Linear problem with no short cuts
LP Intro An optimization method Either minimization or maximization What decisions do we have control over (Decision variables) Without constraints/limitations we could drive profits to positive infinity or costs to zero (negative infinity) But, we have constraints that we need to abide to We will model the problem and then hand over to excel to solve Wyndor Two decisions: How many units to make of Product 1 How many units to make of Product 2 Constraints or limitations: Production time available in Plant 1 (4 hours) Production time available in Plant 2 (12 hours) Production time available in Plant 3 (18 hours) Max(3000 * units made of Product 1) + (5000 * units made of Product 2) Constraints: Production time available in Plant 1 (4 hours)(1 * units made of Product 1) + (0 * units made of Product 2) <= 4="" production="" time="" available="" in="" plant="" 2="" (12="" hours)="" (0="" *="" units="" made="" of="" product="" 1)="" +="" (2="" *="" units="" made="" of="" product="" 2)=""><= 12="" production="" time="" available="" in="" plant="" 3="" (18="" hours)="" (3="" *="" units="" made="" of="" product="" 1)="" +="" (2="" *="" units="" made="" of="" product="" 2)=""><= 18="" nonnegativity="" x1:="" how="" many="" units="" to="" make="" of="" product="" 1="" x2:="" how="" many="" units="" to="" make="" of="" product="" 2="" max="" 3000x1="" +="" 5000x2="" constraints:="" production="" time="" available="" in="" plant="" 1="" (4="" hours)="" 1x1="" +="" 0x2=""><= 4="" production="" time="" available="" in="" plant="" 2="" (12="" hours)="" 0x1="" +="" 2x2=""><= 12="" production="" time="" available="" in="" plant="" 3="" (18="" hours)="" 3x1="" +="" 2x2=""><= 18="" nonnegativity="" x1="" x2="" 10="" 20="" max="" 3000="" 5000="" constraints:="" production="" time="" available="" in="" plant="" 1="" (4="" hours)="" 1="" 0=""><= 4="" production="" time="" available="" in="" plant="" 2="" (12="" hours)="" 0="" 2=""><= 12="" production="" time="" available="" in="" plant="" 3="" (18="" hours)="" 3="" 2=""><= 18="" nonnegativity="" manufacturing="" decision="" variables:="" x1:="" how="" many="" units="" to="" make="" of="" product="" 1="" x2:="" how="" many="" units="" to="" make="" of="" product="" 2="" x3:="" how="" many="" units="" to="" make="" of="" product="" 3="" max="" 50x1="" +="" 20x2="" +="" 25x3="" constraints:="" available="" time="" on="" milling="" machine="" (500="" hours)="" 9x1="" +="" 3x2="" +="" 5x3=""><= 500="" available="" time="" on="" lathe="" (350="" hours)="" 5x1="" +="" 4x2="" +="" 0x3=""><= 350="" available="" time="" on="" grinder="" (150="" hours)="" 3x1="" +="" 0x2="" +="" 2x3=""><= 150="" max="" demand="" for="" product="" 3="" (20)="" 0x1="" +="" 0x2="" +="" 1x3=""><= 20="" nonnegativity="" x1="" x2="" x3="" 26="" 0="" 55="" 0="" 20="" max="" 50="" 20="" 25="" constraints:="" available="" time="" on="" milling="" machine="" (500="" hours)="" 9="" 3="" 5=""><= 500="" available="" time="" on="" lathe="" (350="" hours)="" 5="" 4="" 0=""><= 350="" available="" time="" on="" grinder="" (150="" hours)="" 3="" 0="" 2=""><= 150="" max="" demand="" for="" product="" 3="" (20)="" 0="" 0="" 1=""><= 20="" nonnegativity="" staffing="" decision="" variables:="" x1:="" how="" many="" people="" work="" monday="" through="" friday="" x2:="" how="" many="" people="" work="" tuesday="" through="" saturday="" x3:="" how="" many="" people="" work="" wednesday="" through="" sunday="" x4:="" how="" many="" people="" work="" thursday="" through="" monday="" x5:="" how="" many="" people="" work="" friday="" through="" tuesday="" x6:="" how="" many="" people="" work="" saturday="" through="" wednesday="" x7:="" how="" many="" people="" work="" sunday="" through="" thursday="" min="" 200x1="" +="" 204x2="" +="" 208x3="" +="" 208x4="" +="" 208x5="" +="" 208x6="" +="" 204x7="" constraints:="" labor="" hours="" needed="" on="" monday="" (80)="" 8x1="" +="" 0x2="" +="" 0x3="" +="" 8x4="" +="" 8x5="" +="" 8x6="" +="" 8x7="">=80 Labor hours needed on Tuesday (85)8X1+8X2+0X3+0X4+8X5+8X6+8X7>=85 Labor hours needed on Wednesday (75)8X1+8X2+8X3+0X4+0X5+8X6+8X7>=75 Labor hours needed on Thursday (80)8X1+8X2+8X3+8X4+0X5+0X6+8X7>=80 Labor hours needed on Friday (90)8X1+8X2+8X3+8X4+8X5+0X6+0X7>=90 Labor hours needed on Saturday (70)0X1+8X2+8X3+8X4+8X5+8X6+0X7>=70 Labor hours needed on Sunday (60)0X1+0X2+8X3+8X4+8X5+8X6+8X7>=60 Nonnegativity X1X2X3X4X5X6X7 5555555 Min200204208208208208204 Constraints: Labor hours needed on Monday (80)8008888>=80 Labor hours needed on Tuesday (85)8800888>=85 Labor hours needed on Wednesday (75)8880088>=75 Labor hours needed on Thursday (80)8888008>=80 Labor hours needed on Friday (90)8888800>=90 Labor hours needed on Saturday (70)0888880>=70 Labor hours needed on Sunday (60)0088888>=60 Nonnegativity Agricultural Decision Variables: X1:How many acres of corn to plant on the Rice farm X2:How many acres of wheat to plant on the Rice farm X3:How many acres of soy beans to plant on the Rice farm X4:How many acres of corn to plant on the Barns farm X5:How many acres of wheat to plant on the Barns farm X6:How many acres of soy beans to plant on the Barns farm X7:How many acres of corn to plant on the Snyder farm X8:How many acres of wheat to plant on the Snyder farm X9:How many acres of soy beans to plant on the Snyder farm Max1000X1+750X2+250X3+1000X4+750X5+250X6+1000X7+750X8+250X9 Constraints: Usable land on Rice farm (400)1X1+1X2+1X3+0X4+0X5+0X6+0X7+0X8+0X9<= 400="" usable="" land="" on="" barns="" farm="" (600)="" 0x1="" +="" 0x2="" +="" 0x3="" +="" 1x4="" +="" 1x5="" +="" 1x6="" +="" 0x7="" +="" 0x8="" +="" 0x9=""><= 600="" usable="" land="" on="" snyder="" farm="" (300)="" 0x1="" +="" 0x2="" +="" 0x3="" +="" 0x4="" +="" 0x5="" +="" 0x6="" +="" 1x7="" +="" 1x8="" +="" 1x9=""><= 300="" acre="" feet="" of="" water="" on="" rice="" farm="" (600)="" 3x1="" +="" 2x2="" +="" 1x3="" +="" 0x4="" +="" 0x5="" +="" 0x6="" +="" 0x7="" +="" 0x8="" +="" 0x9=""><= 600="" acre="" feet="" of="" water="" on="" barns="" farm="" (800)="" 0x1="" +="" 0x2="" +="" 0x3="" +="" 3x4="" +="" 2x5="" +="" 1x6="" +="" 0x7="" +="" 0x8="" +="" 0x9=""><= 800="" acre="" feet="" of="" water="" on="" snyder="" farm="" (375)="" 0x1="" +="" 0x2="" +="" 0x3="" +="" 0x4="" +="" 0x5="" +="" 0x6="" +="" 3x7="" +="" 2x8="" +="" 1x9=""><= 375="" max="" acres="" of="" corn="" (600)="" 1x1="" +="" 0x2="" +="" 0x3="" +="" 1x4="" +="" 0x5="" +="" 0x6="" +="" 1x7="" +="" 0x8="" +="" 0x9=""><= 600="" max="" acres="" of="" wheat="" (500)="" 0x1="" +="" 1x2="" +="" 0x3="" +="" 0x4="" +="" 1x5="" +="" 0x6="" +="" 0x7="" +="" 1x8="" +="" 0x9=""><= 500="" max="" acres="" of="" soy="" beans="" (325)="" 0x1="" +="" 0x2="" +="" 1x3="" +="" 0x4="" +="" 0x5="" +="" 1x6="" +="" 0x7="" +="" 0x8="" +="" 1x9=""><= 325="" nonnegativity="" x1="" x2="" x3="" x4="" x5="" x6="" x7="" x8="" x9="" 5="" 10="" 15="" 0="" 20="" 25="" 0="" 30="" 35="" 0="" 40="" 0="" 45="" max="" 1000="" 750="" 250="" 1000="" 750="" 250="" 1000="" 750="" 250="" constraints:="" usable="" land="" on="" rice="" farm="" (400)="" 1="" 1="" 1="" 0="" 0="" 0="" 0="" 0="" 0=""><= 400="" 0="" usable="" land="" on="" barns="" farm="" (600)="" 0="" 0="" 0="" 1="" 1="" 1="" 0="" 0="" 0=""><= 600="" 0="" usable="" land="" on="" snyder="" farm="" (300)="" 0="" 0="" 0="" 0="" 0="" 0="" 1="" 1="" 1=""><= 300="" 0="" acre="" feet="" of="" water="" on="" rice="" farm="" (600)="" 3="" 2="" 1="" 0="" 0="" 0="" 0="" 0="" 0=""><= 600="" acre="" feet="" of="" water="" on="" barns="" farm="" (800)="" 0="" 0="" 0="" 3="" 2="" 1="" 0="" 0="" 0=""><= 800="" acre="" feet="" of="" water="" on="" snyder="" farm="" (375)="" 0="" 0="" 0="" 0="" 0="" 0="" 3="" 2="" 1=""><= 375="" max="" acres="" of="" corn="" (600)="" 1="" 0="" 0="" 1="" 0="" 0="" 1="" 0="" 0=""><= 600="" max="" acres="" of="" wheat="" (500)="" 0="" 1="" 0="" 0="" 1="" 0="" 0="" 1="" 0=""><= 500="" max="" acres="" of="" soy="" beans="" (325)="" 0="" 0="" 1="" 0="" 0="" 1="" 0="" 0="" 1=""><= 325="" nonnegativity="" proportionality="" decision="" variables:="" x1:="" how="" many="" acres="" of="" corn="" to="" plant="" on="" the="" rice="" farm="" x2:="" how="" many="" acres="" of="" wheat="" to="" plant="" on="" the="" rice="" farm="" x3:="" how="" many="" acres="" of="" soy="" beans="" to="" plant="" on="" the="" rice="" farm="" x4:="" how="" many="" acres="" of="" corn="" to="" plant="" on="" the="" barns="" farm="" x5:="" how="" many="" acres="" of="" wheat="" to="" plant="" on="" the="" barns="" farm="" x6:="" how="" many="" acres="" of="" soy="" beans="" to="" plant="" on="" the="" barns="" farm="" x7:="" how="" many="" acres="" of="" corn="" to="" plant="" on="" the="" snyder="" farm="" x8:="" how="" many="" acres="" of="" wheat="" to="" plant="" on="" the="" snyder="" farm="" x9:="" how="" many="" acres="" of="" soy="" beans="" to="" plant="" on="" the="" snyder="" farm="" max="" 1000x1="" +="" 750x2="" +="" 250x3="" +="" 1000x4="" +="" 750x5="" +="" 250x6="" +="" 1000x7="" +="" 750x8="" +="" 250x9="" constraints:="" usable="" land="" on="" rice="" farm="" (400)="" 1x1="" +="" 1x2="" +="" 1x3="" +="" 0x4="" +="" 0x5="" +="" 0x6="" +="" 0x7="" +="" 0x8="" +="" 0x9=""><= 400="" usable="" land="" on="" barns="" farm="" (600)="" 0x1="" +="" 0x2="" +="" 0x3="" +="" 1x4="" +="" 1x5="" +="" 1x6="" +="" 0x7="" +="" 0x8="" +="" 0x9=""><= 600="" usable="" land="" on="" snyder="" farm="" (300)="" 0x1="" +="" 0x2="" +="" 0x3="" +="" 0x4="" +="" 0x5="" +="" 0x6="" +="" 1x7="" +="" 1x8="" +="" 1x9=""><= 300="" acre="" feet="" of="" water="" on="" rice="" farm="" (600)="" 3x1="" +="" 2x2="" +="" 1x3="" +="" 0x4="" +="" 0x5="" +="" 0x6="" +="" 0x7="" +="" 0x8="" +="" 0x9=""><= 600="" acre="" feet="" of="" water="" on="" barns="" farm="" (800)="" 0x1="" +="" 0x2="" +="" 0x3="" +="" 3x4="" +="" 2x5="" +="" 1x6="" +="" 0x7="" +="" 0x8="" +="" 0x9=""><= 800="" acre="" feet="" of="" water="" on="" snyder="" farm="" (375)="" 0x1="" +="" 0x2="" +="" 0x3="" +="" 0x4="" +="" 0x5="" +="" 0x6="" +="" 3x7="" +="" 2x8="" +="" 1x9=""><= 375="" max="" acres="" of="" corn="" (600)="" 1x1="" +="" 0x2="" +="" 0x3="" +="" 1x4="" +="" 0x5="" +="" 0x6="" +="" 1x7="" +="" 0x8="" +="" 0x9=""><= 600="" max="" acres="" of="" wheat="" (500)="" 0x1="" +="" 1x2="" +="" 0x3="" +="" 0x4="" +="" 1x5="" +="" 0x6="" +="" 0x7="" +="" 1x8="" +="" 0x9=""><= 500="" max="" acres="" of="" soy="" beans="" (325)="" 0x1="" +="" 0x2="" +="" 1x3="" +="" 0x4="" +="" 0x5="" +="" 1x6="" +="" 0x7="" +="" 0x8="" +="" 1x9=""><= 325="" proportionality:="" rice="" &="" barns="" 600x1="" +="" 600x2="" +="" 600x3="" -="" 400x4="" -="" 400x5="" -="" 400x6="" +="" 0x7="" +="" 0x8="" +="" 0x9="0" proportionality:="" barns="" &="" snyder="" 0x1="" +="" 0x2="" +="" 0x3="" +="" 300x4="" +="" 300x5="" +="" 300x6="" -="" 600x7="" -="" 600x8="" -="" 600x9="0" nonnegativity="" x1="" x2="" x3="" x4="" x5="" x6="" x7="" x8="" x9="" 5="" 10="" 15="" 0="" 20="" 25="" 0="" 30="" 35="" 0="" 40="" 0="" 45="" max="" 1000="" 750="" 250="" 1000="" 750="" 250="" 1000="" 750="" 250="" constraints:="" usable="" land="" on="" rice="" farm="" (400)="" 1="" 1="" 1="" 0="" 0="" 0="" 0="" 0="" 0=""><= 400="" 0="" usable="" land="" on="" barns="" farm="" (600)="" 0="" 0="" 0="" 1="" 1="" 1="" 0="" 0="" 0=""><= 600="" 0="" usable="" land="" on="" snyder="" farm="" (300)="" 0="" 0="" 0="" 0="" 0="" 0="" 1="" 1="" 1=""><= 300="" 0="" acre="" feet="" of="" water="" on="" rice="" farm="" (600)="" 3="" 2="" 1="" 0="" 0="" 0="" 0="" 0="" 0=""><= 600="" acre="" feet="" of="" water="" on="" barns="" farm="" (800)="" 0="" 0="" 0="" 3="" 2="" 1="" 0="" 0="" 0=""><= 800="" acre="" feet="" of="" water="" on="" snyder="" farm="" (375)="" 0="" 0="" 0="" 0="" 0="" 0="" 3="" 2="" 1=""><= 375="" max="" acres="" of="" corn="" (600)="" 1="" 0="" 0="" 1="" 0="" 0="" 1="" 0="" 0=""><= 600="" max="" acres="" of="" wheat="" (500)="" 0="" 1="" 0="" 0="" 1="" 0="" 0="" 1="" 0=""><= 500="" max="" acres="" of="" soy="" beans="" (325)="" 0="" 0="" 1="" 0="" 0="" 1="" 0="" 0="" 1=""><=325 proportionality: rice & barns600600600-400-400-400000=0 proportionality: barns & snyder000300300300-600-600-600=0 nonnegativity proportion of land planted on rice farm = proportion of land planted on barns farm = proportion of land planted on snyder farm proportion of land planted on rice farm = proportion of land planted on barns farm proportion of land planted on barns farm = proportion of land planted on snyder farm proportion of land planted on rice farm = proportion of land planted on snyder farm old rule of math: if a = b and b = c then a = c proportion of land planted on rice farm = proportion of land planted on barns farm (amount of land planted on rice farm)/(amount of land on rice farm) = (amount of land planted on barns farm)/(amount of land on barns farm) (corn planted on rice farm + wheat planted on rice farm + soy beans planted on rice farm)/400 = (corn planted on barns farm + wheat planted on barns farm + soy beans planted on barns farm)/600 (x1 + x2 + x3)/400 = (x4 + x5 + x6)/600 600(x1 + x2 + x3) = 400(x4 + x5 + x6) 600x1 + 600x2 + 600x3 = 400x4 + 400x5 + 400x6 600x1 + 600x2 + 600x3 - 400x4 - 400x5 - 400x6 = 0 600x1 + 600x2 + 600x3 - 400x4 - 400x5 - 400x6 + 0x7 + 0x8 + 0x9 = 0 proportion of land planted on barns farm = proportion of land planted on snyder farm (amount of land planted on barns farm)/(amount of land on barns farm) = (amount of land planted on snyder farm)/(amount of land on snyder farm) (corn planted on barns farm + wheat planted on barns farm + soy beans planted on barns farm)/600 = (corn planted on snyder farm + wheat planted on snyder farm + soy beans planted on snyder farm)/300 (x4 + x5 + x6)/600 = (x7 + x8 + x9)/300 300(x4 + x5 + x6) = 600(x7 + x8 + x9) 300x4 + 300x5 + 300x6 = 600x7 + 600x8 + 600x9 300x4 + 300x5 + 300x6 - 600x7 - 600x8 - 600x9 = 0 0x1 + 0x2 + 0x3 + 300x4 + 300x5 + 300x6 - 600x7 - 600x8 - 600x9 = 0 assignment 1 problem: a chocolate maker has contracted to operate a small candy counter in a fashionable store. to start with, the selection of offerings will be intentionally limited. the counter will offer a regular mix of candy made up of equal parts of cashews, raisins, caramels, and chocolates, and a deluxe mix that is one-half cashews and one-half chocolates, which will be sold in one-pound boxes. in addition, the candy counter will offer individual one-pound boxes of cashews, raisins, caramels, and chocolates. a major attraction of the candy counter is that all candies are made fresh at the counter. however, storage space for supplies and ingredients is limited. bins are available that can hold the amounts shown in the following table: ingredientcapacity (pounds/day) cashews120 raisins200 caramels100 chocolates160 in order to present a good image and to encourage purchases, the counter will make at least 20 boxes of each type of product each day. assume that demand of the candies is such that regardless of candy mix all candy will be sold every day. formulate the problem in order to maximize profit. the profit per box for the various items is as follows: itemprofit per box regular$.80 deluxe$.90 cashews$.70 raisins$.60 caramels$.50 chocolates$.75 deliverable 1 (10 points): 1) give a typed formulation with decision variables clearly defined and all constraints clearly defined. 2) solve the formulation in solver and save the sensitivity report. 3) using sensitivity analysis: if you had $6.00 to spend and it cost $1.50 to increase capacity of cashews by one pound, $1.20 to increase capacity of raisins by one pound, $0.60 to increase capacity of caramels by one pound, and $0.30 to increase capacity of chocolates by one pound; how would you spend the $6.00 to maximize your return and what would be your net return? explain your answer. deliverable 2 (10 points): the chocolate maker is considering broadening his candy selection by adding two candies to his selection: the regular candy with marshmallow topping and the deluxe candy with marshmallow topping. he will add 0.05 pounds of marshmallow topping on one pound of each of the regular or deluxe candies to make the new candies. since he is not sure whether to add these candies he will not require a certain amount be made per day. he has space to add 10 pounds of marshmallow per day and the profit for the marshmallow regular is $1.00/pound and the marshmallow deluxe is $1.05/pound. also, the production of the marshmallow regular cannot be greater than the total production of the regular and deluxe candies. formulate the problem in order to maximize profit. requirements: 1) give a typed formulation with decision variables clearly defined and all constraints clearly defined. 2) solve the formulation in solver. problem: a chocolate maker has contracted to operate a small candy counter in a fashionable store. to start with, the selection of offerings will be intentionally limited. the counter will offer a regular mix of candy made up of equal parts of cashews, raisins, caramels, and chocolates, and a deluxe mix that is one - half cashews and one - half chocolates, which will be sold in one - pound boxes. in addition, the candy counter will offer indi vidual one - pound boxes of cashews, raisins, caramels, and chocolates. a major attraction of the candy counter is that all candies are made fresh at the counter. however, storage space for supplies and ingredients is limited. bins are available that can ho ld the amounts shown in the following table: ingredient capacity (pounds/day) cashews 120 raisins 200 caramels 100 chocolates 160 in order to present a good image and to encourage purchases, the counter will make at least 20 boxes of each type of product each day. assume that demand of the candies is such that regardless of candy mix all candy will be sold every day. formulate the problem in order to maximize profit. the profit per box for the various items is as follows: item profit per box regular $.80 deluxe $.90 cashews $.70 raisins $.60 caramels $.50 chocolates $.75 deliverable 1 (10 points): 1) give a typed formulation with decision variables clearly defined and all cons traints clearly defined. 2) solve the formulation in solver and save the sensitivity report. 3) using sensitivity analysis: if you had $6.00 to spend and it cost $1.50 to increase capacity of cashews by one pound, $1.20 to increase capacity of raisins by one p ound, $0.60 to increase capacity of caramels by one pound, and $0.30 to increase capacity of chocolates by one pound; how would you spend the $6.00 to maximize your return and what would be your net ret urn? explain your answer. deliverable 2 (10 points): the chocolate maker is considering broadening his candy selection by adding two candies to his selection: the regular candy with marshmallow topping and the deluxe candy with marshmallow topping. he will add 0.05 pounds of marshmallow topping on one pound of each of the regular or deluxe candies to make the new candies. since he is not sure whether to add these candies he will not require a certain amount be made per day. he has space to add 10 pounds of marshmallow per day and the profit for the marshmallow regular is $1.00/pound and the marshmallow deluxe is $1.05/pound. also, the production of the marshmallow regular cannot be greater than the total production of the regular and deluxe candies. problem: a chocolate maker has contracted to operate a small candy counter in a fashionable store. to start with, the selection of offerings will be intentionally limited. the counter will offer a regular mix of candy made up of equal parts of cashews, raisins, caramels, and chocolates, and a deluxe mix that is one-half cashews and one-half 325="" proportionality:="" rice="" &="" barns="" 600="" 600="" 600="" -400="" -400="" -400="" 0="" 0="" 0="0" proportionality:="" barns="" &="" snyder="" 0="" 0="" 0="" 300="" 300="" 300="" -600="" -600="" -600="0" nonnegativity="" proportion="" of="" land="" planted="" on="" rice="" farm="Proportion" of="" land="" planted="" on="" barns="" farm="Proportion" of="" land="" planted="" on="" snyder="" farm="" proportion="" of="" land="" planted="" on="" rice="" farm="Proportion" of="" land="" planted="" on="" barns="" farm="" proportion="" of="" land="" planted="" on="" barns="" farm="Proportion" of="" land="" planted="" on="" snyder="" farm="" proportion="" of="" land="" planted="" on="" rice="" farm="Proportion" of="" land="" planted="" on="" snyder="" farm="" old="" rule="" of="" math:="" if="" a="b" and="" b="c" then="" a="c" proportion="" of="" land="" planted="" on="" rice="" farm="Proportion" of="" land="" planted="" on="" barns="" farm="" (amount="" of="" land="" planted="" on="" rice="" farm)/(amount="" of="" land="" on="" rice="" farm)="(Amount" of="" land="" planted="" on="" barns="" farm)/(amount="" of="" land="" on="" barns="" farm)="" (corn="" planted="" on="" rice="" farm="" +="" wheat="" planted="" on="" rice="" farm="" +="" soy="" beans="" planted="" on="" rice="" farm)/400="(Corn" planted="" on="" barns="" farm="" +="" wheat="" planted="" on="" barns="" farm="" +="" soy="" beans="" planted="" on="" barns="" farm)/600="" (x1="" +="" x2="" +="" x3)/400="(X4" +="" x5="" +="" x6)/600="" 600(x1="" +="" x2="" +="" x3)="400(X4" +="" x5="" +="" x6)="" 600x1="" +="" 600x2="" +="" 600x3="400X4" +="" 400x5="" +="" 400x6="" 600x1="" +="" 600x2="" +="" 600x3="" -="" 400x4="" -="" 400x5="" -="" 400x6="0" 600x1="" +="" 600x2="" +="" 600x3="" -="" 400x4="" -="" 400x5="" -="" 400x6="" +="" 0x7="" +="" 0x8="" +="" 0x9="0" proportion="" of="" land="" planted="" on="" barns="" farm="Proportion" of="" land="" planted="" on="" snyder="" farm="" (amount="" of="" land="" planted="" on="" barns="" farm)/(amount="" of="" land="" on="" barns="" farm)="(Amount" of="" land="" planted="" on="" snyder="" farm)/(amount="" of="" land="" on="" snyder="" farm)="" (corn="" planted="" on="" barns="" farm="" +="" wheat="" planted="" on="" barns="" farm="" +="" soy="" beans="" planted="" on="" barns="" farm)/600="(Corn" planted="" on="" snyder="" farm="" +="" wheat="" planted="" on="" snyder="" farm="" +="" soy="" beans="" planted="" on="" snyder="" farm)/300="" (x4="" +="" x5="" +="" x6)/600="(X7" +="" x8="" +="" x9)/300="" 300(x4="" +="" x5="" +="" x6)="600(X7" +="" x8="" +="" x9)="" 300x4="" +="" 300x5="" +="" 300x6="600X7" +="" 600x8="" +="" 600x9="" 300x4="" +="" 300x5="" +="" 300x6="" -="" 600x7="" -="" 600x8="" -="" 600x9="0" 0x1="" +="" 0x2="" +="" 0x3="" +="" 300x4="" +="" 300x5="" +="" 300x6="" -="" 600x7="" -="" 600x8="" -="" 600x9="0" assignment="" 1="" problem:="" a="" chocolate="" maker="" has="" contracted="" to="" operate="" a="" small="" candy="" counter="" in="" a="" fashionable="" store.="" to="" start="" with,="" the="" selection="" of="" offerings="" will="" be="" intentionally="" limited.="" the="" counter="" will="" offer="" a="" regular="" mix="" of="" candy="" made="" up="" of="" equal="" parts="" of="" cashews,="" raisins,="" caramels,="" and="" chocolates,="" and="" a="" deluxe="" mix="" that="" is="" one-half="" cashews="" and="" one-half="" chocolates,="" which="" will="" be="" sold="" in="" one-pound="" boxes.="" in="" addition,="" the="" candy="" counter="" will="" offer="" individual="" one-pound="" boxes="" of="" cashews,="" raisins,="" caramels,="" and="" chocolates.="" a="" major="" attraction="" of="" the="" candy="" counter="" is="" that="" all="" candies="" are="" made="" fresh="" at="" the="" counter.="" however,="" storage="" space="" for="" supplies="" and="" ingredients="" is="" limited.="" bins="" are="" available="" that="" can="" hold="" the="" amounts="" shown="" in="" the="" following="" table:="" ingredient="" capacity="" (pounds/day)="" cashews="" 120="" raisins="" 200="" caramels="" 100="" chocolates="" 160="" in="" order="" to="" present="" a="" good="" image="" and="" to="" encourage="" purchases,="" the="" counter="" will="" make="" at="" least="" 20="" boxes="" of="" each="" type="" of="" product="" each="" day.="" assume="" that="" demand="" of="" the="" candies="" is="" such="" that="" regardless="" of="" candy="" mix="" all="" candy="" will="" be="" sold="" every="" day.="" formulate="" the="" problem="" in="" order="" to="" maximize="" profit.="" the="" profit="" per="" box="" for="" the="" various="" items="" is="" as="" follows:="" item="" profit="" per="" box="" regular="" $.80="" deluxe="" $.90="" cashews="" $.70="" raisins="" $.60="" caramels="" $.50="" chocolates="" $.75="" deliverable="" 1="" (10="" points):="" 1)="" give="" a="" typed="" formulation="" with="" decision="" variables="" clearly="" defined="" and="" all="" constraints="" clearly="" defined.="" 2)="" solve="" the="" formulation="" in="" solver="" and="" save="" the="" sensitivity="" report.="" 3)="" using="" sensitivity="" analysis:="" if="" you="" had="" $6.00="" to="" spend="" and="" it="" cost="" $1.50="" to="" increase="" capacity="" of="" cashews="" by="" one="" pound,="" $1.20="" to="" increase="" capacity="" of="" raisins="" by="" one="" pound,="" $0.60="" to="" increase="" capacity="" of="" caramels="" by="" one="" pound,="" and="" $0.30="" to="" increase="" capacity="" of="" chocolates="" by="" one="" pound;="" how="" would="" you="" spend="" the="" $6.00="" to="" maximize="" your="" return="" and="" what="" would="" be="" your="" net="" return?="" explain="" your="" answer.="" deliverable="" 2="" (10="" points):="" the="" chocolate="" maker="" is="" considering="" broadening="" his="" candy="" selection="" by="" adding="" two="" candies="" to="" his="" selection:="" the="" regular="" candy="" with="" marshmallow="" topping="" and="" the="" deluxe="" candy="" with="" marshmallow="" topping.="" he="" will="" add="" 0.05="" pounds="" of="" marshmallow="" topping="" on="" one="" pound="" of="" each="" of="" the="" regular="" or="" deluxe="" candies="" to="" make="" the="" new="" candies.="" since="" he="" is="" not="" sure="" whether="" to="" add="" these="" candies="" he="" will="" not="" require="" a="" certain="" amount="" be="" made="" per="" day.="" he="" has="" space="" to="" add="" 10="" pounds="" of="" marshmallow="" per="" day="" and="" the="" profit="" for="" the="" marshmallow="" regular="" is="" $1.00/pound="" and="" the="" marshmallow="" deluxe="" is="" $1.05/pound.="" also,="" the="" production="" of="" the="" marshmallow="" regular="" cannot="" be="" greater="" than="" the="" total="" production="" of="" the="" regular="" and="" deluxe="" candies.="" formulate="" the="" problem="" in="" order="" to="" maximize="" profit.="" requirements:="" 1)="" give="" a="" typed="" formulation="" with="" decision="" variables="" clearly="" defined="" and="" all="" constraints="" clearly="" defined.="" 2)="" solve="" the="" formulation="" in="" solver.="" problem:="" a="" chocolate="" maker="" has="" contracted="" to="" operate="" a="" small="" candy="" counter="" in="" a="" fashionable="" store.="" to="" start="" with,="" the="" selection="" of="" offerings="" will="" be="" intentionally="" limited.="" the="" counter="" will="" offer="" a="" regular="" mix="" of="" candy="" made="" up="" of="" equal="" parts="" of="" cashews,="" raisins,="" caramels,="" and="" chocolates,="" and="" a="" deluxe="" mix="" that="" is="" one="" -="" half="" cashews="" and="" one="" -="" half="" chocolates,="" which="" will="" be="" sold="" in="" one="" -="" pound="" boxes.="" in="" addition,="" the="" candy="" counter="" will="" offer="" indi="" vidual="" one="" -="" pound="" boxes="" of="" cashews,="" raisins,="" caramels,="" and="" chocolates.="" a="" major="" attraction="" of="" the="" candy="" counter="" is="" that="" all="" candies="" are="" made="" fresh="" at="" the="" counter.="" however,="" storage="" space="" for="" supplies="" and="" ingredients="" is="" limited.="" bins="" are="" available="" that="" can="" ho="" ld="" the="" amounts="" shown="" in="" the="" following="" table:="" ingredient="" capacity="" (pounds/day)="" cashews="" 120="" raisins="" 200="" caramels="" 100="" chocolates="" 160="" in="" order="" to="" present="" a="" good="" image="" and="" to="" encourage="" purchases,="" the="" counter="" will="" make="" at="" least="" 20="" boxes="" of="" each="" type="" of="" product="" each="" day.="" assume="" that="" demand="" of="" the="" candies="" is="" such="" that="" regardless="" of="" candy="" mix="" all="" candy="" will="" be="" sold="" every="" day.="" formulate="" the="" problem="" in="" order="" to="" maximize="" profit.="" the="" profit="" per="" box="" for="" the="" various="" items="" is="" as="" follows:="" item="" profit="" per="" box="" regular="" $.80="" deluxe="" $.90="" cashews="" $.70="" raisins="" $.60="" caramels="" $.50="" chocolates="" $.75="" deliverable="" 1="" (10="" points):="" 1)="" give="" a="" typed="" formulation="" with="" decision="" variables="" clearly="" defined="" and="" all="" cons="" traints="" clearly="" defined.="" 2)="" solve="" the="" formulation="" in="" solver="" and="" save="" the="" sensitivity="" report.="" 3)="" using="" sensitivity="" analysis:="" if="" you="" had="" $6.00="" to="" spend="" and="" it="" cost="" $1.50="" to="" increase="" capacity="" of="" cashews="" by="" one="" pound,="" $1.20="" to="" increase="" capacity="" of="" raisins="" by="" one="" p="" ound,="" $0.60="" to="" increase="" capacity="" of="" caramels="" by="" one="" pound,="" and="" $0.30="" to="" increase="" capacity="" of="" chocolates="" by="" one="" pound;="" how="" would="" you="" spend="" the="" $6.00="" to="" maximize="" your="" return="" and="" what="" would="" be="" your="" net="" ret="" urn?="" explain="" your="" answer.="" deliverable="" 2="" (10="" points):="" the="" chocolate="" maker="" is="" considering="" broadening="" his="" candy="" selection="" by="" adding="" two="" candies="" to="" his="" selection:="" the="" regular="" candy="" with="" marshmallow="" topping="" and="" the="" deluxe="" candy="" with="" marshmallow="" topping.="" he="" will="" add="" 0.05="" pounds="" of="" marshmallow="" topping="" on="" one="" pound="" of="" each="" of="" the="" regular="" or="" deluxe="" candies="" to="" make="" the="" new="" candies.="" since="" he="" is="" not="" sure="" whether="" to="" add="" these="" candies="" he="" will="" not="" require="" a="" certain="" amount="" be="" made="" per="" day.="" he="" has="" space="" to="" add="" 10="" pounds="" of="" marshmallow="" per="" day="" and="" the="" profit="" for="" the="" marshmallow="" regular="" is="" $1.00/pound="" and="" the="" marshmallow="" deluxe="" is="" $1.05/pound.="" also,="" the="" production="" of="" the="" marshmallow="" regular="" cannot="" be="" greater="" than="" the="" total="" production="" of="" the="" regular="" and="" deluxe="" candies.="" problem:="" a="" chocolate="" maker="" has="" contracted="" to="" operate="" a="" small="" candy="" counter="" in="" a="" fashionable="" store.="" to="" start="" with,="" the="" selection="" of="" offerings="" will="" be="" intentionally="" limited.="" the="" counter="" will="" offer="" a="" regular="" mix="" of="" candy="" made="" up="" of="" equal="" parts="" of="" cashews,="" raisins,="" caramels,="" and="" chocolates,="" and="" a="" deluxe="" mix="" that="" is="" one-half="" cashews="" and="">