Question

Gandhi Clothing Company produces shirts and pants. Each shirt requires 2 sq yd of cloth, each pair of pants, 3. During the next two months, the following demands for shirts and pants must be met (on time): month 1—10 shirts, 15 pairs of pants; month 2—12 shirts, 14 pairs of pants. During each month, the following resources are available: month 1—90 sq yd of cloth; month 2—60 sq yd. (Cloth that is available during month 1 may, if unused during month 1, be used during month 2.) During each month, it costs $4 to make an article of clothing with regular-time labor and$8 with overtime labor. During each month, a total of at most 25 articles of clothing may be produced with regular-time labor, and an unlimited number of articles of clothing may be produced with overtime labor. At the end of each month, a holding cost of $3 per article of clothing is assessed. Formulate an LP that can be used to meet demands for the next two months (on time) at minimum cost. Assume that at the beginning of month 1, 1 shirt and 2 pairs of pants are available.

Solution

Verified

Step 1

1 of 2

Clothing company produces shirts and pants. Each shirt reuqires 22 square yards for production while pants require 3.3. For first month demands are 1010 shirts and 1515 pants while for the month two demands are 1212 shirts and 1414 pants. Therefore we introduce ds1=10,ds2=12ds_1=10,ds_2=12 and dp1=15,dp2=14.dp_1=15,dp_2=14. Those demands must be met on time. In the first month 9090 square yards of cloth is available while in second month 6060 sq yd is available. However, if unused, cloth from the first month can be used in the second month. During each month production in regular time costs 4$4\$ while in overtime it costs8$.8\$. During each month at most 2525 articles of clothing may be produced during reuglar time labor. Unlimited amount of clothing can be produced during the overtime labor. Obviously it doesn't represent the problem for minimization since we want to minimize the cost of the production. Similarly we could argue that we could skip regular time labor and jump immediately to overtime, but this will easily turn out not to be optimal. Holding cost is assesed at the end of the month with price of 3$3\$ per article of clothing. Also let us say that at the beggining of month there is11shirt and22pairs of pants in the inventory.Notice that overtime labour is possible, therefore we introducexp1,xp2,xs1xp_1,xp_2,xs_1andxs2xs_2as a decision variables that represent number of pants or shirts made in overtime labour in their respective month. Let us define the following:$ s0=1,st=st1+St+xstdst,s_0=1,s_t=s_{t-1}+S_t+xs_t-ds_{t}, p0=2,pt=pt1+Pt+xptdpt,p_0=2,p_t=p_{t-1}+P_t+xp_t-dp_t, wherewhereS_tandandP_tarenumberofshirtsandpantsmadeinthemonthare number of shirts and pants made in the montht(respectively).Also(respectively). Alsods_tandanddp_tisdemandforshirtsandpantsinmonthis demand for shirts and pants in montht.$Those demands have to be met, ofcourse. \ Since at most2525articles of clothing can be produced by regular time labour, we simply have:$ S1+P125,S2+P225.S_1+P_1\leq25, S_2+P_2\leq25. NowletusformulateanLPthatminimizesthecostofproductionthatmeetsdemandsintime.LetNow let us formulate an LP that minimizes the cost of production that meets demands in time. Letzbetheobjectivefunction.Itfollows:be the objective function. It follows: minz=4(P1+P2+S1+S2)+8(xp1+xp2+xs1+xs2)+3(s1+s2+p1+p2).\min z=4(P_1+P_2+S_1+S_2)+8(xp_1+xp_2+xs_1+xs_2)+3(s_1+s_2+p_1+p_2). $We are left to model the constraints which make sure we are using no more than9090sq yd during month11and no more than6060sq yd during month$2.2S1+2xs1+3P1+3xp1902S_1+2xs_1+3P_1+3xp_1\leq90 2S2+3P2+2xs2+3xp260+90(2S1+2xs1+3P13xp1).2S_2+3P_2+2xs_2+3xp_2\leq60+90-(2S_1+2xs_1+3P_1-3xp_1).$ Therefore we are done formulating an LP.

Create an account to view solutions

By signing up, you accept Quizlet's Terms of Service and Privacy Policy
Continue with GoogleContinue with Facebook

Create an account to view solutions

By signing up, you accept Quizlet's Terms of Service and Privacy Policy
Continue with GoogleContinue with Facebook

Recommended textbook solutions

Introduction to Operations Research 10th Edition by Frederick S. Hillier

Introduction to Operations Research

10th EditionFrederick S. Hillier
Verified solutions
Operations Research: Applications and Algorithms 4th Edition by Wayne L Winston

Operations Research: Applications and Algorithms

4th EditionWayne L Winston
1,261 solutions
STAT: Behavioral Sciences 2nd Edition by Gary Heiman

STAT: Behavioral Sciences

2nd EditionGary Heiman
A Mathematical Look at Politics 1st Edition by Daniel H. Ullman, E. Arthur Robinson Jr.

A Mathematical Look at Politics

1st EditionDaniel H. Ullman, E. Arthur Robinson Jr.

Related questions