ModelINPUTS# LoadsInterimPer-Mile Rates for CarriersDestinationStateRequiredStopsMilesABCTIRSTLASTMRSTNESTPSSTAtlantaGA40612$999$0.88$1.15$0.87$0.95$1.05EverettMA13612$999$1.18$1.27$1.39$1.35$1.28EphrataPA30190$999$3.42$1.73$1.71$1.82$2.00RiverviewMI50383$0.79$1.01$1.25$0.96$0.95$1.11CarsonCA123063$999$0.80$0.87$999$1.00$999ChambleeGA10429$999$1.23$1.61$1.22$1.33$1.47RosevilleMN13600$1.24$1.13$1.89$1.32$1.41$1.41HanoverPA10136$999$4.78$2.23$2.39$2.26$2.57SparksNV202439$999$1.45$999$1.20$999$999ParsippanyNJ11355$999$1.62$1.36$1.39$1.03$1.76EffinghamIL50570$0.87$0.87$1.25$0.87$0.90$1.31KearnyNJ70324$999$2.01$1.54$1.53$1.28$1.95Minimum Charge per Truckload$350$400$350$300$350$300Interim Stop-off Charge$50$75$50$35$50$50Available Pulls (Trucks)487734Minimum Load Commitment176004WORK AREACostsPer-Load Transportation Costs for CarriersDestinationABCTIRSTLASTMRSTNESTPSSTAtlanta$611,388.00$538.56$703.80$532.44$581.40$642.60Everett$611,538.00$947.16$927.24$955.68$976.20$933.36Ephrata$189,810.00$649.80$350.00$324.90$350.00$380.00Riverview$350.00$400.00$478.75$367.68$363.85$425.13Carson$3,060,037.00$2,600.40$2,764.81$3,060,007.00$3,163.00$3,060,037.00Chamblee$428,571.00$527.67$690.69$523.38$570.57$630.63Roseville$894.00$903.00$1,284.00$897.00$996.00$996.00Hanover$135,864.00$650.08$350.00$325.04$350.00$349.52Sparks$2,436,561.00$3,536.55$2,436,561.00$2,926.80$2,436,561.00$2,436,561.00Parsippany$354,695.00$650.10$532.80$528.45$415.65$674.80Effingham$495.90$495.90$712.50$495.90$513.00$746.70Kearny$323,676.00$651.24$498.96$495.72$414.72$631.80Decision VariablesCarrierDestinationABCTIRSTLASTMRSTNESTPSSTAtlanta000400Everett000001Ephrata001002Riverview400100Carson010000Chamblee010000Roseville010000Hanover000001Sparks000200Parsippany000010Effingham050000Kearny005020# Loads# LoadsCarrierAssignedDestinationDeliveredABCT4Atlanta4IRST8Everett1LAST6Ephrata3MRST7Riverview5NEST3Carson1PSST4Chamblee1Roseville1Hanover1Sparks2Parsippany1Effingham5Kearny7Node ConstraintsCarrierInflow=OutflowDestinationInflow=OutflowABCT=Atlanta=IRST=Everett=LAST=Ephrata=MRST=Riverview=NEST=Carson=PSST=Chamblee=Roseville=Hanover=Sparks=Parsippany=Effingham=Kearny=Arc Constraints# LoadsAvailable# Loads# LoadsCarrierCommitment<Assigned<PullsDestinationDelivered=RequiredABCT≤≤Atlanta=IRST≤≤
: =H19
: =F59Everett
: =F60
: =D5=
: =D6LAST≤≤Ephrata=MRST≤≤Riverview=NEST≤≤Carson=PSST≤≤Chamblee=Roseville=Hanover=Sparks=Parsippany=Effingham=Kearny=Objective FunctionMinimize Costs
: =SUMPRODUCT(C27:H38,C44:H55)
: =C59
: =C60
: =SUM(C44:C55)
: =SUM(D44:D55)
: =SUM(C44:H44)
: =SUM(C45:H45)
: =F59
: =F60
: =G20
: =H20
: =C59
: =C60
: =G19
Your_AnswersQuestionsGradingQuestion 1a) Suppose LAST were to charge $0.80 per mile rather than $1.15 for trips to Atlanta. In the new optimal assignment, which loads would be assigned to LAST? (Provide the names of cities (two cities) and the number of loads for each city.)QuestionScoreAnswer: Name of the first city (in no particular order)Model vali ...
3. : =SUM(C45:H45)
: =F59
: =F60
: =G20
: =H20
: =C59
: =C60
: =G19
Your_AnswersQuestionsGradingQuestion 1a) Suppose LAST
were to charge $0.80 per mile rather than $1.15 for trips to
Atlanta. In the new optimal assignment, which loads would be
assigned to LAST? (Provide the names of cities (two cities) and
the number of loads for each city.)QuestionScoreAnswer: Name
of the first city (in no particular order)Model
validation12Question 1-aCity 1: Name0.25Answer: Number of
loads for the first city.City 1: Loads0.25City 2:
Name0.25Answer: Name of the second city.City 2:
Loads0.25Total cost0.5Answer: Number of loads for the second
city.Carrier0.5Question 1-bYes/No0.5Explanation0.5What
would be the total cost?Question 1-cAdditional cost for
Mohawk0.5Answer: Round your answer to two decimal
places.LAST's revenue0.5Question 23Question 3Shadow
price2Which carrier would not have all of its trucks
assigned?Cost decrease2Answer:Question 43Total26.00b)
Suppose ABCT doubled its interim stop-off charge. Would this
change its optimal load assignment?Answer: Yes or NoWhy or
why not?Answer: Explain here.c) Suppose LAST could
negotiate its contract so that its minimum commitment is 7
4. loads. How much more would Mohawk pay in total
transportation costs under this contract?Answer: Round your
answer to two decimal places.How much revenue would LAST
receive under this contract?Answer: Round your answer to two
decimal places.Question 2Suppose IRST could negotiate its
contract so that it was guaranteed all the loads destined for
Ephrata. How much more would Mohawk pay in total
transportation costs under this contract?
Hint
Change the per-mile rates to Ephrata for the other carriers to
$999.Answer: Round your answer to two decimal
places.Question 3Cells E90:G90 contains the constraint which
guarantees that the number of loads assigned to ABCT is no
more than its number of available trucks. What is the shadow
price for this constraint?Answer: Round your answer to two
decimal places.How much would Mohawk’s total transportation
costs decrease if ABCT had another available truck?Answer:
Round your answer to two decimal places.Question 4In the
inputs section of the model shown in Figure 3, the value 999
was entered as the per-mile transportation cost for any carrier
that did not service a particular region. Why can’t we simply
leave these cells blank?Answer: Explain here.
GradingQuestionsPoint allocatedYour answerCorrect
answerScoreNoteFor Model ValidationOriginalModel
validation12$0.00$21,893.54020612$999$0.88$1.15$0.87$0.95
$1.0540612$999$0.88$1.15$0.87$0.95$1.05Question 1-aCity 1:
Name0.250Atlanta0Only one set of answers is
graded.23612$999$1.18$1.27$1.39$1.35$1.2813612$999$1.18$
1.27$1.39$1.35$1.28City 1:
Loads0.2504020190$999$3.42$1.73$1.71$1.82$2.0030190$999
$3.42$1.73$1.71$1.82$2.00City 2:
Name0.250Kearny020383$0.79$1.01$1.25$0.96$0.95$1.115038
3$0.79$1.01$1.25$0.96$0.95$1.11City 2:
Loads0.25030223063$999$0.80$0.87$999$1.00$999123063$99
9$0.80$0.87$999$1.00$999City 2:
Name0.250Kearny020429$999$1.23$1.61$1.22$1.33$1.4710429
5. $999$1.23$1.61$1.22$1.33$1.47City 2:
Loads0.2503023600$1.24$1.13$1.89$1.32$1.41$1.4113600$1.2
4$1.13$1.89$1.32$1.41$1.41City 1:
Name0.250Atlanta020136$999$4.78$2.23$2.39$2.26$2.5710136
$999$4.78$2.23$2.39$2.26$2.57City 1:
Loads0.25040202439$999$1.45$999$1.20$999$999202439$999
$1.45$999$1.20$999$999Total
cost0.5$0.00$22,185.44021355$999$1.62$1.36$1.39$1.03$1.76
11355$999$1.62$1.36$1.39$1.03$1.76Carrier0.50IRST020570$
0.87$0.87$1.25$0.87$0.90$1.3150570$0.87$0.87$1.25$0.87$0.
90$1.31Question 1-
bYes/No0.50No020324$999$2.01$1.54$1.53$1.28$1.9570324$9
99$2.01$1.54$1.53$1.28$1.95Explanation0.50Because in the
optimal assignment, ABCT receives 4 loads for Riverview,
which does not require any interim stops.Minimum Charge per
Truckload$350$400$350$300$350$300Minimum Charge per
Truckload$350$400$350$300$350$300Question 1-cAdditional
cost for Mohawk0.5$0.00$21.450Interim Stop-off
Charge$50$75$50$35$50$50Interim Stop-off
Charge$50$75$50$35$50$50LAST's
revenue0.5$0.00$3,194.800Available Pulls
(Trucks)487734Available Pulls (Trucks)487734Question
23$0.00$1,027.700Minimum Load
Commitment176004Minimum Load
Commitment176004Question 3Shadow price2$0.00-$23.680Cost
decrease2$0.00$23.680Question 430Excel interprets blank cells
as zero, so there would be no mileage charge for carriers on
routes which they don’t service – this would make these carriers
very attractive candidates for those routes. The 999 value is
supposed to make them very unattractive
candidates.Total26.00Your Score0.000.00%
Based on a case study from Practical Management Science:
SpreadsheetModeling and Applications, Winston and Albright,
Duxbury Press, 1997, p. 208.
6. MOTOR CARRIER SELECTION AT MOHAWK PAPER MILLS
Mohawk Paper Mills Inc., located in Cohoes, New York,
produces up to 200 tons of paper per day to fill orders from 250
customers. The mill’s products include: fine papers (used in
printing applications such as magazines and annual reports),
bleached paperboard (used in packaging such as milk cartons),
and kraft paper (used for corrugated boxes and decorative
laminates such as Formica). The mill is served by six carriers,
and twenty shipments per day are not unusual. The carriers will
generally accept shipments to any destination that they serve,
subject to daily volume commitments and equipment
availability. Each carrier has a different and somewhat complex
rate structure.
Rob Wood is the mill’s load planner. Each morning, Rob
reviews the previous day’s production to divide large orders and
consolidate less-than-truckload (LTL) orders into truckload
quantities. Orders typically weigh between 1000 and 15,000
pounds, and a truck can generally pull a trailer with 46,000 to
48,000 pounds of paper. Rob plans the consolidated loads so
that each truck makes no more than four stops.
After loads are planned, they are turned over to Ed Wong, the
transportation planner. Ed’s job is to assign carriers to loads so
as to minimize the total shipping cost. He has a contract for
each carrier that gives the rates to each destination served. The
rates include a mileage charge, a stop-off charge, and a
minimum charge per truckload.
Tables 1 – 3 show a typical problem that Ed must solve. In this
day’s scenario, there are 32 loads that must be sent on 12
different routes. Some of the routes contain interim stops
before the final destination. When multiple loads must be sent
to the same destination, the load assignment may be split among
different carriers. The destinations and load requirements are
listed in Table 1.
7. Table 1: Routes and Loads
Destination
State
# Loads Required
Interim Stops
Miles
Atlanta
GA
4
0
612
Everett
MA
1
3
612
Ephrata
PA
3
0
190
Riverview
MI
5
0
383
Carson
CA
1
2
3063
Chamblee
GA
1
0
9. loads must be assigned to IRST. Minimum charges per
truckload, interim stop-off charges, numbers of available trucks,
and the minimum load commitments for each carrier are
displayed in Table 2.
Table 2: Carrier Information
Carrier
ABCT
IRST
LAST
MRST
NEST
Minimum Charge per Truckload
$350
$400
$350
$300
$350
Interim Stop-off Charge
$50
$75
$50
$35
$50
Available Pulls (Trucks)
4
8
7
10. 7
3
Minimum Load Commitment
1
7
6
0
0
The per-mile transportation charges to each destination for each
of the six carriers are displayed in Table 3. Not every carrier
services every destination.
Table 3: Per-Mile Carrier Rates
Per-Mile Rates for Carriers
Destination
State
ABCT
IRST
LAST
MRST
NEST
Atlanta
GA
**
$0.88
$1.15
$0.87
$0.95
13. **
$2.01
$1.54
$1.53
$1.28
The mileages in Table 1 represent the total number of miles for
the trip from the Cohoesmill to the final destination, including
any intermediate stops. The total charge is calculated as
follows. Suppose that the Roseville, MN trip is assigned to
carrier IRST. The transportation cost would be
600(1.13)+3(75)=$903. Stop-off charges only apply to
intermediate stops and not the final destination. If the cost
calculated this way had been less than IRST’s minimum
truckload charge of $400, the cost would have been $400.
Solution
Ed can formulate his assignment task as a network flow
problem, which can then be translated to a linear program. The
graph for this problem is shown in Figure 1. The six nodes on
the left side of the graph represent the carriers; the twelve
nodes on the right side of the graph represent the destinations.
The flow on an arc entering a carrier node represents the
number of loads assigned to that carrier. The minimum flow on
this arc is the existing commitment to the carrier, and the
maximum flow is the number of available drivers from that
14. carrier.
The flow on an arc connecting a carrier node to a destination
node represents the number of loads for that destination
assigned to that carrier. The minimum flow on this arc is zero,
and there is no limit on the maximum flow. (We could limit the
maximum flow by the number of available drivers, but the flow-
balance constraint for the carrier node makes this redundant.)
The per-unit flow cost on the arc is the cost of sending a load to
the destination using the carrier, computed from the information
in Tables 1-3.
The flow on an arc leaving a destination node represents the
number of loads transported to that destination. The minimum
and maximum flow on this arc is the number of loads that must
be sent to the destination. (We assume every load must be
delivered.)
Figure 1: Mohawk’s Carrier Assignment Problem
To translate the graph shown in Figure 1 to a linear program, we
first assign a decision variable to every arc. The decision
variables are shown in Figure 2.
The objective function is to minimize the sum of the decision
variables shown in Figure 2 multiplied by the corresponding
15. per-unit flow costs shown in Figure 1. There are two types of
constraints. First, each node in the graph has a flow balance
constraint, which guarantees that inflow equals outflow. (The
inflow is the sum of the decision variables corresponding to
arcs entering the node, and the outflow is the sum of the
decision variables corresponding to arcs leaving the node.) For
example, the balance constraint for the 3.LAST carrier node is:
X3 = X3A + X3B + … + X3L.
The balance constraint for the D.Riverview destination node is:
X1D + X2D + … + X6D = XD.
Next, each arc in the graph has a pair of flow constraints
determined by its upper and lower bounds. For example, the
constraints for the arc entering the 5.NEST carrier node are:
0 ≤ X5 ≤ 3.
Figure 2: Decision Variables
Developing the Mohawk Carrier Assignment Model
Open the file titled “MohawkShell,” which is an incomplete
version of the model shown in Figures3– 6.The cells shaded
blue in Figure 3 are user inputs.Enter these numbers directly; no
equations are needed. Notice that the value 999 has been
entered as the per-mile transportation cost for any carrier that
does not service a particular destination.
16. Figure 3: Inputs Section of the Excel Model
Figure 4 shows each carrier’s total transportation costs for
thedestinations it services. The total costs include the per-mile
charges, stop-off costs, and minimum charges as described
earlier. (Validation hint: Notice that LAST’s costs for Ephrata
and Hanover equal its minimum charge per truckload.)
Figure 4: Calculations of Per-Load Transportation Costs for
each Carrier-Route Combination
Figure 5 displays the portion of the worksheet containing the
decision variables. (The decision variablecells are shaded
yellow, and the values shown in the figure are the optimal
solution.) As shown in Figure 2, there is one variable for each
arc in the network. The decision variables corresponding to the
arcs entering the carrier nodes (X1,…,X6) are contained in cells
C59:C64. The decision variables corresponding to the arcs
connecting the carrier nodes to the destination nodes (Xij,
decision variables corresponding to the arcs leaving the
destination nodes (XA,…,XL) are contained in cells F59:F70.
Figure 6 shows the portion of the worksheet with the constraints
and objective function. The flow-balance constraints for the
carrier nodes are contained in cells B74:E79; those for the
destination nodes are in cells G74:J85. The arc constraints for
17. the arcs entering the carrier nodes are contained in cells
B90:G95; those for the arcs leaving the destination nodes are in
cells I90:L101. The constraints for the arcs connecting the
carrier and destination nodes are handled by the non-negativity
constraints in the Solver Options window.
Figure 5: Decision Variables
Figure 6: Constraints and Objective Function
Finish the formatting for the shell as shown in Figures 3– 6.
Enter the problem data as shown in the figures. Add the
objective function and set up the constraints. Set up the Solver
Parameters window as shown in Figure 7.Remember to check
“Make Unconstrained Variables Non-Negative” and use
“Simplex LP” in Solver.
Figure 7: Solver ParametersWindow
Validating the Mohawk Carrier Selection Model
Change the number of loads required for Ephrata and Riverview
to 1 each, and run Solver. Now the optimal carrier assignments
should be as shown in Figure 8, with total cost $20,390.95.After
validation, restore the number of loads required for Ephrata and
18. Riverview to the original values.
Figure 8: Validation of the Model
Using the Mohawk Carrier Assignment Model
1. Restore the input parameters to their original values after
each step.
1. Suppose LAST were to charge $0.80 per mile rather than
$1.15 for trips to Atlanta. In the new optimal assignment,
which loads would be assigned to LAST? What would be the
total cost? Which carrier would not have all of its trucks
assigned?
b) SupposeABCT doubled its interim stop-off charge. Would
this change its optimal load assignment? Why or why not?
c) Suppose LAST could negotiate its contract so that its
minimum commitment is 7 loads. How much more would
Mohawk pay in total transportation costs under this contract?
How much revenue would LAST receive under this contract?
1. Suppose IRST could negotiate its contract so that it was
guaranteed all the loads destined for Ephrata. How much more
would Mohawk pay in total transportation costs under this
contract?
2. Cells E90:G90 contains the constraint which guarantees that
the number of loads assigned to ABCT is no more than its
number of available trucks. What is the shadow price for this
19. constraint? How much would Mohawk’s total transportation
costs decrease if ABCT had another available truck?
3. In the inputs section of the model shown in Figure 3, the
value 999 was entered as the per-mile transportation cost for
any carrier that did not service a particular region. Why can’t
we simply leave these cells blank?
Page 1 of 7
6
Page 7 of 7
QuizENTER YOUR NAMESECTIONACADEMIC
INTEGRITYCERTIFYMaximum points14I have neither
received nor given any assistance on this quiz from
anyone.Your score0GradeFollow the instructions in red font.
Complete melon-shaded cells and answer questions 1 ~
3.OriginDestinationDistanceAvg. SpeedDelayTotal TimeDec.
Var.Dec. Var.City
CodesCodeCode(miles)(mph)(hours)(hours)Label(0/1)CodeCity
12122510.002.39X120.11Amarillo13359590.006.08X130.12Lub
bock21122510.002.39X210.13Ft.
Worth24345510.006.76X240.14El
Paso25167510.003.27X250.15Abilene31359590.006.08X310.16
21. (City 1) to San Antonio (City 7)?How long does it take to
drive? (enter a number with two decimal places)Question 3.
Suppose the driver must stop in El Paso (City 4) along the
way,What is the fastest route from Amarillo (City 1) to San
Antonio (City 7)?How long does it take to drive? (enter a
number with two decimal places)
