I N F O R M STransactions on Education

Vol. 12, No. 2, January 2012, pp. 78–88ISSN 1532-0545 (online)

http://dx.doi.org/10.1287/ited.1110.0076© 2012 INFORMS

The Effectiveness of Using a Web-Based Appletto Teach Concepts of Linear Programming:

An Experiment in Active Learning

Christine T. KyddDepartment of Business Administration, Alfred Lerner College of Business and Economics, University of Delaware,

Newark, Delaware 19716, chriskyd@udel.edu

The graphical solution method to a two-variable linear program (LP) provides valuable insights aboutthe general nature of multivariable linear programming models. As a result, introductory operations

research/management science textbooks typically present a graphical solution method to a two-variable LP as aprelude to the presentation of more complex problems. Construction of a two-dimensional feasible region com-bined with iso-profit (or iso-cost) lines on a blackboard or overhead projector can be tedious, at best. Even withtools such as PowerPoint, that include drawing tools, it is difficult to show students what happens in a graphicalLP as constraint lines and iso-profit lines shift around on a graph. To overcome this hurdle, this paper presentsa Web-based Java script applet that was used by both instructors and students to graphically illustrate/learnfundamental concepts of LP models. It then describes the results of a study that compares student performanceon exams of those who did use the applet versus those who did not. Results show that the students who usedthe applet to learn about LP concepts performed significantly better than those who did not. Implications forusing such active learning techniques and models in the classroom are discussed.

Key words : linear programming; active learning; sensitivity analysis; Web-based applet; linear programgraphical solution

History : Received: June 2010; accepted: September 2010.

IntroductionThe graphical method of solving linear program-ming problems is not a realistic applied solutionapproach because the method is limited to two-variable problems. However, the graphical techniqueis notably valuable from an instructional standpointbecause the technique pictorially illustrates numerousaspects of the more complex, algebra-based, solutionalgorithm. Therefore, numerous introductory opera-tions research/management science textbooks illus-trate a graphical solution to a two-variable problem(Anderson et al. 2008, Render et al. 2009, Stevenson2009, and Taylor 2010). Specifically, a graphical solu-tion to a linear program provides a student with intu-itive visual aids to facilitate their understanding ofsuch concepts as a basic feasible solution, the feasi-ble region, an unbounded solution, degeneracy, slack,surplus, binding constraints, nonbinding constraints,right-hand side ranges, and objective function of coef-ficient ranges.

Construction of a two-dimensional feasible regioncombined with iso-profit (or iso-cost) lines on a

blackboard or overhead projector can be tedious, atbest. Graphically guiding students through specificconcepts (i.e., basic feasible solutions, or right-handside ranges) borders on the impossible when usingsuch traditional media. One possibility is to show agraphical representation on an overhead transparencyand physically move lines on the graph. For this, theinstructor needs a unique slide for each constraintline, where the constraint line in question is shownand manipulated manually. Within programs such asPowerPoint or Excel, there are drawing tools avail-able, but even with these tools, it is very difficult tocapture the dynamic nature of a linear program onPowerPoint or Excel slides. To overcome these hur-dles, this paper presents a Web-based Java applet thatcan be used by both instructors to graphically illus-trate fundamental concepts and by students to prac-tice determining the effect of changes to constraintlines on the optimal solution. The applet has theadvantage of allowing students to actively move con-straints and the objective function line within a stan-dard problem (by clicking and dragging) to facilitatetheir understanding of numerous LP concepts and to

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experiment with changes within a graphical represen-tation of a linear program.

Other Web-Based AppletsSeveral other Web-based applets exist related to thegraphical solution of a linear program. Visual LinProg (Vanderbei 2010) is a tool that allows studentsto solve their own general linear program (LP) prob-lems. Once formulated, the tool allows the student towatch the solution process step by step, which is veryuseful for illustrating the simplex algorithm. How-ever, the student cannot move constraint lines aroundon the graph to see the effect of changes in the con-straints or objective function. To view the effect ofchanges, new constraints would have to be entered.

The Intermediate College Algebra tool (ExploringLinear Programming) (Green 2010), the GraphicalSimplex Algorithm (2D) (Zhang 2010), LP Explorer 1.0(Hall and Baird 2002), and The McGill LP tool (cgm)(Shepard 2010) are other existing Web-based appletsthat allow students to learn about linear programs byproviding some type of solution, and in some cases,a graph. However each one has one or more limita-tions regarding student experimentation with modelparameters. None allow students to manipulate theright-hand sides of the constraints in order to see theeffect of changes to these values on the feasible regionor the solution. None allow students to change theslopes of the objective function coefficients to findupper and lower bounds on the objective functioncoefficients.

Animated Linear Programming Applet (Wright2010) is a Java applet that appears to be the mostclosely related to the applet described in this paper.This applet does allow the user to “click and drag”constraints across the feasible region for two-variableproblems. However, the Animated Linear Program-ming Applet appears to present complete results forjust one previously defined standard problem. Theapplet does not allow for a user-defined problem tobe solved. When the authors attempted to input a dif-ferent problem, errors occurred in the input table ofthe Animated Linear Programming Applet.

A Web-Based Applet asan Active Learning TechniqueActive learning has been used in several different dis-ciplines to engage students in the learning process(Gardner 2008, Umble and Umble 2004, Prince 2004,Cook and Hazelwood 2002). It is basically defined asany instructional method that asks students to com-plete activities and to think about the activities asthey are being done (Bonwell and Eison 1991). Thusthere is an active component to the learning. Propo-nents of the use of active learning in the classroomsuggest that active learning techniques get students

more involved than they would normally be simplywatching an instructor solve a problem or listening toa lecture (Auster and Wylie 2006, Whiting 2006). Inthe case of learning about LP, it is extremely impor-tant for students to interact with the problem itselfand with the graphical representation of the problemin order to fully understand how the different compo-nents of an LP work together. Students can formulateLP problems and then solve them using any numberof LP computer packages, such as LINDO (LINDOSystems 2010). However these programs only showthem how to structure the inputs of the model andthen allows them to look at the solution printed out incomputer format. So it helps them set up the inputsand see the outputs, but it does not enhance under-standing of what happens in the middle. One way tohelp students truly understand the intricacies of anLP solution is to allow the student to be able to inter-act with the graphical representation of the problem.

The applet described here was designed to allowstudents to interact with a graphical representationof the problem. It lets the student input the param-eters of an LP problem and then to manipulate thegraphical representation/solution of that problem. Byactually having the ability to slide constraint andobjective function lines around on the graph, the stu-dent is able to see the immediate effect on the solutionof having more or fewer resources available (by mov-ing the constraint lines) or of changing the prices orprofits of the products (by changing the slope of theobjective function line). It is an interactive graph thatengages the student by demonstrating how changesto the inputs change the output. The applet presentedhere is a good example of an active learning technicaltool that engages students and provides them with aneffective learning environment.

Description of the LP Java AppletThe applet presented here (Kydd 2010) was originallydeveloped for the purpose of teaching sensitivityanalysis on a graphical solution to a linear pro-gram. It is intuitive and instructive to show studentswhat happens on a graph when finding bounds onresources so that they can have a picture in theirminds of what these bounds are all about. Howeveronce the applet was completed, it was obvious that itcould also be used for showing initial feasible solu-tions, finding the optimal solution and also doingthe sensitivity analysis, as originally conceptualized.Thus, each of these categories is discussed brieflybelow.

Finding the Feasible Region and Optimal SolutionStudents can first see the graphical representation ofa linear program using this applet. The constraintline coefficients and right-hand side (RHS) values,

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Figure 1 Graphical Representation of a Linear Program

Constraint lineand objectivefunctioncoefficientsFeasible

spaceshaded

Objectivefunctionline

along with the objective function coefficients can allbe entered into the spaces on the right side of theapplet. Then the lines are shown on the graph, andalso the feasible space is shaded in gray (see Fig-ure 1). This lets the students see immediately whattheir problem looks like in graphical form.

They can then click and drag the objective functionline away from the origin until it touches the last fea-sible point, thereby showing them which intersectionpoint is the optimal solution (for a standard maxi-mization problem) (see Figure 2).

In this applet, a very helpful feature is that everyintersection point on the graph is circled, and as the

Figure 2 Graphical Representation of a Linear Program Showing Optimal Solution Point

Objectivefunction line

Optimalsolutionpoint

lines move, so do the circles, thus highlighting atall times the intersection of the two lines in ques-tion. This feature helps the students to see theseintersection points, as well as helps the instructorto draw attention to the intersection points. As weknow, it is at these intersection points where virtu-ally all the action is in LP! So when the objectivefunction line hits the intersection point of the redand pink lines, the students have found the opti-mum. The value of Z at this point is shown atthe right (see Figure 3), so they know immediatelywhat the objective function is equal to at the optimalpoint.

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Figure 3 Value of Objective Function at Optimum

Value ofobjectivefunction atoptimum

Finding the Upper and Lower Bounds on theResources (Constraints)Once the optimal solution has been identified on thegraph, students can then be shown how to completesensitivity analysis for each resource or constraint.Here again, it is very helpful to have each intersec-tion point highlighted by a circle. For each constraintline, the students can slide the line both away fromthe origin and toward the origin to determine wherethe upper and lower bounds are for that line (orresource). When sliding the line away from the ori-gin, the intersection point will itself slide either upor down the other line that it intersects with to givethe optimal solution. For example, in this problem theoptimal solution occurs at the intersection of the redand pink lines. To find the upper bound for pink,students need to slide the pink line out and to theright. While doing this, the circle at the intersectionbetween the red and pink lines slides down the redline (see Figure 4).

When it finally hits the next intersection point atthe x-axis, the upper bound point for pink has beendetermined. If the pink line is moved further in thatdirection, the intersection of red and pink is no longeroptimal because the point has moved out of the fea-sible space. Because the student is sliding the lineand can watch where the circle, or intersection point,moves to, he or she can easily see where the upperbound occurs. At the same time, the numerical valueof the upper bound can be seen at the right as thenew RHS for that constraint equation. Thus, studentscan determine where the bound occurs and what thevalue is. They are also able to determine the allow-able increase for that resource. In this case, the upper

Figure 4 Graphical Representation of Bounds on Constraint Lines

Originaloptimal point

Optimalintersection

point

bound for the pink constraint line is 750, which canbe seen at the right (see Figure 5).

The same thing applies for finding the lower boundbut in the opposite direction. Each constraint linecan be moved in turn to determine upper and lowerbounds.

Seeing the Effect of Changing the Slope of theObjective FunctionThe final part of sensitivity analysis that can be showneasily through use of this applet is the effect of chang-ing the slope of the objective function on the optimalsolution. In this case, the student can grab one endof the objective function line and change the slopeby dragging the endpoint to a new location (see Fig-ure 6). Then the new optimal solution can be deter-mined by again sliding the objective function lineaway from the origin to the last feasible intersectionpoint (see Figure 7).

Bounds on the objective function coefficients canbe determined by changing the slope of the objectivefunction line until it is the same as each of the twoconstraint lines in turn that originally intersected togive the optimum. When the objective function lineis coincident with one or the other of these two lines,then one of the objective function coefficients willshow as having changed in the right side window ofthe applet. This new value will be the upper or lowerbound on the objective function coefficient value (seeFigure 8). For example, if the objective function lineis coincident with the red line, then the coefficientsare shown as 2.67 and 4. With these coefficients, theobjective function line now has the same slope as thered line (slope = −1/105 = −2/3).

MethodA study was completed regarding the LP appletdescribed here using students in six sections of anundergraduate operations management course in one

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Figure 5 Upper Bound Value for Constraint Line

New RHS valuefor pink line

(upper bound)

Figure 6 New Slope of Objective Function

Change slopeof objectivefunction

Figure 7 Finding New Optimal Solution Based on Modified OF Slope

Slide new OFline out to findnew optimalpoint

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Figure 8 Finding Bounds on OF Coefficients

Equalslopes

Objective functionline coincident withred line

semester at a large university in the eastern UnitedStates. In two of the sections, chosen because theauthor was the instructor, the above-described LPapplet was used both during the class lectures on LPand outside of class for student practice. Specifically,the applet was used during class lectures to demon-strate feasible solutions and the feasible region, theoptimal solution, and ranges on the right-hand sidevalues. The ranges were shown with the applet bysliding each constraint line away from the origin andthen toward the origin in turn, until the line movedoutside of the feasible region. The applet was alsomade available to the students in these same twosections to use outside of class. The students wereprovided with the Web address of the applet andinstructed to practice finding the bounds on their owntime. No data was collected as to how many studentsactually used the applet outside of class; however apractice problem for use with the applet was sug-gested to these students and several asked questionsabout this problem during office hours when the stu-dents were preparing for the exam. In the other foursections, the use of overhead transparencies and man-ually sliding “pencil” constraint lines around on thegraph illustrated the same concepts. The same prac-tice problem was given to these students to practiceon with paper and pencil in preparation for the exam.

There were between 45 and 55 students in eachof the six sections of the course. Although studentscould not be randomly assigned to sections, the stu-dents in each section were very similar in terms oftheir basic characteristics. All of the students in this

study were either juniors or seniors in college, and allof them were pursuing a major in business adminis-tration. Ages ranged from 19 to 22 across all sections.There were no notable differences among the six sec-tions in terms of gender. This course (Introduction toOperations Management) was a required course forall of the students in this study. All of the studentsin this study had the same prerequisites for the oper-ations management course. LP is not taught in anyother class, so the seniors did not have any unfairadvantage here.

Although all sections of the course were not taughtby the same instructor, the specific material involvingLP and sensitivity analysis (two 75-minute lectures)was taught by the same instructor in all six sections.Thus, the way in which material was presented acrosssections, with the exception of using the applet, wasbasically the same.

An exam was administered in each of the six sec-tions, which consisted of the same set of questionson LP and sensitivity analysis. For purposes of thisstudy, the questions examined the student’s knowl-edge about LP in general, and specifically aboutgraphical sensitivity analysis, i.e., finding upper andlower bounds on constraint lines, or ranging. The spe-cific exam questions can be seen in the appendix; part(e) is the ranging question. The questions center onan LP problem and solution that was given to the stu-dents in the exam. Computer output and a graphicalrepresentation of the problem were also provided tothe students.

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ResultsTo examine whether student performance on an examwas impacted by use of the LP applet described inthis paper, student scores were recorded for (1) thefull exam, (2) the complete LP question on the exam,and (3) the score on just the LP ranging part of theLP question on the exam. To standardize the scoresacross the control group versus the group exposed tothe applet, percentages of each problem correct wereused as the main dependent variable. Other analyseswere also conducted on points earned on each part ofthe exam in order to verify the findings in this study.

The data was grouped by control versus experi-mental group; thus there were 203 students in thecontrol group (four sections) and 108 students in theexperimental group (two sections). The two sets ofdata were tested as to whether the variances on meanpercentage correct were equal or not. For the entireLP question, the variances were found to be signif-icantly different (F = 006845; p = 000109) for the twogroups (thus, a t-test on the difference between meansassuming unequal variances was used here). For theLP ranging part of the LP question, the variances werenot found to be significantly different (F = 00802; p =

00091) for the two groups (thus, a t-test on the differ-ence between means assuming equal variances wasused here).

The major test completed was a comparison ofmeans between the two groups to determine whetherstudents in the experimental group performed bet-ter than those in the control group on the LP rang-ing question and also on the entire LP question. Forthe LP ranging question, those students who were inthe experimental group were found to perform sig-nificantly better than those in the control group (seeTable 1(a)). The experimental mean was 0.518 whereasit was only 0.315 in the control group showing thatuse of the LP applet in preparation for this partic-ular question (on ranging) appeared to have a posi-tive impact on percentage of the question correct. Forthe entire LP question, there was no significant dif-ference in mean scores between those students whowere in the experimental group versus those in thecontrol group (see Table 1(b)). The experimental meanpercentage was 0.526 and the control group meanwas 0.535. So although interaction with the LP appletappeared to help students understand ranging betterthan their counterparts who had not been exposedto the applet, it did not seem to help them under-stand other aspects of LP problems better. Becausethis result might seem to indicate that the experimen-tal group performed worse than the control groupon the nonranging part of the LP question, the meanscores on only the nonranging part of the LP ques-tion were compared. Although the average numberof points scored was slightly higher for the control

Table 1(a) Mean Percent Score for Ranging Part of LP Question

Control Experimental t-value Significance

Mean 00315 00518 −4007 0000003Variance 00161 00201Sample size 203 108

Table 1(b) Mean Percent Score for Full LP Question

Control Experimental t-value Significance

Mean 00535 00526 00224 n.s.Variance 00092 00134Sample size 203 108

Table 1(c) Mean Score for LP Question Minus Ranging Part of LPQuestion

Control Experimental t-value Significance

Mean 170192 140962 10974 n.s.Variance 820918 1590068Sample size 203 108

group, there was no significant difference between themeans (see Table 1(c)).

To further explore the data collected in this study,additional analyses were run to determine the effectof student performance on other parts of the exam onperformance on the ranging question. The first regres-sion model was specified as Y = a+b1∗X1+b2∗X2+

6 additional terms, where Y was the score on the LPranging question, X1 represented the student’s gradeon the LP question excluding the LP ranging ques-tion, X2 represented the student’s grade on the examexcluding the LP question, and each of the six classsections was represented by a 0-1 dummy variable.The results (Table 2) showed that the regression wassignificant overall at the p < 00001 level (R2 = 0033).The two variables, X1 and X2, were both significantat the p < 00001 level whereas none of the dummy

Table 2 Ranging Score vs. (LP-Ranging) Score and (Exam-LP) Score

ANOVA df SS MS F Significance F

Regression 8 310700281 3830785 230801 3.1014E−28Residual 303 515830759 180428Total 311 816540039

Coefficients Standard error t stat P -value

Intercept 50435 10507 30607 000004LP − ranging 00109 00012 80924 4.407E−17Exam − LP 00162 000202 80020 2.326E−14Sect 10 10143 00954 10198 00232Sect 11 10449 00914 10586 00114Sect 12 20063 00932 20213 000276Sect 13 00025 00925 000265 00979Sect 14 00767 00963 00796 00426Sect 15 00720 00834 00864 00388

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variables representing section was significant. Thus,although none of the individual sections had anyimpact on how the students performed on the rang-ing question of the broader LP question, both thescore on the LP question minus the LP ranging ques-tion and the score on the exam minus the LP ques-tion were significant in accounting for the variance inthe dependent variable (performance on LP rangingquestion alone). What this tells us is that both per-formance on the LP question minus the LP Rangingquestion and performance on the exam minus the LPquestion helped to explain performance on the rang-ing question. So performance on various other partsof the exam is linked to performance on the rangingquestion, which makes sense because students shouldperform consistently across all parts of the exam.

The second regression model was similar to the oneabove but without X1 and including a variable X3that represented a dummy variable for control versusexperimental group. The model was Y = a+ b1∗X2+

b2 ∗ X3, where X2 again represented the student’sgrade on the LP question excluding the LP rangingquestion, and Y again represented the score on the LPranging question. The results (Table 3) showed thatthe regression was significant overall at the p < 00001level (R2 = 0025). The variable X2 was significant atthe p < 0001 level. The variable X3 was significant atjust over the 0.05 level (p < 00058). This suggests thatthe score on the LP ranging question could partiallybe attributed to score on the LP question minus theLP Ranging question and also partially to the exper-imental treatment. This makes sense in that if a stu-dent understands LP in general, then that might helpthe student score higher on the ranging part of theproblem. Also it shows that the control versus experi-mental dummy variable was mildly significant in pre-dicting performance on the ranging question. Thisresult supports the hypothesis regarding how the useof the LP applet led to higher percentages correct onthe ranging question. This result is also supported bythe t-test on the difference between the mean per-cent scores for the ranging question as reported earlier(difference between the means was significant at thep < 00001 level).

Table 3 Ranging Score vs. (LP-Ranging) Score and Condition(Control vs. Experimental)

ANOVA df SS MS F Significance F

Regression 2 112910778 6450889 490634 2.7838E−19Residual 292 317990804 1300130Total 294 510910582

Coefficients Standard error t stat P -value

Intercept 00141 00405 00349 0072704LP − ranging 00175 000184 90498 8.053E−19C vs. E −00980 00516 10899 000589

The student’s grade on the overall exam should notmatter here because there were other topics coveredon other parts of the exam that had nothing to dowith linear programming or the LP applet describedin this paper. To check this, the values for X1 (examscore excluding LP question score) were first normal-ized. An analysis of variance (ANOVA) was then car-ried out on the normalized mean scores for the examscore minus the LP question for all sections. Therewere no significant differences across sections. Thesix classes of students tested approximately the sameon the parts of the exam that were not on LP. Thusthe students in the experimental group sections werenot different (i.e., smarter) in terms of performanceon other parts of the exam than the control groupsections.

Finally to test for differences within treatmentgroups (control sections versus experimental sections)on percentage correct on the ranging part of the LPquestion, the entire LP question and the entire exam,several ANOVAs were run. There were no significantdifferences among the four sections that were in thecontrol group for any of these variables. Thus, thesefour sections were similar on all three variables. Simi-larly for the two sections that were in the experimen-tal group, there were no significant differences on anyof these same three variables.

Analysis of variance tests were run on the controlgroup sections (four sections) and separately on thetwo experimental sections for the normalized valuesfor X1 (exam score − LP question). There were nosignificant differences among the four sections thatwere in the control group for this variable. There werealso no significant differences across the two sectionsin the experimental group. This demonstrates thatwithin the control versus experimental groups, therewere no differences on performance on the parts ofthe exam that were not on linear programming acrosssections in each group.

DiscussionThe LP applet used in this study was found to behelpful for student understanding of certain aspectsof linear programs. Results of this study showed thatstudents who are high performing will do well on allparts of an exam, and vice versa (performance on theranging question was strongly related to performanceon other parts of the exam, and on performance onother parts of the LP problem). However the stu-dents also performed better on the ranging part ofthe LP exam question when they had been shown theconcept using the applet compared to a pencil-and-paper demonstration. These results show the valueof utilizing a computer-based tool as an active learn-ing technique. Because the students first watched and

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then used the LP applet to complete ranging tasks,it appears that their interaction with the LP appletaided learning. However, it is interesting that use ofthe LP applet did not improve learning overall for thetopic of linear programming. Both the control groupand the experimental group performed equally welloverall on the LP problem. One would think that ifa student gained a greater understanding of rangingin LP (one of the more difficult aspects to under-stand regarding an LP solution) based on using the LPapplet developed here, then the student would alsounderstand LP solutions better in general, becauseall of the different concepts are related to each other,particularly within a linear programming graphicalsolution.

There are a few possible explanations for theseresults. The first is that the focus of the LP appletin this setting was to teach students about rangingin particular. It is true that the applet was used todemonstrate other concepts as well, such as an initialfeasible solution, corner points, and redundant con-straints. However, the primary focus was on the stu-dent’s ability to determine upper and lower boundson resources (i.e., ranging) for a specific problem solu-tion. One particular problem was suggested to thestudents for practice on ranging, and so if they usedthe LP applet primarily for this problem, then theiruse of the applet would have been primarily (andperhaps only) to practice ranging. Data on frequencyof use of the applet and also what students used itto practice were not collected, because students hadaccess to the applet outside of class. In future studiessuch data should be collected and also perceptions bythe students as to whether the applet helped illustratebasic LP concepts to them or not.

The second possible reason why use of the LPapplet did not appear to improve general under-standing of linear programming problems is that theinstructor who taught linear programming to all sec-tions of the class included in this study has beenteaching these concepts for over 20 years. Perhaps theinstructor’s experience in presenting graphical solu-tions to LP problems outweighs the value of the LPapplet. Explanations and examples used to illustrateall parts of an LP solution were identical in both thecontrol and experimental groups. The only differencewas that in the control group, the instructor usedslides and an overhead projector to show the graphi-cal solution technique instead of the LP applet. Thusfuture studies of the effect of using interactive tools onlearning and understanding should probably be com-pleted across instructors with different levels of expe-rience to see if the LP applet could potentially helpinstructors with less experience in presenting difficulttopics. Also it is possible that the novelty of having aguest lecturer in the control group classes caused the

students to pay closer attention than they might have.Additional focus could have led to a greater under-standing of LP problems in general in this group.

Future research on the use of computer-basedteaching tools should focus on a few factors. First itneeds to be determined whether it is the tool itself, theinteraction between the tool and student (the activelearning part of the process), or the way in which theinstructor uses the tool that impacts student learning.There have been numerous anecdotal stories regard-ing teaching tools that have been very successfulunder the hands of a particular instructor, but whenothers try to use it, the effect is not as positive. Alsothe applet itself can be improved and extended sothat students can use it to practice all aspects of solv-ing an LP problem. Extensions to this applet includebuilding in additional functions to make it more userfriendly, and expanding the number of constraints itcan handle.

This applet represents one more technical, Web-based teaching tool that could be very useful ingetting difficult mathematical concepts across tostudents, particularly to those who are not planningto go into a technical field but who have to learn thebasics of an applied field such as operations manage-ment. By getting the students engaged with a toolsuch as this, it appears that “active learning” methodsor tools can help students learn. Benefits to using thisapplet include not only illustrating important pointsin the classroom setting, but also allowing students towork with the graphical representation on their ownoutside of class. To date, it appears that this appletcan have very positive effects on student understand-ing and learning.

Appendix. LP Exam QuestionsThe Ohio Creek Ice Cream Company is planning itsproduction output for next week. Demand for Ohio Creekpremium and light ice cream continues to outpace the com-pany’s production capacities. Ohio Creek earns a profitof $100 per hundred gallons of premium and $100 perhundred gallons of light ice cream. Three resources usedin ice cream production are in short supply for nextweek: the capacity of the mixing machine, the amount ofhigh-grade milk, and the quantity of sweetener available.After accounting for required maintenance time, the mix-ing machine will be available for 140 hours next week. Onehundred gallons of premium ice cream requires 0.3 hoursof mixing, 90 gallons of milk, and 10 cups of sweetener.One hundred gallons of light ice cream requires 0.5 hoursof mixing, 70 gallons of milk and five cups of sweetener.Only 28,000 gallons of high-grade milk and only 3,000 cupsof sweetener will be available for next week.

A linear programming formulation for this problem isgiven:

Max Z = 100P + 100L

subject to 003P + 005L≤ 140 hours

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90P + 70L≤ 281000 gallons

10P + 5L≤ 31000 cups

P1L≥ 00

A computer-generated solution to this problem is pro-vided.

Please answer the following questions regarding thisproblem.

(a) How many hundreds of gallons of premium and lightice cream should the Ohio Creek Co. produce next week?How much profit will they make?

(b) How much of each of the resources is left over at theoptimal solution? Verify algebraically the amount of high-grade milk that is left over.

(c) Suppose that there is a sudden jump in the demandfor the light ice cream, and the profit margin for Light dou-bles. Using the graph, determine what the optimal solutionwould now be. Explain.

(d) Suppose that Ohio Creek finds out that it can leaseadditional mixing time next week, but it will cost $75 perhour. Should the firm lease additional mixing hours or not?If so, how many hours should it purchase? If not, explainwhy not.

(e) From the graph only, determine the upper and lowerbounds for the mixing machine hours within which theoptimal intersection point remains the same. Show thepoints that you use to find these bounds.

(f) If Ohio Creek can obtain an additional 2,000 gallonsof milk at no cost, how much additional profit can it make?Explain your answer.

(g) Ohio Creek is considering introducing a new, super-diet type of ice cream that uses 0.6 hours of mixing time,75 gallons of milk and no sweetener per hundred gallons ofice cream. It will bring in a profit of $100, the same as theother two brands. Should Ohio Creek produce the new dietice cream? Why or why not? Show all work and explaincompletely.

(h) Show the complete LP formulation for this problemthat includes the new super-diet ice cream along with thepremium and light.

Computer-generated solution to the LP problem

Objective function value(1) 35,000.00

Variable Value Reduced cost

P 175.000000 0.000000L 175.000000 0.000000

Row Slack or surplus Dual prices

(2) 0.000000 83.333336(3) 0.000000 0.833333(4) 375.000000 0.000000No. Iterations = 2

Ranges in which the basis is unchanged:OBJ coefficient ranges

Variable Current Allowable Allowable

Coef Increase Decrease

P 100.000000 28.571428 39.999996L 100.000000 66.666664 22.222223

Right-hand side ranges

Current Allowable Allowable

Row RHS Increase Decrease

2 1400000000 600000000 3600000003 2810000000000 215710428711 8140000000004 310000000000 Infinity 3750000000

300 467311

280

400

600

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