A production department requires different numbers of full-time employees on different days of the week. The number of full-time employees required on each day is given (see Table). Union rules state that each full-time employee must work five consecutive days and then two days off. For example, if an employee works Monday to Friday, he/she will be off on Saturday and Sunday. Suppose that the production department wants to meet its daily requirements using only full-time employees. Formulate an LP that the production department can use to minimize the number of full time employees who must be hired. Day # of Full-Time Employees Required Monday 17 Tuesday 13 Wednesday 15 Thursday 19 Friday 14 Saturday 16 Sunday 11 a) Suppose that the number of workers needed on day i is di . Let wi be the actual number of workers on day i. Formulate an LP where the “cost” of having too many workers on day i is fi (wi – di). b) Suppose that the production department wants to minimize the maximum of the surpluses on each day (max (w1 – d1, w2 – d2,…, w7 – d7)). Formulate an LP. c) Suppose that the production department wants to ensure that at least 30% of the workers have Sunday off. Formulate a constraint for this case. Sunday. Suppose that the production department wants to meet its daily requirements using onlyfull-time employees. Formulate an LP that the production department can use to minimize thenumber of full time employees who must be hired.# of Full-Time Employees Required17DayMondayTuesdayWednesdayThursdayFridaySaturdaySunday131519141611Incorrect Formulation:X;: the number of employees working on day iMinimize Z=x1 + x2 + X3 + X4 + Xs + X6 + X717 ב s.t.xX2 2 13X3 2 15X4 2 19Xs 2 14X6 2 16X72 11Xị 2 0, vi = 1,2,...,7In this formulation, objective function is not the number of full-time employees. The current objectivefunction counts each employee five times, not once. For example, each employee who starts work onMonday works Monday to Friday and is included in x1, x., x3, X4 and xs. The key to corectlyformulating this problem is to realize that the production department's primary decision is not howmany people are working each day but rather how many people begin work on each day of the week.Correct Formulation:Xị : the number of employees beginning work on day iMinimize Z=x, +x, +x, +x, +X, +x, +X,+*,+x, +*,X, +X, +x, +X, +X, +x, +X +x, +x, +x, +*, + x, + x, + x, + x,x, +x, +x, +X,+x,X, + X, + X, + x, + x, 211x 20, Vi = ,2,.,7+X, +x, + x, +x,217+ x, +x, + x,2 13x+x,2 15*X,2 192 14s.t.2 16

For the above question, we can assume the variables as follows: Let X1 be the number of workers who…

Answered: Formulate an LP

Practical Management Science

6th Edition

ISBN:9781337406659

Author:WINSTON, Wayne L.

Publisher:WINSTON, Wayne L.

Chapter2: Introduction To Spreadsheet Modeling

Section: Chapter Questions

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A production department requires different numbers of full-time employees on different days of the week. The number of full-time employees required on each day is given (see Table). Union rules state that each full-time employee must work five consecutive days and then two days off. For example, if an employee works Monday to Friday, he/she will be off on Saturday and Sunday. Suppose that the production department wants to meet its daily requirements using only full-time employees. Formulate an LP that the production department can use to minimize the number of full time employees who must be hired. Day # of Full-Time Employees Required Monday 17 Tuesday 13 Wednesday 15 Thursday 19 Friday 14 Saturday 16 Sunday 11 a) Suppose that the number of workers needed on day i is di . Let wi be the actual number of workers on day i. Formulate an LP where the “cost” of having too many workers on day i is fi (wi – di). b) Suppose that the production department wants to minimize the maximum of the surpluses on each day (max (w1 – d1, w2 – d2,…, w7 – d7)). Formulate an LP. c) Suppose that the production department wants to ensure that at least 30% of the workers have Sunday off. Formulate a constraint for this case.

Sunday. Suppose that the production department wants to meet its daily requirements using only
full-time employees. Formulate an LP that the production department can use to minimize the
number of full time employees who must be hired.
# of Full-Time Employees Required
17
Day
Monday
Tuesday
Wednesday
Thursday
Friday
Saturday
Sunday
13
15
19
14
16
11
Incorrect Formulation:
X;: the number of employees working on day i
Minimize Z=x1 + x2 + X3 + X4 + Xs + X6 + X7
17 ב s.t.x
X2 2 13
X3 2 15
X4 2 19
Xs 2 14
X6 2 16
X72 11
Xị 2 0, vi = 1,2,...,7
In this formulation, objective function is not the number of full-time employees. The current objective
function counts each employee five times, not once. For example, each employee who starts work on
Monday works Monday to Friday and is included in x1, x., x3, X4 and xs. The key to corectly
formulating this problem is to realize that the production department's primary decision is not how
many people are working each day but rather how many people begin work on each day of the week.
Correct Formulation:
Xị : the number of employees beginning work on day i
Minimize Z=x, +x, +x, +x, +X, +x, +X,
+*,+x, +*,
X, +
X, +x, +
X, +X, +x, +
X +x, +x, +x, +
*, + x, + x, + x, + x,
x, +x, +x, +X,+x,
X, + X, + X, + x, + x, 211
x 20, Vi = ,2,.,7
+X, +x, + x, +x,217
+ x, +x, + x,2 13
x+x,2 15
*X,2 19
2 14
s.t.
2 16

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