9 introduction to operations research problem solving frederick s hillier

b The goal of this study was "to improve the throughput performance of existing andnew manufacturing systems through coordinated efforts in three areas: modeling andalgorithms, data coll

TABLE OF CONTENTS SOLUTIONS TO END-OF-CHAPTER PROBLEMS AND CASES

CHAPTER 23 Additional Special Types of Linear Programming Problems 23-1

in convincing investors and employees about the benefits of the changes Consequently,

"today FedEx is one of the nation's largest integrated, multi-conveyance freight carriers"[p 32]

CHAPTER 2: OVERVIEW OF THE OPERATIONS RESEARCH

MODELING APPROACH

2.1-1.

(a) The rise of electronic brokerage firms in the late 90s was a threat against full-servicefinancial service firms like Merrill Lynch Electronic trading offered very low costs,which were hard to compete with for full-service firms With banks, discount brokers andelectronic trading firms involved, the competition was fierce Merrill Lynch needed anurgent response to these changes in order to survive

(b) "The group's mission is to aid strategic decision making in complex businesssituations through quantitative modeling and analysis" [p.8]

(c) The data obtained for each client consisted of "data for six categories of revenue, fourcategories of account type, nine asset allocation categories, along with data on number oftrades, mutual fund exchanges and redemptions, sales of zero coupon bonds, andpurchases of new issues" [p 10]

(d) As a result of this study, two main pricing options, viz., an asset-based pricing optionand a direct online pricing option were offered to the clients The first targeted the clientswho want advice from a financial advisor The clients who would choose this optionwould be charged at a fixed rate of the value of their assets and would not pay for eachtrade The latter pricing option was for the clients who want to invest online and who donot want advice These self-directed investors would be charged for every trade

(e) "The benefits were significant and fell into four areas: seizing the marketplaceinitiative, finding the pricing sweet spot, improving financial performance, and adoptingthe approach in other strategic initiatives" [p.15]

2.1-2.

(a) This study arose from GM's efforts to survive the competition of the late 80s Variousfactors, including the rise of foreign imports, the increase in customer expectations andthe pricing constraints, forced GM to close plants and to incur large financial losses.While trying to copy Japanese production methods directly, GM was suffering from

"missing production targets, working unscheduled overtime, experiencing high scrapcosts, and executing throughput-improvement initiatives with disappointing results" [p.7] The real problems were not understood and the company was continuously losingmoney while the managers kept disagreeing about solutions

(b) The goal of this study was "to improve the throughput performance of existing andnew manufacturing systems through coordinated efforts in three areas: modeling andalgorithms, data collection, and throughput-improvement processes" [p 7]

(c) The data collection was automated by using programmable logic controllers (PLCs).The software kept track of the production events including "machine faults and blockingand starving events" [p 13] and recorded their duration The summary of this data wasthen transferred to a centralized database, which converted this to workstation-

2.1-3.

2.1-4.

The financial benefits that resulted from this study include savings of $40 million in 2001and of $5 million in 2002 The savings for any major disruption have been between $1and $5 million The new system enabled Continental Airlines to operate in an efficientand cost-effective manner in case of disruptions The time to recover and the costsassociated with disruptions are reduced What-if analysis allowed the company to

(a) Swift & Company operates in an industry that involves highly skilled labor, manyproduction pathways and perishable products To generate profit, the company needs tomake an efficient use of every single animal procured Before this study, Swift was notable to meet the shipping deadlines and as a result of this, it was forced to offerdiscounts The consequences of this practice included highly reduced profits, inaccurateforecasts and very low reliability The company had to find a way to come up with thebest product mix and to survive in this business defined by volatility and velocity

(b) The purpose of the scheduling models is "to fix the production schedule for the nextshift and to create a projection of short order" [p 74] They generate shift-level and dailyschedule for 28 days The capable-to-promise (CTP) models "determine whether a plantcan ship a requested order-line-item quantity on the requested date and time given theavailability of cattle and constraints on the plants' capacity during the 90-day modelhorizon" [p 75] The starting inventory, committed orders, and production schedulegenerated by the CTP models are inputs to the available-to-promise (ATP) models Every

15 minutes, the ATP models determine the unsold production of each shift and alert thesalespeople to undesirable inventory levels

(c) The company now uses 45 optimization models

(d) As a result of this study, the key performance measure, namely the weekly sold position has increased by 22% The company can now allocate resources to theproduction of required products rather than wasting them The inventory resulting fromthis approach is much lower than what it used to be before Since the resources are usedeffectively to satisfy the demand, the production is sold out The company does not need

percent-to offer discounts as often as before The cuspercent-tomers order earlier percent-to make sure that theycan get what they want by the time they want This in turn allows Swift to operate evenmore efficiently The temporary storage costs are reduced by 90% The customers arenow more satisfied with Swift With this study, Swift gained a considerable competitiveadvantage The monetary benefits in the first years was $12.74 million, including theincrease in the profit from optimizing the product mix, the decrease in the cost of lostsales, in the frequency of discount offers and in the number of lost customers The mainnonfinancial benefits are the increased reliability and a good reputation in the business

2.2-3.

2.2-4.

(b) The ultimate purpose of this study was "to improve competitiveness by improvingcustomer service, increasing sales and margins, and reducing obsolescence andinventories" [p 38] To achieve this, the project team aimed at designing a collaborative-planning (CP) process that would improve trust and collaboration between partners andaccelerate decision making.

(c) "The algorithm can generate feasible plans within seconds In fact, the calculation ofthe plan is hardly noticeable to the people participating in the weekly CP meeting Thespeed of the algorithm also allows planners to compute multiple plans during themeeting, creating an interactive planning environment The software environment alsoprovides strong problem-solving support, used extensively during the CP meetings Onesuch capability is called backward pegging It exploits the one-to-one relationshipbetween the storage of an end item in some future period and a constraining stock onhand or scheduled receipt of one or more upstream items Thus, the backward-peggingmechanism makes the actual material bottlenecks in the network visible" [p 41-42].(d) The four steps of the collaborative-planning process are gathering data, deciding,escalating and deploying

(e) This study allowed the companies to solve complex problems quickly, to exploitprofitable opportunities and to enhance trust within the supply chain The information isnow conveyed to other parties in a shorter time and more accurately As a result of this,the companies can have accurate information about the availability of material atdifferent stages This results in the reduction of inventory and obsolescence as well as theability to respond promptly to the changes in market conditions The benefit fromdecreasing inventory and obsolescence is around $5 million per year in total.Nonfinancial benefits include enhanced flexibility and reliability throughout the chain

2.3-2.

CHAPTER 3: INTRODUCTION TO LINEAR PROGRAMMING

3.1-1.

Swift & Com pany solved a series of LP problems to identify an optim al productionschedule The f irst in this series is the sc heduling model, which generates a shift-levelschedule for a 28-day horizon The objective is to minimize the difference of the totalcost and the revenue The total cost incl udes the operating costs and the penalties f orshortage and capacity violation The constr aints include carcass availability, production,inventory and demand balance equations, and limits on the production and inventory Thesecond LP problem solved is that of capable -to-promise m odels This is basically thesame LP as the first one, but excludes c oproduct and inventory The third type of LPproblem arises from the available-to-prom ise models The objective is to m aximize thetotal available production subject to production and inventory balance equations

As a result of this study, the key perform ance measure, namely the weekly percent-soldposition has increased by 22% The com pany can now allocate resources to theproduction of required products rather than wasting them The inventory resulting fromthis approach is m uch lower than what it us ed to be before Since the resources are usedeffectively to satisfy the dem and, the production is sold out The com pany does not need

to offer discounts as often as before The cu stomers order earlier to m ake sure that theycan get what they want by the tim e they want This in turn allows Swif t to operate evenmore efficiently The tem porary storage co sts are reduced by 90% The custom ers arenow more satisfied with Swift W ith this study, Swift gained a considerable com petitiveadvantage The m onetary benefits in the first years was $12.74 m illion, including theincrease in the profit from optimizing the pr oduct mix, the decrease in the cost of lostsales, in the frequency of discount offers a nd in the num ber of lost custom ers The mainnonfinancial benefits are the increased reliability and a good reputation in the business

3.1-2.

(a) (b)

Resource Usage per Unit of Activity

(c) Optimal Solution: Ð[ ß EÑ œ ÐB B Ñ œ Ð'ß "Þ&ч", ‡# and T œ "#"&‡

(d) From Sensitivity Analysis in IOR Tutorial, the allowable range f or the prof it perwood-framed window is between '(Þ& and infin ity As long as all the other param etersare fixed and the profit per wood-fram ed window is larger than $'(Þ&, the solution found

in (c) stays optimal Hence, when it is $"#! instead of $")!, it is still optimal to produce' wood-framed and "Þ& aluminum-framed windows and this results in a total profit of

$)&& However, when it is decreased to $ '!, the optimal solution is to m ake #Þ'( framed and aluminum-framed windows The total profit in this case is $% &#!

(a) Let be the num ber of units of produc t to produce and be the num ber of unitsB" " B#

of product to produce Then the problem can be formulated as follows:#

ma ximize T œ B  #B" #

subject to B  $B Ÿ #!!" #

#B  #B Ÿ $!!

(b) Optimal Solution: ÐB B Ñ œ Ð"#&ß #&ч", ‡# and T œ "(&‡

,

B   ! B   !" #(b) Optimal Solution: ÐB B Ñ œ Ð'!!ß $!!ч", ‡# and ^ œ $'!!‡

(c) The relevant two equations are $B  #B œ #%!!" # and #B œ "#!!" , so B œ '!!" and

(b) Optimal Solution: ÐB B Ñ œ Ð$#!!ß "#!!ч", ‡# and T œ #*#!‡

3.1-11.

(a) Let be the number of units of product produced for B3 3 3 œ "ß #ß $

maximize ^ œ & B  # B  #&B0 " 0 # $

subject to *B  $B  &B Ÿ &!!" # $

&B  %B" # Ÿ $&!

$B"  #B Ÿ "&!$

, , B B B   !" # $(b)

3.1-12.

Case 1: - œ !# (vertical objective line)

If -  !" , the objective value increases as increases, so B" B œ Ð ß !ч ""

the line FG is optimal

Case 3: -  !# (objective line with slope --", objec tive value increases as the line is

#shifted down)

(b) Optimal Solution: ÐEß FÑ œ ÐB ß B Ñ œ Ð#Î$ß #Î$ч" ‡# and T œ $Þ$$‡

(c) We have to solve #E  F œ # and E  #F œ # By subtracting the second equationfrom the first one, we obtain E  F œ !, so E œ F Plugging this in the first equation,

we get # œ #E  F œ $E, hence E œ F œ #Î$

3.2-2.

(a) TRUE (e.g., maximize D œ B  %B" #)

(b) TRUE (e.g., maximize D œ B  $B" #)

(c) FALSE (e.g., maximize D œ B  B" #)

3.2-3.

(a) As in the W yndor Glass Co problem , we want to find the optim al levels of twoactivities that com pete for limited resources Let and be the f raction purchased ofB" B#the partnership in the first and second friends venture respectively

Resource Usage per Unit of Activity

(c) Optimal Solution: (B ß B Ñ œ Ð#Î$ß #Î$ч" ‡# and T œ '!!!‡

Proportionality: It is fair to assum e that the am ount of work and m oney spent and theprofit earned are directly proportional to the fraction of partnership purchased in eitherventure

Additivity: The profit as well as time and money requirements for one venture should notaffect neither the profit nor tim e and m oney requirem ents of the other venture Thisassumption is reasonably satisfied

Divisibility: Because both friends will allow purchase of any fraction of a full

partnership, divisibility is a reasonable assumption

Certainty: Because we do not know how accurate the profit estimates are, this is a m oredoubtful assumption Sensitivity analysis should be done to take this into account

Divisibility: It is not justified, since activity levels are not allowed to be fractional

Certainty: It is reasonable, since the data provided is accurate

3.4-1.

In this study, linear program ming is used to im prove prostate cancer treatm ents Thetreatment planning problem is form ulated as an MIP problem The variables consist ofbinary variables that represent whether seed s were placed in a location or not and thecontinuous variables that denote the deviati on of received dose from desired dose Theconstraints involve the bounds on the dose to each anatom ical structure and variousphysical constraints Two m odels were st udied The first m odel aim s at finding themaximum feasible subsystem with the binary variables while the second one minimizes aweighted sum of the dose deviations with the continuous variables

With the new system , hundreds of m illions of dollars are saved and treatm ent outcomeshave been m ore reliable The side effects of the treatm ent are considerably reduced and

as a result of this, postoperation costs d ecreased Since planning can now be done justbefore the operation, pretreatment costs decreased as well The number of seeds required

is reduced, so is the cost of procuring th em Both the quality of care and the quality oflife after the operation are im proved The au tomated computerized system significantlyeliminates the variability in quality Moreove r, the speed of the system allows theclinicians to efficiently handle disruptions

3.4-2.