Formulating duality problems
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "", "advice": "\n", "rulesets": {"ruleset0": []}, "extensions": [], "variables": {"c": {"name": "c", "group": "Ungrouped variables", "definition": "repeat(random(-15..15),9)", "description": "", "templateType": "anything"}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "repeat(random(-15..15),9)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "repeat(random(-5..5),9)", "description": "", "templateType": "anything"}, "randindex": {"name": "randindex", "group": "Ungrouped variables", "definition": "random(0..3)", "description": "", "templateType": "anything"}, "a20": {"name": "a20", "group": "Ungrouped variables", "definition": "random(-1..1)", "description": "", "templateType": "anything"}, "a21": {"name": "a21", "group": "Ungrouped variables", "definition": "random(-1,1)", "description": "", "templateType": "anything"}, "a22": {"name": "a22", "group": "Ungrouped variables", "definition": "random(-1..1)", "description": "", "templateType": "anything"}, "a23": {"name": "a23", "group": "Ungrouped variables", "definition": "random(1..2)", "description": "", "templateType": "anything"}, "aa1": {"name": "aa1", "group": "Ungrouped variables", "definition": "repeat(random(-5..5),9)", "description": "", "templateType": "anything"}, "mm": {"name": "mm", "group": "Ungrouped variables", "definition": "random(\"Minimize\",\"Maximize\")", "description": "", "templateType": "anything"}, "mmd": {"name": "mmd", "group": "Ungrouped variables", "definition": "if(mm=\"Maximize\",\"Minimize\",\"Maximize\")", "description": "", "templateType": "anything"}, "aa2": {"name": "aa2", "group": "Ungrouped variables", "definition": "repeat(random(-5..5),9)", "description": "", "templateType": "anything"}, "aa3": {"name": "aa3", "group": "Ungrouped variables", "definition": "repeat(random(-5..5),9)", "description": "", "templateType": "anything"}, "aa4": {"name": "aa4", "group": "Ungrouped variables", "definition": "repeat(random(-5..5),9)", "description": "", "templateType": "anything"}, "a24": {"name": "a24", "group": "Ungrouped variables", "definition": "random(-1..3)", "description": "", "templateType": "anything"}, "eqx": {"name": "eqx", "group": "Ungrouped variables", "definition": "repeat(random(\"<=\",\">=\",\"=\"),9)", "description": "", "templateType": "anything"}, "eq2": {"name": "eq2", "group": "Ungrouped variables", "definition": "random(\"<=\",\">=\",\"=\")", "description": "", "templateType": "anything"}, "eq3": {"name": "eq3", "group": "Ungrouped variables", "definition": "random(\"<=\",\">=\",\"=\")", "description": "", "templateType": "anything"}, "eq4": {"name": "eq4", "group": "Ungrouped variables", "definition": "random(\"<=\",\">=\",\"=\")", "description": "", "templateType": "anything"}, "eq5": {"name": "eq5", "group": "Ungrouped variables", "definition": "random(\"<=\",\">=\",\"=\")", "description": "", "templateType": "anything"}, "eq6": {"name": "eq6", "group": "Ungrouped variables", "definition": "random(\"<=\",\">=\",\"=\")", "description": "", "templateType": "anything"}, "eq0les": {"name": "eq0les", "group": "Ungrouped variables", "definition": "if(mm=\"Minimize\",if(eqx[0]=\"<=\",1,0),if(eqx[0]=\">=\",1,0))", "description": "", "templateType": "anything"}, "eq1les": {"name": "eq1les", "group": "Ungrouped variables", "definition": "if(mm=\"Minimize\",if(eqx[1]=\"<=\",1,0),if(eqx[1]=\">=\",1,0))", "description": "", "templateType": "anything"}, "eq2les": {"name": "eq2les", "group": "Ungrouped variables", "definition": "if(mm=\"Minimize\",if(eqx[2]=\"<=\",1,0),if(eqx[2]=\">=\",1,0))", "description": "", "templateType": "anything"}, "eq3les": {"name": "eq3les", "group": "Ungrouped variables", "definition": "if(mm=\"Minimize\",if(eqx[3]=\"<=\",1,0),if(eqx[3]=\">=\",1,0))", "description": "", "templateType": "anything"}, "eq4les": {"name": "eq4les", "group": "Ungrouped variables", "definition": "if(mm=\"Minimize\",if(eqx[4]=\"<=\",1,0),if(eqx[4]=\">=\",1,0))", "description": "", "templateType": "anything"}, "eq5les": {"name": "eq5les", "group": "Ungrouped variables", "definition": "if(mm=\"Minimize\",if(eqx[5]=\"<=\",1,0),if(eqx[5]=\">=\",1,0))", "description": "", "templateType": "anything"}, "eq6les": {"name": "eq6les", "group": "Ungrouped variables", "definition": "if(mm=\"Minimize\",if(eqx[6]=\"<=\",1,0),if(eqx[6]=\">=\",1,0))", "description": "", "templateType": "anything"}, "eq0gr": {"name": "eq0gr", "group": "Ungrouped variables", "definition": "if(mm=\"Minimize\",if(eqx[0]=\">=\",1,0),if(eqx[0]=\"<=\",1,0))", "description": "", "templateType": "anything"}, "eq1gr": {"name": "eq1gr", "group": "Ungrouped variables", "definition": "if(mm=\"Minimize\",if(eqx[1]=\">=\",1,0),if(eqx[1]=\"<=\",1,0))", "description": "", "templateType": "anything"}, "eq2gr": {"name": "eq2gr", "group": "Ungrouped variables", "definition": "if(mm=\"Minimize\",if(eqx[2]=\">=\",1,0),if(eqx[2]=\"<=\",1,0))", "description": "", "templateType": "anything"}, "eq3gr": {"name": "eq3gr", "group": "Ungrouped variables", "definition": "if(mm=\"Minimize\",if(eqx[3]=\">=\",1,0),if(eqx[3]=\"<=\",1,0))", "description": "", "templateType": "anything"}, "eq4gr": {"name": "eq4gr", "group": "Ungrouped variables", "definition": "if(mm=\"Minimize\",if(eqx[4]=\">=\",1,0),if(eqx[4]=\"<=\",1,0))", "description": "", "templateType": "anything"}, "eq5gr": {"name": "eq5gr", "group": "Ungrouped variables", "definition": "if(mm=\"Minimize\",if(eqx[5]=\">=\",1,0),if(eqx[5]=\"<=\",1,0))", "description": "", "templateType": "anything"}, "eq6gr": {"name": "eq6gr", "group": "Ungrouped variables", "definition": "if(mm=\"Minimize\",if(eqx[6]=\">=\",1,0),if(eqx[6]=\"<=\",1,0))", "description": "", "templateType": "anything"}, "eq0e": {"name": "eq0e", "group": "Ungrouped variables", "definition": "if(eqx[0]=\"=\",1,0)", "description": "", "templateType": "anything"}, "eq1e": {"name": "eq1e", "group": "Ungrouped variables", "definition": "if(eqx[1]=\"=\",1,0)", "description": "", "templateType": "anything"}, "eq2e": {"name": "eq2e", "group": "Ungrouped variables", "definition": "if(eqx[2]=\"=\",1,0)", "description": "", "templateType": "anything"}, "eq3e": {"name": "eq3e", "group": "Ungrouped variables", "definition": "if(eqx[3]=\"=\",1,0)", "description": "", "templateType": "anything"}, "eq4e": {"name": "eq4e", "group": "Ungrouped variables", "definition": "if(eqx[4]=\"=\",1,0)", "description": "", "templateType": "anything"}, "eq5e": {"name": "eq5e", "group": "Ungrouped variables", "definition": "if(eqx[5]=\"=\",1,0)", "description": "", "templateType": "anything"}, "eq6e": {"name": "eq6e", "group": "Ungrouped variables", "definition": "if(eqx[6]=\"=\",1,0)", "description": "", "templateType": "anything"}, "m1": {"name": "m1", "group": "Ungrouped variables", "definition": "if(eq1les = 1, 1, if(eq1gr = 1, 2, 3))", "description": "", "templateType": "anything"}, "m2": {"name": "m2", "group": "Ungrouped variables", "definition": "if(eq2les = 1, 1, if(eq2gr = 1, 2, 3))", "description": "", "templateType": "anything"}, "m0": {"name": "m0", "group": "Ungrouped variables", "definition": "if(eq0les = 1, 1, if(eq0gr = 1, 2, 3))", "description": "", "templateType": "anything"}, "m3": {"name": "m3", "group": "Ungrouped variables", "definition": "if(eq3les = 1, 1, if(eq3gr = 1, 2, 3))", "description": "", "templateType": "anything"}, "m4": {"name": "m4", "group": "Ungrouped variables", "definition": "if(eq4les = 1, 1, if(eq4gr = 1, 2, 3))", "description": "", "templateType": "anything"}, "m5": {"name": "m5", "group": "Ungrouped variables", "definition": "if(eq5les = 1, 1, if(eq5gr = 1, 2, 3))", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["c", "a1", "b", "randindex", "a20", "a21", "a22", "a23", "aa1", "mm", "mmd", "aa2", "aa3", "aa4", "a24", "eqx", "eq2", "eq3", "eq4", "eq5", "eq6", "eq0les", "eq1les", "eq2les", "eq3les", "eq4les", "eq5les", "eq6les", "eq0gr", "eq1gr", "eq2gr", "eq3gr", "eq4gr", "eq5gr", "eq6gr", "eq0e", "eq1e", "eq2e", "eq3e", "eq4e", "eq5e", "eq6e", "m1", "m2", "m0", "m3", "m4", "m5"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "The following LP problem is given:
\n{mm}: $\\simplify {{c[0]}x_1+{c[1]}x_2+{c[2]}x_3+{c[3]}x_4+{c[4]}x_5} $
\nSubject to:
\n$\\simplify {{a1[0]}x_1+{a1[1]}x_2+{a1[2]}x_3+{a1[3]}x_4+{a1[4]}x_5} ~ \\var{eqx[0]} ~ \\var{b[0]}$
\n$\\simplify {{a20}x_1+{a21}x_2+{a22}x_3+{a23}x_4+{a24}x_5} ~ \\var{eqx[1]} ~ \\var{b[1]}$
\n$\\simplify {{aa1[0]}x_1+{aa1[1]}x_2+{aa1[2]}x_3+{aa1[3]}x_4+{aa1[4]}x_5} ~ \\var{eqx[2]} ~ \\var{b[2]}$
\n$\\simplify {{aa2[0]}x_1+{aa2[1]}x_2+{aa2[2]}x_3+{aa2[3]}x_4+{aa2[4]}x_5} ~ \\var{eqx[3]} ~ \\var{b[3]}$
\n$\\simplify {{aa3[0]}x_1+{aa3[1]}x_2+{aa3[2]}x_3+{aa3[3]}x_4+{aa3[4]}x_5} ~ \\var{eqx[4]} ~ \\var{b[4]}$
\n$\\simplify {{aa4[0]}x_1+{aa4[1]}x_2+{aa4[2]}x_3+{aa4[3]}x_4+{aa4[4]}x_5} ~ \\var{eqx[5]} ~ \\var{b[5]}$
\n$x_1,x_2,x_3,x_4,x_5 \\ge 0.$
\n\nThe dual problem is formulated as follows:
\n{mmd}: $\\simplify {{b[0]}y_1+{b[1]}y_2+{b[2]}y_3+{b[3]}y_4+{b[4]}y_5+{b[5]}y_6 }$
\nSubject to:
\n$\\simplify { {a1[0]}y_1 + {a20}y_2 + {aa1[0]}y_3 + {aa2[0]}y_4 + {aa3[0]}y_5 + {aa4[0]}y_6 >= {c[0]} }$
\n$\\simplify { {a1[1]}y_1 + {a21}y_2 + {aa1[1]}y_3 + {aa2[1]}y_4 + {aa3[1]}y_5 + {aa4[1]}y_6 >= {c[1]}}$
\n$\\simplify { {a1[2]}y_1 + {a22}y_2 + {aa1[2]}y_3 + {aa2[2]}y_4 + {aa3[2]}y_5 + {aa4[2]}y_6 >= {c[2]}}$
\n$\\simplify { {a1[3]}y_1 + {a23}y_2 + {aa1[3]}y_3 + {aa2[3]}y_4 + {aa3[3]}y_5 + {aa4[3]}y_6 >= {c[3]}}$
\n$\\simplify { {a1[4]}y_1 + {a24}y_2 + {aa1[4]}y_3 + {aa2[4]}y_4 + {aa3[4]}y_5 + {aa4[4]}y_6 >= {c[4]}}$
\n\nFind the range/sign of each dual variable (put in the boxes \"1\" for \"$\\le 0$\", \"2\" for \"$\\ge 0$\" and \"3\" for \"$any ~ number$\"):
\n\n
($~y_1, ~y_2,~ y_3,~ y_4,~ y_5,~ y_6~$): [[0]]
\n
\n\n", "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "matrix([m0,m1,m2,m3,m4,m5])", "correctAnswerFractions": false, "numRows": 1, "numColumns": "6", "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Marie Nicholson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1799/"}, {"name": "Timur Zaripov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3272/"}, {"name": "Musa Mammadov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4417/"}]}]}], "contributors": [{"name": "Marie Nicholson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1799/"}, {"name": "Timur Zaripov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3272/"}, {"name": "Musa Mammadov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4417/"}]}