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CHAPTER 14: Optimization Modeling: Applications
MULTIPLE CHOICE
- Which of the following does not represent a broad class of applications of linear programming models? a. Blending models b. Financial portfolio models c. Logistics models d. Set covering models e. Forecasting models
ANS: E PTS: 1 MSC: AACSB: Analytic
- Many organizations must determine how to schedule employees to provide adequate service. If we assume that an organization faces the same situation each week, this is referred to as a. static scheduling problem b. dynamic scheduling problem c. transportation scheduling problem d. All of these options
ANS: A PTS: 1 MSC: AACSB: Analytic
- Workforce scheduling problems are often integer programming models, which means that they have: a. an integer objective function b. integer decision variables c. integer constraints d. all of these options
ANS: C PTS: 1 MSC: AACSB: Analytic
- A common characteristic of integer programming models is that they: a. are easy to solve graphically b. produce the same answer and standard linear programming models c. often produce multiple optimal solutions d. all of these options
ANS: C PTS: 1 MSC: AACSB: Analytic
- Which of the following is true regarding multiple optimal solutions? a. All solutions have the same values for the decision variables b. All solutions have the same value for the objective function c. All solutions have the same shadow prices d. All of these options
ANS: B PTS: 1 MSC: AACSB: Analytic
- Many organizations must determine how to schedule employees to provide adequate service. If we assume that an organization faces the same situation each week, this is referred to as a. static scheduling problem b. dynamic scheduling problem c. transportation scheduling problem d. All of these options
ANS: A PTS: 1 MSC: AACSB: Analytic
- Rounding the solution of a linear programming to the nearest integer values provides a(n) a. integer solution that is optimal b. integer solution that may be neither feasible nor optimal c. feasible solution that is not necessarily optimal d. infeasible solution
ANS: B PTS: 1 MSC: AACSB: Analytic
- Which of the following statements are false? a. Solver does not offer a sensitivity report for models with integer constraints b. Solver’s sensitivity report is not suited for questions about multiple input changes c. Solver’s sensitivity report is used primarily for questions about one-at-a time changes to input d. None of these options
ANS: D PTS: 1 MSC: AACSB: Analytic
- If refers to the number of hours employee works in week , then to indicate that the number of working hours of 4 employees in week 3 should not exceed 160 hours, we must have a constraint of the form a. b. c. d.
ANS: B PTS: 1 MSC: AACSB: Analytic
- Which of the following statements is a type of constraint that is often required in blending problems? a. Integer constraint b. Binary constraint c. Quality constraint d. None of these options
ANS: C PTS: 1 MSC: AACSB: Analytic
- The constraints in a blending problem can be specified in a valid way and still lead to which of the following problems? a. Unboundedness b. Infeasibility c. Nonlinearity d. None of these options
ANS: C PTS: 1 MSC: AACSB: Analytic
- To specify that must be at most 75% of the blend of , , and , we must have a constraint
of the form a. b. c.
a. warehouses b. geographic locations c. flows d. capacities
ANS: C PTS: 1 MSC: AACSB: Analytic
- In formulating a transportation problem as linear programming model, which of the following statements are correct? a. There is one constraint for each supply location b. There is one constraint for each demand location c. The sum of decision variables out of a supply location is constrained by the supply at that location d. The sum of decision variables out of all supply locations to a specific demand location is constrained by the demand at that location e. All of these options
ANS: E PTS: 1 MSC: AACSB: Analytic
- In a transshipment problem, shipments a. can occur between any two nodes (suppliers, demanders, and transshipment locations) b. cannot occur between two supply locations c. cannot occur between two demand locations d. cannot occur between a transshipment location and a demand location e. cannot occur between a supply location and a demand location
ANS: A PTS: 1 MSC: AACSB: Analytic
- Transportation and transshipment problems are both considered special cases of a class of linear programming problems called a. minimum cost problems b. minimum cost network flow problems c. supply locations network problems d. demand locations network problems
ANS: B PTS: 1 MSC: AACSB: Analytic
- A minimum cost network flow model (MCNFM) has the following advantage relative to the special case of a simple transportation model: a. a MCNFM does not require capacity restrictions on the arcs of the network b. the flows in a general MCNFM don’t all necessarily have to be from supply locations to demand locations c. a MCNFM is generally easier to formulate and solve d. All of these options
ANS: B PTS: 1 MSC: AACSB: Analytic
- In a typical minimum cost network flow model, the nodes indicate a. roads b. rail lines c. geographic locations d. rivers
ANS: C PTS: 1 MSC: AACSB: Analytic
- The flow balance constraint for each transshipment node, in a minimum cost network flow model, takes the form a. (^) Flow in Flow out + Net supply b. Flow out Flow in + Net supply c. Flow in = Flow out d. Flow out Flow in + Net supply e. Flow in Flow out + Net demand
ANS: C PTS: 1 MSC: AACSB: Analytic
- In a minimum cost network flow model, the flow balance constraint for each supply node takes the form a. Flow in Flow out + Net supply b. Flow out Flow in + Net demand c. Flow in = Flow out d. Flow out Flow in + Net supply e. (^) Flow in Flow out + Net demand
ANS: D PTS: 1 MSC: AACSB: Analytic
- In a minimum cost network flow model, the flow balance constraint for each demand node takes the form a. Flow out Flow in + Net supply b. Flow in Flow out + Net demand c. Flow in = Flow out d. (^) Flow in Flow out + Net demand e. (^) Flow out Flow in + Net demand
ANS: B PTS: 1 MSC: AACSB: Analytic
- In aggregate planning models, which of the following statements are correct? a. The number of workers available influences the possible production levels b. We allow the workforce level to be modified each month through the hiring and firing of workers c. We eventually allow demand to be backlogged; that is, demand need not be met on time d. All of these options
ANS: D PTS: 1 MSC: AACSB: Analytic
- Any integer program involving 0 – 1 variables with constraint(s) is called a knapsack problem. a. three b. two c. one d. zero
ANS: C PTS: 1 MSC: AACSB: Analytic
- The binary variables in the fixed cost models correspond to: a. the number of units or products produced b. the total profit c. the amount of labor hours d. a process for which a fixed cost occurs
ANS: D PTS: 1 MSC: AACSB: Analytic
- In nonlinear models, which of the following statements are correct?
ANS: F PTS: 1 MSC: AACSB: Analytic
- In blending problems, if a quality constraint involves a quotient, then the problem will be nonlinear.
ANS: T PTS: 1 MSC: AACSB: Analytic
- Logistics problems are problems of finding the least expensive way to transport products from their origin to their destination.
ANS: T PTS: 1 MSC: AACSB: Analytic
- If all the supplies and demands for a transportation model are integers, then the optimal Solver solution may or may not have integer-valued shipments.
ANS: F PTS: 1 MSC: AACSB: Analytic
- In transportation problems, shipping costs are often nonlinear due to quantity discounts.
ANS: T PTS: 1 MSC: AACSB: Analytic
- In a transportation problem, if it costs $4 per item to ship up to 200 items between cities, and $2 per item for each additional item, the proportionality assumption of LP is satisfied.
ANS: F PTS: 1 MSC: AACSB: Analytic
- The transportation model is a special case of the minimum cost network flow model (MCNFM).
ANS: T PTS: 1 MSC: AACSB: Analytic
- In transportation problems, shipments between supply points or between demand points are possible.
ANS: F PTS: 1 MSC: AACSB: Analytic
- In transportation problems, the three sets of input numbers that are required are capacities, demands and flows.
ANS: F PTS: 1 MSC: AACSB: Analytic
- In network models of transportation problems, arcs represent the routes for getting a product from one node to another.
ANS: T PTS: 1 MSC: AACSB: Analytic
- A good shipping plan uses as many cheap routes as possible, but ultimately is constrained by capacities and demands.
ANS: F PTS: 1 MSC: AACSB: Analytic
- Transshipment points are locations where goods neither originate nor end up, but goods are allowed to enter such points to be shipped out to their eventual destinations.
ANS: T PTS: 1 MSC: AACSB: Analytic
- The flows in a general minimum cost network flow model (MCNFM) do all necessarily have to be from “left to right”; that is, from supply points to demand points.
ANS: F PTS: 1 MSC: AACSB: Analytic
- In an optimized network flow model (MCNFM), all the available capacity will be used.
ANS: F PTS: 1 MSC: AACSB: Analytic
- In aggregate planning models, the number of workers available influences the possible production levels.
ANS: T PTS: 1 MSC: AACSB: Analytic
- Aggregate planning models are usually implemented through a rolling planning horizon.
ANS: T PTS: 1 MSC: AACSB: Analytic
- In aggregate planning models, we can model backlogging of demand by allowing a month’s inventory to be negative.
ANS: T PTS: 1 MSC: AACSB: Analytic
- A 0-1 variable, also called a binary variable, is a variable that must equal 0 or 1.
ANS: F PTS: 1 MSC: AACSB: Analytic
- Any integer programming problem involving 0-1 variables with only one constraint is called a knapsack problem.
ANS: T PTS: 1 MSC: AACSB: Analytic
- In a set-covering model, each member of a given set (set 1) must be “covered” by an acceptable member of another set (set 2). The objective of such problems is to minimize the number of elements in set 2 that are needed to cover all the elements in set 1.
ANS: T PTS: 1 MSC: AACSB: Analytic
- A nonlinear programming problem (NLP) is an optimization problem in which the objective function and/or the constraints are not linear functions of the decision variables.
ANS: T PTS: 1 MSC: AACSB: Analytic
- When we solve a nonlinear programming problem (NLP), it is very possible that Solver will obtain the wrong answer.
ANS: T PTS: 1 MSC: AACSB: Analytic
- For some types of integer programming problems, their LP relaxation solutions are optimal.
ANS: T PTS: 1 MSC: AACSB: Analytic
- A local optimal solution is better than all nearby solutions, but a solution far away might be better than it.
- An auto company must meet (on time) the following demands for cars: 5000 in quarter 1; 3000 in quarter 2; 6000 in quarter 3; 2000 in quarter 4. At the beginning of quarter 1, there are 500 autos in stock. The company has the capacity to produce at most 3600 cars per quarter. At the beginning of each quarter, the company can change production capacity. It costs $125 to increase quarterly production capacity by one unit. It also costs $60 per quarter to maintain each unit of production capacity (even if it is unused during the current quarter). The variable cost of producing a car is $2400. A holding cost of $200 per car is assessed against each quarter’s ending inventory. It is required that at the end of quarter 4, plant capacity must be at least 5000 cars. Determine how to minimize the total cost incurred during the next four quarters.
ANS: Initial inventory 500 Initial capacity per quarter 3600
Cost to increase capacity by 1 unit $ Cost to maintain 1 unit of capacity $ Variable cost of producing a car $2, Holding cost per car in inventory $
Quarter 1 2 3 4 Increase in capacity 900 0 500 0 Production 4500 4000 5000 2000
Capacity 4500 4500 5000 5000
Required final capacity 5000
On hand after production 5000 4000 6000 2000
Demand 5000 3000 6000 2000 End inventory 0 1000 0 0
Production cost $37,200, Holding cost $200, Cost of increasing capacity $175, Cost of maintaining capacity $1,140, Total cost $38,715,
PTS: 1 MSC: AACSB: Analytic
- A company manufactures two products. If it charges price for product , it can sell units of
product , where and. It costs the company $20 to produce a unit of product 1 and $65 to produce a unit of product 2. How many units of each product should the company produce, and what prices should it charge, to maximize its profit?
ANS: Demand for each product is of form , where:
Product 1 Product 2
a 55 75 b -3 2 c 2 -
Unit cost $20 $
Prices $75.00 $112.
Production quantities 55.00 0
Demand 55.00 0
Total cost $1,100. Total revenue $4,125. Total profit $3,025.
PTS: 1 MSC: AACSB: Analytic
- Chemical Bank is attempting to determine where its assets should be invested during the current year. At present, $800,000 is available for investment in bonds, home loans, auto loans, and personal loans. The annual rate of return on each type of investment is known to be the following: bonds, 12%, home loans, 18%, auto loans, 15%, personal loans, 22%. To ensure that the banks portfolio is not too risky, the bank’s investment manager has placed the following restrictions on the bank portfolio: - No more than 30% of the total amount invested may be in personal loans - The amount invested in home loans cannot exceed the amount invested in auto loans - The amount invested in personal loans cannot exceed the amount invested in bonds.
Determine how the bank can maximize the annual return on its investment portfolio.
ANS: Annual rates Bonds Home loans Auto loans Personal loans 12% 18% 15% 24%
Maximum percent in personal loans 30%
Amounts invested Bonds Home loans Auto loans Personal loans $240,000 $160,000 $160,000 $240,
Annual return $139,
PTS: 1 MSC: AACSB: Analytic
PTS: 1 MSC: AACSB: Analytic
- A statistician is currently trying to maximize his profit in the bond market. Four bonds are available for purchase and sale at the bid and ask prices shown in the table below. The statistician can buy up to 1300 units of each bond at the ask price or sell up to 1300 units of each bond at the bid price. During each of the next three years the person who sells a bond will pay the owner of the bond the cash payments that are also shown in the table below. The statistician’s goal is to maximize his revenue from selling bonds less his payments for buying bonds, subject to the constraint that after each year’s payments are received, his current cash position (due only to cash payments from bonds and not purchases or sales of bonds) is nonnegative. His current cash position can depend on past coupons and that cash accumulated at the end of each year earns 12% annual interest. Determine how to maximize net profit from buying and selling bonds, subject to the constraints previously described. Bid (for selling) and ask (for buying) prices of bonds Bond 1 Bond 2 Bond 3 Bond 4 Bid $1,000 $99 $980 $ Ask $1,020 $1,015 $1,002 $ Cash payments from seller to buyer Bond 1 Bond 2 Bond 3 Bond 4 Year 1 $120 $100 $90 $ Year 2 $140 $130 $110 $ Year 3 $1,300 $1,320 $1,290 $1,
ANS:
Bond 1 Bond 2 Bond 3 Bond 4 Bid $1,000 $99 $980 $ Ask $1,020 $1,015 $1,002 $
Bond 1 Bond 2 Bond 3 Bond 4 Year 1 $120 $100 $90 $ Year 2 $140 $130 $110 $ Year 3 $1,300 $1,320 $1,290 $1,
Interest rate for cash 12.000%
Bond 1 Bond 2 Bond 3 Bond 4 Buys 1300 943 0 1300
Maximum 1300 1300 1300 1300
Sells 0 0 1300 1300
Maximum 1300 1300 1300 1300
Cash in Cash out Net cash
Accumulated cash with interest Year 0 (right after buy, sells) $2,522,000 $2,522,000 $0 $ Year 1 $341,266 $208,000 $133,266 $133,
Year 2 $421,546 $260,000 $161,546 $310, Year 3 $4,637,311 $3,380,000 $1,257,311 $1,605,
Cash at end of year 3 $1,605,
PTS: 1 MSC: AACSB: Analytic
- Assume that you are given the following means, standard deviations, and correlations for the annual return on three stocks.
Stock 1 Stock 2 Stock 3 Mean return 0.15 0.18 0. Stdev. of return 0.18 0.28 0.
Correlation matrix Stock 1 Stock 2 Stock 3 Stock 1 1.00 0.62 0. Stock 2 0.62 1.00 0. Stock 3 0.72 0.39 1.
The correlation between stocks 1 and 2 is 0.62, between stocks 1 and 3 is 0.72, and between stocks 2 and 3 is 0.39. You have $12,000 to invest and can invest no more than 55% of your money in any single stock. Determine the minimum variance portfolio that yields an expected annual return of at least 0.
ANS: Stock 1 Stock 2 Stock 3 Mean return 0.15 0.18 0. Stdev of return 0.18 0.28 0.
Correlation matrix Stock 1 Stock 2 Stock 3 Stock 1 1 0.62 0. Stock 2 0.62 1 0. Stock 3 0.72 0.39 1
Stocks Stock 1 Stock 2 Stock 3 Total Available Dollars $6,600 $4,824 $576 $12,000 = $12, Fraction 0.55 0.402028 0.
Maximum 0.55 0.55 0.
Actual Required 0.165899 0.
Standard deviations times fractions invested Stock 1 Stock 2 Stock 3 0.099 0.112568 0.
Portfolio variance 0.
If NY, then LA constraint NY LA 0 1
At most two open warehouses constraint
open Max
2 2
Either Tampa or LA constraint
open Min
1 1
Cost of opening warehouses $1, Cost of production and shipping $5,
Total cost $6,
PTS: 1 MSC: AACSB: Analytic
- A company is considering investing a total amount of $2.50 million in four bonds. The expected annual return, the worst-case annual return on each bond, and the “duration” of each bond are given in the table below.
Bond 1 Bond 2 Bond 3 Bond 4 Expected 16% 11% 14% 19% Worst case 7% 9% 11% 10% Duration 4 5 8 10
The duration of a bond is a measure of the bond’s sensitively to interest rates. The company wants to maximize the expected return from its bond investments, subject to the following constraints:
- The worst-case return of the bond portfolio must be at least 90%.
- The average duration of the portfolio must be at most 7
- Because of diversification requirements, at most 35% of the total amount invested in a single bond.
Determine how the company can maximize the expected return on its investment.
ANS: Returns from bonds
Bond 1 Bond 2 Bond 3 Bond 4 Expected 16% 11% 14% 19%
Worst case 7% 9% 11% 10% Duration 4 5 8 10
Minimum worst case return 9% Maximum average duration 7 Maximum percent in single bond 35%
Amounts invested Bond 1 Bond 2 Bond 3 Bond 4 Total Available $875,000 $250,000 $500,000 $875,000 $2,500,000 $2,500,
Max single $875,000 $875,000 $875,000 $875,
Worst case constraint Duration constraint Actual $226,250 Actual $17,500,
Required $225,000 Required $17,500,
Expected return $403,
PTS: 1 MSC: AACSB: Analytic
- An oil delivery truck contains five compartments, holding up to 2800, 2900, 1200, 1800, and 3200 gallons of fuel, respectively. The company must deliver three types of fuel (super, regular, and unleaded) to a customer. The demands, penalty per gallon short, and the maximum allowed shortage are shown in the table below. Each compartment of the truck can carry only one type of gasoline. Determine how to load the truck in a way that minimizes shortage costs.
Demand Cost (Gallon short) Maximum shortage allowed Super 3000 $9 400 Regular 4000 $7 400 Unleaded 5000 $5 400
ANS:
Data on customer demands
Cost/gallon short
Max shortage (in 100s) Super $9 4 Regular $7 4 Unleaded $5 4
Compartments used for various types of gasoline (1 if used, 0 if not) Compt 1 Compt 2 Compt 3 Compt 4 Compt 5 Super 0 1 0 0 0 Regular 1 0 1 0 0 Unleaded 0 0 0 1 1 Sum 1 1 1 1 1
Max 1 1 1 1 1
Amount withdrawn (in $1000s) $36 $0 $0 $0 $0 $0 $
Logical upper bound $36 $0 $0 $0 $0 $0 $
Cash requirement (in $1000s) $6 $7 $10 $3 $8 $3 $
Cash on hand at end of day $35 $28 $18 $15 $7 $4 $
Meet requirements on time 0 0 0 0 0 0 0
Summary of costs (in dollars) Withdrawal cost $8. Opportunity cost $24. Total cost $32.
PTS: 1 MSC: AACSB: Analytic
NARRBEGIN: SA_76_ A post office requires different numbers of full-time employees on different days of the week. The number of full-time employees required each day is given in the table below.
Mon Tue Wed Thu Fri Sat Sun 20 16 18 22 17 19 14
Union rules state that each full-time employee must work five consecutive days and then receive two days off. The post office wants to meet its daily requirements using only full-time employees. Its objective is to minimize the number of full-time employees that must be hired. NARREND
- (A) Use Solver to formulate and solve the post office’s problem.
(B) Suppose the post office has 30 full-time employees and is not allowed to hire or fire any employees. Determine a schedule that maximizes the number of weekend days off received by the employees.
ANS: (A) Number starting their five-day shift on various days Mon 6 Tue 7 Wed 0 Thu 9 Fri 0 Sat 5 Sun 0
Number working on various days (along top) who started their shift on various days (along side) Mon Tue Wed Thu Fri Sat Sun Mon 6 6 6 6 6
Tue 7 7 7 7 7 Wed 0 0 0 0 0 Thu 9 9 9 9 9 Fri 0 0 0 0 0 Sat 5 5 5 5 5 Sun 0 0 0 0 0 Totals Available^20 18 18 22 22 21
Min required 20 16 18 22 17 19 14
Total employees^27
(B)
Starting day of 5-day shift
Mon. Tue. Wed. Thu. Fri. Sat. Sun.
Total Total employed Number starting 11 5 2 12 0 0 0 30 = 30
Number working on each day 23 16 18 30 30 19 14
Minimal number required each day 20 16 18 22 17 19 14
Weekend days-off 22 5 0 0 0 0 0
Total weekend days-off 27
PTS: 1 MSC: AACSB: Analytic
NARRBEGIN: SA_78_ A company supplies goods to three customers, each of whom requires 50 units. The company has two warehouses. In warehouse 1, 75 units are available, and in warehouse 2, 55 units are available. The costs of shipping one unit from each warehouse to each customer are shown in the table below.
To From Customer 1 Customer 2 Customer 3 Warehouse 1 $20 $40 $ Warehouse 2 $15 $35 $ Not shipped (shortage) $100 $90 $
There is a penalty for each unsatisfied customer unit of demand – with customer 1, a penalty cost of $100 is incurred; with customer 2, $90; and with customer 3, $115. NARREND
- Determine how to minimize the sum of shortage and shipping costs.
ANS:
Unit shipping To
