58
68
163
149
192
Determine the location that minimizes the sum of the distances from the distribution facility to the dealers.
Determine the location that minimizes the maximum distance from the distribution facility to any of the dealers.
 
1.4  
Collecting Credit Card Debt A bank offers a credit card that can be used in various locations. The bank's analysts believe that the percentage P of accounts receivable collected by t months after credit is issued increases at a decreasing rate. Historical data suggest the following function:

The average credit issued in any one month is $125 million, and historical experience suggests that for new credit issued in any month, collection efforts cost $1 million per month.
Determine the number of months that collection efforts should be continued if the objective is to maximize the net collections (dollars collected minus collection costs). Allow for fractional months.
Under the optimal policy in (a), what percentage of accounts receivable should be collected?
 
1.5  
Allocating Plant Output A firm owns five manufacturing plants that are responsible for the quarterly production of an industrial solvent. The production process exhibits diseconomies of scale. At plant p, the cost of making x thousand pounds of the solvent is approximated by the quadratic function f(x) = (1/cp)x2. The parameters cp are plant dependent, as shown in the table.
p
1
2
3
4
5
cp
3
6
4
8
5
 Open table as spreadsheet
The quarterly volume requirement is 50,000 pounds.
How should production be allocated among the five plants in order to minimize the total cost of meeting the volume requirement?
 
1.6  
Determining Production Lot Sizes Four products are routed through a machining center that is notorious for its delays. Each product has had stable demand for some time, so that average weekly demand is predictable over a 3–6 month time frame. However, in the short run, demand fluctuates a great deal, and the load at the machining center varies considerably. The production control system dictates the lot size for each of the products. These quantities are shown, along with other relevant information, in the following table.
 Open table as spreadsheet
Product no.
Demand (weekly)
Setup (hours)
Run time(hours/1000)
Lot size
1
100
3
30
100
2
500
15
45
500
3
50
6
75
100
4
250
24
150
1500
With the current lot sizes, the machining center is running at a utilization of about 76%, but long lead times, sometimes over 2 weeks, have discouraged production planners from increasing its load. (A week contains 120 productive hours.) In the past, lead times spiraled out of control when utilization grew to around 80%.
A lead time model for this problem has been constructed on a spreadsheet.[2] The model permits the user to select lot sizes and thereby influence the average lead time through the bottleneck work center. The lead time prediction is based on advanced modeling techniques, but the details of the model are not of primary importance.
What is the shortest possible lead time, and what lot sizes achieve this value?
 
1.7  
Resolving a Construction Dilemma A library building is about to undergo some renovations that will improve its structural integrity. As part of the process, a number of steel beams will be carried through the existing bookcases from a broad, open area around the entry point. The central aisle between the bookcases is 10 feet wide, while the side aisles (which run perpendicular to the central aisle) are 6 feet wide. The renovation will require that steel beams be carried through the stacks, down the main aisle and turning into the smaller aisles.
What is the longest steel beam that can be carried horizontally through this space to a construction point along the outer walls?
 
1.8  
Selecting the Number of Warehouses The customers of a particular company are located throughout an area comprised of S square miles, and they are serviced from k warehouses. On average, the distance in miles between a warehouse and a customer is given by the formula (S/k)0.5. The annual capital cost of building a warehouse is $40,000 and the annual operating cost of running a warehouse is $60,000. Annual shipping costs average $1 per mile per customer.
Suppose that the current market size is 250,000 customers, spread out over an area of 500 square miles. What is the optimal number of warehouses for the firm to operate?


2.3  
Workforce Scheduling The Operations Manager at the Metropolis National Bank has a staffing problem. Demand for clerical staff varies throughout the day, but 24hour coverage is necessary because the bank handles a number of international transactions. A recent study has shown how many clerical workers are needed each hour in the course of the day, as shown below. (Hour 1 is from midnight to 1 am.)
Hour
1
2
3
4
5
6
7
8
9
10
11
12
Staff
4
3
2
2
3
5
6
6
9
10
10
10
Hour
13
14
15
16
17
18
19
20
21
22
23
24
Staff
12
12
8
6
7
7
7
6
5
4
4
4
 Open table as spreadsheet
Under current labor policies, clerical workers may be assigned to any one of six shifts, some of which overlap. The shifts and salary costs are as follows.
 Open table as spreadsheet
Shift
Daily cost
2 am–10 am
$160
6 am–2 pm
$145
10 am–6 pm
$148
2 pm–10 pm
$154
6 pm–2 am
$156
10 pm–6 am
$160
Provide the operations manager with a schedule that will deploy enough staff to meet the hourly requirements at the minimum daily total cost.
In the optimal schedule, how many hours are overstaffed?
 
2.4  
Selecting a Portfolio A portfolio manager has developed a list of six investment alternatives for a multiyear horizon. These are: Treasury bills, Common stock, Corporate bonds, Real estate, Growth funds, and Savings and Loans. These investments and their various financial factors are described below. In the table, the length represents the estimated number of years required for the annual rate of return to be realized. The annual rate of return is the expected rate over the multiyear horizon. The risk coefficient is a subjective estimate representing the manager's appraisal of the relative safety of each alternative, on a scale of 10. The growth potential is also a subjective estimate of the potential increase in value over the horizon.
 Open table as spreadsheet
 	
Portfolio data
Alternative
TB
CS
CB
RE
GF
SL
Length
4
7
8
6
10
5
Annual return (%)
6
15
12
24
18
9
Risk coefficient
1
5
4
8
6
3
Growth potential (%)
0
18
10
32
20
7
The manager wishes to maximize the annual rate of return on a $3 million portfolio, subject to the following restrictions.
The weighted average length should not exceed 7 years.
The weighted average risk coefficient should not exceed five.
The weighted average growth potential should be at least 10 percent.
The investment in real estate should be no more than twice the investment in stocks and bonds (i.e. in CS, CB, and GF) combined.
What is the optimal return (as a percentage) and the optimal allocation of investment funds?
What is the marginal rate of return? In other words, what would be the return on the next dollar invested, if there were one more dollar in the portfolio?
For additional investment beyond the original $3 million, how will the optimal allocation change?
 
2.5  
Oil Blending An oil company produces three brands of oil: Regular, Multigrade, and Supreme. Each brand of oil is composed of one or more of four crude stocks, each having a different lubrication index. The relevant data concerning the crude stocks are as follows.
 Open table as spreadsheet
Crude
Lubrication
Cost
Supply per day
stock
index
($/barrel)
(barrels)
1
20
7.10
1000
2
40
8.50
1100
3
30
7.70
1200
4
55
9.00
1100
Each brand of oil must meet a minimum standard for a lubrication index, and each brand thus sells at a different price. The relevant data concerning the three brands of oil are as follows.
 Open table as spreadsheet
Brand
Minimum lubrication index
Sellingprice($/barrel)
Dailydemand (barrels)
Regular
25
8.50
2000
Multigrade
35
9.00
1500
Supreme
50
10.00
750
Determine an optimal outputplan forasingle day, assuming thatproduction can be either sold or else stored at negligible cost.
The daily demand figures are subject to alternative interpretations. Investigate the following:
The daily demands represent potential sales. In other words, the model should contain demand ceilings (upper limits). What is the optimal profit?
The daily demands are strict obligations. In other words, the model should contain demand constraints that are met precisely. What is the optimal profit?
The daily demands represent minimum sales commitments, but all output can be sold. In other words, the model should permit production to exceed the daily commitments. What is the optimal profit?
 
2.6  
Coffee Blending and Sales Hill-O-Beans Coffee Company blends four component beans into three final blends of coffee: one is sold to luxury hotels, another to restaurants, and the third to supermarkets for store-label brands. The company has four reliable bean supplies: Argentine Abundo, Peruvian Colmado, Brazilian Maximo, and Chilean Saboro. The table below summarizes the very precise recipes for the final coffee blends, the cost and availability information for the four components, and the wholesale price per pound of the final blends. The percentages indicate the fraction of each component to be used in each blend.
 Open table as spreadsheet
Component (pounds)
Hotel
Rest
Market
Cost per pound
Max. weekly availability
Abundo
20%
35%
10%
$0.60
40,000
Colmado
40%
15%
35%
$0.80
25,000
Maximo
15%
20%
40%
$0.55
20,000
Saboro
25%
30%
15%
$0.70
45,000
Wholesale price per pound
$1.25
$1.50
$1.40
 	 
The processor's plant can handle no more than 100,000 lb per week, and Hill-O-Beans would like to operate at capacity, if possible. Selling the final blends is not a problem, although the Marketing Department requires minimum production levels of 10,000, 25,000, and 30,000 lb, respectively, for the hotel, restaurant and market blends.
To maximize weekly profit, how many pounds of each component should be purchased?
How would the optimal profit change if there were a 1000-lb increase in the availability of Abundo beans? Colmado? Maximo? Saboro?
 
2.7  
Production Planning for Components Rummel Electronics produces two PC cards, a modem and a network adapter. Demand for these two products exceeds the amount that the firm can make, but there are no plans to increase production capacity in the short run. Instead, the firm plans to use subcontracting.
The two main stages of production are fabrication and assembly, and either step can be subcontracted for either type of card. However, the company policy is not to subcontract both steps for either product. (That is, if modem cards are fabricated by a subcontractor, then they must be assembled in house.) Components made by subcontractors must pass through the shipping and receiving departments, just like components made internally. At present, the firm has 5200 hours available in fabrication, 3600 in assembly and 3200 in shipping/inspection. The production requirements, in hours per unit, are given in the following table:
 Open table as spreadsheet
Product/mode
Fabrication
Assembly
Shipping
Modem, made entirely in-house
0.35
0.16
0.08
Network, made entirely in-house
0.47
0.15
0.12
Modem, fabricated by sub
-
0.18
0.10
Network, fabricated by sub
-
0.16
0.15
Modem, assembled by sub
0.35
-
0.09
Network, assembled by sub
0.47
-
0.14
The direct material costs for the modem cards are $3.25 for manufacturing and $0.50 forassembly; for network cards, the costs are $6.10 and $0.50. Subcontracting the manufacturing operation costs $5.35 for modem cards and $8.50 for network cards. Subcontracting the assembly operation costs $1.50 for either product. Modem cards sell for $20, and network cards sell for $28. The firm's policy, for each product, is that at most 40% of the units produced can have subcontracted fabrication, and at most 70% of the units can have subcontracted assembly.
Determine the production and subcontracting schedule that maximizes profits. How many units of each product should be sold, in the optimal plan? What total volume should the subcontractor handle?
Which department capacities limit the manufacturing volume? If 100 hours of overtime could be scheduled, which department(s) should be allocated the overtime? Explain.
 
2.8  
Production Planning for Automobiles The Auto Company of America (ACA) produces four types of cars: subcompact, compact, intermediate, and luxury. ACA also produces trucks and vans. Vendor capacities limit total production capacity to at most 1,200,000 vehicles per year. Subcompacts and compacts are built together in a facility with a total annual capacity of 620,000 cars. Intermediate and luxury cars are produced in another facility with capacity of 400,000; and the truck/van facility has a capacity of 275,000. ACA's marketing strategy requires that subcompacts and compacts must constitute at least half of the product mix for the four car types. Profit margins, market potential, and fuel efficiencies are summarized below.
 Open table as spreadsheet
Type
Profit margin ($/vehicle)
Potential sales(in 000s)
Fuel efficiency(MPG)
Subcompact
150
600
40
Compact
225
400
34
Intermediate
250
300
15
Luxury
500
225
12
Truck
400
325
20
Van
200
100
25
The Corporate Average Fuel Efficiency (CAFE) standards require an average fleet fuel efficiency of at least 27 MPG. ACA would like to use a linear programming model to understand the implications of government and corporate policies on its production plans.
What is the optimal annual profit for ACA?
How much would annual profit drop if the fuel efficiency requirement were raised to 28 MPG?
 
2.9  
Production Planning with Environmental Constraints You are the Operations Manager of Lovejoy Chemicals, Inc., which produces five products in a common production facility that will be subject to proposed Environmental Protection Agency (EPA) limits on particulate emissions. For each product, Lovejoy's sales potentials (demand levels that Lovejoy can capture) are expected to remain relatively flat for at least the next five years. Relevant data for each product are as follows (note: T denotes tons).
 Open table as spreadsheet
Product
Sales potential(T/year)
Variable costs ($/T)
Revenues($/T)
Particulate emissions (T/T produced)
A
2000
700
1000
0.0010
B
1600
600
800
0.0025
C
1000
1000
1500
0.0300
D
1000
1600
2000
0.0400
E
600
1300
1700
0.0250
Your production facility rotates through the product line because it is capable of producing only one product at a time. The production rates differ for the various products due to processing needs. It takes 0.3 hours to make one ton of A, 0.5 hours for B, and one hour each to make a ton of C, D, or E. The facility can be operated up to 4000 hours each year.
The EPA is proposing a "bubble policy" for your industry. In this form of regulation, imagine that a bubble encloses the manufacturing facility, and only total particulates that escape the bubble are regulated. This sort of policy replaces historical attempts by the EPA to micromanage emissions within a firm, and it allows Lovejoy to make any changes it wishes, provided the total particulate emissions from its facility are kept below certain limits. The current proposal is to phase-in strict particulate emissions limits over the next five years. These limits on total particulate emissions are shown in the table below.
 Open table as spreadsheet
Year
1
2
3
4
5
Allowable emissions(T/year)
unlimited
80
60
40
20
One strategy for satisfying these regulations is to adjust the product mix, cutting back on production of some products if necessary. Lovejoy wishes to explore this strategy before contemplating the addition of new equipment.
Determine the maximum profit Lovejoy can achieve from its product line in the coming year (Year 1).
By solving a series of models corresponding to the imposition and tightening of the emissions limit in future years, determine Lovejoy's maximum profits in each of Years 2-5.
Consider the emissions limit that applies in Year 4. Determine how much Lovejoy should be willing to pay at that time to be allowed emissions of one extra ton of particulates above the limit.
 
2.10  
Cargo Loading You are in charge of loading cargo ships for International Cargo Company (ICC) at a major East Coast port. You have been asked to prepare a loading plan for an ICC freight ship bound for Africa. An agricultural commodities dealer would like to transport the following products aboard this ship.
 Open table as spreadsheet
Commodity
Tons available
Volume per ton(cu. ft)
Profit per ton($)
1
4000
40
70
2
3000
25
50
3
2000
60
60
4
1000
50
80
You can elect to load any and/or all of the available commodities. However, the ship has three cargo holds with the following capacity restrictions.
 Open table as spreadsheet
Cargohold
Weight capacity (tons)
Volume capacity(cu. ft)
Forward
3000
100,000
Center
5000
150,000
Rear
2000
120,000
More than one type of commodity can be placed in the same cargo hold. However, because of balance considerations, the weight in the forward cargo hold must be within 10 percent of the weight in the rear cargo hold, and the center cargo hold must be between 40 percent and 60 percent of the total weight on board.
What is the maximum profit and the loading plan that achieves it? What is the optimal total weight to be loaded? What is the optimal total volume to be loaded?
Suppose each one of the cargo holds could be expanded. Which holds and which forms of expansion (weight or volume) would allow ICC to increase its profits on this trip, and what is the marginal value of each form of expansion?
 
2.11  
Computer Center Staffing You are the Director of the Computer Center for Gaillard College and responsible for scheduling the staffing of the center. It is open from 8 am until midnight. You have monitored the usage of the center at various times of the day and determined that the following numbers of computer consultants are required.
 Open table as spreadsheet
Time of day
Minimum number of consultants required to be on duty
8 am–noon
4
Noon-4 pm
8
4am–8pm
10
8 am–midnight
6
Two types of computer consultants can be hired: full-time and part-time. The full-time consultants work for eight consecutive hours in any of the following shifts: morning (8 am–4 pm), afternoon (noon-8 pm), and evening (4 pm-midnight). Full-time consultants are paid $14 per hour.
Part-time consultants can be hired to work any of the four shifts listed in the table. Part-time consultants are paid $12 per hour. An additional requirement is that during every time period, at least one full-time consultant must be on duty for every part-time consultant on duty.
Determine a minimum-cost staffing plan for the center. In your solution, how many consultants will be paid to work full time and how many will be paid to work part time? What is the minimum cost?
After thinking about this problem for a while, you have decided to recognize meal breaks explicitly in the scheduling of full-time consultants. In particular, full-time consultants are entitled to a one-hour lunch break during their eight-hour shift. In addition, employment rules specify that the lunch break can start after three hours of work or after four hours of work, but those are the only alternatives. Part-time consultants do not receive a meal break. Under these conditions, find a minimum-cost staffing plan. What is the minimum cost?
 
2.12  
Make or Buy A sudden increase in the demand for smoke detectors has left Acme Alarms with insufficient capacity to meet demand. The company has seen monthly demand from its retailers for its electronic and battery operated detectors rise to 20,000 and 10,000, respectively, and Acme wishes to continue meeting demand. Acme's production process involves three departments: Fabrication, Assembly, and Shipping. The relevant quantitative data on production and prices are summarized below.
 Open table as spreadsheet
Department
Monthly hours available
Hours/unit (electronic)
Hours/unit (battery)
Fabrication
2000
0.15
0.10
Assembly
4200
0.20
0.20
Shipping
2500
0.10
0.15
Variable
 	
$18.80
$16.00
cost/unit
 	 	 
Retail price
 	
$29.50
$28.00
The company also has the option to obtain additional units from a subcontractor, who has offered to supply up to 20,000 units per month in any combination of electronic and battery operated models, at a charge of $21.50 per unit. For this price, the subcontractor will test and ship its models directly to the retailers without using Acme's production process.
What is the maximum profit and the corresponding make/buy levels? (This is a planning model, and fractional decisions are acceptable.)
Trace the effects of increasing Fabrication capacity by 10 percent. How will the optimal make/buy mix change? How will the optimal profit change?
 
2.13  
Leasing Warehouse Space Cox Cable Company needs to lease warehouse storage space for five months at the start of the year. It knows how much space will be required in each month, and it can purchase a variety of lease contracts to meet these needs. For example, it can purchase one-month leases in each month from January to May. It can also purchase two-month leases in January through April, three-month leases in January through March, four-month leases in January and February, or a five-month lease in January. In total, there are 15 possible leases it could use. It must decide which leases to purchase, and how many square feet to purchase on each lease.
Since the space requirements differ month-to-month, it may be economical to lease only the amount needed each month on a month-by-month basis. On the other hand, the monthly cost for leasing space for additional months is much less than for the first month, so it may be desirable to lease the maximum amount needed for the entire five months. Another option is the intermediate approach of changing the total amount of space leased (by adding a new lease and/or having an old lease expire) at least once but not every month. Two or more leases for different terms can begin at the same time.
The space requirements (in square feet) and the leasing costs (in dollars per thousand square feet) are given in the tables below.
 Open table as spreadsheet
Month
Space requirements
Lease length
Lease cost
Jan
15,000
1 month
$280
Feb
10,000
2 months
450
Mar
20,000
3 months
600
April
5000
4 months
730
May
25,000
5 months
820
The task is to find a leasing schedule that provides the necessary amounts of space at the minimum cost.
Determine the optimal leasing schedule and the optimal total cost.
Trace the effects of increasing the space required for January. How will the leasing schedule change? How will the total cost change?
 
2.14  
Production Planning The Kim Camera Company produces four different camera models, known as C1–C4. Each model can be made by two different methods. The manual method requires work in the Fabrication, Assembly, and Test departments, while the automated method combines the Assembly and Test operations in one department. The first table below describes the price and cost features of the camera models, along with marketing information on the range of possible sales in the coming month. Because model C1 is delivered to one large retailer under a long-term contract, a threshold demand quantity of 1500 units must be met. For the other models, there is flexibility in how much demand to meet, up to a ceiling that represents maximum possible sales.
 Open table as spreadsheet
 	
C1
C2
C3
C4
Price
125
175
200
135
Manual cost
110
160
155
125
Auto cost
100
112
150
90
Manual margin
15
15
45
10
Auto margin
25
63
50
45
Sales max.
3000
2500
2000
3200
Sales min.
1500
0
0
0
The next table provides data on the various departments at the firm, consisting of the time per camera required in each department and the number of hours available in each department during the month.
 Open table as spreadsheet
 	
C1
C2
C3
C4
Hours
Manual fab.
3
5
4
4
16,000
Manual asy.
8
12
10
9
30,000
Manual test
2
3
5
2
15,000
Auto fab.
5
6
7
4
24,000
Auto asy/test
4
5
8
5
20,000
What production plan will maximize profit for Kim Camera?
How would the solution in (a) change if there were no threshold requirement for Camera C1?
 
2.15  
Make/Buy Planning The CammTex Fabric Mill is in the process of deciding on a production schedule. It wishes to know how to weave the various fabrics it will produce during the coming quarter. The sales department has confirmed orders for each of the 15 fabrics that are produced by CammTex. These quarterly demands are given in the table below. Also tabulated is the variable cost for each fabric. The mill operates continuously during the quarter: 13 weeks, 7 days a week, and 24 hours a day.
CammTex uses two types of looms: dobbie and regular. Dobbie looms can make all fabrics, and they are the only looms that can weave certain fabrics such as plaids. The production rate for each fabric on each type of loom is also given in the table. (If the production rate is zero, the fabric cannot be woven on that type of loom.) CammTex has 90 regular looms and 15 dobbie looms.
Fabrics woven at CammTex proceed to the finishing department in the mill and are then sold. Any fabrics not woven in the mill because of limited capacity are subcontracted to an outside producer and sold at the selling price. The cost of purchasing from the subcontractor is also given in the table.
 Open table as spreadsheet
Fabric
Demand(yd)
Dobbie(yd/hr)
Regular (yd/hr)
Mill cost($/yd)
Sub. cost($/yd)
1
16,500
4.653
0.00
0.6573
0.80
2
52,000
4.653
0.00
0.555
0.70
3
45,000
4.653
0.00
0.655
0.85
4
22,000
4.653
0.00
0.5542
0.70
5
76,500
5.194
5.313
0.6097
0.75
6
110,000
3.767
3.809
0.6153
0.75
7
122,000
4.055
4.185
0.6477
0.80
8
62,000
5.208
5.232
0.488
0.60
9
7500
5.208
5.232
0.5029
0.70
10
69,000
5.208
5.232
0.4351
0.60
11
70,000
3.652
3.733
0.6417
0.80
12
82,000
4.007
4.185
0.5675
0.75
13
10,000
4.291
4.439
0.4952
0.65
14
380,000
5.208
5.232
0.3128
0.45
15
62,000
4.004
4.185
0.5029
0.70
What is minimum total cost of production and purchasing for CammTex?
Which fabrics should be made at the mill and which should be purchased? For those made at the mill, which loom types should be assigned to their production?
 
2.16  
Production Scheduling The Seaboch Tire Company produces four lines of tires: the Economy, the Glass-belted, the Snow and the Radial tire. The problem it faces is to schedule two shifts of production during the last quarter of the calendar year. The production process primarily involves the use of vulcanization, fabrication, and plastometer equipment, but the limiting resource is the availability of the vulcanization machines. The four tires require different amounts of time at vulcanization, as tabulated below.
 Open table as spreadsheet
Tire
Econ.
Glass
Snow
Radial
Hours/tire
4.5
5.0
5.5
6.0
A sales forecast is available, breaking down predicted sales (in thousands) by tire type and by month.
 Open table as spreadsheet
Sales
Oct
Nov
Dec
Econ.
8
7
6
Glass
18
16
18
Snow
4
15
15
Radial
6
5
8
In addition, the number of hours of vulcanizing time (in thousands), for each shift and for each month, is also known.
 Open table as spreadsheet
 	
Oct
Oct
Nov
Nov
Dec
Dec
Shif
1
2
1
2
1
2
Hours
110
100
130
120
120
115
The labor cost of operating the vulcanizing machines is $10 per hour during the first shift and $12 per hour during the second shift. The other relevant cost is storage: It costs $4 per month for storage and handling at the warehouse, regardless of tire type. This cost is incurred if there is not enough labor capacity to meet demand in the month when it occurs.
What production plan will minimize cost and meet demand at Seaboch Tire?
How would the solution in (a) change if sales for each tire in December were 10 percent higher?
Case: JetGreen[*]
JetGreen flies three airplanes, using a "hub and spoke" flight schedule between Houston and three cities, Chicago, Miami, and Phoenix. These three cities are the "spokes" connected by the Houston "hub." Once each day, the three airplanes fly from the spoke cities to Houston.
Exhibit 2.1: Price for Each Passenger Route
 Open table as spreadsheet
 	 	
Destination
 	 	
Houston
Chicago
Miami
Phoenix
 	
Houston
–
$197
$110
$125
 	
Chicago
$190
–
$282
$195
Origin
Miami
$108
$292
–
$238
 	
Phoenix
$110
$192
$230
–
They arrive nearly simultaneously at Houston, then connecting passengers change aircraft during a one-hour layover, and the three airplanes return to their starting cities. One set of six flights (3 inbound to Houston and 3 outbound) is called a bank. A bank can serve passengers flying on 12 different routes: three inbound direct routes (Chicago or Miami or Phoenix into Houston), three outbound direct routes (Houston to Chicago or Miami or Phoenix), and six routes requiring two flights each (Chicago-Miami, Chicago-Phoenix, Miami-Phoenix, Miami-Chicago, Phoenix-Chicago, and Phoenix-Miami).
JetGreen charges a regular price for a one-way ticket on each route. Exhibit 2.1 shows the regular prices. Following a well-established policy, JetGreen offers a discount to senior travelers. The ticket price for a senior traveler is 90 percent of the regular price, rounded down to the next smaller integer number of dollars. (For example, on the Houston-Phoenix route, the senior ticket price is $112.) The marginal cost of flying a passenger on each route is virtually zero.
Each of JetGreen's three airplanes contains 260 seats. Exhibit 2.2 shows demand for the routes in a bank at the regular price, and Exhibit 2.3 shows the demand from seniors (at the discounted price). These figures apply to the times at which JetGreen flies, and they show that passenger demand exceeds airplane capacity on every flight segment. For example, on the flight from Miami to the Houston hub, the total regular demand is the sum of demands for three passenger routes, (Miami to Houston or Chicago or Phoenix), totaling 72 + 105 + 68 = 245 passengers (from the third row of Exhibit 2.2). For seniors, the comparable figure is 6 + 15 + 8 = 29, and the total is 245 + 29 = 274. Because only 260 passengers can travel on the Miami-Houston flight, at least 14 passengers represent lost demand.
When the total demand for a particular flight is larger than the available capacity, an airline can decide whether to accept or reject an offer to buy a ticket for a particular route. Controlling sales in this way to maximize revenue is called revenue management. For example, JetGreen may decide to sell large numbers of tickets for the Miami-Houston and Miami-Chicago routes, but might severely restrict sales of the Miami-Phoenix tickets. Given the data above, JetGreen might sell tickets to 78 Miami-Houston passengers, 120 Miami-Chicago passengers, and only 62 Miami-Phoenix passengers, thus filling all 260 seats on the Miami-Houston flight. All 14 lost demands would then come from the Miami-Phoenix route.
Exhibit 2.2: Regular Demand During One Bank
 Open table as spreadsheet
 	 	
Destination
 	 	
Houston
Chicago
Miami
Phoenix
 	
Houston
-
123
80
110
Origin
Chicago
130
–
98
88
 	
Miami
72
105
–
68
 	
Phoenix
115
90
66
-
Exhibit 2.3: Senior Demand During One Bank
 Open table as spreadsheet
 	 	 	
Destination
 	 	
Houston
Chicago
Miami
Phoenix
 	
Houston
-
12
7
10
Origin
Chicago
15
–
10
13
 	
Miami
6
15
–
8
 	
Phoenix
12
8
5
-
Assuming that the various demands in Exhibits 2.2 and 2.3 are known, JetGreen wants to determine the number of tickets it should sell to regular and senior passengers on each route.
[*]Adapted from case material written by Professor Rob Shumsky of Dartmouth College.

3.4  
Distributing a Product The Lincoln Lock Company manufactures a commercial security lock at plants in Atlanta, Louisville, Detroit, and Phoenix. The unit cost of production at each plant is $35.50, $37.50, $37.25, and $36.25, and the annual capacities are 18,000, 15,000, 25,000, and 20,000, respectively. The locks are sold through wholesale distributors in seven locations around the country. The unit shipping cost for each plant-distributor combination is shown in the following table, along with the forecasted demand from each distributor for the coming year.
 Open table as spreadsheet
 	
Tacoma
SanDiego
Dallas
Denver
St Louis
Tampa
Baltimore
Atlanta
2.50
2.75
1.75
2.00
2.10
1.80
1.65
Louisville
1.85
1.90
1.50
1.60
1.00
1.90
1.85
Detroit
2.30
2.25
1.85
1.25
1.50
2.25
2.00
Phoenix
1.90
0.90
1.60
1.75
2.00
2.50
2.65
Demand
5.500
11.500
10.500
9.600
15.400
12.500
6.600
Determine the least costly way of shipping locks from plants to distributors.
Show the network diagram corresponding to the solution in (a). Thatis, label eachof the arcs in the solution and verify that the flows are consistent with the given information.
Suppose that the unit cost at each plant were $10 higher than the original figure. What change in the optimal distribution plan would result? What general conclusions can you draw for transportation models with nonidentical plant-related costs?
 
3.5  
Repositioning Supply The American Rent-a-Car Company has eight outlets in a metropolitan area. American operates under a policy that calls for a specific "target" percentage of all available cars to be located at each outlet at the start of each day. These percentages are summarized in the following table.
 Open table as spreadsheet
Outlet
1
2
3
4
5
6
7
8
Percentage
20
10
20
5
10
20
5
10
For example, if 50 cars are available, 10 should be at outlet 1 at the start of the day. At the end of a day, if the current distribution of cars does not comply with the targets, American employees drive the cars overnight from outlet to outlet so that the new distribution meets the specified targets. The distance between each pair of outlets is given in the following table.

At the end of a particular day, American finds that the 100 cars currently available are distributed at the outlets as follows.
Outlet
1
2
3
4
5
6
7
8
Cars
4
14
5
17
22
7
10
21
 Open table as spreadsheet
Given this distribution of cars, find a schedule for minimizing the total distance traveled during the overnight redistribution of the cars.
Show the network diagram corresponding to the solution in (a). That is, label each of the arcs in the solution and verify that the flows are consistent with the given information.
 
3.6  
Designing a Distribution System The Krotzer Company manufactures and distributes meters used to measure electric power consumption. The company started with a small production plant in El Paso and gradually built a customer base throughout Texas. A distribution center was established in Ft Worth, and later, as the business expanded, a second distribution center was established in Santa Fe. The El Paso plant was expanded when the company began marketing its meters in Arizona, California, Nevada, and Utah. With the growth of the West Coast business, the company opened a third distribution center in Las Vegas and just two years ago opened a second production plant in Sacramento.
Manufacturing costs differ between the company's production plants. The cost of each meter produced at the El Paso plant is $10.50. The Sacramento plant uses newer and more efficient equipment, and as a result, its manufacturing costs come to only $10.00 per unit.
The quarterly production capacity is 30,000 meters at the older El Paso plant and 20,000 meters at the Sacramento plant. No shipments are allowed from the Sacramento plant to the Ft. Worth distribution center.
Due to the firm's rapid growth, little attention has been paid to the efficiency of the distribution system, but company management has decided that it is time to address this issue. The cost of shipping a meter from each of the two plants to each of the three distribution centers is shown in the following table.
 Open table as spreadsheet
 	
Distribution center
Plant
Ft Worth
Santa Fe
Las Vegas
El Paso
3.20
2.20
4.20
Sacramento
-
3.90
1.20
The company serves nine customer zones from the three distribution centers. The forecast for the number of meters needed in each customer zone for the next quarter is shown in the following table.
 Open table as spreadsheet
Customer zone
Demand (meters)
Dallas
6300
San Antonio
4880
Wichita
2130
Kansas City
1210
Denver
6120
Salt Lake City
4830
Phoenix
2750
Los Angeles
8580
San Diego
4460
The cost per unit of shipping from each distribution center to each customer zone is given in the following table. Note that some distribution centers do not serve certain customer zones because the costs would be prohibitive.
 Open table as spreadsheet
 	
Customer zone
DC
Dal
SA
Wich
KC
Den
SLC
Pho
LA
SD
FW
0.30
2.10
3.10
4.40
6.00
-
-
-
-
SF
5.20
5.40
4.50
6.00
2.70
4.70
3.40
3.30
2.70
LV
-
-
-
-
5.40
3.30
2.40
2.10
2.50
In the current distribution system, demand at the Dallas, San Antonio, Wichita, and Kansas City customer zones is satisfied by shipments from the Ft Worth distribution center. In a similar manner, the Denver, Salt Lake City, and Phoenix customer zones are served by the Santa Fe distribution center, and the Los Angeles and San Diego customer zones are served by the Las Vegas distribution center. The El Paso plant supplies the Ft. Worth and Santa Fe distribution centers, while the Sacramento plant supplies the Las Vegas distribution center.
You have been called in to make recommendations for improving the distribution system, and, in particular, to address the following issues.
If the company does not change its current distribution strategy, what will the distribution system cost be for the following quarter?
Suppose that the company is willing to consider dropping the distribution center limitations. In other words, customer zones would not necessarily be assigned to unique distribution centers, and distribution centers would not necessarily be assigned to unique plants. With this added flexibility, by how much could costs be reduced?
In the foreseeable future, the company anticipates moderate growth of about 20 percent in demand. Suppose this growth is met using the current routes and expanding plant capacity as needed. What plant capacities would be required? What would be the total system cost?
Relative to the cost in part (c), how much could both distribution flexibility and plant capacity save in annual expenses? What plant capacities would be required?
 
3.7  
Oil Distribution Texxon Oil Distributors, Inc., has three active oil wells in a west Texas oil field. Well 1 has a capacity of 93 thousand barrels per day (TBD), Well 2 can produce 88 TBD, and Well 3 can produce 95 TBD. The company has five refineries along the Gulf Coast, all of which have been operating at stable demand levels. In addition, three pump stations have been built to move the oil along the pipelines from the wells to the refineries. Oil can flow from any one of the wells to any of the pump stations, and from any one of the pump stations to any of the refineries, and Texxon is looking for a minimum cost schedule. The refineries' requirements are as follows.
Refinery
R1
R2
R3
R4
R5
Requirement (TBD)
30
57
48
91
48
 Open table as spreadsheet
The company's cost accounting system recognizes charges by the segment of pipeline that is used. These daily costs are given in the tables below, in dollars per thousand barrels.
 Open table as spreadsheet
To
 	
Pump A
Pump B
Pump C
 	
Well 1
1.52
1.60
1.40
From
Well 2
1.70
1.63
1.55
 	
Well 3
1.45
1.57
1.30
 Open table as spreadsheet
To
 	
R1
R2
R3
R4
R5
 	
Pump A
5.15
5.69
6.13
5.63
5.80
From
Pump B
5.12
5.47
6.05
6.12
5.71
 	
Pump C
5.32
6.16
6.25
6.17
5.87
What is the minimum cost of providing oil to the refineries? Which wells are used to capacity in the optimal schedule?
Show the network diagram corresponding to the solution in (a). That is, label each of the arcs in the solution and verify that the flows are consistent with the given information.

6.5  
Catering Logistics Jessica's Catering Service bakes and delivers lasagnas to parties and group meetings. In a typical week, Jessica has around 40 orders. Each order involves a specific amount of the required lasagna in pounds. A small group would need about 20 lb, whereas a large group would need almost triple that amount. The following table shows the different customer orders that have come in this week, grouped into eight categories by weight.
 Open table as spreadsheet
Job type
1
2
3
4
5
6
7
8
Weight (lbs)
20
25
30
35
40
45
50
55
Number
10
5
4
7
3
9
2
1
Jessica has an inventory of 25 six-lb trays and 18 ten-lb trays. Although she appears to have enough capacity in her trays, she would like to plan her orders so that the amount of excess lasagna is kept to a minimum.
How many six-lb trays and ten-lb trays should Jessica use for each of the orders?

	
Reservation Scheduling Roth Auto Rentals, a car rental company specializing is SUVs, is making up a schedule for the next weekend's demands. The peak demand period occurs on the weekend, when Roth may not have enough SUVs to meet demand. The customer demands that have been logged in are listed below.
 Open table as spreadsheet
Days
Customers
Fri–Mon
1
Fri–Sat
4
Fri–Sun
5
Sat–Sun
4
Sat–Mon
3
Sun–Sun
2
The rental cost depends on which days the contract covers.
Days
FSSM
FS
FSS
SS
SSM
Sun
Rate
119.95
69.95
99.95
74.95
89.95
39.95
 Open table as spreadsheet
Roth Auto Rentals carries only one type of vehicle and expects to have 10 SUVs available for rental over the weekend.
What is the maximum revenue that can be generated from the list of orders?
In the optimal solution of (a), what percentage of customer demand is satisfied?
In the optimal solution of (a), what percentage of dollar demand is satisfied?
Answer the set of three questions above for fleet sizes of 11–16.

6.8  
Scheduling Reservations Reed's Rent-a-Car is a traditional auto rental company facing the problem of assigning vehicles to weekend demands. However, Reed's distinguishes rentals by car type. Its fleet consists of three compact (C) cars, five mid-size (M) cars and three full-size (F) cars. The customer demands that have been logged in are listed below.
 Open table as spreadsheet
Days
C
M
F
Fri–Mon
0
1
0
Fri–Sat
1
2
1
Fri–Sun
2
2
1
Sat–Sun
1
3
0
Sat–Mon
3
0
0
Sun–Sun
0
1
1
The rental rates depend on how many days the contract covers. Prices for compact cars are shown below. Mid-size cars carry a 10 percent premium, and full-size cars carry a 20 percent premium.
Days
1
2
3
4
Rate
39.95
74.95
99.95
119.95
 Open table as spreadsheet
Assume Reed's were to prohibit a customer who ordered one size from renting another size. What is the maximum revenue that can be generated from the list of orders?
Assume Reed's were to permit a customer to substitute a larger size for any order, but with no change in price. What is the maximum revenue that can be generated from the list of orders?
In the optimal solution of (b), what percentage of dollar demand is satisfied?

8.6  
Pricing with Dependent Demands Covington Motors is a car dealership that specializes in the sales of sport utility vehicles and station wagons. Due to its reputation for quality and service, Covington has a strong position in the regional market, but demand is somewhat sensitive to price. After examining the new models, Covington's marketing consultant has come up with the following demand curves.

The dealership's unit costs are $17,000 for SUVs and $14,000 for wagons. Each SUV requires 2 hours of prep labor, and each wagon requires 3 hours of prep labor. The current staff can supply 320 hours of labor.
Determine the profit-maximizing prices for SUVs and Wagons. (Round off any fractional demands.)
What demand levels will result from the prices in (a)?
What is the marginal value of dealer prep labor?
 
8.7  
Pricing with Interdependent Demands Covington Motors sells sport utility vehicles and station wagons in a price-sensitive market. Its marketing consultant has rethought the simple demand curves first proposed (in the previous exercise) and now wants to recognize the interaction of the two markets. This gives rise to a revised pair of demand curves for SUVs and wagons, as shown below.

The dealership's unit costs are $17,000 and $14,000 per unit, respectively. Each SUV requires 2 hours of prep labor, and each Wagon requires 3 hours of prep labor. The current staff can supply 320 hours of labor. Covington Motors wants to maximize its profits from the SUVs and Wagons that it acquires for its stock.
Determine the profit-maximizing prices for SUVs and Wagons. (Ignore the fact that these prices may induce fractional demands.)
What sales levels will result from the prices in (a)?
What is the marginal value of dealer prep labor?