1 Consultant hiring problem: Modeling on AMPL

2 Outline of modeling process 1.Carefully read the problem to get a general idea 2.Analyze each paragraph to extract all relevant data and information: a)Identify given data and define corresponding sets, parameters. b)Identify the goal of the problem and define a corresponding objective function (in this stage might be in English; will get a linear function after defining all necessary data and variables). c)The goal can be achieved by making certain decisions. Define decision variables when those are obviously implied by the problem statement. More variables might be defined later. d)Identify the constraints of the system and state them in English (will express in linear functions after defining all relevant data and variables).

3 Outline of modeling process 3. Put everything together to get a coherent model. In this stage, all the constraints and the objective function are translated into linear functions. Might need to define new data and variables to accomplish it. 4. Solve by an IP software (AMPL in our case). The software output might prompt what are the deficiencies of the model and how those can be fixed.

4 Relevant Information from Paragraph 1 set I; # set of projects param profit{i in I}; #profits from projects excluding consulting costs

5 Possible data for Paragraph 1 parameters and sets set I:= A B C D; param profit[*] := A250 B300 C200 D400 ;

6 Relevant Information from Paragraph 2 set J; # set of consultants param cost{j in J}; # weekly cost of consultant j set ATTR; # set of K attributes param cons_attr{j in J, k in ATTR}; # is 1 if consultant j possesses attribute k, 0 otherwise

7 Possible data for Paragraph 2 parameters and sets set J:= Tom Jim Ann Tony; param cost[*] := Tom25 Jim36 Ann28 Tony30;

8 Possible data for Paragraph 2 parameters and sets (cont.) set ATTR:= C++ IP Linux; param cons_attr[*,*] : C++IPLinux:= Tom 101 Jim110 Ann011 Tony101;

9 Relevant Information from Paragraph 3 param proj_attr{i in I, k in ATTR}; # number of consultants needed to possess attribute k in project I # Constraint 1: # number of consultants possessing attribute k in project i # should be at least proj_attr[i,k] if project i is pursued # Provision: A consultant can fulfill many needs of a project.

10 Possible data for Paragraph 3 parameters and sets param proj_attr[*,*] :C++IPLinux:= A212 B111 C201 D210 ;

11 Relevant Information from Paragraph 4 # Constraint 2: # each consultant can work only on one project at a time param b{i in I}; # first week of project i param e{i in I}; # last week of project I Possible data for Paragraph 4 parameters and sets param: be := A24A24 B13B13 C35C35 D23 ;

12 Relevant Information from Paragraph 5 var proj{i in I} binary; # is 1 if project i is pursued var assign{j in J, i in I} binary; # is 1 if consultant j is assigned to project i # Constraint 3: # consultants can work only on projects that are pursued

13 Proceeding to next stage We are done with Stage 2, extracting relevant data and information from each paragraph. In Stage 3, we will express the objective function and the constraints in terms of defined variables and parameters.

14 Objective Function maximize Profit:# profit from projects – consultant costs sum{i in I}profit[i]*proj[i] - sum{j in J, i in I}cost[j]*(e[i]-b[i]+1)*assign[j,i];

15 Constraint 1 Information we have from Stage 2: # Constraint 1: # number of consultants possessing attribute k in project i # should be at least proj_attr[i,k] if project i is pursued # Provision: A consultant can fulfill many needs of a project. The corresponding constraint in the model: subject to Number_of_Consultants {i in I, k in ATTR}: sum{j in J} cons_attr[j,k] * assign[j,i] >= proj_attr[i,k] * proj[i];

16 Constraint 2 Information we have from Stage 2: # Constraint 2: # each consultant can work only on one project at a time Getting the corresponding constraint in the model: #First need to define new set and parameter for time. set WEEKS:= min{i in I}b[i]..max{i in I}e[i]; # the range of the weeks when the projects are planned param up{w in WEEKS, i in I}:= if b[i]<=w<=e[i] then 1 else 0; # this parameter is 1 if project i is planned to be up in week w #Then the constraint is: subject to one_project_at_a_time {j in J, w in WEEKS}: sum{i in I} up[w,i] * assign[j,i] <= 1 ;

17 Constraint 3 Information we have from Stage 2: # Constraint 3: # consultants can work only on projects that are pursued The corresponding constraint in the model: subject to only_pursued_projects {i in I, j in J}: assign[j,i] <= proj[i];

18 Summarizing Put all the sets, parameters, variables, constraints and the objective function together to get a complete model. The complete model along with a possible data set is given in the handout.