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OPIM101 – Decision Analysis Instructor: FENG Guiyun

NOTE THAT THE REVIEW QUESTIONS ARE FROM MIDTERM EXAMS OF LAST YEARS. SOME OF THE REVIEW QUESTIONS MAY NOT BE NOT COVERED BY OUR CLASS. PLEASE IGNORE THESE QUESTIONS IF THE TOPIC IS NOT COVERED IN OUR CLASS.

True/False Questions: 1. In a linear programming problem, the constraints must be linear, but the objective function may be nonlinear. FALSE: Objective function should also be linear. 2. If the isoprofit line is not parallel to a constraint in a LP with bounded feasible region, then the solution must be unique. TRUE 3. An LP model can have exactly two optimal solutions. FALSE: An LP model can have more than one optimal solution. However, if that is the case, there will always be an infinite number of alternative solutions, namely all the points lying on the line segment connecting any two different optimal solutions will be optimal. 4. The following two objective functions for an LP model are equivalent. That is, if they are both used, one at a time, to solve a problem with the same constraints, the optimal values for 𝑥" and 𝑥# will be the same in both cases. max 2𝑥" + 3𝑥# min −2𝑥" − 3𝑥# TRUE. 5. In Excel, selecting ‘Simplex LP’ will prevent the linear programming problem from having alternative solutions. FALSE. That option only guarantees that the final solution is optimal (not unique) and that we can obtain the sensitivity analysis. It can only be selected if the underlying model is linear. 6. Decision variables may also be called parameters. FALSE: Parameters are uncontrollable variables

OPIM101 – Decision Analysis Instructor: FENG Guiyun

7. Model variables can be controllable or uncontrollable. TRUE 8. The optimal solution to a bounded feasible linear programming problem must always lie on a constraint. TRUE 9. In a linear program, the constraints must be linear, but the objective function may be nonlinear. FALSE: Objective function should also be linear

10. Anytime we have an isoprofit line that is parallel to a non-redundant constraint for a feasible LP, we have alternative optimal solutions. FALSE: It depends on the improvement direction, iso-profit line may not intersect with the parallel constraint. 11. Blending problems arise when one must decide which of two or more ingredients is to be chosen to produce a product. FALSE: can mix ingredients 12. In the linear programming formulation of the assignment problem, all constraint coefficients are equal to one. TRUE 13. In a transportation problem, each destination must be supplied by one and only one source. FALSE: each destination can be supplied more than one source 14. An integer programming solution for a maximization problem can never produce a greater profit objective than the LP solution to the same problem. TRUE 15. In a linear program formulation, a constraint requiring an integer outcome may not be included in the constraint set of the formulation. FALSE: x=2 can be a legitimate constraint for any problems

OPIM101 – Decision Analysis Instructor: FENG Guiyun

16. Requiring an integer solution to a linear programming problem does not increase the size of the feasible region. TRUE

OPIM101 – Decision Analysis Instructor: FENG Guiyun

Multiple Choice Questions: 1. Adding a constraint to a linear programming (maximization) problem may result in, but is not necessarily limited to, a. a decrease in the value of the objective function. b. an increase in the value of the objective function. c. no change to the value of the objective function. d. either a or c depending on the constraint. e. either b or c depending on the constraint. ANSWER: D

2. If one changes one of the objective coefficients in the objective function of an LP, a. the feasible region will change. b. the slope of the iso-profit or iso-cost line may change. c. the optimal solution to the LP is sure to no longer be optimal. d. all of the above e. none of the above ANSWER: B 3. Assume that the shadow price for a resource in a linear programming (maximization) problem is $20 and the allowable increase and decrease are 50 and 120 units respectively. The current amount of resource available is 100 units. If someone offers to sell you additional resource at $25 per unit, how much should you buy? a. None b. 50 units c. 120 units d. 100 units e. The question cannot be answered with the given information ANSWER : A 4. Which of the following statement is TRUE? a. Any optimal solution (if exists) to a bounded linear programming model always occurs at a corner point of the feasible region. b. Any optimal solution (if exists) to an integer programming model always occurs at a corner point of the feasible region. c. Both a and b. d. Neither a nor b.

OPIM101 – Decision Analysis Instructor: FENG Guiyun

ANSWER: D a is not true because of alternative optimal solutions. 5. Which one of the following statement is TRUE for an LP problem? a. Non-negativity constraints restrict the decision variables to positive values. b. A maximization problem typically requires finding the maximum value of an objective while simultaneously maximizing usage of the resource constraints. c. If the feasible region is unbounded, then the LP problem is unbounded. d. If the iso-profit (or iso-cost) line is parallel to one of the constraints, we will have multiple optimal points. e. None of the above ANSWER: E

6. Which of the following constraints are not linear so cannot be included as a constraint in a linear programming problem? a. 2𝑥" + 𝑥# − 3𝑥( ≥ 50 b. 2𝑥" + 𝑥# ≥ 60 " c. 4𝑥" − 𝑥# = 75 ( d.

(01 2#0 3 4(05 01 203 20 5

≤ 0.9

e. b and d ANSWER: B 7. The coefficients of the variables in the constraints that represent the amount of resources needed to produce one unit of the variable are called a. technological coefficients. b. objective coefficients. c. shadow prices. d. reduced costs. e. none of the above ANSWER: A

8. If you remove a binding constraint from a (maximization) LP model, a. neither the feasible region nor the optimal objective function value will change; b. the feasible region will increase, but the objective function value will not change; c. the feasible region and the objective function value will increase; d. the feasible region will increase, but the objective function value will decrease. ANSWER: C

OPIM101 – Decision Analysis Instructor: FENG Guiyun

9. In a maximization LP model, assume that the objective function coefficient of a variable increases by an amount that is larger than the Allowable Increase mentioned in the Sensitivity Analysis. Then we can say that: a. the optimal production mix and profit stay the same, b. the optimal production mix changes and the profit increases, c. the optimal production mix stays the same but the profit increases, d. both (2) and (3) are possible. ANSWER: B 10. When appropriate, the optimal solution to a maximization linear programming problem can be found by graphing the feasible region and (a) finding the profit at every corner point of the feasible region to see which one gives the highest value. (b) moving the isoprofit lines towards the origin in a parallel fashion until the last point in the feasible region is encountered. (c) locating the point that is highest on the graph. (d) none of the above (e) all of the above ANSWER: A 11. If two corner points tie for the best value of the objective function in an LP, then (a) (b) (c) (d) (e)

the solution is infeasible. there are alternative optimal solutions. the problem is unbounded. the problem has a unique optimal solution. none of the above

ANSWER: B 12. Sensitivity analyses are used to examine the effects of changes in (a) (b) (c) (d) (e) ANSWER: D

contribution rates (e.g., prices) for each decision variable. technological coefficients. available resources. all of the above none of the above

OPIM101 – Decision Analysis Instructor: FENG Guiyun

13. The condition when there is no solution that satisfies all the constraints is called (a) (b) (c) (d) (e)

boundedness. redundancy. optimality. dependency. none of the above

ANSWER: E, infeasibility

14. In order for a linear programming problem to have a unique solution, the solution must exist (a) (b) (c) (d) (e)

at the intersection of the non-negativity constraints. at the intersection of a non-negativity constraint and a resource constraint. at the intersection of the objective function and a constraint. at the intersection of two or more constraints. none of the above

ANSWER: D 15. In order for a linear programming problem to have multiple solutions, the solution must exist (a) (b) (c) (d) (e)

at the intersection of the non-negativity constraints. on a non-redundant constraint parallel to the objective function. at the intersection of the objective function and a constraint. at the intersection of three or more constraints. none of the above

ANSWER: B 16. When formulating transportation LP problems, constraints usually deal with the (a) (b) (c) (d) (e) ANSWER: E

number of items to be transported. shipping cost associated with transporting goods. distance goods are to be transported. number of origins and destinations. capacities of origins and requirements of destinations.

OPIM101 – Decision Analysis Instructor: FENG Guiyun

17. A type of integer programming is (a) (b) (c) (d)

pure. mixed. zero-one. all of the above

ANSWER: D 18. An integer programming (maximization) problem was first solved as a linear programming problem, and the objective function value (profit) was $253.67. The two decision variables (X, Y) in the problem had values of X = 12.45 and Y = 32.75. If there is a single optimal solution, which of the following must be true for the optimal integer solution to this problem? (a) (b) (c) (d) (e)

X = 12 Y = 32 X = 12 Y = 33 the objective function value must be less than $253.67 the objective function value will be greater than $253.67 none of the above

ANSWER: C 19. Solving an integer programming problem by rounding off answers obtained by solving it as a linear programming problem (using simplex) to values in the feasible region, we find that (a) the values of decision variables obtained by rounding off are always very close to the optimal values. (b) the value of the objective function for a maximization problem will likely be less than that for the simplex solution. (c) the value of the objective function for a minimization problem will likely be less than that for the simplex solution. (d) all constraints are satisfied exactly. (e) none of the above ANSWER: B

20. Consider the following 0-1 integer programming problem: Minimize Subject to:

20X +36Y + 24Z 2X + 4Y + 3Z ≥ 7 12X + 8Y + 10Z ≥ 25 X, Y, Z must be 0 or 1

OPIM101 – Decision Analysis Instructor: FENG Guiyun

If we wish to add the constraint that X must be positive, and that only Y or Z but not both can be positive, how would the additional constraint(s) be written? (a) (b) (c) (d) (e)

X + Y + Z ≤ 3, Y + Z ≤1 X ≤1, Y + Z = 1 X ≤ 2, and Y ≤ 2, and Z ≤ 2 X = 1, Y + Z ≤1 none of the above

ANSWER: D 21. A feasible solution to a linear programming problem a. must be corner point of the feasible region. b. must satisfy all of the problem's constraints simultaneously. c. needs not satisfy all of the constraints. d. must give the maximum possible profit. e. must give the minimum possible cost. ANSWER: B

OPIM101 – Decision Analysis Instructor: FENG Guiyun

Quantitative Questions: Question 1: Sourcing Problem Alfonso, the owner of the corner shop down the road from your apartment, has learnt that you are pursuing a degree in business. He asks whether you can help him decide where to order his stock from, how much to order of the different items he carries. He has categorized his stock keeping units into dairy products (milk, yoghurt, etc) and sweets. His small shop has a limited shelf space of 35 meters available to those products. From past experience, Alfonso knows that his demand for the different categories will be around 1000 units of dairy products, 800 sweets. Since all these products are perishable, he needs to restock every week. The shelf space required per unit of each product is 3 cm for a unit of dairy product and 1 cm for a unit of sweet. As a part of his marketing strategy, Alfonso purchases from three different suppliers, A, B and C. The price per item and the capacity of the van varies for each supplier as outlined in the following table. (Assume that each product unit – be it dairy product or sweet – takes up one unit of van capacity.) Because of the contractual agreements between

A B C

Dairy Products Sweets $0.50 $2.00 $2.05 $0.40 $2.15

Van Capacity 600 500 600

Alfonso resells one unit of dairy product for $0.95 and one unit of sweet for $3. Alfonso devotes his time in processing (unpacking and placing them to the shelf) the orders that he receives from his suppliers. Because of differences in packaging, the order processing times are different for each product and each supplier. The following table summarizes the time required for one unit of product.

A B C

Dairy Products 0.03 minutes 0.02 minutes 0.04 minutes

Sweets 0.02 minutes 0.015 minutes 0.05 minute

In a week, Alfonso does not want spend more than half working day for order processing (Alfonso works on average 10 hours each day). Alfonso asks your advice on the product mix and order placement. He informs you that this week he can spare up to $2,300 to order new stock.

OPIM101 – Decision Analysis Instructor: FENG Guiyun

a. Formulate the problem as a linear program. Clearly state the decision variables, objective function and constraints. Xij = # of units of product I ordered from supplier j, i = 1 for D, 2 for S; j = 1 for A, 2 for B, 3 for C max 0.95 (X11+X12+X13) + 3(X21+X22+X23) - (0.5X11+ 0.45X12+ 0.40X13+ 2X21+ 2.05X22+ 2.15X23) s.t. 0.03(X11+X12+X13) + 0.01(X21+X22+X23) = 35 (shelf space) X11+X12+X13 ≤ 1000 (demand) X21+ X22+ X23 ≤ 800 (demand) X11+ X21 =600 (van capacity) X12+ X22 =500 (van capacity) X13+ X23 =600 (van capacity) 0.5X11+ 0.45X12+ 0.40X13+ 2X21+ 2.05X22+ 2.15X23 ≤ 2300 (budget) 0.03X11+ 0.02X12+ 0.04X13+ 0.02X21+ 0.015X22+ 0.05X23 ≤ 300 (order processing) Xij ≥ 0

Question 2: Street Vendor A keen entrepreneur, you have decided to augment your meager student income with a little street vending corner store catering to the visitors of the Formula 1 races. After buying a license to set up a stall at a street corner, you have S$ 160 left to invest in buying your stock. You are thinking of selling chilled drinks, ice creams, and fresh fruits. The cost and resale price of each item is as follows: Item Drink Ice cream Fruit

Purchase Sales price price S$ 0.50 S$ 1.25 S$ 1.00 S$ 1.80 S$ 0.75

S$ 1.60

OPIM101 – Decision Analysis Instructor: FENG Guiyun

You have a cooler that has a capacity of 200 units (assume that each item takes up one unit capacity). In order to prevent your ice cream from melting too fast, fruit should make up at most 50% of the content of your cooler Per hour, you estimate that demand for snacks (ice cream and fruit jointly) will be 40 units. Demand for drinks is unlimited. – you are a busy student with presentations and midterm exams coming up!

a. Formulate the problem. Assume that the number of drinks is D , ice creams I and fruit F . Then we can formulate the following profit maximization problem: max

0.75D + 0.8I + 0.85F subject to 0.5D + I + 0.75F £ 160 D + I + F = 200 F £ 100 I + F £ 160 D, I , F ³ 0

b. Find the optimal product mix. In order to be able to solve this problem graphically, we need to reduce it to two variables. We can use the cooler capacity constraint to remove one variable from the problem. We will substitute for D = 200 - I - F . This yields the following problem formulation: max 0.05I + 0.1F + 150 subject to 0.5I + 0.25F £ 60 F £ 100 I + F £ 160 I, F ³ 0

Based on this problem formulation, we obtain the following graph (constraints as marked, feasible region is shaded, two iso-profit lines):

OPIM101 – Decision Analysis Instructor: FENG Guiyun

Fruit

250 Budget

200

Demand Cooler

150 100 50 0 0

50

100

150 Ice Cream 200

We can read off the solution on the graph, namely where the iso-profit line is about to leave the feasible region, at the intersection of the cooler constraint and the demand constraint. This means that F = 100 , I = 60 and thus D = 40 , for a total profit of S$ 163. c. Would you change your affect your profit? Why (not)?

if your budget increased by S$20? How would that

From the graphical solution, we can see that the budget constraint is a non-binding constraint. Hence, a further increase in the budget will not affect our optimal product mix or our profit.

d. Would you change your product mix if you were allowed to fill up to 60% of the cooler with fruit? How would that affect your profit? Why (not)? The constraint on the total amount of fruit we can carry is a binding constraint. Hence, an increase in the amount of fruit (the most profitable item) we are allowed to carry, will increase our profit. To be more specific, we can read off from the graph that the new product mix will contain means that F = 120 , I = 40 and thus D = 40 , for a total profit of S$ 164. e. You have two friends who are interested to join you in your little scheme. You have agreed that they will get (sales revenue minus cost of goods sold). In exchange, you provide the cart and the vending license. Both together are At each location, your estimate of the demand per hour is the same for each location. You decide to reinvest some of your profit from the previous day, and your

OPIM101 – Decision Analysis Instructor: FENG Guiyun

purchasing budget (for drinks, ice cream and fruit) is now S$ 280. Your parents are willing to loan you the amount required for the additional carts if you decide to involve your friends. As before, the carts need to be full when you set out from home in order to maintain a cool temperature, , and you and your friends will be working for four hours. What is your objective? What are your decision variables? What are your constraints? Please be precise in your answer (no formulation required). Afterwards write down the optimization problem. Objective: profit maximization: revenue from own and friends’ sales minus cost of goods sold and cost of carts Decision Variables:

- product mix for each cart - decision whether to include friends or not (2 binary variables)

Constraints:

- budget - demand in each location - full cooler in each location - fruit constraint in each location Let us introduce the index i Î {1,2,3} to indicate my stock and my two friends’ stock respectively. Let us also introduce two binary variables, x2 , x3 which indicate whether I include the first friend and/or the second friend respectively. Then I can write down the following optimization problem: max 0.75 D1 + 0.80 I 1 + 0.85 F1

+ 0.5 (0.75 ( D2 + D3 )+ 0.80 (I2 + I3 )+ 0.85 ( F2 + F3 )) - 40( x2 + x3 ) D 1 + I 1 + F1 = 200 subject to D i + I i + Fi = 200x i , i = {2,3} Fi £ 80 0.50( D1 + D2 + D3 ) + I1 + I 2 + I 3 + 0.75( F1 + F2 + F3 ) £ 280 I i + Fi £ 160 D i ,I i ,Fi ³ 0 Question 3: Advertisement Management The manager of a department store is attempting to decide on the types and amounts of advertising the store should use. He has invited representatives from the local radio station, television station, and newspaper to make presentations in which they describe their audiences. The television station representative indicates that a TV commercial, which<...